Paper Title: Precoding Based Downlink OAM-MIMO Communications with Rate Splitting.







































IEEE Transactions on Broadcasting, VOL. 69, NO. 4, DECEMBER 2023
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals


















Precoding Based Downlink OAM-MIMO Communications with Rate Splitting
    Ruirui Chen,
        Jinyang Lin,
        Beibei Zhang,
        Yu Ding,
        Keyue Xu
        
This work is supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20200650, in part by National Natural Science Foundation of China under Grant 62071472, in part by China Postdoctoral Science Foundation under Grant 2019M660133, in part by Program for “Industrial IoT and Emergency Collaboration” Innovative Research Team in CUMT under Grant 2020ZY002, in part by Fundamental Research Funds for the Central Universities under Grant 2019QNB01 and 2020ZDPY0304. (Corresponding author: Ruirui Chen.)
Ruirui Chen, Jinyang Lin, Beibei Zhang, Yu Ding and Keyue Xu are with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China (emails: rrchen@cumt.edu.cn, linjinyang@cumt.edu.cn, lb15060029@cumt.edu.cn, yding@cumt.edu.cn, kyxu@cumt.edu.cn). Beibei Zhang is also with Jiangsu Automation Research Institute, Lianyungang, 222061, China.






    August 5, 2024
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ ABSTRACT 
Orbital angular momentum (OAM) and rate splitting (RS) are the potential key techniques for the future wireless communications. As a new orthogonal resource, OAM can achieve the multifold increase of spectrum efficiency to relieve the scarcity of the spectrum resource, but how to enhance the privacy performance imposes crucial challenge for OAM communications.
RS technique divides the information into private and common parts, which can guarantee the privacies for all users.
In this paper, we integrate the RS technique into downlink OAM-MIMO communications, and study the precoding optimization to maximize the sum capacity. First, the concentric uniform circular arrays (UCAs) are utilized to construct the downlink transmission framework of OAM-MIMO communications with RS. Particularly, users in the same user pair utilize RS technique to obtain the information and different user pairs use different OAM modes. Then, we derive the OAM-MIMO channel model, and formulate the sum capacity maximization problem. Finally, based on the fractional programming, the optimal precoding matrix is obtained to maximize the sum capacity by using quadratic transformation.
Extensive simulation results show that by using the proposed precoding optimization algorithm, OAM-MIMO communications with RS can achieve higher sum capacity than the traditional communication schemes.




Orbital angular momentum (OAM), rate splitting (RS), MIMO, precoding, uniform circular array.















§ INTRODUCTION



 §.§ Background


The rapid growing demand of spectrum efficiency in the big data era brings great challenges to existing traditional communication technologies <cit.>.

Due to the scarcity of the spectrum resource, it becomes increasingly important to search for the new orthogonal resource <cit.>.
Therefore, orbital angular momentum (OAM), which can be considered as a new orthogonal resource, provides a new solution to the multifold increase of spectrum efficiency for the future wireless communications <cit.>.
The vortex waves with different OAM modes are orthogonal to each other, and thus the infinite OAM modes can be used to significantly improve the spectrum efficiency in theory <cit.>.
However, vortex waves with non-integer OAM modes are difficult to decode, and vortex waves with high OAM modes result in serious attenuation and beam divergence, which limits the user access number of the OAM communications <cit.>.
Therefore, improving the access capability is necessary for the application of OAM communications into the multi-user scenario.

To serve large number of users, OAM communications can utilize the space division multiple access (SDMA) technique or the non-orthogonal multiple access (NOMA) technique <cit.>.
By using SDMA technique, each user directly decodes the information by treating undesired signals as interferences, which leads to the rate saturation even with excess transmit power when the transmit antenna number is insufficient or a resource block is used by multiple users <cit.>.
It is difficult for SDMA technique to support the future high-load traffic scenarios, where there are high-rate services caused by many users at the same time <cit.>.
NOMA technique performs successive interference cancellation (SIC) at the receiver, which can realize the information transmission at the same time and frequency <cit.>.
In NOMA transmission, only the last user can guarantee the private information, which is not decoded by other users <cit.>.


Rate splitting (RS) technique combines the advantages of SDMA and NOMA by the flexible balance between the interference and desired signal <cit.>.
By dividing the transmission information into private message and common message , RS technique can serve multiple users at the same time and frequency <cit.>. All users use SIC technique to remove the common message and decode their own private message <cit.>.
Thus, RS technique decodes part of the interference and treats part of the interference as noise <cit.>, which avoids completely treating interference as noise and decoding interference <cit.>.
In addition, RS technique can guarantee the private transmission of all users and be applied to high-load traffic scenarios for serving large number of users <cit.>.

In this paper, we optimize the precoding matrix to maximize the sum capacity of the downlink OAM-MIMO communications with RS, which can guarantee the privacy performance for all users.



 §.§ Related Works

In this paper, RS technique is introduced to the downlink OAM-MIMO communications, and precoding matrix is optimized to maximize the sum capacity.
We will divide previous contributions related with this work into two aspects, i.e., precoding design for wireless communications with RS and beam steering design for OAM communications.



  §.§.§ Precoding Design for Wireless Communications with RS

As early as 1981, T. Han and K. Kobayashir proposed the RS technique <cit.>.
Based on linear precoding and SIC at the receiver, RS technique decodes part of the interference and treats the remaining part of the interference as noise <cit.>.
It outperforms SDMA and NOMA techniques in terms of network load and user privacy <cit.>.
Dividing message into a private part and a common part is the key of the precoding design for wireless communications with RS <cit.>.
In <cit.>, RS technique was proposed to be utilized in the multigroup multicast precoding design.
The authors of <cit.> investigated RS technique to improve the energy efficiency of the multi-user multicast MISO system.

Furthermore, RS technique achieves higher energy efficiency and requires less channel state information feedback <cit.>.
The work in <cit.> indicates that RS technique is more robust to the impact factors such as channel disparity, channel orthogonality, network load and channel quality.
These results demonstrate that the RS technique obtains better performance compared with other access techniques.




  §.§.§ Beam Steering Design for OAM Communications



Although vortex wave carrying OAM is a promising beyond 5G technique <cit.>, there are still some technical challenges for the practical application of OAM communications.
One challenge is that OAM communications require perfect alignment between the transmit and receive antenna arrays <cit.>. For OAM narrow-band communication system, the authors in <cit.> drawn a conclusion that if the antennas are not aligned, the system performance quickly deteriorates. To solve this problem, the authors of <cit.> studied the effect of misalignment on the performance of OAM broadband system and proposed a beam steering scheme for the misaligned multi-mode OAM communications. In <cit.>, the authors proposed a hybrid position phase beam steering (HPPBS) scheme to overcome the performance deterioration caused by the misalignment of uniform circular array (UCA) based OAM communications. The authors of <cit.> investigated the effect of the misalignment on channel capacity, and the proposed beam steering method can avoid the performance deterioration caused by misalignment. The proposed beam steering method in <cit.> can eliminate inter-mode interferences induced by the misalignment error between the transmit and receive UCAs.
In misaligned and incomplete receive UCA scenario, a partial receive scheme based on beam steering was proposed for the non-ideal OAM-MIMO communications to address the performance degradation caused by OAM beam divergence and misalignment <cit.>. Considering the size and cost of UCA, the authors in <cit.> developed an efficient technique for beam steering of arbitrary sizes of circular arrays based on OAM modes. For OAM communications, the authors of <cit.> presented a novel practical realization of field eigenmode beamforming in the azimuthal domain. To avoid multiple radio frequency chains and costly analog devices, a beam steering method was proposed in <cit.> to generate arbitrary-order OAM mode.
The authors of <cit.> utilized the spherical conformal array antenna to achieve beam steering of vortex waves, which can be applied in OAM communications.
In <cit.>, the authors proposed beam steering method to solve the aperture alignment of the transmitter and receiver in OAM communications. Through superposition of vortex waves with different OAM modes, the authors of <cit.> achieved single and multiple beam steerings in the transverse direction.



 §.§ Motivations


To enhance the privacy performance of OAM communications, we are motivated to study the downlink OAM-MIMO communications with RS, where multiple vortex wave signals are simultaneously transmitted and privacy transmission can be guaranteed for all users. Then, the challenges are discussed as follows.

  
  * Limited OAM Mode: Not all OAM modes can be used to achieve the high-capacity OAM communications. For the non-integer OAM mode, it is difficult to decode the vortex wave signal due to the nonuniform phase distribution. Furthermore, OAM communications with the high-order OAM mode suffers from severe divergence and attenuation. In practice, OAM communications mainly utilize the low-order integer OAM mode, thus dissatisfying the large number of user access requirements, which seriously limits its application in the multi-user scenario.
  
  * Lack of Precoding Algorithm: The vortex wave signal is transmitted by the UCA, each element of which generates the phase shift to produce different OAM modes. Then, the transmission information is divided into private and common parts. The integration of RS technique into OAM communications complicates the transmission framework, which results in the difficult formulation of sum capacity optimization problem. Furthermore, it is difficult to obtain the optimal precoding matrix for non-convex sum capacity maximization problem.




 §.§ Main Contributions


To address the challenges, the sum capacity optimization problem is formulated for the downlink OAM-MIMO communications with RS. However, the formulated optimization problem is non-convex. We utilize the fractional programming (FP) with quadratic transformation to obtain the optimal precoding matrix.

The main contributions of this paper are summarized as follows.

  
  * By integrating RS into OAM communications, different user pairs use different OAM modes and users in the same user pair utilize RS technique to obtain the information. The transmission framework is presented for the downlink OAM-MIMO communications with RS, where the UCA is used to transmit the vortex wave signal.
  
  * The channel model is derived based on the UCA. The non-convex optimization problem is formulated to maximize the sum capacity of the downlink OAM-MIMO communications with RS. Based on the FP, the optimal precoding matrix is obtained by using the quadratic transformation.
  
  * Simulation results demonstrate the capacity of users in the same user pair and analyze the impact of the distance on the downlink OAM-MIMO communications with RS. Furthermore, RS technique is superior to the other traditional access schemes.


 The rest of this paper is organized as follows. Section II gives the system model. In Section III, the transmission framework is proposed for the downlink OAM-MIMO communications with RS. Section IV derives the channel model and formulates the sum capacity maximization problem. In Section V, we derive the optimal precoding matrix based on the FP. Section VI gives the simulation results to evaluate the proposed precoding optimization algorithm for the downlink OAM-MIMO communications with RS. Finally, the paper concludes with Section VII.



§ SYSTEM MODEL


This paper considers the downlink OAM-MIMO communications with RS, which consist of multiple user devices (UDs) and a base station (BS). The UDs are denoted by UD_i, where i = { 1, 1pt 1pt 1pt 2, … ,2k - 1,2k, … ,2K - 1,2K}. Based on RS technique, the transmission information is partitioned into private message and common message. The UDs should be divided into pairs, which can achieve better privacy performance. In this paper, we choose two UDs as one pair. The reason is that the two user pair is the most general case for the reader to understand our work and easy to be realized in the practical wireless communications. Note that although two user pair is chosen, our work can be extended to the user pair with more users. Furthermore, we choose the two users, which have the nearest distance, as one user pair.

Thus, the k-th user pair (UD_2k-1, UD_2k) can complete the wireless communications by dividing their information into private message and common message. The information of UD_2k-1 and UD_2k can be respectively partitioned into {F_2k-1^p,F_2k-1^c} and {F_2k^p,F_2k^c}. Furthermore, the private messages F_2k-1^p and F_2k^p are encoded separately into transmit signals x̂_2k-1 and x̂_2k, which are respectively decoded by UD_2k-1 and UD_2k. The common messages F_2k-1^c and F_2k^c are encoded into transmit signal x̂_2k - 1,2k, which are decoded by both UD_2k-1 and UD_2k. The k-th user pair (UD_2k-1, UD_2k) can use the carrier with OAM mode l_k. In other words, each user pair uses one OAM mode. From the theoretical perspective, there exist infinite user pairs due to infinite OAM modes.

OAM communications utilize the UCA to transmit/receive vortex wave signal in general. The UCA based system model is demonstrated in Fig. 1. Each UD is equipped with one UCA to receive the vortex wave signal. BS has the transmit concentric UCAs, which include different antenna elements on different UCAs with the same center. In the downlink OAM-MIMO communications with RS, BS assigns the k-th transmit UCA to serve the k-th user pair (UD_2k-1, UD_2k). The k-th transmit UCA consists of M_k antenna elements, which are denoted by m_k ∈M={1,2, … ,M_k}. The center of the k-th transmit UCA is at the location (0, 0, 0) in the 3D cylindrical coordinate system. The z-axis is the line across the center of transmit concentric UCAs and perpendicular to the transmit concentric UCAs. The receive UCA of UD_i has N_i antenna elements, which are represented by n_i∈N = {1,2, … ,N_i}. The distance from the i-th UCA center of UD_i to the k-th transmit UCA center of BS is denoted by d_k,i.





§ TRANSMISSION FRAMEWORK OF DOWNLINK OAM-MIMO COMMUNICATIONS WITH RS

In the downlink OAM-MIMO communications with RS, the k-th UCA transmits the RS signal x_k = p_2k - 1^px̂_2k - 1^p + p_2k^px̂_2k^p + p_2k - 1,2k^cx̂_k^c to the k-th user pair (UD_2k - 1, UD_2k) by using the carrier with OAM mode l_k, where p_2k - 1,2k^c is the precoding vector of common message, p_2k - 1^p and p_2k^p are respectively the precoding vectors of private messages for UD_2k - 1 and UD_2k.
Moreover, the total transmit power is limited by

    p_2k - 1,2k^c_2^2 + p_2k - 1^p_2^2 + p_2k^p_2^2 ≤P_,

where p_2k^p_2^2 = ∑_m_k = 1^M_k∑_n_2k = 1^N_ip^p_m_k,n_2k_2^2, p_2k - 1^p_2^2 = ∑_m_k = 1^M_k∑_n_2k - 1 = 1^N_i
p^p_m_k,n_2k - 1_2^2,
p_2k - 1,2k^c_2^2  = ∑_m_k = 1^M_k∑_n_k = 1^N_i  p^c_m_k,n_k_2^2.
·_2^2 is the second-order norm, P_ denotes the maximum transmit power,
p^p_m_k,n_2k-1 represents the precoding power amplitude of the private message transmitted from the m_k-th antenna element to the n_2k-1-th antenna element of UD_2k-1,
p^p_m_k,n_2k represents the precoding power amplitude of the private message transmitted from the m_k-th antenna element to the n_2k-th antenna element of UD_2k
and p^c_m_k,n_k represents the precoding power amplitude of the common information transmitted from m_k-th antenna element to n_k-th antenna element of all users. The precoding matrix can be expressed as

    P = [ [    p_2k - 1^p        p_2k^p p_2k - 1,2k^c ]].



Multiple OAM modes are used to transmit different RS signals to different user pairs. By utilizing the carrier with different OAM modes, there is no inter-channel interference at the same frequency and time slot.
The RS signal x_k is simultaneously transmitted by M_k antenna elements, which is the vortex wave signal transmitted by the carrier with OAM mode l_k. For the m_k-th antenna element, the transmit vortex wave signal is expressed as

    X _m_k = ∑_l_k∈ L( x_k)e^j( ψ _m_k + η _m_k)l_k


    = ∑_l_k∈ L( x_k)e^j( 2π (m_k - 1)/M_k + η_m_k)l_k ,




where l_k denotes the OAM mode utilized by the k-th user pair (UD_2k-1, UD_2k), and belongs to the OAM mode set L = {l_k|k ∈ [1,K]}. In (<ref>), ψ _m_k = 2π (m_k - 1)/M_k represents the basic angle of the transmit UCA antenna element. ψ _m_k + η_m_k denotes the azimuth angle that is the angular position between the x-axis on the plane z=0 and m_k-th antenna element. Furthermore, η_m_k is the angle between respective antenna element and the phase angle.

The receive vortex wave signal goes through the OAM channel with zero mean circular symmetric complex gaussian (ZMCSCG) noise. By using the SIC technique, UD_2k-1 and UD_2k respectively decode their private signals x̂_2k-1^p and x̂_2k^p. To obtain x̂_k^c, UD_2k-1 and UD_2k will decode the common message. The OAM signals received by UD_2k-1 and UD_2k are respectively written as


    [                                                                                                       y_2k-1 =; ∑_n_2k-1 = 1^N_2k-1(( ∑_m_k = 1^M_k∑_l_k∈ LX _m_kh_m_k,n_2k - 1e^j( 2π (m_k - 1)/M_k + η _m_k)l_k) + z_n_2k-1) ]

 
 and
 
    [                                                                                                y_2k =; ∑_n_2k = 1^N_2k(( ∑_m_k = 1^M_k∑_l_k∈ L^X _m_kh_m_k,n_2ke^j( 2π (m_k - 1)/M_k +η _m_k)l_k) + z_n_2k), ]

where h_m_k,n_2k - 1 and h_m_k,n_2k - 1 denote the channel coefficients from the m_k-th antenna element of the k-th transmit UCA to n_2k - 1-th antenna element of the UD_2k - 1 receive UCA and n_2k-th antenna element of the UD_2k receive UCA, respectively; z_n_2k-1 and z_n_2k represent the ZMCSCG noise with zero mean and variance σ^2 received by the n_2k-1-th antenna element of the UD_2k-1 receive UCA and n_2k-th antenna element of the UD_2k receive UCA, respectively.



§ CHANNEL MODEL AND CAPACITY MAXIMIZATION PROBLEM

The channel coefficient from the m_k-th antenna element of the k-th transmit UCA to n_i-th antenna element of the UD_i receive UCA is derived as

    h_m_k,n_i = A_k,n_ie^ - j(B_k,n_isin (ψ _m_k + η_m_k - φ _n_i - η _n_i + ζ _k,n_i)),

where

    A_k,n_i=   βλ/4π√(d_k,i^2 + R_k^2 + r_i^2)e^ - j2π/λ√(d_k,i^2 + R_k^2 + r_i^2)
       e^j2π r_i^d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/λ√(d_k,i^2 + R_k^2 + r_i^2)

and


    B_k,n_i= 
       2πR_k√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ ))/λ√(d_k,i^2 + R_k^2 + r_i^2).


For the UD_i receive UCA, φ _n_i = 2π (n_i - 1)/N_i is the basic angle of the n_i-th antenna element. β denotes the phase rotation error and attenuation caused by antenna property. R_k is the radius of the k-th transmit UCA. r_i represents the radius of the UD_i receive UCA. λ denotes the wavelength. η _n_i denotes the angle between the UD_i receive UCA and initial phase of transmit UCA. θ represents the angle between the x-axis and the central line projection of the transmit UCA and UD_i receive UCA. ζ _k,n_i is defined as

    sinζ _k,n_i =
       r_i - d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ ))

and

    cosζ _k,n_i =
       d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )),

where ϕ _i is the angle between the z-axis and the central connection from the transmit UCA to the UD_i receive UCA.

For the private message, the signal-to-interference-plus-noise ratios (SINRs) of UD_2k-1 and UD_2k are respectively expressed as

    [                                                                                                              γ _    m_k,   n_2k - 1^p =;                    | h_    m_k,    n_2k - 1^T  p_    m_k,    n_2k - 1^p|^2/(∑_n_2i∈N∑_m_i∈M| h_    m_k,     n_2k - 1^T  p_m_i,n_2i^p|^2; +∑_[            n_2i - 1∈N; n_2i - 1∉    n_2k - 1 ]∑_[      m_i∈M; m_i∉   m_k ]| h_    m_k,    n_2k - 1^T  p_m_i,n_2i - 1^p|^2  + σ ^2) ]

and
 
    [                                                                                           γ _    m_k,    n_2k^p =;   | h_    m_k,    n_2k^T  p_    m_k,    n_2k^p|^2/(∑_n_2i - 1∈N∑_m_i∈M| h_    m_k,    n_2k^T  p_m_i,n_2i - 1^p|^2; +∑_[        n_2i∈N; n_2i∉    n_2k ]∑_[       m_i∈M; m_i∉    m_k ]| h_    m_k,    n_2k^T  p_m_i,n_2i^p|^2  + σ ^2) ]

For the common part, the SINRs of UD_2k-1 and UD_2k can be respectively written as
 
    [                                                                                                                                                       γ _    m_k,   n_2k - 1^c =;                                                         | h_   m_k,   n_2k - 1^T  p_   m_k,   n_k^c|^2/  (σ ^2 +∑_n_2i - 1∈N∑_m_i∈M| h_   m_k,   n_2k - 1^T  p_m_i,n_2i - 1^p|^2; + ∑_n_2i∈N∑_m_i∈M| h_   m_k,   n_2k - 1^T  p_m_i,n_2i^p|^2 + ∑_[             n_i ∈N; n_i ∉     n_2k - 1 ]∑_[      m_i∈M; m_i∉   m_k ]| h_   m_k,   n_2k - 1^T  p_m_i, n_i ^c|^2) ]

 and
 
    [                                                                                                                                            γ _   m_k,   n_2k^c =;                                                 | h_   m_k,   n_2k^T  p_   m_k,    n_k^c|^2/(σ ^2 +∑_n_2i - 1∈N∑_m_i∈M| h_   m_k,    n_2k^T  p_m_i,n_2i - 1^p|^2; + ∑_n_2i∈N∑_m_i∈M| h_   m_k,   n_2k^T  p_m_i,n_2i^p|^2 + ∑_[         n_i ∈N; n_i ∉     n_2k ]∑_[      m_i∈M; m_i∉   m_k ]| h_   m_k,   n_2k^T  p_m_i, n_i^c|^2 ) ]


The private capacities of UD_2k-1 and UD_2k are respectively written as
 
    C_2k - 1^p = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2( 1 + γ _   m_k,   n_2k - 1^p/M_kτ _q_2k - 1^2)

 and
 
    C_2k^p = ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + γ _   m_k,   n_2k^p/M_kτ _q_2k^2).

The common capacities of UD_2k-1 and UD_2k are respectively derived as
 
    C_2k - 1^c = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2( 1 + γ _   m_k,   n_2k - 1^c/M_kτ _q_2k - 1^2)

 and
 
    C_2k^c = ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + γ _   m_k,   n_2k^c/M_kτ _q_2k^2),

where τ _q_2k-1^2 and τ _q_2k^2 are respectively the eigenvalues of channel matrix from the BS to UD_2k-1 and UD_2k.

Moreover, the ranks of the channel matrix are Q_2k-1≤R_2k-1 = min (N_2k-1,M_k) and Q_2k≤R_2k = min (N_2k,M_k). The downlink OAM-MIMO communications with RS will broadcast the common message to UD_2k-1 and UD_2k, and thus the common capacity is derived as C_2k - 1,2k^c = min{  C_2k-1^c,  C_2k^c}, which guarantees that UD_2k-1 and UD_2k can receive the common message. The common capacities of UD_2k-1 and UD_2k are respectively c_2k-1 and c_2k, which satisfy the constraint c_2k-1 + c_2k≤C_2k - 1,2k^c. In this paper, we consider the full rank channel matrix (i.e., Q_2k-1 = R_2k-1 and Q_2k = R_2k).
The capacities of UD_2k-1 and UD_2k are respectively derived as follows
 
    C_2k-1^RS = C_2k-1^p + c_2k-1

 and
 
    C_2k^RS = C_2k^p + c_2k.

The sum capacity of the downlink OAM-MIMO communications with RS is expressed as
 
    C_sum^RS = C_2k-1^p + C_2k^p + c_2k-1 + c_2k.

The optimization problem is to maximize the sum capacity C_sum^RS.



§ FRACTIONAL PROGRAMMING BASED PRECODING OPTIMIZATION ALGORITHM


The FP with quadratic transformation is introduced to obtain the optimal precoding matrix for the sum capacity maximization problem.

The private and common capacities of UD_2k-1 and UD_2k can be respectively transformed in the following.

The private capacity of UD_2k - 1 is derived as

    C_2k - 1^p    =∑_[ n_2k - 1∈N;      m_k∈M ] C_  m_k,  n_2k - 1^p
        =∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ̅_  m_k,  n_2k - 1^p/M_kτ _q_2k - 1^2),

where

    γ̅_  m_k,  n_2k - 1^p = (a_  m_k,  n_2k - 1^p)^T(b_  m_k,  n_2k - 1^p)^ - 1a_  m_k,  n_2k - 1^p.

a_    m_k,    n_2k - 1^p and b_    m_k,    n_2k - 1^p can be respectively defined as

    a_    m_k,    n_2k - 1^p = h_    m_k,    n_2k - 1  p_    m_k, n_2k - 1^p

and

    [                                                      b_  m_k,  n_2k - 1^p =   h_  m_k,  n_2k - 1^(∑_n_2i∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T; + ∑_[           n_2i - 1∈N; n_2i - 1∉   n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ] p_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T )h_  m_k,  n_2k - 1^T+σ ^2. ]

The private capacity of UD_2k is expressed as

    C_2k^p    = ∑_[ n_2k∈N;  m_k∈M ] C_  m_k,  n_2k^p
          =∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ̅_  m_k,  n_2k^p/M_kτ _q_2k^2),

where

    γ̅_  m_k,  n_2k^p = (a_  m_k,  n_2k^p)^T(b_  m_k,  n_2k^p)^ - 1a_  m_k,  n_2k^p.

a_   m_k,   n_2k^p and b_   m_k,   n_2k^p can be respectively defined as

    a_   m_k,   n_2k^p = h_   m_k,   n_2k^  p_   m_k,   n_2k^p

and

    [                           b_  m_k,  n_2k^p = h_  m_k,  n_2k^(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T; + ∑_[       n_2i ∈N; n_2i ∉   n_2k ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_2i^p(p_m_i,n_2i^p)^T )h_  m_k,  n_2k^T + σ ^2. ]


The common capacity of UD_2k - 1 is derived as

    C_2k - 1^c   = ∑_[ n_2k - 1∈N;      m_k∈M ] C_  m_k,  n_2k - 1^c
            = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ̅_  m_k,  n_2k - 1^c/M_kτ _q_2k - 1^2),

where

    γ̅_  m_k,  n_2k - 1^c = (a_  m_k,  n_2k - 1^c)^T(b_  m_k,  n_2k - 1^c)^ - 1a_  m_k,  n_2k - 1^c.

a_    m_k,    n_2k - 1^c and b_    m_k,    n_2k - 1^c can be respectively defined as

    a_    m_k,    n_2k - 1^c = h_    m_k,    n_2k - 1^  p_    m_k, n_k^c

and

    b_  m_k,  n_2k - 1^c =σ ^2+ h_  m_k,  n_2k - 1^(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T
    
     + ∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T
    
    +∑_[          n_i ∈N; n_i ∉  n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_i^c(p_m_i,n_i^c)^T )h_  m_k,  n_2k - 1^T .

The common capacity of UD_2k can be derived as

    C_2k^c    = ∑_[ n_2k∈N;  m_k∈M ] C_  m_k,  n_2k^c
         =∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ̅_  m_k,  n_2k^c/M_kτ _q_2k^2),

where

    γ̅_  m_k,  n_2k^c = (a_  m_k,  n_2k^c)^T(b_  m_k,  n_2k^c)^ - 1a_  m_k,  n_2k^c.

a_    m_k,    n_2k^c and b_    m_k,    n_2k^c are respectively defined as

    a_  m_k,  n_2k^c = h_  m_k,  n_2k^p_  m_k,  n_k^c

and

    b_  m_k,  n_2k^c = σ ^2+  h_  m_k,  n_2k(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T
                                   + ∑_n_2i∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T
                    + ∑_[           n_i∈N; n_i ∉  n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_i^c(p_m_i,n_i^c)^T )h_  m_k,  n_2k - 1^T.


Then, the sum capacity maximization problem can be rewritten as problem P1.

    P1:maxmize _P C_sum^RS


    [ s.t.C1: C_2k - 1^p≤∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ _  m_k,  n_2k - 1^p/M_kτ _q_2k - 1^2) ]


    [ C2: C_2k^p≤∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ _  m_k,  n_2k^p/M_kτ _q_2k^2) ]


    [ C3: C_2k - 1,2k^c≤∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 +γ _  m_k,  n_2k - 1^c/M_kτ _q_2k - 1^2) ]


    [ C4: C_2k - 1,2k^c≤∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ _  m_k,  n_2k^c/M_kτ _q_2k^2) ]


    [ C5:∑_n_2k - 1∈N∑_m_k∈Mp_m_k,n_2k - 1^p_2^2  + ∑_n_2k∈N∑_m_k∈Mp_m_k,n_2k^p_2^2;                                           + ∑_n_k∈N∑_m_k∈Mp_m_k,n_k^c_2^2≤P_T ]

where C1 and C2 are the constraints on the private capacities of  UD_2k - 1 and UD_2k, respectively; C3 and C4 are the constraints on the common capacities of UD_2k - 1 and UD_2k, respectively; C5 is the total transmit power constraint.

The precoding matrix P is a non-empty matrix. According to the FP in <cit.>, problem P1 can be converted to the convex problem P2.

    P2:maxmize _PC̅_sum^RS


    [                                                                                                                      s.t. C1: C_2k - 1^p≤; ∑_[ n_2k-1∈N;    m_k∈M ]∑_[ q_2k - 1∈Q_2k - 1 ]log_2( 1 +τ _q_2k - 1^2/M_k( 2 Re{( y_    m_k,    n_2k - 1^p)^Ta_    m_k,    n_2k - 1^p}..;                                                     . .  - ( y_    m_k,    n_2k^p)^Tb_    m_k,    n_2k - 1^p( y_    m_k,    n_2k - 1^p))) ]


    [                                                                                                     C2:C_2k^p≤; ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + τ _q_2k^2/M_k( 2 Re{( y_    m_k,    n_2k^p)^Ta_    m_k,    n_2k^p}..;                                  . .  - ( y_    m_k,    n_2k^p)^Tb_    m_k,    n_2k^p( y_    m_k,    n_2k^p))) ]


    [                                                                                                                        C3: C_2k - 1,2k^c≤; ∑_[ n_2k-1∈N;    m_k∈M ]∑_[ q_2k - 1∈Q_2k - 1 ]log_2( 1 +τ _q_2k - 1^2/M_k( 2 Re{( y_    m_k,    n_2k - 1^c)^Ta_    m_k,    n_2k - 1^c}..;                                                 . .  - ( y_    m_k,    n_2k - 1^c)^Tb_    m_k,    n_2k - 1^c( y_    m_k,    n_2k - 1^c))) ]


    [                                                                                             C4: C_2k - 1,2k^c≤; ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + τ _q_2k^2/M_k( 2 Re{( y_    m_k,    n_2k^c)^Ta_    m_k,    n_2k^c}..;                                  . .  - ( y_    m_k,    n_2k^c)^Tb_    m_k,    n_2k^c( y_    m_k,    n_2k^c))) ]


    [ C5:∑_n_2k - 1∈N∑_m_k∈Mp_m_k,n_2k - 1^p_2^2  + ∑_n_2k∈N∑_m_k∈Mp_m_k,n_2k^p_2^2;                                           + ∑_n_k∈N∑_m_k∈Mp_m_k,n_k^c_2^2≤P_T ]


    C6:C̅_sum^RS≤ C_2k - 1^p + C_2k^p + C_2k - 1,2k^c

where C̅_sum^RS is the sum capacity C_sum^RS transformed from FP optimization algorithm, and the auxiliary variables y_    m_k,    n_2k - 1^p, y_    m_k,    n_2k^p and y_    m_k,    n_2k - 1^c, y_    m_k,    n_2k^c can be written as

    y_    m_k,    n_2k - 1^p = ( b_    m_k,    n_2k - 1^p*)^ - 1a_    m_k,    n_2k - 1^p*


    y_    m_k,    n_2k^p = ( b_    m_k,    n_2k^p*)^ - 1a_    m_k,    n_2k^p*


    y_    m_k,    n_2k - 1^c = ( b_    m_k,    n_2k - 1^c*)^ - 1a_    m_k,    n_2k - 1^c*


    y_    m_k,    n_2k^c = ( b_    m_k,    n_2k^c*)^ - 1a_    m_k,    n_2k^c*

The iterations of b_    m_k,    n_2k - 1^p*, b_    m_k,    n_2k^p*, b_    m_k,    n_2k - 1^c*, b_    m_k,    n_2k^c* and a_    m_k,    n_2k- 1^p*, a_    m_k,    n_2k^p*, a_    m_k,    n_2k - 1^c*, a_    m_k,    n_2k^c* are the auxiliary variables calculated by substituting the precoding matrix obtained from the previous iteration into (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), (<ref>), (<ref>), (<ref>), respectively. Then, we calculate C̅_sum^RS* at each iteration through iterative convex approximation. The iteration stops when C̅_sum^RS converges to the convergence threshold Γ. By substituting the optimal precoding matrix into (<ref>), we can obtain the maximum sum capacity C_sum^RS. The FP based precoding optimization algorithm is summarized as follows.
[0.25]8.9cm0.5pt
FP based precoding optimization algorithm
1) Initialize P^*, C̅_sum^RS*;
2) Repeat steps 3) - 7);
3) Update (<ref>), (<ref>), (<ref>), (<ref>) to obtain y_   m_k,   n_2k-1^p,    y_   m_k,   n_2k^p,
y_   m_k,   n_2k-1^c and y_   m_k,   n_2k^c;
4) By taking the fixed y_   m_k,   n_2k-1^p, y_   m_k,   n_2k^p, y_   m_k,   n_2k-1^c,    y_   m_k,   n_2k^c
and P^* into the problem P2, P and C̅_sum^RS are   obtained
by convex approximation;
5) Set P = P^*;
6) When C̅_sum^RS -C̅_sum^RS*≤Γ, go to step 8);
7) Set C̅_sum^RS* = C̅_sum^RS, go to step 3);
8) Substitute the current P into the problem P1 to obtain
   the maximum sum capacity C_sum^RS.
[0.25]8.9cm0.5pt



§ SIMULATION RESULTS


For the downlink OAM-MIMO communications with RS, the FP based precoding optimization algorithm is demonstrated in simulation results.


The transmit UCA consists of 3 antenna elements. The antenna element number of UD receive UCA is set to be 4.
The attenuation and phase rotation error caused by antenna property is β=4π.
The radius of the transmit UCA is R_k = kλ and the radius of the user UCA is r_i = nλ.
The wavelength is set to be λ=0.01.

The angle between the x-axis and the central line projection from the transmit UCA to the receive UCA is θ  = 0. The angle between the z-axis and the central connection from the transmit UCA to the receive UCA is set to be ϕ _i = 0.

Table <ref> presents the four cases of mode combinations for the downlink OAM-MIMO communications with RS. There are 4, 3, 3 and 2 antenna elements in transmit UCAs, respectively. The antenna element numbers of UD receive UCAs are respectively set to be 4, 4, 5 and 4.



The capacity of different OAM mode combination is presented in Fig. <ref>. It can be observed that the sum capacity is a monotonic decreasing function of the transmission distance. In addition, the proposed OAM communications with case 1 obtains the highest sum capacity. This is due to the fact that the array antenna combination of case 1 has the largest number of independent OAM channels. In other words, vortex wave beams with different OAM modes are equivalent to multiple independent OAM channels, which provides high sum capacity.



Figure <ref> plots the sum capacity versus the transmission distance with multiple OAM modes.

We can see that the sum capacity monotonically decreases with the increase of transmission distance.
Furthermore, compared to the traditional communications with RS, the proposed OAM communications with different user pairs can obtain higher capacity. This is because the proposed OAM communications with multiple OAM modes can simultaneously transmit multiple signals by using multiple independent OAM channels. In addition, the channel condition between BS and UD decreases slowly as the transmission distance increases.




The variation of UD_2k-1 capacity with the transmission distance is illustrated in Fig. <ref>.
We can see that the sum capacity is a monotonic increasing function of the total transmit power. Furthermore, compared with the traditional wireless communications, OAM-MIMO communications with RS can achieve the
highest capacity. This is due to the utilization of multiple OAM modes,
which enables different vortex wave beams to transmit multiple signals independently.
Figure <ref> demonstrates the UD_2k capacities of different wireless communications. From Fig. <ref>, it can be observed that TDMA communications can achieve higher capacity than the other three wireless communications if the total transmit power is relatively small. When the total transmit power is large, OAM-MIMO communications with RS can obtain the highest capacity among the four wireless communications.











Figure <ref> illustrates the performance comparison of OAM communications with multiple OAM modes. We can observe that the proposed OAM communications with case 1 achieves higher sum capacity than the other three cases. This is because the array antenna combination of case 1 produces the largest number of independent channels. Multiple vortex wave beams with various OAM modes simultaneously transfer multiple OAM signals to the user pairs by utilizing multiple independent OAM channels. Each OAM channel carries different OAM signal to the user pair. The more the OAM modes are, the higher the sum capacity is.




§ CONCLUSIONS


To enhance the privacy performance, we integrated RS technique into OAM communications and constructed the transmission framework of the downlink OAM-MIMO communications with RS. Different user pairs employed different OAM modes and users in the same user pair used RS technique to obtain the information. Based on concentric UCA, the OAM-MIMO channel model and the capacity were derived for the downlink OAM-MIMO communications with RS. The simulation experiments were conducted to show that the proposed precoding optimization algorithm can achieve higher sum capacity compared with the traditional schemes.

99
1
W. Cheng, W. Zhang, H. Jing, et al, “Orbital angular momentum for wireless communications," IEEE Wireless Communications, vol. 26, no. 1, pp. 100-107, Feb. 2019.

2a
H. Jing, W. Cheng, W. Zhang, et al, “Optimal UCA design for OAM based wireless backhaul transmission," ICC 2020 - 2020 IEEE International Conference on Communications (ICC), pp. 1-6, 2020.

2
A. Trichili, K. Park, M. Zghal, et al, “Communicating using spatial mode multiplexing: potentials, challenges, and perspectives," IEEE Communications Surveys & Tutorials, vol. 21, no. 4, pp. 3175-3203, 2019.

3a
R. Lyu, W. Cheng, B. Shen, et al, “OAM-SWIPT for IoE-driven 6G," IEEE Communications Magazine, vol. 60, no. 3, pp. 19-25, Mar. 2022.

3
Y. Lei, Y. Yang, Y. Wang, et al, “Throughput performance of wireless multiple-input multiple-output systems using OAM antennas," IEEE Wireless Communications Letters, vol. 10, no. 2, pp. 261-265, Feb. 2021.

4
R. Chen, H. Zhou, M. Moretti, et al., “Orbital angular momentum waves: generation, detection, and emerging applications," IEEE Communications Surveys & Tutorials, vol. 22, no. 2, pp. 840-868, 2020.

5
R. Lyu, W. Cheng and W. Zhang, “Modeling and performance analysis of OAM-NFC systems," IEEE Transactions on Communications, vol. 69, no. 12, pp. 7986-8001, Dec. 2021.

6
H. Jing, W. Cheng, Z. Li, et al., “Concentric UCAs based low-order OAM for high capacity in radio vortex wireless communications," Journal of Communications and Information Networks, vol. 3, no. 4, pp. 85-100, Dec. 2018.

7
W. Long, R. Chen, M. Moretti, et al., “Joint spatial division and coaxial multiplexing for downlink multi-user OAM wireless backhaul," IEEE Transactions on Broadcasting, vol. 67, no. 4, pp. 879-893, Dec. 2021.

8
A. Amin and S. Shin, “Channel capacity analysis of non-orthogonal multiple access with OAM-MIMO system," IEEE Wireless Communications Letters, vol. 9, no. 9, pp. 1481-1485, Sep. 2020.

9
H. Yin and H. Liu, “Performance of space-division multiple-access (SDMA) with scheduling," IEEE Transactions on Wireless Communications, vol. 1, no. 4, pp. 611-618, Oct. 2002.

10
S. Thoen, L. Perre, M. Engels, et al., “Adaptive loading for OFDM/SDMA-based wireless networks," IEEE Transactions on Communications, vol. 50, no. 11, pp. 1798-1810, Nov. 2002.

11
L. Dai, B. Wang, Y. Yuan, et al., “Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends," IEEE Communications Magazine, vol. 53, no. 9, pp. 74-81, Sep. 2015.

12
N. Zhao, D. Li, M. Liu, et al., “Secure transmission via joint precoding optimization for downlink MISO NOMA," IEEE Transactions on Vehicular Technology, vol. 68, no. 8, pp. 7603-7615, Aug. 2019.

13
R. Chen, F. Cheng, J. Lin, et al., “Performance analysis of rate splitting multiple access based vortex wave communications," IEEE Wireless Communications Letters, vol. 11, no. 8, pp. 1570-1574, Aug. 2022.

14
D. Yu, J. Kim and S. Park,  “An efficient rate-splitting multiple access scheme for the downlink of C-RAN systems," IEEE Wireless Communications Letters, vol. 8, no. 6, pp. 1555-1558, Dec. 2019.

15
Y. Mao, B. Clerckx and V. Li, “Energy efficiency of rate-splitting multiple access, and performance benefits over SDMA and NOMA," 2018 15th International Symposium on Wireless Communication Systems (ISWCS), pp. 1-5, 2018.

16
Z. Lin, M. Lin, T. Cola, et al., “Supporting IoT with rate-splitting multiple access in satellite and aerial-integrated networks," IEEE Internet of Things Journal, vol. 8, no. 14, pp. 11123-11134, Jul. 2021.

17
J. Zhang, B. Clerckx, J. Ge, et al., “Cooperative rate splitting for MISO broadcast channel with user relaying, and performance benefits over cooperative NOMA," IEEE Signal Processing Letters, vol. 26, no. 11, pp. 1678-1682, Nov. 2019.

18
D. Li, Z. Yang, N. Zhao, et al., “Precoding optimization assisted secure transmission for rate-splitting multiple access," ICC 2022 - IEEE International Conference on Communications, pp. 673-678, Aug. 2022.

19
Y. Mao, B. Clerckx and V. Li, “Rate-splitting for multi-antenna non-orthogonal unicast and multicast transmission: Spectral and energy efficiency analysis," IEEE Transactions on Communications, vol. 67, no. 12, pp. 8754-8770, Dec. 2019.





20
Te Han and K. Kobayashi, “A new achievable rate region for the interference channel," IEEE Transactions on Information Theory, vol. 27, no. 1, pp. 49-60, Jan. 1981.

21
Y. Mao, B. Clerckx, and V. Li, “Rate-splitting multiple access for downlink communication systems: bridging, generalizing, and outperforming SDMA and NOMA," EURASIP journal on wireless communications and networking, vol. 2018, no. 1, p. 133, May 2018.

22
B. Clerckx, Y. Mao, R. Schober, et al., “Rate-splitting unifying SDMA, OMA, NOMA, and multicasting in MISO broadcast channel: a simple two-user rate analysis," IEEE Wireless Communications Letters, vol. 9, no. 3, pp. 349-353, Mar. 2020.

23
B. Clerckx, H. Joudeh, C. Hao, et al., “Rate splitting for MIMO wireless networks: a promising PHY-layer strategy for LTE evolution,"  IEEE Communications Magzine, vol. 54, no. 5, pp. 98-105, May 2016.

24
H. Joudeh and B. Clerckx, “Rate-splitting for max-min fair multigroup multicast beamforming in overloaded systems," IEEE Transactions on Wireless Communications, vol. 16, no. 11, pp. 7276-7289, Nov. 2017.

25
H. Chen, D. Mi, B. Clerckx, et al., “Joint power and subcarrier allocation optimization for multigroup multicast systems with rate splitting," IEEE Transactions on Vehicular Technology, vol. 69, no. 2, pp. 2306-2310, Feb. 2020.

26
M. Dai, B. Clerckx, D. Gesbert, et al., “A rate splitting strategy for massive MIMO with imperfect CSIT," IEEE Transactions on Wireless Communications, vol. 15, no. 7, pp. 4611-4624, Jul. 2016.

27
O. Tervo, L. Trant, S. Chatzinotas, et al., “Multigroup multicast beamforming and antenna selection with ratesplitting in multicell systems," 2018 IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 1-5, Jun. 2018.

28
S. Christensen, R. Agarwal, E. De Carvalho, et al., “Weighted sum-rate maximization using weighted MMSE for MIMOBC beamforming design," IEEE Transactions on Wireless Communications, vol. 7, no. 12, pp. 4792-4799, Dec. 2008.

29
M. Vazquez, M. Caus, and A. Perez-Neira, “Rate splitting for mimo multibeam satellite systems," WSA 2018: 22nd International ITG Workshop on Smart Antennas, pp. 1-6, Mar. 2018.

30
H. Joudeh and B. Clerckx, “Robust transmission in downlink multiuser MISO systems: a rate-splitting approach," IEEE Transactions on Signal Processing, vol. 64, no. 23, pp. 6227-6242, Dec. 2016.



31
Z. Zhang, Y. Xiao, Z. Ma, et al, “6G wireless networks: vision, requirements, architecture, and key technologies," IEEE Vehicular Technology Magazine, vol. 14, no. 3, pp. 28-41, Sept. 2019.

32
P. Yang, Y. Xiao, M. Xiao, et al, “6G wireless communications: vision and potential techniques," IEEE Network, vol. 33, no. 4, pp. 70-75, Aug. 2019.

33
G. Xie, L. Li, Y. Ren, et al, “Performance metrics and design considerations for a free-space optical orbital-angular-momentum-multiplexed communication link," Optica, vol. 2, no. 4, pp. 357-365, 2015.

34
Y. Zhang, W. Feng and N. Ge, “On the restriction of utilizing orbital angular momentum in radio communications," 2013 8th International Conference on Communications and Networking in China (CHINACOM), pp. 271-275 2013.

35
Z. Tian, R. Chen, W. Long, et al, “Broadband beam steering for misaligned multi-mode OAM communication systems," Journal of Systems Engineering and Electronics, vol. 32, no. 4, pp. 779-788, Aug. 2021.

36
W. Yu, B. Zhou, Z. Bu, et al, “UCA based OAM beam steering with high mode isolation," IEEE Wireless Communications Letters, vol. 11, no. 5, pp. 977-981, May 2022.

37
R. Chen, H. Xu, M. Moretti, et al, “Beam steering for the misalignment in UCA-based OAM communication systems," IEEE Wireless Communications Letters, vol. 7, no. 4, pp. 582-585, Aug. 2018.

38
R. Chen, W. Long, X. Wang, et al, “Multi-mode OAM radio waves: generation, angle of arrival estimation and reception with UCAs," IEEE Transactions on Wireless Communications, vol. 19, no. 10, pp. 6932-6947, Oct. 2020.

39
Y. Xiu, Y. Wang, P. Shi, et al, “Maximization of millimeter-wave LoS OAM link using beam steering and partial receiving," 2022 16th European Conference on Antennas and Propagation (EuCAP),  pp. 1-5, 2022.

40
M. Klemes, H. Boutayeb and F. Hyjazie, “Orbital angular momentum (OAM) modes for 2-D beam-steering of circular arrays," 2016 IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), pp. 1-5, 2016.

41
S. Zheng, Y. Chen, Z. Zhang, et al, “Realization of beam steering based on plane spiral orbital angular momentum wave," IEEE Transactions on Antennas and Propagation, vol. 66, no. 3, pp. 1352-1358, Mar. 2018.

42
R. Chen, M. Zou, X. Wang, et al, “Generation and beam steering of arbitrary-order OAM with time-modulated circular arrays," IEEE Systems Journal, vol. 15, no. 4, pp. 5313-5320, Dec. 2021.

43
S. Yu, N. Kou, J. Jiang, et al, “Beam steering of orbital angular momentum vortex waves with spherical conformal array," IEEE Antennas and Wireless Propagation Letters, vol. 20, no. 7, pp. 1244-1248, Jul. 2021.

44
Q. Song, Y. Wang, K. Liu, et al. “Beam steering for OAM beams using time modulated circular arrays," Electronics Letters, vol. 54, no. 17, pp. 1017-1018, 2018.

45
N. Qasem, A. Alamayreh, J. Rahhal, “Beam steering using OAM waves generated by a concentric circular loop antenna array," Wireless Networks, vol. 27, no. 4, pp. 2431-2440, 2021.

46
K. Shen and W. Yu, “Fractional programming for communication systems-part I: power control and beamforming,"  IEEE Transactions on Signal Processing, vol. 66, no. 10, pp. 2616-2630, May, 2018.


[
    < g r a p h i c s >
]Ruirui Chen
received the Ph.D. degree in military communication from Xidian University, Xi'an, China, in 2018. He is currently a lecturer with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. He is also a master supervisor of electronic information in China University of Mining and Technology. His research interests focus on vortex electromagnetic wave communications, MIMO communications and UAV communications. Email: rrchen@cumt.edu.cn.


[
    < g r a p h i c s >
]Jinyang Lin
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2021. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in vortex wave communication. Email: linjinyang@cumt.edu.cn.



[
    < g r a p h i c s >
]Beibei Zhang
received the M.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2011. He is currently a Senior Engineer with Jiangsu Automation Research Institute, Lianyungang, China and also working towards the Ph.D. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His research interests include wireless sensor networks and vortex wave communications. Email: lb15060029@cumt.edu.cn.




[
    < g r a p h i c s >
]Yu Ding
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2022. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in UAV communications and vortex wave communication. Email: yding@cumt.edu.cn.



[
    < g r a p h i c s >
]Keyue Xu
received the B.S. degree in communication engineering from Suzhou University of Science and Technology, Suzhou, China, in 2022. She is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. Her current research interests focus on in intelligent communications and vortex wave communication. Email: kyxu@cumt.edu.cn.





 Paper Title: Illumination Design for Joint Imaging and Wireless Power Transfer Systems.


	
	DiM-Gesture: Co-Speech Gesture Generation with Adaptive Layer Normalization Mamba-2 framework
    Fan Zhang^ 0000-0002-9534-1777,
Naye Ji ^() ^ 0000-0002-6986-3766,
Fuxing Gao^ 0009-0008-5586-4734,
Bozuo Zhao ^ 0009-0008-2152-0087,
Jingmei Wu ^ 0009-0000-9221-8784,
Yanbing Jiang ^ 0009-0003-3020-598X,
Hui Du^ 0000-0001-6737-9420, 
Zhenqing Ye^ 0009-0003-4341-2734, 
Leyao Yan ^ 0009-0008-8797-176,
Jiayang Zhu ^ 0009-0005-3072-651X,
WeiFan Zhong^ 0009-0006-3251-1392,
Xiaomeng Ma ^ 0009-0008-0112-5354 


Fan Zhang, Naye Ji, Zhenqing Ye, Jiayang Zhu, WeiFan Zhong, Leyao Yan, Yanbing Jiang, Hui Du, Fuxing Gao, are with the College of Media Engineering, Communication University of Zhejiang, China; (e-mail: fanzhang@cuz.edu.cn;jinaye@cuz.edu.cn; 1767641184@qq.com; zjy190127 26077@foxmail.com; 2633252395@qq.com; yanleyao8@gmail.com; duhui@ cuz.edu.cn; fuxing@cuz.edu.cn;)

Jingmei Wu, Xiaomeng Ma, are with the School of Broadcast Announcing Arts, Communication University of Zhejiang; (e-mail: 20190095@cuz.edu.cn; 1256914539@qq.com)
Zhao Bozuo is with Changjiang Academy of Art and Design, Shantou University, China (e-mail: bzzhao@stu.edu.cn)

    August 5, 2024
================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

empty
	empty
	
	
	


§ ABSTRACT  


This paper presents a novel concept termed Integrated Imaging and Wireless Power Transfer (IWPT), wherein the integration of imaging and wireless power transfer functionalities is achieved on a unified hardware platform. IWPT leverages a transmitting array to efficiently illuminate a specific Region of Interest (ROI), enabling the extraction of ROI's scattering coefficients while concurrently providing wireless power to nearby users. The integration of IWPT offers compelling advantages, including notable reductions in power consumption and spectrum utilization, pivotal for the optimization of future 6G wireless networks. 
As an initial investigation, we explore two antenna architectures: a fully digital array and a digital/analog hybrid array. Our goal is to characterize the fundamental trade-off between imaging and wireless power transfer by optimizing the illumination signal. With imaging operating in the near-field, we formulate the illumination signal design as an optimization problem that minimizes the condition number of the equivalent channel. To address this optimization problem, we propose an semi-definite relaxation-based approach for the fully digital array and an alternating optimization algorithm for the hybrid array. 
Finally, numerical results verify the effectiveness of our proposed solutions and demonstrate the trade-off between imaging and wireless power transfer.







Index terms— Imaging, wireless power transfer, near field, integrated imaging and wireless power transfer (IWPT).
	

	
	
	

	

§ INTRODUCTION


Leveraging high-frequency bands and extensive antenna arrays is anticipated to empower 6G systems to predominantly function within the radiation near-field region, offering unparalleled communication and sensing capabilities, flexibility, and resolution <cit.>. Within this framework, notable research contributions have underscored the potential of holographic imaging <cit.>, a technique widely utilized in various environmental sensing applications such as security inspections and medical diagnoses. Typically integrated into internet of things (IoE) devices for decision-making processes, these applications face limitations imposed by device size and hardware costs, often constrained by battery capacities <cit.>. Radio frequency-based wireless power transfer (WPT) emerges as a promising technology for wirelessly charging such devices, presenting a viable solution to address energy scarcity concerns <cit.>.


Traditionally, imaging and WPT have been explored as separate domains. Imaging tasks have conventionally relied on technologies like microwaves or visible light, including lidars <cit.>, with limited exploration on repurposing wireless communication systems for imaging unfamiliar objects within a designated Region of Interest (ROI) <cit.>. However, in the context of forthcoming wireless systems, the allure of near-field imaging, particularly through the utilization of extremely large Massive MIMO (XL-MIMO) and high-frequency bands, has garnered attention. Initial investigations have delved into the beamforming challenges intrinsic to this approach <cit.>, hinting at a future where imaging could seamlessly integrate into the IoE sensor network without necessitating a dedicated imaging infrastructure.
On the other hand, the concept of radio wave-powered devices has been a focal point in pioneering discussions <cit.>. Notably, within the realm of wireless communication, there is a growing interest in amalgamating WPT with existing systems, such as simultaneous wireless information and power transfer (SWIPT) <cit.> and integrated sensing and wireless power transfer (ISWPT) <cit.>. These technologies aim to streamline communication and power delivery at the transmitter's end to curtail device expenses. Despite these advancements, the integration of imaging and WPT remains unexplored due to the traditionally distinct nature of these fields and their operational paradigms.



To exploit higher array gain in such scenarios, precoding and combining techniques play a crucial role. The most adaptable solution for an array of radiating elements is the fully digital architecture, where each antenna element connects to a dedicated Radio Frequency (RF) chain, enabling precise beam control. However, the scalability of this approach poses challenges due to the substantial costs associated with expensive RF components and high power consumption <cit.>. To address the complexity of fully digital architectures, hybrid solutions have emerged, blending traditional antenna technologies with the integration of multiple antennas or panels to an RF chain through dedicated analog circuitry <cit.>.
These hybrid architectures optimize resources by employing fewer RF chains relative to antenna elements. They achieve this by combining low-dimensional digital processing with high-dimensional analog precoding techniques, such as phase-shifter networks <cit.>. Such methodologies find applications in various large-scale antenna technologies, including near-field communications <cit.> and wideband communications <cit.>.





In this paper, we propose a novel concept that merges imaging and WPT into a unified hardware platform, terming the envisioned scenario as integrated imaging and wireless power transfer (IWPT). The proposed IWPT system is envisioned for deployment in common IoE settings that necessitate imaging functionalities, like security inspections and nondestructive testing, enabling the simultaneous operation of imaging and charging sensors. The integration of imaging and WPT within a single system presents several advantages. Firstly, it streamlines device costs and optimizes spectrum resources compared to conventional separate designs, as WPT can be achieved by utilizing the imaging beams, obviating the necessity for additional power beam configurations. Moreover, the continuous coverage provided by the illuminating beam over the ROI ensures a stable energy supply to devices within this area. Additionally, harvesting energy from the illumination signals enhances the overall energy efficiency of the platform. Furthermore, the proximity of charge sensors to the ROI typically situates them within the radiation near-field range of the transmitter, which amplifies the efficiency of WPT due to the near-field propagation mechanism, thereby offering additional benefits to the system's energy transfer capabilities.





In the envisioned IWPT scenario, both imaging and WPT functionalities are consolidated within a singular hardware platform, operating on a unified waveform. As these functions utilize the same antenna and transmission power resources, a crucial performance trade-off emerges between imaging and WPT. This trade-off is fundamentally characterized by optimizing the beamforming vector of the transmitted signal within the IWPT system.
The primary contributions of this communication can be succinctly summarized as follows:


  * To our knowledge, we are the first to propose the concept of IWPT, which aims to enhance cost-effectiveness and improve spectrum and energy efficiency by leveraging a shared hardware platform, spectrum, and power resources for two functions. This aligns with the anticipated development trajectory of future 6G networks. In our investigation, we explore two array architectures for this novel concept: the fully digital array and the hybrid array.

  * Additionally, we employ the condition number of the equivalent channel as a metric to evaluate the quality of imaging, providing an effective measure for performance assessment. We analyze the trade-off between imaging and WPT by optimizing the transmit beamforming vectors. Although the beamforming design problem is non-convex, we tackle it iteratively using the semidefinite relaxation approach and successive convex approximation method. Furthermore, for the challenge of coupling digital and analog precoding in the hybrid array, we develop an alternating optimization algorithm to address this issue.

  * We provide numerical results to demonstrate the fundamental performance trade-off between imaging and WPT. Numerical results also verify the effectiveness of the proposed beamforming designs for the novel IWPT system.




Notation: Scalar variables, vectors and matrices are represented with lower letters, lower bold letters, and capital bold letters, respectively (e.g., x, x, and X, respectively). The term ℂ^N × N denotes a complex space of dimension N × N. ( ·)^H denotes the Hermittian operators. For a matrix R, R≽ 0 means that R is positive semidefinite. 


The rest of this paper is organized as follows: Sec. <ref> introduces the considered array architecture, the signal model and the performance metrics of two functions. In Sec. <ref>, the design of beamforming design scheme is discussed, while Sec. <ref> details numerical simulations. Finally, Sec. <ref> provides concluding remarks.

	






§ SYSTEM MODEL


In this section, we provide a detailed description of the propsed ISWPT system. We begin by outlining the signal and channel model in Section <ref>. Following that, in Sec. <ref>, we introduce the two array architectures utilized throughout this paper. Subsequently, we introduce the imaging and WPT metrics in <ref> and formulate the beamforming design problem for ISWPT systems in <ref>.



 





 §.§ Transmitted Signal and Channel Model





We consider a joint imaging and wireless power transfer system as illustrated in Fig <ref>, wherein the transmitting and receiving discrete antenna arrays (TX/RX) are colocated and equipped with N antennas, the n-th antenna is located at a_n = [ x^ a_n, y^ a_n, z^ a_n ], n = 1,2, ⋯, N. 
The TX illuminates the ROI with the illumination signal and serves M energy receivers simultaneously, where the m-th energy receiver is located at p_m = [ x^ p_m, y^ p_m, z^ p_m ], m = 1,2, ⋯, M. 
The RX observes the reflected signal of the illumination signal from the ROI. The ROI is divided into a grid of K square cells of size Δ, whose positions are r_k = [ x^ r_k, y^ r_k, z^ r_k ], k = 1,2, ⋯, K. The k-th cell is characterized by a scattering coefficient defined as γ_k, related to the radar cross section (RCS) of the scatterer included in the cell (if any), which have RCS_k = |γ_k|^2 2λ/4π. If the k-th cell is empty, then γ_k = 0. The magnitude of the scattering coefficient |γ_k| is upper bounded by the maximum RCS from a scatterer of area Δ^2, which corresponds to the RCS of a perfect electric conductor (PEC) having the same area given by RCS_max = Δ^2 2λ/4π. Therefore, we have |γ_k| ≤Δ.


Given that energy receivers can harvest energy from the illumination signal, there is no necessity for designing dedicated WPT signals. In this context, we contemplate employing solely the illumination signal to facilitate both imaging and WPT functionalities. Consequently, the transmitted baseband signal can be represented as x = [ x_1, x_2, ⋯, x_N ]^T ∈ℂ^N × 1.
To optimize WPT efficiency, we make the assumption that the transmitter (TX) consistently transmits signals at the maximum power level, such that ‖ x‖^2 = P_ t, where P_ t signifies the total transmit power budget.


Assuming both imaging and WPT occur in the radiating near-field, i.e., the distance, d, between the TX/RX and the energy receiver or ROI should ensure that d < 2D^2/λ, where D and λ representing the array diameter and the signal wavelength, respectively. Define H_ T∈ℂ^K × N, H_ R∈ℂ^N × K and G∈ℂ^N × M as the imaging transmission channel matrix, the imaging receive channel matrix, and the power transmission channel matrix, respectively. Since the TX and RX is colocated, we have H_ R =  H^T_ T.
Then, under the line of sight (LOS) channel condition, the element of the imaging transmission could be expressed as

    H_ T[k,n] =  H_ R[n,k] = F(Θ_k,n)λ/4π d_k,ne^-j2π/λd_k,n,

and the element of the power transmission channel matrix could be expressed as

    G[m,n] = F(Θ_m,n)λ/4π d_m,ne^-j2π/λd_m,n,

where d_k,n and d_m,n are the distance between the corresponding antenna and the ROI cell or the energy receiver, F(Θ_k,n) and F(Θ_m,n) are the radiation profile, evaluated in the direction of arrival (i.e., Θ_k,n and Θ_m,n), which modeled as <cit.>. For convenience, in the following study, we assume the channel information is known.








 §.§ Array Architecture






We consider two types of antenna architectures to generate x, depicted in Fig. <ref>, that are introduced separately in the following subsections. For convenience, the antennas of all architectures are placed in a rectangular arrangement with N_ d rows and N_ e columns, with an overall of N=N_ dN_ e elements.



  
Fully digital architecture 


The implementation of fully digital antennas, where all processing is conducted in the digital domain, is widely preferred for its remarkable flexibility and the extensive freedom it offers for beamforming <cit.>, notwithstanding the potential for increased costs when deploying large-scale arrays.
In architectures employing fully digital antennas, each antenna element is linked to a dedicated Radio Frequency (RF) chain, as shown in Fig. <ref>. The transmitted baseband signal, denoted as x∈ℂ^N × 1, is directly conveyed to the antenna via the RF chain subsequent to digital precoding.



  
Hybrid architecture
In our investigation, we center our attention on a partially connected hybrid array configuration, as shown in Figure <ref>. Within this arrangement, the quantity of Radio Frequency (RF) chains, designated as N_ d, is less than the overall count of antenna elements. Each digital signal output links to an RF chain, and subsequently, each RF chain output connects to the transmit antennas through a partially connected analog beamforming network reliant on phase shifters. This hybrid array architecture adeptly balances complexity and performance, offering a cost-efficient resolution for expansive antenna arrays.


The array is specifically partitioned into N_ d linear sub-arrays, each consisting of N_ e elements uniformly arranged. Each linear sub-array is managed by an autonomous phase-shifter network associated with a single RF chain. Consequently, the signal transmitted by the hybrid array is given by

    x= Q  w.

Here w∈ℂ^ N_ d× 1 is the the digital precoding output for the hybrid array, and the matrix Q∈ℂ^N_ d× N represents analog precoding of the hybrid array that satisfies 

    Q_n, (i-1) N_ e+l=  q_i, l∈ℱ    i=n, 
     0     i ≠ n.

where q_i, l denotes the phase of phase-shifter, which satisfies

    q_i, l∈ℱ≜{e^j ϕ_i,l|ϕ_i,l∈[0,2 π]},       ∀ i, l.


Note that we did not distinguish the transmission signals of the two array architectures, but unified them as x because they follow the same constraints. The difference is that fully digital antennas can directly design x, while hybrid arrays require a joint design scheme for the digital and analog domains to obtain x as (<ref>). 








 §.§ Imaging and WPT Metric



According to the process of the illumination signal reflection as mentioned, the received signal on RX can be expressed as

    y =  H_ RΓ H_ T x +  n,

where Γ∈ℂ^K × K is a diagonal matrix with the diagonal element Γ_k,k=γ_k, n is is an additive white Gaussian noise with variance σ^2.
Let γ = [ γ_1, γ_2, ⋯, γ_K]^T, we have Γ = diag(γ), our aim is to estimate γ from the received signal, and (<ref>) could be rewritten as 

    y =  H_ Rdiag(  H_ T x) γ +  n.


Without losing generality, we assume that N>K, and define H =  H_ Rdiag(  H_ T x), H can be equivalent regarded as a MIMO channel, then estimate γ from (<ref>) becomes a conventional signal recovery problem. Using the common least squares method, the estimated scattering coefficients can be expressed as

    γ̂ =  H^† y,

where H^† is the pseudo-inverse of the equivalent channel matrix H.
Once G_ T and G_ R are fixed, H would also determined by the illumination signal, so it is possible to improve the estimation accuracy of γ̂ in imaging by design x.

To enhance the accuracy of estimation in (<ref>), one direct strategy involves scrutinizing the mean square error (MSE), defined as 𝔼‖γ - γ̂‖^2. However, estimating γ can often present challenges due to ill-posedness <cit.>, particularly when H T and H R exhibit rank deficiency in practical near-field scenarios, leading to a rank-deficient composite channel matrix H. The potential ill-posed nature of the problem renders the analysis of MSE overly intricate for signal design purposes. Instead, in this paper, we choose the condition number of the equivalent channel as the metric of imaging.


The condition number provides insight into the efficient characterization of the near field MIMO channel, in general, can be defined as

    cond = λ_max/λ_min,

where λ_max and λ_min are the maximum and minimum eigenvalue of H, respectively.
When the condition number of a channel approaches 1, the channel is deemed well-conditioned, indicating that its spatial characteristics are in an optimal state. Conversely, a channel tends towards infinite condition numbers when it is rank deficient. Consequently, our objective is to minimize the condition number of the equivalent channel through the design of the illumination signal. This optimization problem can be formulated as

    min _ x cond
         s.t.  ‖ x‖^2 = P_ t.


For WPT operation, the received power of the m-th user is contingent on the transmitted signal (here is the illumination signal), which can be represented as 

    P_m ( R) =ζ g_m R g^H_m,

where 0<ζ<1 is the constant energy conversion efficiency, and R =  x x^H denotes the covariance matrix of the illumination signal, g_m ∈ℂ^1 × N is the m-th row of G, denotes the wireless channel between the transmitter and the m-th energy receiver. 

In general, the goal of WPT is to maximize the sum-harvested power, while adhering to the total transmit power constraint, which can be mathematically formulated as the following maximization problem: 

    max _ x ∑_m=1^ M P_m (  R ) = ζtrace( G R G^H ) 
         s.t.   R =  x x^H, 
               ‖ x‖^2 = P_ t.


Following <cit.>, we can obtain a closed-form optimal solution of (<ref>) directly, which expressed as 

    x^∗ = √(P_ t) u_1 (  G),

where u_1 (  G) denotes the eigenvector corresponding to the maximal eigenvalue of the matrix G. 





 §.§ Problem Formulation
 


The result derived from (<ref>) is improbable to represent the optimal solution of (<ref>), indicating the presence of an inherent trade-off between imaging and WPT performance in our systems. In this paper, we would like to characterize this fundamental performance trade-off between imaging and WPT by optimizing the illumination signal x. Mathematically, our interested beamforming design problem is formulated as 

    min _ x cond
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               ‖ x‖^2 = P_ t.

Where E_ r is the set received power threshold that satisfies E_ r∈[0,E_max], here E_max is the achievable maximum received power calculated as E_max = ζtrace( G R^∗ G^H ), and R^∗ is the corvirance matrix of x^∗ from (<ref>). 
The extreme case, e.g.,  E_ r=0 or E_ r=E_max corresponds to the scenario of only imaging or only WPT. 

Problem (<ref>) serves as a comprehensive optimization challenge applicable across all antenna architectures. In the forthcoming section, our focus will revolve around investigating the precoding scheme aimed at acquiring x that minimizes (<ref>) within diverse antenna architectures.










§ JOINT BEAMFORMING DESIGN 


In this section, we study beam precoding design to achieve the performance trade-off between imaging and WPT. We first explicitly express the relationship between the objective function of our optimization problem and the optimization variables in Sec.<ref>. Subsequently, we delve into resolving the aforementioned issue within the framework of a fully digital array in Section <ref>, followed by the derivation of the optimal hybrid phase shifter configuration in Section <ref>.








 §.§ Problem Reformulation
 


Problem (<ref>) is hard to solve due to the absence of a direct relationship between x and the condition number as defined by (<ref>). Inspired by <cit.>, the problem of minimizing the condition number of H can be equivalent to minimizing the trace of the product of H and its transpose. This equivalence arises from the fact that minimizing the condition number is akin to minimizing the maximum eigenvalue λ_max. Specifically, in accordance with the definition of the condition number, we have

    cond≤(λ_max/λ_min)^2.

Thus the minimization problem can be regarded as minimize λ_max^2. Further more, for the semidefinite matrix H H^H, we have

    λ_max^2 < trace(  H H^H ).

Based on (<ref>) and (<ref>), problem (<ref>) can be equivalent formulated as



    min _ x trace(  H H^H )   
         s.t.   H =  H_ Rdiag(  H_ T x) 
               ζtrace( G R G^H ) ≥ E_ r
               ‖ x‖^2 = P_ t.



In problem (<ref>), the optimization variable x is tied to the objective function through (<ref>). However, the diagonalization process in (<ref>) continues to pose challenges in assessing the influence of x on the objective function. To address this issue, we employ a series of matrix operations to eliminate the constraint (<ref>) in problem (<ref>):

Firstly, the objective function of (<ref>) can be regarded as

    trace(  H H^H ) = vec(  H)^H vec(  H),

where vec( ·) denotes the vector stretching operation on matrix, and according to (<ref>), we have

    vec(  H) = (  I_N ⊗ H_ R) vec( diag(  H_ T x) ).

Substituting (<ref>) into (<ref>), so we have

    trace(  H H^H )    = vec( diag(  H_ T x) )^H (  I_N ⊗ H_ R)^H  (  I_N ⋯
          ⊗ H_ R) vec( diag(  H_ T x) )
       (a)=| x^H  H_ T^H  H_ R^H  H_ R H_ T x|,

where (a)= comes from that the calculation related to zero elements of vec( diag(  H_ T x) ) in above equation can be ignored. 

Therefore, the problem (<ref>) can be rewritten as



    min _ x | x^H  H_ T^H  H_ R^H  H_ R H_ T x|
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               ‖ x‖^2 = P_ t.







 §.§ Optimal Digital Beamforming Design



Problem (<ref>) is non-convex due to the quadratic equality constraint (<ref>), rendering it intricate to solve directly. To tackle this issue, we use the SDR approach <cit.> to transform (<ref>). According to the definition of R, (<ref>) can be rewritten as an equivalent quadratic semidefinite programming (QSDP) problem with rank-1 constraints:

    min _ R trace(  H_ T^H  H_ R^H  H_ R H_ T R)  
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               trace( R) = P_ t,  
               rank( R) = 1, 
                R≽ 0.


Due to the non-convex rank-one constraint, the problem (<ref>) is still a non-convex optimization problem. Next, we use the difference-of-convex (DC) programming to handle this rank-one constraint. Specifically, for the semi-definite matrix R, the rank-1 constraint can be equivalent expressed as 

    trace( R) = ‖ R‖_2.

Therefore, we can transform the rank-1 constraint of R into a penalty term to the objective function of the problem (<ref>), and the transformed problem is formulated as

    min _ R trace(  H_ T^H  H_ R^H  H_ R H_ T R) 
             + η( trace( R) - ‖ R‖_2 )  
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               trace( R) = P_ t,   
                R≽ 0.

where ζ≫ 0 is a penalty factor related to the rank-1 constraint. (<ref>) is still non-convex due to the presence of a 2-norm ‖ R‖_2 in the objective function, so we utilize the SCA method to write a first order Taylor expansion of ‖ R‖_2 as its lower bound:

    ‖ R‖_2    ≥‖ R^(t)‖_2 + trace(  u_max^(t) u_max^(t)H ( R -  R^(t)) ) 
       ≜ f(  R^(t),  R),

where R^(t) is the feasible point of the first order Taylor expansion in the t-th iteration, u_max^(t) is the eigenvector corresponding to the largest singular value of the matrix R^(t). Hence, by replacing ‖ R‖_2 with f(  R^(t),  R) in the t-th iteration, (<ref>) can be approximately transformed into

    min _ R trace(  H_ T^H  H_ R^H  H_ R H_ T R) 
             + η( trace( R) - f(  R^(t),  R ) )  
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               trace( R) = P_ t,   
                R≽ 0.


R^(t) is iteratively updated. In each iteration, R^(t) is taken from the solution of problem (<ref>) in the previous iteration, and the resulting R^(t) would iteratively close to the optimal solution of (<ref>). Meanwhile, we set that R^(1) =  R^∗ to ensure the initial satisfies all constraints of (<ref>).
The resulting problem (<ref>) is a typical QSDP problem, which is convex and can be solved directly using existing convex optimization solvers such as CVX <cit.>. 

Once R^(t) reaches convergence, it can be approximated as a rank 1 matrix, i.e., the optimal solution of problem (<ref>) would be found. The jointly optimal beam x^opt for imaging and WPT could be calculated from the eigendecomposition of R^(t) directly, which expressed as   

    x^⋆ = λ_max^(t) u_max^(t),

where λ_max^(t) is the largest singular value of the matrix R^(t), since the transmit power constraint, we have λ_max^(t) = √(P_ t), thus (<ref>) could be also expressed as x^⋆ = √(P_ t) u_max^(t). 

The worst case complexity to solve the QSDP problem (<ref>) is 𝒪( N^6.5log (1 / ϵ) ) with the primal-dual interior-point algorithm <cit.>, where ϵ is the solution accuracy. We assume that the number of iterations required for the SCA process to reach convergence is K, thus the whole complexity of the beamforming design is expressed as 𝒪( K N^6.5log (1 / ϵ) ).



 §.§ Joint Analog and Digital Design for Hybrid array



As outlined in Section <ref>, in the scenario of a fully digital array, the optimal beamforming design can be directly acquired by addressing problem (<ref>). Conversely, in the context of a hybrid array, considering (<ref>), the beamforming design predicament necessitates a collective precoding design spanning the digital and analog domains. Analogous to (<ref>), the optimization conundrum (<ref>) under the hybrid array scenario can be reformulated as

    min _ Q, w | w^H Q^H  T Qw|
         s.t.  ζtrace( G R G^H ) ≥ E_ r
               ‖ Qw‖^2 = P_ t
               (<ref>), (<ref>).

Where T =  H_ T^H  H_ R^H  H_ R H_ T, and R =  Qw w^H Q^H. 

Since Problem (<ref>) involves coupled optimization variables and unit modulus constraints, it is non-convex.

To address this, we aim to derive a hybrid solution for Problem (<ref>) that closely approximates the fully-digital solution of another problem, denoted as Problem (<ref>). This approach is quite often employed for optimizing hybrid analog/digital systems. Essentially, our objective is to determine a hybrid precoding mapping that closely resembles the fully-digital precoding in the context of the Frobenius norm.

Specifically, let x^⋆∈ℂ^N × 1 be the result of the optimal digital beamforming design by solving problem (<ref>). The resulting surrogate optimization problem is given by

    min _ Q, w ‖ x^⋆ -  Qw‖^2 
         s.t.  ‖ w‖^2 = P_ t/N_ d
               (<ref>), (<ref>).

Where the power constraint in (<ref>) comes from the ‖ Qw‖^2 = P_ t in (<ref>), according to (<ref>) and (<ref>), we have ‖ Qw‖^2 = trace( w^H Q^H Qw) = N_ dtrace( w^H w).

We tackle (<ref>) using the alternating optimization. In particular, for a given Q, the digital precoding result w^⋆ which minimizes (<ref>) is stated in the following least-squares solution:

    w^⋆ = (  Q^H  Q)^-1 Q^H  x^⋆ =  Q^H  x^⋆/N_ d,

the least-squares solution in (<ref>) dose not satisfy the power constraint in (<ref>). Nonetheless, once w and Q are tuned to optimize (<ref>), we can update the digital precoder by multiplying a factor, i.e., w^⋆ = √(P_ t) w^⋆/√(N_ d)‖ w^⋆‖.

When optimizing Q with a fixed w, we leverage the partially connected structure constraint (<ref>) and the unit-modulus constraint (<ref>) on the matrix Q. Through these constraints, a closed-form solution for Q emerges. Specifically, when w is given, the phase shifter coefficient matrix Q that minimizes (<ref>) can be attained as per the following proposed theorem.


 
For a given w, the matrix Q which minimizes (<ref>) is

    Q^⋆_n, (i-1) N_ e+l=  q^⋆_i, l∈ℱ    i=n, 
     0     i ≠ n.

here q^⋆_i, l is the corresponding phase shifter coefficient which given by 

    q^⋆_i, l = e^j( ∠x^⋆_(i-1) N_ e+l - ∠ w_i).

where ∠· denotes the angle of complex value.


 
The proof is given in Appendix <ref>.


The result of Theorem. <ref> indicates that the joint precoding result of the partially connected hybrid array can not perfectly match the result of a fully digital scheme. This discrepancy arises due to the insufficient number of RF chains to impeccably regulate the signal amplitudes at the antennas within the context of the considered partially connected architecture. This disparity isn't a consequence of the chosen scheme but rather stems from the inherent distortion induced by the structural constraints of partially connected hybrid arrays. Consequently, this mismatch engenders a suboptimal trade-off in performance for the hybrid array concerning IWPT compared to the fully digital array.

Finally, the overall resulting configuration algorithm for joint digital and analog precoding for the hybrid array is summarized as the following algorithm.






 

In Algorithm <ref>, it can be seen that Q only needs one update to achieve phase matching based on Theorem. <ref>, while the update of w is only related to Q by (<ref>), so the algorithm only needs two iterations to converge.
In addition, since the update process only involves closed form solution calculation, the complexity of Algorithm <ref> mainly comes from the calculation of x^⋆, which be expressed as 𝒪( K N^6.5log (1 / ϵ) ).












	
	
	




§ NUMERICAL EVALUATIONS







 §.§ Simulation Parameters
 
In this section, we provide numerical results to demonstrate the fundamental trade-off between imaging and WPT under a novel IWPT scenario. 
In our experiments, the TX/RX is set at configured as uniform squared arrays, with the array's reference position situated at (0,0,0) m on the Y Z-plane. The region of ROI forms a rectangle centered at (2,0,0) m and is parallel to the Y Z-plane. We direct the TX energy towards three power users concurrently, positioned at (1.5, 1, 1) m, (1, -1.5, 0) m, (1.5, -1, 0) m, respectively.Given these specifications, the maximum received power for WPT can be computed as E_max. Furthermore, the other simulation parameters are given in Table <ref>.





The numerical indicators of imaging performance are represented by root mean square error (RMSE), calculated as

    RMSE = √(1/N_ t∑_t = 1^N_ t‖γ̂^(t) - γ‖^2 ),

where N_ t is the number of Monte Carlo experiments, which set as N_ t = 200 in this paper, and γ̂^(t) is the estimate in the t-th experiment.





 §.§ Numerical Results


To assess the imaging efficiency of the proposed beamforming scheme, we initially establish the scenario where γ_k = Δ when the k-th cell of the Region of Interest (ROI) contains a scatter to be detected; otherwise, γ_k = 0. The image under consideration for estimation is illustrated in Fig. <ref>. Our objective is to estimate the scattering coefficients γ, which represent the original image, under various illumination signal beamforming strategies.

In our results, following cases are considered: 1) Random: In this scenario, the illumination signal is generated as a random signal, serving as a fundamental performance benchmark. The estimated outcome is depicted in Fig. <ref>; 2) Imaging only: The beamforming process focuses solely on optimizing imaging performance. The corresponding illumination is derived by solving (<ref>) with E_ r=0. The resulting estimate is illustrated in Fig. <ref>; 3) WPT only: Here, the illumination signal is directly obtained via (<ref>). The beamforming aims at maximizing the total received power of the power users exclusively. The estimated outcome is shown in Fig. <ref>; 4) IWPT with fully digital array: The optimal signal beam for imaging and Wireless Power Transfer (WPT) is determined by solving (<ref>) from the proposed beamforming design scheme, with the power threshold set at E_ r = 0.15E_max. The estimated result is displayed in Fig. <ref>; 5) IWPT with hybrid array: The signal beam is computed using (<ref>), where the digital and analog precoding are obtained from the proposed Algorithm <ref>.

The random illumination scheme demonstrates a fundamental ability to recover the original image from the received signal overall, albeit with some residual distortion. In contrast, the imaging-only scenario exhibits the most effective image restoration, showcasing how optimizing the equivalent channel conditions can significantly enhance imaging performance, while the WPT-only estimation yields the most distorted results, underscoring the impact of WPT considerations on imaging fidelity. Furthermore, the estimation outcomes of the IWPT scenario with a fully digital array closely resemble those of the imaging-only scenario. This alignment underscores the efficacy of our proposed beamforming optimization design. Conversely, the performance of the IWPT scheme with the hybrid array falls between that of the fully digital setup and the random scheme. This placement corroborates our earlier analysis in Sec. <ref>, but it should be emphasized that despite this, the hybrid array outperformed the random scheme while significantly cutting down the required number of RF chains by a factor of 13 compared to the all-digital array.




In Fig.<ref>, we illustrate the trade-off between imaging and Wireless Power Transfer (WPT). For detailed insights, Fig.<ref> and Fig. <ref> depict the correlation between the condition number and received power, as well as the Root Mean Square Error (RMSE) and received power, respectively. The comparison points "Random," "Imaging only," and "WPT only" are derived from averaging 200 Monte Carlo experiments within the mentioned context. It is evident that both the condition number of the equivalent channel and the RMSE of imaging increase as the received power of users rises, indicating a fundamental trade-off between imaging and WPT. Enhancing performance in one aspect typically comes at the expense of the other. Our proposed joint optimal scheme can achieve two extremes—the imaging-only scenario and the WPT-only scenario—and offers a flexible trade-off by adjusting the power threshold. As previously discussed, the performance of the hybrid array setup is suboptimal compared to the fully digital array case, with noticeable fluctuations in the lines as the power threshold increases. This variability occurs as the trade-off gradually shifts towards WPT, suggesting that a certain level of distortion may benefit imaging and vice versa. Nonetheless, this phenomenon does not alter the overarching trend of the trade-off performance.



The comparison between Fig. <ref> and Fig. <ref> reveals a positive correlation between the condition number and the Root Mean Square Error (RMSE) of imaging, particularly noticeable in the hybrid array scenario where the RMSE results exhibit similar fluctuations as the condition number. Despite a non-linear relationship, the RMSE results are indirectly influenced by optimizing the condition number. Given the complexity introduced by the near-field environment, analyzing imaging RMSE directly can be intricate, making the condition number a simpler yet effective optimization target. 
Furthermore, our joint optimization strategy based on imaging and WPT indicates that even under high power constraints, the imaging performance remains superior to that of random illumination. In Fig. <ref>, the power gating for improved imaging surpasses 12.2 μW, while in Fig. <ref>, it reaches 14 μW, showcasing the substantial potential of combining imaging with WPT.







The hybrid array design achieves a reduction in RF chains at the expense of a certain performance trade-off. In Fig.<ref>, we delve into the correlation between the number of RF chains in the hybrid array and the condition number. Keeping other simulation parameters consistent with Sec. <ref>, we vary the number of antenna rows for TX/RX , i.e., N_ d, and calculate the corresponding condition number. As N_ d changes, the total antenna count N also changes, allowing us to compare the results with a fully digital array setup having an identical antenna configuration, the later requiring 13× more RF chains than the hybrid array. We examine two scenarios: imaging only (i.e., the power threshold E_ r=0) and IWPT with E_ r = 0.15E_ max. The trend reveals that the condition number decreases with an increase in RF chains (or antenna numbers in the fully digital array), leading to a gradual reduction in the performance gap between the hybrid and fully digital arrays to a stable level. This suggests that the performance of the hybrid array hinges on that of the fully digital array it emulates, with a persistent performance gap due to RF chain reduction. 
The performance gap between hybrid arrays and fully digital arrays is narrower in the imaging-only scenario compared to the IWPT scenario. This difference is a consequence of the inherent trade-off between imaging and WPT, which exacerbates the mismatch when aligning the imaging beam with the hybrid array.







	
	
	




§ CONCLUSIONS



In this paper, we introduced the concept of Integrated Imaging and Wireless Power Transfer (IWPT), which combines imaging and wireless power transfer functionalities on a unified hardware platform. IWPT utilizes a transmitting array to efficiently illuminate a specific Region of Interest (ROI), enabling the extraction of the ROI's scattering coefficients while simultaneously providing wireless power to nearby users. We characterized the fundamental trade-off between imaging and wireless power transfer of IWPT system by optimizing the illumination signals, under both the fully digital array and hybrid array architectures. With imaging operating in the near-field, we formulated the illumination signal design as an optimization problem aimed at minimizing the condition number of the equivalent channel. To solve this optimization problem, we proposed a semi-definite relaxation-based approach for the fully digital array and an alternating optimization algorithm for the hybrid array. Our numerical results validate the effectiveness of our proposed solutions and highlight the trade-off between imaging and wireless power transfer. These findings underscore the potential of IWPT systems for practical applications, demonstrating their capability to enhance imaging precision and power transfer efficiency in future wireless networks.






	
	propositionsection 
	lemmasection 
	corollarysection 
	remarksection 
	equationsection	
	
	
		

	

§ PROOF OF THEOREM <REF>

		

To drop the non-convex structure constraint (<ref>), we define that q = vec(  Q) and V =  w^T ⊗ I_N. The objective function of (<ref>) can be rewritten as

    ‖ x^⋆ -  Qw‖^2 = ‖ x^⋆ -  V q‖^2

where q is the modified version of q obtained by removing all the zero elements of q and thus q∈ℂ^N × 1, V is the modified version of V obtained by removing the elements having the same index as the zero elements of q. By substituting (<ref>) into (<ref>) with given w, (<ref>) can be rewritten as a problem about optimizing q which given by

    min _ q ‖ x^⋆ -  V q‖^2 
         s.t.  q_n∈ℱ.

where q_n denotes the n-th element of q. In addition, according to the definition V, we obverse that V is a diagonal matrix with the diagonal elements vector expressed as [w_1  l_N_ e^T, ⋯, w_N_ d l_N_ e^T]^T.
Therefore, (<ref>) can be futher decomposed into N_ d subproblems, each with an identical structure. In particular, each subproblem individually designs the weighting coefficients for a single RF chains, with the i-th subproblem in scalar form given by

    min _ q_n | x_n^⋆ - w_i  q_n |^2 
         s.t.  q_n∈ℱ, 
               n = (i-1)N_ e,⋯,iN_ e.


Since q_n is only related to the phase, the solution to problem (<ref>) is q_n = e^j( ∠x^⋆_n - ∠ w_i).
Therefore, according to the mapping relationship between q_n and q_i,l, i.e., q_i,l = q_(i-1)N_ e+l, the optimal phase shifter coefficient q^⋆_i, l is given by

    q^⋆_i, l = e^j( ∠x^⋆_(i-1) N_ e+l - ∠ w_i).

This proves the theorem.













	
	
	IEEEtran
	
	
	
	
	


 Paper Title: Ground-to-UAV and RIS-assisted UAV-to-Ground Communication Under Channel Aging: Statistical Characterization and Outage Performance.



































Ground-to-UAV and RIS-assisted UAV-to-Ground Communication Under Channel Aging: Statistical Characterization and Outage Performance
     
	Thanh Luan Nguyen, Georges Kaddoum, Tri Nhu Do, Zygmunt J. Haas,




    
    
	T.-L. Nguyen and G. Kaddoum are with the Department of Electrical Engineering, the École de Technologie Supérieure (ÉTS), Université du Québec, Montréal, QC H3C 1K3, Canada. 
    G. Kaddoum is also with Artificial Intelligence & Cyber Systems Research Center, Lebanese American University (emails: thanh-luan.nguyen.1@ens.etsmtl.ca, georges.kaddoum@etsmtl.ca).

	T. N. Do is with Electrical Engineering Department, Polytechnique Montréal, Montréal, QC, Canada (email: tri-nhu.do@polymtl.ca).
	
	Z. J. Haas is with Department of Computer Science, University of Texas at Dallas, TX 75080, USA, and also with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA (e-mail: haas@cornell.edu).

    Received August 5, 2024
==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
    This paper studies the statistical characterization of ground-to-air (G2A) and reconfigurable intelligent surface (RIS)-assisted air-to-ground (A2G) communications in RIS-assisted UAV networks under the impact of channel aging. 
    A comprehensive channel model is presented, which incorporates the time-varying fading, three-dimensional (3D) mobility, Doppler shifts, and the effects of channel aging on array antenna structures. 
We provide analytical expressions for the G2A signal-to-noise ratio (SNR) probability density function (PDF) and cumulative distribution function (CDF), demonstrating that the G2A SNR follows a mixture of noncentral χ^2 distributions. The A2G communication is characterized under RIS arbitrary phase-shift configurations, showing that the A2G SNR can be represented as the product of two correlated noncentral χ^2 random variables (RVs). Additionally, we present the PDF and the CDF of the product of two independently distributed noncentral χ^2 RVs, which accurately characterize the A2G SNR's distribution. 
    Our paper confirms the effectiveness of RISs in mitigating channel aging effects within the coherence time. 
Finally, we propose an adaptive spectral efficiency method that ensures consistent system performance and satisfactory outage levels when the UAV and the ground user equipments are in motion.



RIS, UAV, Channel Aging, Channel Characterization, Outage Probability, Law of Total Cumulance 




§ INTRODUCTION

Integrated terrestrial and non-terrestrial networks (TNTNs) have emerged as a pivotal innovation in the realm of wireless systems, particularly for the advent of six-generation (6G) networks <cit.>.
    The traditional non-terrestrial networks (NTNs), previously limited to ground infrastructure coverage, have seen a myriad of new applications emerge in beyond fifth-generation (B5G) and 6G networks when integrated with terrestrial communication networks <cit.>.
This integration is envisioned to offer widespread, seamless, and high-speed connectivity for a multitude of device types and applications, such as high-altitude platforms, unmanned aerial vehicles (UAVs), different types of ground base stations (BSs) and both stationary and mobile ground user equipments (GUEs).
    In this context, three-dimensional (3D) flying UAV platforms, such as Dà-Jiāng Innovations drones, play a critical role in enhancing the capabilities of TNTNs by providing flexible, mobile platforms that can be rapidly deployed to extend network coverage and capacity in disadvantaged or remote areas <cit.>. 
    
Authorized by the Federal Aviation Administration (FAA), civilian drones are governed by a flight ceiling of 400 feet (≈ 120 m) above ground level. 
    
    Additionally, these drones are permitted to fly at speeds of up to 100 mph (≈ 40 m/s) <cit.>. To optimize their performance, UAVs can communicate over several frequency bands, including the 2 GHz <cit.>, 5.8 GHz, and 5030-5091 MHz bands. 
    In the context of TNTNs, UAVs are recognized for their cost-effectiveness and adaptability to offer ubiquitous coverage to support a diverse range of applications <cit.>. 
Specifically, the integration of UAVs with cutting-edge technologies, such as Terahertz (THz) communication, massive multiple-input multiple-output (mMIMO) communications, and reconfigurable intelligent surfaces (RISs) are instrumental in facilitating high-performance wireless communications, even in disaster-stricken or remote areas <cit.>.
    However, the precision of UAV tracking, and the ground-to-air (G2A) and air-to-ground (A2G) communications can be affected by position deviation issues due to wobbling and antenna jittering caused by body vibrations <cit.>.
Maintaining direct Line-of-Sight (LoS) to the GUE facilitates consistent connections, whereas Non-Line-of-Sight (NLoS) communication may require relay stations for reliable connectivity.
    Unfortunately, UAVs might operate in complex and unpredictable propagation environments, with A2G links occasionally blocked by trees or high-rise buildings <cit.>.
In addition, the mobility of UAVs introduces a phenomenon termed channel aging due to UAV’s variations in the G2A and A2G channels over time. 
    This phenomenon causes a mismatch between the actual channel state information (CSI) and the estimated CSI, rendering the estimated CSI outdated even if the channel acquisition was perfect, which degrades the performance of UAV-integrated wireless systems <cit.>.


Recent research works have widely studied the deployment of RISs in various environments, such as on the exteriors of high-rise buildings, to improve A2G communications <cit.>.
    Using planar arrays of many low-cost passive elements, which can range from a few to thousands of elements <cit.>, RISs can intelligently manipulate the direction of the impinging electromagnetic waves, thus improving coverage and reliability, especially when direct LoS links are scarce.
However, RIS operation requires perfect CSI for optimal phase shift configuration, posing a challenge in dynamic communication environments like UAV-integrated systems. 
    Next, we shed light on some recent works that are relevant to our research.

    
    The study in <cit.> considered networks of cellular-connected UAV swarms and a massive MIMO BS, where the authors developed a closed-form expression for the signal-to-interference-plus-noise ratio that determines the uplink spectral efficiency (SE) of RIS-assisted A2G channel. 
In <cit.>, the authors studied multi-hop multi-RIS-aided systems and proposed characterizing the true distribution of the RIS-assisted channel's magnitude by a double Nakagami-m distribution. 
    This approach is similar to the generalized-K distribution used to describe the behavior of a more straightforward system with a single RIS and a single-hop system in <cit.>. 
Under the Rayleigh fading assumption, the authors in <cit.> derived the SNR's exact distribution in RIS-assisted communication systems without LoS, following which the Central Limit Theorem (CLT) was adopted to approximate the derived results to noncentral χ^2 distribution.
    In <cit.>, which considered a single-antenna UAV serving multiple GUE with the aid of multiple RISs under independent and not necessarily identically distributed (non-IID) Rician fading with imperfect and outdated CSI, the authors applied the CLT to characterize the A2G signal-to-noise ratio (SNR) by a noncentral chi-squared (χ^2) distribution.
While this noncentral χ^2 distribution is mentioned in <cit.> and <cit.>, the degrees of freedom (d.o.f.) are set to 2, which oversimplifies the distribution. Specifically, the noncentral χ^2  distribution, with 2k d.o.f., noncentrality 2λ, and scale /2, is equivalent to the generalized κ-μ distribution through a parameter transformation. This highlights a research gap that requires further study for more accurate characterization.
    In another approach, <cit.> examined the effects of channel aging on RIS-enhanced massive MIMO systems with mobile UEs, where SE optimization under imperfect CSI was presented. The research also suggested that performance degradation due to channel aging can be mitigated by adjusting the frame duration or increasing the number of RIS elements.
Moreover, the authors in <cit.> evaluated the performance of RIS-assisted communications, accounting for hardware imperfections and channel aging due to GUE movement. 
    In <cit.>, the authors considered a UAV-mounted RIS system and characterized the end-to-end (e2e) SNR using the α-μ distribution, where the CLT is adopted for asymptotically large numbers of reflecting elements, e.g., up to 64^2 elements. 
However, the works in <cit.> and <cit.> did not consider UAV systems. In addition, although UAV systems are considered in <cit.>, the impact of channel aging was not addressed in any of the references.


While RIS-assisted UAV wireless communication has gained significant attention in the literature, there is limited analysis of the statistical characterization of G2A and RIS-assisted A2G communications under channel aging. Moreover, most works do not take into account the size of the RISs when the number of reflecting element grows.
    This research gap has resulted in a lack of tools to facilitate the systematic design, such as how large can the SE be achieved without compromising the end-to-end outage probability (eOP) when the UAV and GUE are in motion.
    
This paper aims to fill the aforementioned gaps by studying the statistical characterization of G2A and RIS-assisted A2G communications under the impact of channel aging. 
    Specifically, we derive the exact distribution, including the probability density function (PDF) and cumulative distribution function (CDF), of the G2A SNR using Laplace transform and inverse Laplace transform. 
For the RIS-assisted A2G communication, we first derive the characteristic function (CF) of the A2G communication under arbitrary phase-shift configurations (PSCs). 
    We then propose characterizing the A2G channel as a complex Gaussian distribution conditioned on the estimated (i.e., delayed) CSI and RIS phase shifts.
The key contributions of this paper can be summarized as follows[Part of this paper was presented at the 2024 IEEE International Conference on Communications <cit.>.]:




    
  * We consider an RIS-assisted UAV system designed to accommodate the three-dimensional (3D) movements of the UAV and GUE. 
    Specifically, our analysis can be used for the BS and RIS with antenna array architectures, with uniform planar array (UPA) as a special case. As a result, our analysis is also applicable for large RISs, where the size of the RIS cannot be neglected. Moreover, we introduce a comprehensive channel model that accounts for time-varying fading, continuous-time mobility, Doppler frequency shifts, and channel aging.
    
    
  * For the G2A communication, we derive a analytically tractable expression for the G2A SNR PDF while the delayed CSI-based maximal ratio combining (MRT) is adopted. Our findings reveal that the G2A SNR follows a mixture of noncentral χ^2 distributions, which is later shown to be directly related to the κ-μ-squared distribution. This important finding enables us to determine the exact G2A SNR CDF without further complex derivations.
    
    
  * The unique technical contribution of our paper lies in our unified characterization of the A2G communication under arbitrary PSC. 
        Specifically, we derive the exact CF of the complex RIS-assisted A2G channel, conditioned on delayed CSI and RIS phase shifts. Then, the conditional A2G channel is found to follow a complex Gaussian distribution when the RIS has sufficiently large number of elements, denoted as N. 
    This leads us to a novel finding that the A2G SNR is characterized by the product of two correlated non-central χ^2 RVs. 
        Simulation of Kullback–Leibler divergence between the exact and characterized A2G SNR is also presented to illustrate the accuracy of the proposed characterization.
    
    
  * We tailor the method of moments to characterize the A2G SNR as the product of two independent noncentral χ^2 RVs. Our approach includes a framework for matching central moments of the noncentral χ^2 distribution, where the law of total cumulance (LTC) is utilized for this purpose. 
        Additionally, we introduce novel and exact PDF and CDF expressions for the product of two non-IID noncentral χ^2 RVs to derive the A2G SNR PDF and CDF.
    Interestingly, our results show that the derived PDF/CDF is surprisingly accurate even with only N = 2^2 reflecting elements.
    
    
  * For the system performance analysis, we examine the eOP under a fixed target SE and its asymptotic behavior. As a case study, we employ random mobility models for the GUE and UAV, specifically the random waypoint mobility (RWM) model and reference point group mobility (RPGM) model, to examine temporal fluctuations in the eOP. 
    Subsequently, we introduce an adaptive SE approach to maintain the eOP at satisfactory levels, ensuring a consistent and reliable connection as the GUE and UAV are in motion.
    


Notations: [·], [·], μ_3[·] denotes expectation, variance, and third central moment operators, respectively; X =^d Y, X ≈^d Y, and X →^d Y indicate that the RVs X and Y are equal, approximately equal in distribution, and X is matched to Y in distribution, respectively.

    



§ SYSTEM MODEL




We consider a UAV-assisted terrestrial network, shown in Fig. <ref>, where 
a single-antenna UAV is serving a single-antenna GUE by relaying the information from the M-antenna BS.
    In this system, the direct BS-GUE and UAV-GUE links are absent due to various Radio Frequency (RF) obstructions <cit.>. 
Hence, we consider a RIS with N reflecting elements, installed on a building facade, to facilitate a virtual LoS link for the A2G communication <cit.>.
    Hereafter, we use , _m, , , _n, and  as subscripts to depict the BS, the BS's antenna m ∈ M≜{ 1, 2, …, M }, the UAV, the RIS, the reflecting (or RIS) element n ∈ N≜{ 1, 2, …, N }, and the GUE, respectively.
In addition, the transmission from BS to GUE is equally divided into two orthogonal time slots as follows. 
    During the first time slot, the BS transmits the unit-energy information signal x_ to the UAV. In the second time slot, the UAV decodes x_ and forwards the re-encoded version x_ to the GUE with the help of the RIS. 

Moreover, we denote 𝐩_(t) ≜[ x_(t), y_(t), z_(t) ]^ T as the 3D Cartesian coordinates of node  with respect to node , where , ∈{, _m, , , _n, }.
    The conversion from the spherical coordinates [ d_(t), θ_(t), φ_(t) ]^ T to the 3D Cartesian coordinates is <cit.>


    [ x_(t); y_(t); z_(t) ]= d_(t) 
    [ cosθ_(t) cosφ_(t); cosθ_(t) sinφ_(t);          sinθ_(t) ]≜ d_(t) 𝐫_(t),


where the unit vector 𝐫_(t), i.e., ‖𝐫_(t) ‖^2 = 1, represents the direction from node  to node , while d_(t), θ_(t) ∈[0, π], and φ_(t) ∈ [0, 2π] denote the distance [m], the elevation angle [rad], and the azimuth angle [rad], between nodes  and , respectively. 
    The conversion from the Cartesian coordinates to the spherical coordinates is given by


    [
        d_, θ_, φ_]^ T
    = [
        √(x^2_ + y^2_ + z^2_),
    
             sin^-1( z_/√(x_^2 + y_^2 + z_^2)),
        cos^-1( y_/x_)
    ]^ T.



We also denote 𝐩_(t) ≜[ x_(t), y_(t), z_(t) ]^ T as node 's Cartesian coordinates at time t with respect to the origin in a predefined global coordinates system, thus 𝐩_(t) =𝐩_(t) - 𝐩_(t). 





 §.§ Time-Varying Fading Channel


    We denote the time-varying channel coefficient from node  to node  at time t as c_(t) ≜√(_(t)) h_(t), where _(t) is the path loss component and h_(t) is the small-scale fading component.
Moreover, we consider that all channels experience Rician fading comprising a line-of-sight (LoS) path and multiple non-LoS (NLoS) paths. Specifically, the channel vectors are modeled as 𝐡_(t)
    =   √(K_(t))/√(K_(t) + 1)𝐡̅_(t)
    + 𝐡_(t)/√(K_(t) + 1) <cit.>, where 𝐡̅_(t) represents the LoS component and 𝐡_(t) represents the NLoS components whose entries are independent and identically distributed (IID) complex Gaussian RVs with zero mean and unit variance. 
    The Rician factor is modeled as a function of the elevation angle as K_(t) = K_0 e^2/πθ_(t) lnK_π/K_0 for LoS scenarios, where K_0 and K_π are environmental coefficients. It is noted that K_(t) = 0 indicates the NLoS scenarios, where the channel experience Rayleigh fading.

When antenna array architectures are employed at the BS, we rewrite 𝐡̅_(t) as <cit.>


    [ 𝐡̅_(t) ]_m = 
        exp[ j 2 π t Δf__m(t) ]
        √(M)α_(𝐫_(t)) ,


where Δf__m(t) is the Doppler frequency shift in the LoS path induced by the movement of the UAV and α_(𝐫) is the normalized array response or normalized steering vector in the direction 𝐫∈ℝ^3. 
    In particular, the vector 𝐫 = 𝐫_(t) specifies the direction of the waveform departing from the BS toward the UAV, or the so-called direction of departure (DoD).
The array steering vector α_(𝐫) in the direction 𝐫∈ℝ^3 can be expressed as <cit.>


    α_(𝐫)
        =
        [ 
            e^j 2π/λ𝐫^ T𝐩__1 _0,  …, e^j 2π/λ𝐫^ T𝐩__M _0]^ T/ √(M),


where 𝐩__m_0 denotes the position of the m^th antenna with respect to the center of the antenna platform. Here, the wave vector in <cit.> relates to the direction vector as 𝐤 = 2π/λ𝐫.

When the BS is equipped with an UPA in the yz-plane with M_H,  M_V and d_H, d_V being the number of elements and element spacing on the y (horizontal) axis, and the number of elements and element spacing on the z (vertical) axis, respectively, we have 
𝐩__m _0 = [ 0, (i__m-1) d_H, (j__m-1) d_V]^ T, for m ∈ M, where i__m = 1, …, M_H and j__m = 1, …, M_V are the corresponding horizontal and vertical subscripts of the M_H× M_V matrix, and where M_H M_V = M. Specifically, i__m and j__m are determined as ( i__m, j__m) = ( mod(m-1, M_H) + 1, ⌊ (m-1)/M_H⌋ + 1 ) ≜ind2sub( [ M_H, M_V], m ) <cit.>. 


Following (<ref>), we rewrite the LoS component of the UAV-RIS and RIS-GUE channel vectors as


    [ 𝐡̅_(t) ]_n
           =  exp[ j 2 π t Δf__n(t) ] 
        √(N)α_(-𝐫__n(t)), 
    [ 𝐡̅_(t) ]_n 
           =  exp[ j 2 π t Δf__n(t) ] 
        √(N)α_(𝐫__n(t)),


where Δ f__n(t) and Δ f__n(t) are the Doppler frequency shifts in the LoS path induced by the movement of the UAV and the GUE, respectively, and -𝐫__n(t) is the direction of the waveform departing from the UAV and arriving at the n^th element of the RIS. The direction -𝐫__n(t) is also referred to as the direction of arrival (DoA). Moreover, 𝐫__n(t) is the DoD of the waveform departing from the n^th element of the RIS toward the GUE, and α_(𝐫)
    =   
    [ 
        e^j 2π/λ𝐫^ T𝐩__1 _0, …, 
        e^j 2π/λ𝐫^ T𝐩__N _0]^ T/ √(N).

When the RIS has an UPA architecture in the yz-plane with N_H, N_V, and d_H, d_V being the number of elements and element spacing on the horizontal axis and the number of elements and element spacing and vertical axis, respectively, we have 
𝐩__n _0 = [ 0, (i__n-1) d_H, (j__n-1) d_V]^ T for n ∈ N, where, for i__n = 1, …, N_H and j__n = 1, …, N_V and ( i__n, j__n)^ T = ind2sub( [ N_H, N_V], n ).





 §.§ Continuous-Time Mobility Model


The GUE's motion with respect to the origin can be modeled by the change of its positions over time as 𝐩_(t, τ) = 𝐩_(t+τ) - 𝐩_(t)∈ℝ^3 <cit.>, where τ [sec] is the delayed duration from the initial observation instant t [sec]. 
The GUE's motion can also be determined based on its average velocity over a period τ, denoted as 𝐯_(t, τ), as


    𝐩_(t, τ) 
        =  𝐩_(t) + τ𝐯_(t, τ).



In various communication scenarios, the UAV is required to move following the GUE's movement to ensure uninterrupted connectivity.
    In this case, the UAV's motion is a function of the GUE's motion, and is modeled as


    𝐩_(t, τ)
        =   𝐩_(t, τ) + Δ𝐩_(t, τ),


where Δ𝐩_(t, τ) ∈ℝ^3 represents the UAV's motion with respect to the GUE's motion.

The mobility of the UAV and GUE causes the A2G channel to vary over time, which leads to the channel aging phenomenon <cit.>. Channel aging causes the CSI to become outdated due to the delay between the estimation time and the CSI transmission time. Moreover, the velocity vectors of the UAV and the GUE, i.e., 𝐯_(t) and 𝐯_(t), also vary over time, which results in a time-varying Doppler frequency.





 §.§ Time-Varying Doppler Frequency


If nodes  and  move with instantaneous velocities 𝐯_(t) ∈ℝ^3 and 𝐯_(t) ∈ℝ^3, respectively, the Doppler shift along the LoS path is expressed as


    Δf_(t) = f_c/c( 𝐯_(t)-𝐯_(t) )^ T𝐫_(t),


where 𝐫_(t) is determined in (<ref>). We note that 𝐯_(t) = 0 (or 𝐯_(t) = 0) implies that node  (or ) is standing still.

We note that (<ref>) is applicable to 3D mobility scenarios, with one-dimensional (1D) mobility being a special case. In this context, the term 𝐯_(t) - 𝐯_(t) ≜𝐯_(t) indicates the relative velocity of node  with respect to node . 
    Moreover, (<ref>) can be equivalently represented by one-dimensional (1D) mobility as Δf_(t) = ±f_c/c‖𝐯_(t) - 𝐯_(t) ‖cosθ_(t), where θ_(t) is determined in (<ref>) and the ± sign is determined by whether the node  is moving towards (+) or away (-) from node , respectively. 
Based on (<ref>), we formulate the maximum Doppler frequency shift across the NLoS components as


    Δf^max_(t) 
        = f_c/c‖𝐯_(t) - 𝐯_(t) ‖.





 §.§ Channel Aging Model



    To measure the CSI aging caused by the Doppler effect, correlation metrics are used to determine the time-varying property of the CSI.
Specifically, the channel state at a discrete time instant t_k can be modeled as <cit.>


    𝐡_[t_k]
        = ρ_[t_k] 𝐡_[0] + ρ̅_[t_k] 𝐟_[t_k],


where 𝐟_[t_k] =^d 𝒞𝒩( 0, 𝐈) represents the innovation component, ρ_[t_k] ∈ [-1, 1] is the temporal  correlation coefficient between the channel realizations at time instances 0 and k, and ρ̅_[t_k] = √(1-ρ_^2[t_k]). 
As in <cit.>, we consider that the channels evolve according to Jakes’ model, thus


    ρ_[t_k] = J_0( 2π t_k T_s Δf^max_(t_k T_s) ),


where J_0(x) is the zeroth-order Bessel function of the first kind and T_s [sec] is the channel sampling duration. 
Since ρ_[t_k] is a Bessel function, either change in the UAV velocity, GUE velocity, or a higher sample index t_k will yield a non-monotonic decrease in the correlation coefficient which oscillates around zero with decreasing magnitude. 

Considering that the channels' estimates at time instant τ are perfect and that these estimates are used as the initial states to get estimates of the channels at all other time instants, then the  channels' estimates at later time instants will be inaccurate <cit.>. 
    Thus, the current channel at time instant t_k can be expressed in terms of the estimated channel at the τ^th time instant as <cit.>


    𝐡_[t_k] 
        =  ρ_[τ-t_k] 𝐡_[τ] + ρ̅_[τ-t_k] 𝐳_[t_k],


where 𝐳_[t_k] ∼𝒞𝒩(0, 𝐈) denotes the independent innovation component that correlates 𝐡_[t_k] and the (outdated) estimated CSI 𝐡_[τ].     
Hereafter, we drop the time indices and denote the delayed CSI as 𝐡_ = 𝐡_[τ] for convenience.


   


 §.§ Ground-to-Air Signal-to-Noise Ratio


In the first time slot, the BS applies the beamforming vector 𝐰_ to steer the desired signal x_ to the UAV, where E[|x_|^2] = 1. Hence, the received signal at the UAV is obtained as


    y_ = 
        √(P__)[
            (ρ_𝐡_ 
                + ρ̅_𝐳_)^ H𝐰_] x_
        + n_,


where P_ [W] denotes the BS's transmit power and n_ is the additive white Gaussian noise (AWGN) with zero mean and variance σ^2_ [W] at the UAV.
    We assume that __1≈__2≈…≈__M = _ since the maximum dimension of the BS's antenna array is significantly smaller than its distance to the UAV.

Moreover, we consider that the BS adopts maximal ratio transmission (MRT) using the estimated CSI, where 𝐰_ = 𝐡_/‖𝐡_‖. 
Hence, the G2A SNR under delayed CSI-based MRT is obtained as


    _
        =  P__/σ_^2| 
            (ρ_𝐡_ + ρ̅_𝐳_ )^ H𝐡_|^2
        ‖𝐡_‖^-2.







 §.§ Air-to-Ground Signal-to-Noise Ratio


In the second time slot, the UAV decodes x_ and forwards the re-encoded version x_ to the RIS.
    Next, the RIS reflects the signal x_ from the UAV to the GUE by intelligently adjusting its phase-shift matrix.
Hence, the received signal at the GUE is obtained as


    y_
        =  √(P_)( 
            ∑_n=1^N√(__n) h__nβ_n √(__n) h__n e^jϑ__n)
        x_ + n_,


where P_ [W] is the UAV's transmit power, n_ is the AWGN with zero mean and variance σ_^2 [W] at the GUE, β_n ∈ℝ and ϑ__n= [0,2π) are the amplitude reflection coefficient and the phase-shift value of the reflecting element n ∈ N, respectively <cit.>. 
    In practice, the number of phase-shifts is limited and constrained by the phase-shift resolution, denoted by Q ≜ 2^b, where b is the number of quantization bits and the value of a phase shift belongs to the set Q = {0, 2π/Q, 4π/Q,...,2π(Q-1)/Q} <cit.>. 
Hence, the SNR received at the GUE is obtained as



    _   =  P_/σ_^2|
            ∑_n=1^N√(__n) h__nβ_n √(__n) h__n e^jϑ__n|^2 
       ≊^(a)_|
            𝐡_^ HΘ_𝐡_|^2, ∀ n ∈ N,


where Θ_≜( β_1 e^jϑ__1, …, β_N e^jϑ__N) represents the phase-shift matrix, 
    _≜P_/σ_^2ℓ__nℓ__n for any n ∈ N when the RIS's maximum dimension being considerably smaller than the distances d__n and d__n <cit.>. It is noted that our analysis is applicable when the RIS dimensions are non-negligible. 
    In this case, we define ℓ_≜1/N( ∑_n=1^N√(ℓ__n)β_n √(ℓ__n))^2 as the average effective A2G path loss, and the terms β_n and _ must be modified to √(ℓ__n)β_n √(ℓ__n)/√(ℓ_) and P_ℓ_/σ_^2, respectively.





§ STATISTICAL CHARACTERIZATION OF G2A AND RIS-ASSISTED A2G COMMUNICATION


In this section, we study the statistical characteristics of the G2A and A2G communications. Specifically, we derive the distribution, i.e., the PDF and CDF, of the G2A SNR. For the A2G communication, we propose to characterize the A2G channel distribution conditioned on the estimated CSI. Then, the PDF and CDF of the A2G SNR are derived.
    Moreover, we focus on the distributions of the LoS communication scenario since the NLoS communication scenario can be analogously obtained by setting the Rician-K factors to zeros <cit.>.





 §.§ Statistical Characterization of G2A Communication


We denote h_≜ (ρ_𝐡_ + ρ̅_𝐳_ )^ H𝐰_ as the effective G2A small-scale fading channel.
    For practical purposes, we consider a reasonable assumption that 𝐰_ and 𝐳_ are statistically independent. 
Subsequently, h_ is characterized as h_=^d 𝒞𝒩( ρ_𝐰_^ H𝐡_, ρ̅_^2 ). 
    We find that h_ exhibits the properties of the Rician fading, where ρ_𝐰_^ H𝐡_ and ρ̅_^2 represent the effective LoS component and the average power of the NLoS components, respectively.
Applying the delayed CSI-based MRT beamforming, h_ is further characterized as


    h_=^d 𝒞𝒩( 
            ρ_‖𝐡_‖, ρ̅_^2 
        ).


    
Under the delayed CSI-based MRT beamforming, 𝐰_^ H𝐡_ follows the noncentral chi-square (χ^2) distribution with scale factor 1/2(K_+1), 2M degrees of freedom (d.o.f.), and noncentrality parameter 2MK_.
    Such a fading falls into the category of fluctuating LoS fading <cit.>. However, <cit.> only considers that the magnitude of the LoS component follows a Nakagami-m distribution, forming the Gamma-shadowed Rician fading channel. Hereafter, we use the notation χ_k^2( λ) to specify the noncentral χ^2 RV with scale 1/2, 2 k d.o.f., and noncentrality parameter 2λ.
The PDF and CDF of χ_k^2( λ) are given by <cit.>


    f_χ^2(k, λ; x) 
           =   e^-x-λ( x/λ)^k-1/2 I_k-1( 2√(λ x)),  x > 0, 
    
    F_χ^2(k, λ; x) 
           =   1 - Q_k( √(2λ), √(2 x)),  x > 0,


where Q_k(a, b) = a^1-k∫_b^∞ x^k e^-x^2+a^2/2 I_v-1(ax) d x denotes the Marcum Q-function. For the noncentral χ^2 distribution with scale factor , denoted as χ_k^2( λ), we adopt the scaling property of PDFs and CDFs as f_χ^2(k, λ; x) = 1/ f_χ^2(k, λ; x/) and 
F_χ^2(k, λ; x) = F_χ^2(k, λ; x/), respectively, for > 0.


The distribution of χ_k^2( λ) is directly related to the κ-μ-squared distribution with fading parameters (Ω, κ, μ). Specifically, the two distributions can be transformed into each other by setting the parameters as follows:


    + To obtain the κ-μ-squared distribution from the noncentral χ^2 distribution, set = Ω/μ (1+κ), λ = μκ, and k = μ.
    
    + To obtain the noncentral χ^2 distribution from the κ-μ-squared distribution, set μ = k, κ = λ/k, and Ω = (k + λ).









 
Under the delayed CSI-based MRT, the PDF of the G2A SNR is formulated as


    f__(x)
           =  e^ -M K_ρ_^2/K_ρ̅_^2+1
        -(K_+1)x/_ (K_ρ̅_^2 + 1) (
            K_+1/K_ρ̅_^2+1)^M
        
         ×∑_i=0^M-1M-1i(ρ̅_^2)^i (ρ_^2)^M-i-1/_^M-i( x/Ξ_)^M-i-1/2
               ×
        I_M-i-1(
             2 √(Ξ_ x)/K_ρ̅_^2+1),  x > 0,


where Ξ_≜ M K_ρ_^2 (K_ +1) _^-1 and γ̅_ = P__/σ_^2. 












For convenience, the proof of (<ref>) is presented in Appendix <ref>, which involves many mathematical manipulations based on the Laplace transform of _.
























Based on (<ref>), we find that the G2A SNR under the delayed CSI-based MRT follows a mixture of M noncentral χ^2 distributions, where the m^th noncentral χ^2 distribution is identically distributed to χ_M-m(λ), for m = 0, 1, …, M - 1, where 
    λ≜ M K_ρ_^2/K_ρ̅_^2 + 1 and
    ≜_K_ρ̅_^2 + 1/K_ + 1.
The mixing proportion of each m^th noncentral χ^2 distribution is 


    ϰ_m ≜M-1m (ρ̅_^2)^m (K_+1)^m/(K_ρ̅_^2+1)^M-1 (ρ_^2)^M-m-1.



Hence, based on the CDF noncentral χ^2 RV's CDF in (<ref>), we formulate the CDF of the G2A SNR as


    F__(x) 
        =   ∑_m=0^M-1ϰ_m 
        F_χ^2( 
            M - m, λ; x/), x > 0.








 §.§ Statistical Characterization of RIS-assisted A2G Communication


    
As outlined in <cit.>, the phase-shift configuration ϑ__n = -∠[ 𝐡_]_n - ∠[ 𝐡_]_n, for all n ∈ N, is optimal in the absence of channel aging.
    However, adopting the aforementioned configurations in the presence of channel aging requires the current CSI 𝐡_ and 𝐡_, which are usually unavailable in practice since z_ and z_ are unknown.
    This inspired us to focus on PSCs that are independent of z_ and z_.

Hereafter, we rewrite the phase-shift matrix as 

Θ_ = Θ_^ Hβ_Θ_,


where
    β_ = ([β_1, β_2, …, β_N]) is the RIS amplitude reflection coefficients matrix,
    Θ_ = ( [e^jϑ__1, …, e^jϑ__N] )
    and Θ_^ H = ( [e^jϑ__1, …, e^jϑ__N] ) are the phase-shift matrices of the UAV-to-RIS and RIS-to-GUE channels, respectively.
Thus, the phase-shift of the reflecting element ∀ n ∈ N is rewritten as ϑ__n = ϑ__n - ϑ__n.    
Hence, we assume that the configured channel vectors 
    μ_≜ (β_)^1/2Θ_ h_ and 
    μ_≜ (β_)^1/2Θ_ h_ are characterized as


    μ_   =^d   
        𝒞𝒩( 
            ρ_  (β_)^1/2Θ_ h_, 
            ρ̅_^2 β_),
    
    μ_   =^d  
        𝒞𝒩( ρ_ (β_)^1/2Θ_^ H h_, 
        ρ̅_^2 β_)
    ,


respectively. Here, (<ref>) and (<ref>) suggest that


    
  * Focusing solely on the phase shifts Θ_ and Θ_ configurations, we can adjust the value at which the channel power is concentrated.
    
  * The reflection coefficient β_, not only influences the value at which the channel power is concentrated, but also shapes the spread or dispersion of the channel's power distribution.



Let μ_≜ (β_)^1/2Θ_ h_ and
    μ_≜ (β_)^1/2Θ_^ H h_, 
the PDF of μ_ conditioned on μ_ is given by


    f_μ_ | μ_(𝐲)
        =   |β_|^-1/π^N ρ̅_^2N 
        e^- 1/ρ̅_^2 
            (𝐲 - ρ_μ_)^ Hβ_^-1 (𝐲 - ρ_μ_),


for 𝐲∈ℂ^N.



  §.§.§ Effective Air-to-Ground Channel

    In the following, we first study the statistical characterization of the effective A2G channel through the RIS.



Denoting Z ≜μ_^ Hμ_ as the effective A2G channel, the CF of Z conditioned on the estimated CSI, i.e., ĥ__n and ĥ__n, n = 1, 2, …, N, and the phase shifts, i.e., ϑ__n, n = 1, 2, …, N, is obtained as































    Φ_Z|μ_, μ_(jω)
        =  ∏_n=1^N(
            1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2
        )^-1
         ×exp{
            - | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2/1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2(
                ρ_^2/ρ̅_^2| ĥ__n|^2  
                + ρ_^2/ρ̅_^2| ĥ__n|^2 
            )
        .
    
                   .
            + j ρ_ρ_( 
                ω^∗ĥ__nβ_n e^jϑ__nĥ__n)/1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2}, ω∈ℂ,






    We first derive Φ_Z_1, Z_2(jω_1, jω_2) in Appendix <ref> as the joint CF of Z_1 = (Z) and Z_2 = (Z). Then, applying the definition of the CF of complex RVs as Φ_Z(jω) = { e^j(ω^∗ Z)} = Φ_Z_1 ,Z_2(jω_1, jω_2), where ω = ω_1 + jω_2, for ω_1 ∈ℝ and ω_2 ∈ℝ, and after some mathematical manipulations, we obtain (<ref>). This thus completes the proof.


The CF of the effective A2G channel in Lemma <ref> provides the following results:


    
  * The joint CF of Z_1 and Z_2, denoted as Φ_Z_1, Z_2(jω_1, jω_2), is obtained as Φ_Z_1, Z_2(jω_1, jω_2) = Φ_Z(j ω_1, -ω_2).
    
  * The CFs of Z_1 and Z_2 are
        Φ_Z_1(jω_1) = Φ_Z_1, Z_2(jω_1, 0) = Φ_Z(j ω_1) and 
        Φ_Z_2(jω_2) = Φ_Z_1, Z_2(0, jω_2) = Φ_Z(- ω_2), respectively.
    
  * We find that Φ_Z_1(jω_1) Φ_Z_2(jω_2) Φ_Z_1, Z_2(jω_1, jω_2), which implies that Z_1 and Z_2 are correlated.
    
  * The effective A2G channel Z can be characterized as the sum of non-IID RVs. Specifically, we can rewrite Z as Z = ∑_n=1^NZ^(n), where Z^(1), …, Z^(N) are non-IID RVs. 
    Comparing with (<ref>), we determine that Z^(n) = h__nβ_n e^jϑ__n h__n.



Let Z_1^(n)≜( Z^(n)) and 
    Z_2^(n)≜( Z^(n)), we propose the following characterizations:


    Z_1^(n)   =^dβ_n
        [ 
              | Z__n + Z__n/√(2) + X_1^(n)|^2 
        .
    
             .
            - | Z__n - Z__n/√(2) + Y_1^(n)|^2
        ]
        ρ̅_ρ̅_/2 , 
    
    Z_2^(n)   =^dβ_n
        [ 
            | Z__n - j Z__n/√(2) + X_2^(n)|^2 
        .
    
             .
            - | Z__n + j Z__n/√(2) + Y_2^(n)|^2
        ]
        ρ̅_ρ̅_/2 ,


where 
    Z__n = ρ_/ρ̅_ĥ__n e^j ϑ__n,
    Z__n = ρ_/ρ̅_ĥ__n e^-j ϑ__n, and 
    X_1^(n), Y_1^(n), X_2^(n), and Y_2^(n) are independent and identically distributed standard complex circularly symmetric Gaussian RVs.



Here, we utilize the notation =^d, rather than ≈^d, due to exact matches between the right-hand sides (RHSs) and left-hand sides (LHSs) of (<ref>) and (<ref>). 
    However, it is important to note that (<ref>) and (<ref>) do not account for correlations between Z_1^(n) and Z_2^(n). Nonetheless, these correlations are insignificant in characterizing the distribution of |Z|^2 when N is sufficiently large.







Under channel aging, for sufficiently large N, the RIS-assisted A2G fading is characterized as

lCl
Z      →^d    
    𝒞𝒩( 
        μ_Z ≜ ρ_ ρ_ 
        h_^H Θ_ h_, 
        σ_Z^2 ≜ 
        ρ_^2 ρ̅_^2 
        ‖β_ 𝐡_ ‖^2
    

                  + ρ̅_^2 ρ_^2 
        ‖β_ 𝐡_ ‖^2
        + ρ̅_^2 ρ̅_^2 
        ‖β_ ‖^2_F
    ). 





We find that


    Z^(n)_1    =^d 
        χ_1^2 ( X_1 ) 
            -  χ_1^2 ( X_2 )/2β_n ρ̅_ρ̅_.


where X_1 ≜1/2| Z__n + Z__n|^2 
    and X_2 ≜1/2| Z__n - Z__n|^2.
    
For a noncentral χ^2 RV modeled as Y =^d χ_k^2(X), the conditional mean and conditional variance of Y given X are (Y|X) = k + X and (Y|X) = 2k+4X, respectively. 
    The total mean of Y is directly computed by taking the expectation of (Y|X) over X as (Y) = ((Y|X)) = k + (X). 
Consequently, the total mean of Z^(n)_1 is computed as


    [ Z^(n)_1 ]
        =  [ X_1 - X_2 ]/2β_n ρ̅_ρ̅_.



The total variance of Y follows the law of total variance as (Y) = ((Y|X)) + ( (Y|X) ) <cit.>. 

Hence, the total variance of Z^(n)_1 is computed as


    [ Z^(n)_1 ] 
        =  2 ( 1 + X_1 + X_2 )
            + ( X_1 - X_2 )/4β_n^2 ρ̅_^2 ρ̅_^2.



Hence, the total mean and total variance of Z_1 are computed as


    (Z_1)    = [ _n=1^N Z_1^(n)]
        =^(a)_n=1^N[ Z_1^(n)]
        = [(μ_Z)], 
    (Z_1) 
           = [ _n=1^N Z_1^(n)] 
        =^(b)_n=1^N[ Z_1^(n)]
    
       = [(μ_Z)] + [σ_Z^2],


respectively, where (a) is based on the linearity of expectation, and (b) is due to the fact that Z^(n)_1 are mutually independent, as shown in (<ref>).

The mean and variance of Z_2 are analogous to those of Z_1 in (<ref>) and (<ref>), respectively, i.e., by simply replacing (μ_Z) with (μ_Z).
    
    According to the CLT, Z →^d 𝒞𝒩( [Z], [Z] ), where [Z] = [μ_Z] and [Z] = [μ_Z] + [σ_Z^2]. However, the CLT necessitates an asymptotically large N. 
This motivates us to explore the relationships σ_Z^2 →[ σ_Z^2 ] and μ_Z →^d  CN( [μ_Z], [μ_Z] ), resulting in (<ref>). In this manner, we conclude the proof of Proposition <ref>.




  §.§.§ Effective Air-to-Ground Signal-to-Noise Ratio



Based on Proposition <ref>, we note that the distribution of |Z|^2 
in the ideal scenario ρ_^2 = ρ_^2 = 1 is simplified as


    |Z|^2 =^d |  h_^ HΘ_ h_|^2,


which can be effectively characterized by the noncentral χ^2 and generalized-K-squared distribution <cit.>.


When ρ_^2 → 0 and ρ_^2 → 0, e.g., the UAV and GUE move with very high speed, the distribution of |Z|^2 is rewritten as |Z|^2 →^d ( ∑_n=1^Nβ_n^2 ) χ_1^2. 
    In this case, the effective A2G channel exhibits Rayleigh-like fading. Hence, configuring the phase shifts Θ_ and Θ_ is ineffective in highly dynamic environments. Nonetheless, increasing the number of reflecting elements can yield considerable performance gains without the need for effective PSCs.

Most importantly, when ρ_^2, ρ_^2 ∈ (0, 1), the A2G SNR can be characterized as


    _→^d_ |Z|^2 →^d_σ_Z^2 
        χ_1^2( |μ_Z|^2/σ_Z^2) 
        ≜_σ_Z^2 γ_Z,


for sufficiently large N.
Here, the scale component σ_Z^2 is the linear combination of non-IID noncentral χ^2 RVs, which follows the generalized noncentral χ^2 distribution <cit.>. 
    While the exact PDF (or CDF) of the generalized noncentral χ^2 distribution remains unknown, the distribution is well characterized by a noncentral χ^2 distribution <cit.>.
    The approach followed to match the true distribution of an arbitrary RV Y to the noncentral χ^2 distribution is presented in Lemmas <ref> and <ref> as follows. 



For Y =^d χ_k^2(X), the mean, the second central moment (variance), and the third central moment of Y are, respectively, presented as



    [Y]    = (k+[X]) , 
    [Y]    = (k + [X] + 2[X]) ^2, 
    μ_3[Y]    = ( 2k+6[X]+6[X] + μ_3[X] ) ^3.









We adopt the LTC <cit.>, where the law of total variance, utilized in the proof of Proposition <ref>, and the third central moment are special cases of LTC.
    Here, the third central moment of Y is derived as <cit.>


    μ_3(Y)    = 3 [ ( Y|X ), ( Y|X ) ] 
    
           
        + [ μ_3(Y|X) ] + μ_3[ ( Y|X ) ],


where the conditional moment and central moments are computed as (Y|X) = (k+X), (Y|X) = (1+2X) ^2, and μ_3(Y|X) = (2+6X) ^3, respectively.
Furthermore, we have [μ_3(Y|X)] = (2+6[X]) ^3, μ_3[(Y|X)] = μ_3[X] ^3, and [(Y|X), (Y|X)] = 2 [X] ^3.
    Plugging the foregoing results into (<ref>) yields (<ref>), completing the proof of Lemma <ref>.




When X is deterministic and X = λ, the mean, variance, and third central moment of Y =^dχ_k^2(X) in Lemma <ref> are obtained by treating [X] = λ, [X] = 0, and μ_3[X] = 0, which yields




    [Y]    = (k+λ) , 
    [Y]    = (k+ 2λ) ^2, 
    μ_3[Y]    = ( 2k+6λ) ^3.







When matching the distribution of Y =^dχ_k^2(X) to the noncentral χ^2 distribution, denoted as Y →^d_Y χ_k_Y^2(λ_Y), the moment-based estimators for the parameters k_Y, λ_Y, and _Y are derived as




  * Case ^2[Y] > 1/2[Y] μ_3[Y] > 3/4^2[Y]:
    
    _Y    = [Y]/[Y][Y] - √(^2[Y] - 1/2[Y] μ_3[Y])/[Y],
                
    λ_Y    = [Y]/_Y - [Y]/_Y, and 
            k_Y = [Y]/_Y - λ_Y.


  * Case ^2[Y] ≤1/2[Y] μ_3[Y] or [Y] μ_3[Y] ≤3/2^2[Y]:
    
    _Y = [Y]/[Y],
            k_Y = ^2[Y]/[Y], and λ_Y = 0,



where [Y], [Y], and μ_3[Y] are derived using (<ref>) if X is an RV or using (<ref>) if X is deterministic.



In order to match the distribution of Y to the noncentral χ^2 distribution, we solve the system of equations presented in (<ref>) for k ≡ k_Y, λ≡λ_Y, and ≡_Y. 
Applying the changes of variables x ←_Y, y ←_Y k_Y, and z ←_Y λ_Y yields a new system of quadratic equations in terms of x, y, and z. Solving this new system yields (<ref>) and (<ref>), which concludes the proof of Lemma <ref>.





The true distribution of the scale component σ_Z^2 can be characterized by the noncentral χ^2 distribution as


    σ_Z^2 =^d_Z χ_N k_ Z^2 ( Nλ_ Z).





The above proposition is based on matching the true distribution of σ_Z^2 to the noncentral χ^2 distribution using the method in Lemma <ref>, for Y ≡σ_Z^2, k_Y ≡ N k_ Z, λ_Y ≡ N λ_ Z, and _Y ≡_Z. It is noted that the mean, variance, and third central moment of σ_Z^2 are derived as




    [ σ_Z^2 ] 
           =  ( 1 - ρ_^2 ρ_^2 ) 
        ‖β_‖_ F^2, 
    μ_k[ σ_Z^2 ] 
           =^(a)[ 
            ( ρ_^2 ρ̅_^2 )^k μ_k( | ĥ__n|^2 )
    
           
            +   ( ρ̅_^2 ρ_^2 )^k μ_k( | ĥ__n|^2 )
        ]
        ‖β_^k ‖_ F^2,



where μ_2(·) ≡(·) and (a) is due to the fact that the k^th central moments of a linear combination of non-IID RVs is written as μ_k(a X + b Y + c) = a^kμ_k(X) + b^k μ_k(Y), for k = 2 and k = 3, where X and Y are non-IID and a, b, and c are constants. Since | ĥ_|^2 follows a noncentral χ^2 distribution, its variance and third central moment are given by (<ref>), with k = 1, λ = K_, and = 1/K_+1.
    This completes the proof of Proposition <ref>.


The second case of Lemma <ref> implies that the true distribution of σ^2_Z can be matched by the χ^2 distribution with 2 N k_ Z d.o.f., which is identical to the Gamma distribution with shape N k_ Z and unit scale. 
In general, the PDF and CDF of a χ^2-distributed RV with 2 k d.o.f. are given by <cit.>


    f_χ^2(k; x)    = 1/Γ(k) x^k-1 e^-x, x > 0, 
    
    F_χ^2(k; x)    = 1/Γ(k)γ(k; x), x > 0,


respectively, where γ(k; x) is the lower incomplete Gamma function <cit.> and Γ(k) is the Gamma function <cit.>. 

We recall from (<ref>) that |Z|^2 →^d σ_Z^2 γ_Z for sufficiently large N. 
    Here, we have [γ_Z] = 1 + [X] + 2 [X] > 1 with X ≡|μ_Z|^2/σ_Z^2, which is obtained from (<ref>). 
Given that [σ_Z^2] is considerably less than 1, as calculated in (<ref>), we find that [γ_Z] is significantly larger than [σ_Z^2].
    Consequently, the statistical properties of |Z|^2 are predominantly influenced by the component γ_Z = χ_1^2( |μ_Z|^2/σ_Z^2) in 
    (<ref>). 
However, the true distribution of |Z|^2 is not solely determined by the individual distributions of σ_Z^2 and γ_Z but also by the correlation between them. 
    When these components are separately matched to other distributions, the existing correlation is overlooked. 
To tackle this problem, we propose the following matching:



    |Z|^2    →^d σ_Z^2 _ R, as N ≫ 1,



subject to [ |Z|^2 ] = [ σ_Z^2 _ R] and [ |Z|^2 ] = [ σ_Z^2 _ R], 
where σ_Z^2 and _ R are non-IID. 
    We highlight that _ R→^d γ_Z as ( σ_Z^2, γ_Z ) = ( |μ_Z|^2 ) - ( σ_Z^2 ) ( |μ_Z|^2/σ_Z^2) → 0.




The dominant component _ R can be characterized by the noncentral χ^2 distribution as


    _ R=^d 
        _ Rχ_k_ R = 1^2( λ_ R).





The mean and variance of |Z|^2 are derived as [ |Z|^2 ] = [ σ_Z^2 ] + [ |μ_Z|^2 ] and 
    [ |Z|^2 ] = [ σ_Z^2 ] + [ | μ_Z |^2 ] + [ σ_Z^4 ] + 2 [ σ_Z^2 ] [ | μ_Z |^2 ], where [ σ_Z^4 ] = [ σ_Z^2 ] + [ σ_Z^2 ].

Since σ_Z^2 and _ R are statistically independent, the mean and variance of _ R are computed as


    [ _ R] 
          =  [ |Z|^2 ]/[ σ_Z^2 ]
       = [ σ_Z^2 ] + [ | μ_Z |^2 ]/[ σ_Z^2 ], 
    [ _ R] 
          =  [ |Z|^2 ] - [ σ_Z^2 ] ^2[ _ R]/[ σ_Z^2 ] + ^2[ σ_Z^2 ],


where the mean and variance of σ_Z^2 are computed using (<ref>). 
Then, denoting μ̂__n≜ĥ__n^∗ĥ__n e^jϑ__n, the mean of |μ_Z|^2 is


    [ |μ_Z|^2 ]
           =  _n=1^Nβ_n^2 ρ_^2 ρ_^2 
            [ | μ̂__n|^2 ] 
        + 2 β_n ρ_^2 ρ_^2
        
         ×_m=n+1^N
        β_m
        ( [
            μ̂__nμ̂__m^∗] ),


where, for L_r(x) denoting the r^th order Laguerre polynomial <cit.>, we have <cit.>

lCl
[ | μ̂__n |^2 r ] 
    =
    L_r(-K_) L_r(-K_)/ [K_+1]^r [K_+1]^r 
    r^2 Γ^2(r). 



Obtaining the exact variance of |μ_Z|^2 is quite complex, which motivates us to adopt the Taylor series to derive an accurate approximation. By adopting the Taylor series, where ∂/∂(z) |z|^2 = 2 (z), and ∂/∂(z)^2 |z|^2 = 2 for a complex-valued variable z, we obtain 



    [ | μ_Z |^2 ]
           →^2[ μ_Z ],
         if [ μ_Z ] ≫[ (μ_Z) ] → 0, 
       → 4 ( [ μ_Z ] )
        [ ( μ_Z ) ], otherwise,


where [ (μ_Z) ] →[ | μ_Z |^2 ] -| [ μ_Z ] |^2 if ( [ μ_Z] ) → 0, otherwise [ (μ_Z) ] →1/2[ [ | μ_Z |^2 ] -( ^2[ μ_Z ] ) ]. Here, 
[ | μ_Z |^2 ] is derived in (<ref>) and | [ μ_Z ] |^2 is derived as


    | [ μ_Z ] |^2
           =  | 
            _n=1^Nβ_n ρ_ρ_
            [ μ̂__n]
        |^2.



It is noted that the value of the expectations in (<ref>) and (<ref>) depend on the PSC at the RIS, i.e., different PSCs yield different expectations.
    Based on the foregoing results, we obtain the mean and variance of _ R in (<ref>) and (<ref>), respectively. 
Afterwards, _ R and λ_ R are derived using (<ref>) and (<ref>) for Y ≡_ R, ≡_ R, and λ≡λ_ R. This yields


    _ R   = [ _ R] - √(^2[ _ R]-[ _ R]), 
    λ_ R   =  1/_ R√(^2[ _ R]-[ _ R]),


which completes the Proof of Proposition <ref>.














Based on Propositions <ref> and <ref>, we characterize the RIS-assisted A2G SNR as the product of two noncentral χ^2 RVs. 
    The PDF and CDF of this distribution can be represented by an infinite series of modified Bessel functions <cit.>, at the cost of significantly increasing computational complexity. 
In Theorem <ref>, we propose another framework to derive the distribution of the RIS-assisted A2G SNR.



Under arbitrary PSCs, for a sufficiently large N, the RIS-assisted A2G SNR can be characterized by the product of two non-IID noncentral χ^2 RVs. 
    In this case, the PDF of the RIS-assisted A2G SNR is obtained as

lCl
f__(z) 
       =      ∑_k_1=A_1^B_1 p_K_1(k_1)
    ∑_k_2=A_2^B_2 p_K_2(k_2)
    ( _R _Z _ )^-1
    

          ×f_K( 
        Nk_Z+k_1, k_R+k_2; z/_R _Z _
    ),
    
    



where p_ K_1(k_1) = (N λ_ Z)^k_1 e^-N λ_ Z/k_1! and 
p_ K_2(k_2) = λ_ R^k_2 e^-λ_ R/k_2! are the 
probability mass functions (PMFs) of Poisson-distributed RVs with means N λ_ Z and λ_ R, respectively. Moreover, 


    f_K(a, α; x) = x^a+ α/2-12 K_a-α(2√(x))/Γ(a) Γ(α) = f_K(α, a; x),


for x > 0 specifies the PDF of the product of two non-IID χ^2-distributed RVs, which is also known as the generalized-K-squared distribution.








































































See Appendix <ref>.



In Theorem <ref>, the coefficients A_i and B_i, where i ∈{1, 2}, are chosen to satisfy ( A_i ≤ K_i ≤ B_i ) → 1.
    Some special instances include A_i = B_i = 0 when ( K_i) = 0 and A_i = 0 for p_ K_i(0) > 0.
For large arguments, i.e., large a (or α), we propose modifying (<ref>) as f_K(a, α; x) →[ a + α+(α-x/a)^2/2 - x/a]
    f_χ^2( α; x/a)/a
for a ≫ 1. 








 
Under arbitrary PSCs, the CDF of the RIS-assisted A2G SNR is obtained by replacing f_K(a, α; x) in (<ref>) with the CDF of the product of two non-IID χ^2 RVs. Specifically, denoting the aforementioned CDF as F_K(a, α; x) = F_K(α, a; x), we obtain

F_K(a, α; x)
    =   1/Γ(a) Γ(α)G^2,1_1,3(
        .
        [       1; a, α, 0 ]| x
    ), x > 0, 
where F_K(a, α; x) →1/Γ(α)γ( α; x/a) = F_χ^2( α; x/a) for a ≫ 1.






 §.§ Phase-Shift Configurations



From (<ref>) and (<ref>) in Proposition <ref>, we assert that the average A2G SNR, derived as


    [ _]
        =   _( [ σ_Z^2 ] + [ | μ_Z |^2 ] ),


is a increasing function of ( [ μ̂__nμ̂__m^∗] ), upper bounded by [ | μ̂__n| ]
    [ | μ̂__m| ]. Thus, configuring phase shifts to increase this value correspondingly improves the A2G SNR.

Considering PSCs where ϑ__n and ϑ__m are statistically independent for m  n, we have [ μ̂__nμ̂__m^∗] = [ μ̂__n] ^∗[ μ̂__m].
    This is the case for the delayed CSI-based PSC (DPSC) where we have ϑ__n = -∠μ̂__n for n ∈ N, which does not exploit phase shift values from other reflecting elements.
It is noted that the DPSC
    yields the maximum average A2G SNR since


    ( [ μ̂__nμ̂__m^∗] )
        =   [ | μ̂__n| ]
        [ | μ̂__m| ].



However, the DPSC comes with the trade-off of increasing computational complexity within the RIS, which becomes particularly significant as the number of RIS elements scales into the hundreds.
In practice, the RIS can utilizes other less CSI-demanding PSCs while minimizing performance degradation.
    For PSCs that disregard the information from NLoS components in the estimated CSI, we have


    [ μ̂__n]
        =   √(K_/K_+1K_/K_+1)h̅__n^∗h̅__nΦ_ϑ__n(j ω|_ω = 1),


where Φ_ϑ__n(j ω) is the CF of the phase shift ϑ__n for n ∈ N and ω∈ℝ.

Three common PSCs that fall into this category are:


    
  + Random PSCs: ϑ__n follows a specified distribution. The most common adopted distribution is the uniform phase-shift within the interval [a, b] ⊆ [-π, π] where we obtain Φ_ϑ__n(jω|_ω = 1) = sin b-sin a/b-a + j cos a-cos b/b-a.
    
    
  + Fixed-value (static) PSCs: ϑ__n is fixed and Φ_ϑ__n(j) = e^jϑ__n for n ∈ N. It is noted that ϑ_1 = … = ϑ__N = 0 depicts the communication scenario without PSCs.
    
    
  + LoS-based PSCs: ϑ__n is a function of the LoS component. Here, ϑ__l = - ∠h̅__n + ∠h̅__n yields the highest expected value. 






 §.§ Performance Analysis


For performance analysis, we consider the end-to-end outage probability (eOP), defined as the probability that end-to-end SNR drops below a fixed target average SNR threshold _th = 2^2 R - 1, where R [bps/Hz] denotes the target spectral efficiency (SE). The eOP is formulated as <cit.>


    P_^out(_th) 
           = {min[ _, _] < _th}
       =^(a)  1 - [ 1- F__(_th) ] 
            [ 1- F__(_th) ] ,


where (a) is the result of _ and _ being non-IID, with F__(x) given in (<ref>), and F__(x) presented in Corollary <ref>.
    
Additionally, when the BS transmits with sufficiently large power, we simplify the G2A SNR CDF as


    F__(x)
        →
        e^ - M K_ρ_^2/K_ρ̅_^2+1(K_+1)^M (ρ̅_^2)^M-1/(K_ρ̅_^2+1)^Mx/_,


for x > 0, and where (x)_n ≜Γ(x+n)/Γ(x) is the Pochhammer symbol. 

Moreover, when N is asymptotically large, e.g., hundreds of reflecting elements, the CLT yields σ_Z^2 →[ σ_Z^2 ]. Denoting 

Ω_≜_[ σ_Z^2 ] =P_/σ_^2 (1 - ρ_^2 ρ_^2) ∑_n=1^Nℓ__nβ_n^2 ℓ__n,

and since the normal (Gaussian) and the noncentral χ^2 distributions are related as
    χ_k_ R^2 ( λ_ R) →^d  N( k_ R + λ_ R, k_ R + 2λ_ R) when λ_ R is sufficiently large, the A2G SNR CDF can be simplified as

lCl
F__(x)
    →1 - Q( x/_R Ω_ - ( k_R + λ_R )/√(k_R + 2λ_R) ), x>0,
 


where Q(x) is the Q-function. It is noted that (<ref>) is applicable even when the maximum dimension of the RIS is non-negligible, e.g., with hundreds of reflecting elements.





§ MOBILITY MODEL










    















In this section, we consider a case study where the GUE follows the Random Waypoint Mobility (RWM) model. The RWP model is adopted to capture the random and unpredictable nature of the GUE's motion <cit.>. 
    Moreover, the mobility of the UAV is characterized by the Reference Point Group Mobility (RPGM) model to introduce a degree of deviation in the UAV's motion relative to the GUE's motion <cit.>.





 §.§ Random Waypoint Mobility (RWM) Model


The RWM model assumes that the GUE is initially moving from a source waypoint 𝐰^_(t) [m] to a destination waypoint 𝐰^_(t) [m].
    Here, the waypoints are statistically IID and uniformly distributed within the same active area _ <cit.>.
During the movement from 𝐰^_(t) to 𝐰^_(t), the GUE's velocity 𝐯_(t) ∈ℝ^3 [m/sec] remains constant and is determined as


    𝐯_(t) 
        =   v_(t) 𝐰^_(t)-𝐰_^(t) /‖𝐰^_(t)-𝐰_^(t) ‖≜ v_(t) 𝐫^_(t),


where v_(t) [m/sec] is the GUE's speed, which is uniformly distributed within the interval [v_min, v_max], where v_min [m/sec] and v_max [m/sec] are the minimum and the maximum GUE speeds, respectively, and
𝐫_^(t) ∈ℝ^3 represents the GUE's Cartesian direction of movement.

    Upon arring at 𝐰^_(t), the GUE pauses (stays still) with a probability p_pause for a duration of τ_pause(t), uniformly distributed within the interval [τ_min, τ_max] [sec], where τ_min and τ_max are the minimum and maximum pause durations, respectively. For a probability of 1 - p_pause, the GUE moves to a new source waypoint 𝐰^_(t+Δτ(t)) = 𝐰^_(t) and destination waypoint 𝐰^_(t+Δτ(t)) following the previously presented procedure.





 §.§ Reference Point Group Mobility (RPGM) Model

In the RPGM model <cit.>, the UAV is initially positioned uniformly within a circular area _ centered at a predefined reference point 𝐩^_(t) ∈ℝ^3. 
    This reference point has the same motion as the GUE, i.e., 𝐩_(t, τ) = 𝐩^_(t, τ) <cit.>. 
However, due to imperfect alignment, the UAV's true motion deviates from the reference point by the motion vector Δ𝐩_(t, τ) following a uniform distribution. 
    Therefore, the time-varying velocity of the UAV is

𝐯_(t) = lim_τ→ 0𝐩_(t, τ)/τ = 𝐯_(t) + Δ𝐯_(t),

where Δ𝐯_(t) [m/sec] is the random deviation in the UAV's velocity compared to the GUE's velocity, Δ v_(t) ≜‖Δ𝐯_ (t) ‖ denotes the UAV's speed deviation, which is uniformly distributed within the interval [Δ v_min, Δ v_max].





 §.§ Adaptive Target Spectral Efficiency

When both the GUE and the UAV are in motion, the eOP widely fluctuate over time under a fixed target SE policy.
    This fluctuation poses a challenge to maintaining a consistent and reliable communication system. 
To combat this, we introduce an adaptive target approach designed to stabilize the eOP.
    This method ensures that the eOP consistently stays below a predetermined threshold L, increasing robustness against the unpredictable mobility of the UAV and GUE.

The adaptive target SE, denoted as R [bps/Hz], is obtained by solving the eOP constraint  
    P^out_𝖾2𝖾( _th) ≤ L for _th, thus R = log_2(1+_th)/2.
Here, we adopt the following steps <cit.>


    i) Solve F__( _) = L for _ with F__(x) given in (<ref>).
    
    ii) Solve F__( _) = L for _ with F__(x) given in (<ref>).
    
    iii) Compute _th = min[ _, _], yielding 


    _th   =  min[
            e^K_ρ̅_^2+1/M K_ρ_^2( 
                K_+1/K_ρ̅_^2+1)^M
            ( ρ̅_^2 )^M-1_ L,
    
                 Ω_( Q^-1(1 - L) √([ _ R]) + [ _ R] )
        ],


where Q^-1(x) is the inverse Q-function, i.e., Q^-1(Q(x)) = x.





§ RESULTS AND DISCUSSIONS


This section presents benchmark results for the developed PDFs and CDFs of G2A and A2G SNRs, alongside the eOP and the target SE, not only to validate the correctness of the introduced Propositions and the derived Theorems, but also to provide a deeper understanding of the system behavior under channel aging.
    We highlight that the results in this section can be replicated using MATLAB. 
Moreover, for the computations of the noncentral χ^2 distributions in Theorem <ref> and Corollary <ref>, we employ the built-in functions 𝚗𝚌𝚡2𝚌𝚍𝚏 and 𝚗𝚌𝚡2𝚙𝚍𝚏; while for χ^2 (or Gamma) distributions, the functions 𝚐𝚊𝚖𝚌𝚍𝚏 and 𝚐𝚊𝚖𝚙𝚍𝚏 are utilized.
    Unless stated otherwise, we present the adopted simulation parameters in Table <ref>. 
Moreover, the BS, UAV, RIS, and GUE's positions in the global coordinate systems (i.e., MATLAB's default coordinate system) are presented in Fig. <ref>.
    For practical purposes, we consider the 3GPP Urban Micro (UMi) standard to model the RIS-to-GUE channels' path loss <cit.>. Moreover, the 3GPP UMi-Airborne (UMi-AV) standard is adopted to model the BS-to-UAV (G2A) and UAV-to-RIS (A2G) path loss <cit.>.
It is noteworthy that at a flight altitude of 120 m adopted in Table <ref>, the LoS probability is 1.0 <cit.>.








































In Fig. <ref>, the simulated histograms and CDFs are obtained using the functions 𝚑𝚒𝚜𝚝𝚘𝚐𝚛𝚊𝚖 and 𝚎𝚌𝚍𝚏, respectively. 
    The comparison shows a strong agreement between the analytical and simulated results for different values of M. 
Moreover, increasing M shifts the G2A SNR's PDF and CDF to the right. Consequently, the G2A average SNR becomes higher, thereby enabling it to meet more stringent target SNR thresholds.



Fig. <ref> illustrates simulated and analytical results of the A2G SNR PDF presented in Theorem <ref>, where P_ = 0 dBm. 
    Here, we assume that the RIS utilizes the delayed-CSI based PSC and the average A2G SNR given in (<ref>).
As N increases, the normalized A2G SNR PDF becomes more concentrated around the mean; specifically, the PDF gets closer to 1 with a decreasing variance. 
    It is noted that the Kullback–Leibler divergence, defined as D_KL=∫_-∞^∞ f__(x) log( f__(x)/f'__(x)) d x <cit.>, in the right figure is produced by using MATLAB's built-in function 𝚛𝚎𝚕𝚊𝚝𝚒𝚟𝚎𝙴𝚗𝚝𝚛𝚘𝚙𝚢. 
As | D_KL| → 0, the A2G SNR's true PDF f__(x) converges to the derived PDF f'__(x), given in (<ref>).
    Notably, the proposed characterizations for the A2G SNR based on Propositions <ref> and <ref> are accurate even when the number of reflecting elements is small, i.e., N = 2^2 elements, with D_KL = 10^-2,



Fig. <ref> shows the RIS-assisted A2G SNR with the PSCs presented in Section <ref>. As can be observed, the A2G SNR's true CDF matches the analytical CDF presented in Corollary <ref> despite different PSCs.
    From (<ref>), we highlight that

_≤P_/σ_^2|
        ∑_n=1^N√(__n)| h__n| β_n √(__n)| h__n|
    |^2, 

where the right-hand side depicts the A2G SNR when the true CSI is known and ϑ__n = - ∠ h__n - ∠ h__n for n ∈ N. We refer to the aforementioned configurations for ϑ__n as the ideal PSC and present the corresponding A2G SNR CDF only as benchmarks.
    Furthermore, Fig. <ref> also demonstrates that a reduction in K_0 from 10 dB to 0 dB weakens the LoS component in both UAV-to-RIS and RIS-to-GUE channels. This attenuation results in a 14% performance  degradation when utilizing LoS-based PSCs as opposed to delayed CSI-based PSCs which yield a  6% loss.



Next, we observe the eOP against the transmission power in Fig. <ref>, where P_ = P_ = P [dBm].
    It is noted that simulated results beyond 10^-4 are unreliable since the number of simulation times is insufficient, e.g., 10^5 runs in our case, hence analytical results are preferred.
The figure also shows that increasing transmission power reduces the eOP, converging to the G2A performance limit, expressed as P^out_𝖾2𝖾(_th) → F__(_th) = (<ref>). Although more transmit antennas can improve this bound, the system encounters the lower bound faster as N increases.
    
    
    



Fig. <ref>(a) reveals that the average A2G SNR decreases as the sample index t_k increases, due to the decreasing correlation coefficients. This decline reaches the first local minimum at t_k = 2.4/2π T_s Δf^max_ = 2.4/2π T_s Δf^max_, which is a decreasing function of the GUE and UAV speed, and corresponds to zero correlation, i.e., ρ_ = ρ_ = 0.

    Interestingly, 
the average A2G SNR is increased to -11.2 dB from -22.2 dB when N is increased from 28^2 to 100^2 reflecting elements. 
    This value is derived as [ _] = [ _ |Z|^2 ] = Ω_ based on the result in Section <ref>.
However, a larger N tapers off the increase in the average A2G SNR, as shown in Fig. <ref>(b) since the RIS's dimension also increases significantly <cit.>.



In Fig. <ref>, we assume that channel estimation is initially performed at t_k = 0, shown for k ≥ 1. For subsequent locations, the average eOP at index k is P̅^out_𝖾2𝖾[k] = 1/k P^out_𝖾2𝖾(t_k, _th), where P^out_𝖾2𝖾(t_k, _th) denotes the eOP at the specific time instant t_k = 5000, for which the UAV and GUE are located at 𝐩_[t_k] and 𝐩_[t_k], respectively.
    The figure illustrates that, with an adaptive target SE, the eOP is maintained below the desired outage level of 10^-4, achieving significant stabilization, even as the GUE speed increases from 10 m/s to 40 m/s.
Without SE adaptation, however, the eOP is subject to considerable fluctuations, compromising communication stability and reliability over time.
    Notably, the adaptive target SE may be unnecessary for the initial ten locations, as the system already performs below the 10^-4 threshold with the fixed target SE set in Table <ref>.





§ CONCLUSION

In this paper, we have developed a comprehensive channel model that captures the time-varying fading, 3D mobility, Doppler shifts, and array antenna structures under the impact of channel aging. 
    Our analytical expressions for the G2A SNR PDF and CDF, based on the noncentral χ^2 distribution, demonstrated that the G2A SNR follows a mixture of noncentral χ^2 distributions. 
The statistical characteristics of the A2G communication is thoroughly studied, revealing that the A2G SNR can be modeled as the product of two independent noncentral χ^2 RVs. 
    Furthermore, we have obtained precise PDF and CDF expressions for the product of two non-IID noncentral χ^2 RVs to derive the A2G SNR distribution. 
Our results revealed that the RIS is effective against channel aging as long as the communication is carried out within a coherence time.
    Lastly, our adaptive SE method ensures stable system performance at satisfactory outage levels.





§ PROOF OF THEOREM <REF>



Applying the Laplace transform of _, we obtain

L__(s)
    =   e^-MK_(1+Δ_1 s)^M-1/(1+Δ_2 s)^M
    e^MK_1+Δ_1 s/1+Δ_2 s, 

where Δ_1 = γ̅_ρ̅_^2 and
    Δ_2 = γ̅_K_ρ̅_^2+1/K_+1. 

Then, the PDF of _ is the inverse Laplace transform of L__(s) from s-domain to x-domain. Hence,


    f__(x)
           = e^-MK_
        e^-x/Δ_2
        (Δ_1)^M-1 (Δ_2)^-M
        e^M K_Δ_1/Δ_2
        e^-x/Δ_3
    
             ×ℒ^-1{s^M-1/(s-Δ_3^-1)^M
            e^ MK_Δ_1/Δ_2 Δ_3 /s-Δ_3^-1;
            s, x
        }, 
        
    
           = e^-MK_
        e^-x/Δ_2
        (Δ_1)^M-1 (Δ_2)^-M
        e^M K_Δ_1/Δ_2
        e^-x/Δ_3
    
             × d^M-1/ d x^M-1 
        e^x/Δ_3ℒ^-1{e^ MK_Δ_1/Δ_2 Δ_3 1/s/s^M;
            s, x
        },


where Δ_3 ≜1/1/Δ_1-1/Δ_2 = Δ_1 Δ_2/Δ_2-Δ_1. Here, the frequency shifting property and the derivative in the x-domain property of the Laplace transform are used to obtain (<ref>) and (<ref>), respectively. 
    Using the Leibnitz's rule for higher order derivatives and <cit.>, the derivative in (<ref>) is derived as


    Ξ(x) 
       =  d^M-1/ d x^M-1 
    e^x/Δ_3 x^M-1/2I_M-1( 2 √(M K_Δ_1/Δ_2 Δ_3 x))/(M K_Δ_1 Δ_2^-1Δ_3^-1)^M-1/2
       = _m=0^M-1M-1m e^x/Δ_3 x^M-m-1/2/( M K_Δ_1 Δ_2^-1Δ_3^-1)^M-m-1/2
             ×
    I_M-m-1(
        2 √(M K_Δ_1 Δ_3 Δ_2^-1 x)).


Substituting (<ref>) into (<ref>) and after some mathematical manipulations, we obtain (<ref>). 





§ PROOF OF LEMMA <REF>

 

From the result in <cit.>, we find that Z_1 and Z_2, conditioned on μ_ and μ_, are non-IID, and are statistically characterized as
    Z_1 =^d𝒩( 
        ρ_(
            μ_^ Hμ_),
        ρ̅_^2/2‖β_μ_‖^2
    ) and
    Z_2 =^d𝒩( 
        ρ_(
            μ_^ Hμ_),
        ρ̅_^2/2‖β_μ_‖^2
    ),






















respectively. Hence, the joint CF of Z_1 and Z_2, conditioned on μ_ and μ_, is therefore expressed as


    Φ_Z_1,Z_2|μ_, μ_(jω_1, jω_2)
        =   exp[
            ρ_(
                jω_1 ( μ_^ Hμ_)
            .
        .
    
         .
        .
            + jω_2 ( μ_^ Hμ_)
        )
            - ω_1^2 + ω_2^2/4ρ̅_^2 
            ‖β_μ_‖^2
    ].



Taking the expectation of (<ref>) over μ_, conditioned on μ_, based on the PDF of μ_ in (<ref>) yields


    Φ_Z_1,Z_2|μ_, μ_(jω_1, jω_2)
    =   |β_|^-1/π^N ρ̅_^2Nexp[ - ρ_^2/ρ̅_^2μ_^ Hβ_^-1μ_]
    
         ×∫_𝐲∈ℂ^Nexp[
            - 𝐲^ HΣ𝐲
            +   ρ_/ρ̅_^2
                2 ( 𝐲^ Hβ_^-1μ_)
    
                       
            + j2 ρ_( 
                ω_1 - jω_2/2𝐲^ Hμ_)
        ]  d𝐲,


where Σ≜ω_1^2+ω_2^2/2ρ̅_^2 β_ + 1/ρ̅_^2β_^-1 is invertible and symmetric. Then, based on <cit.>, we obtain the following solution for the complex Gaussian integrals


    |Σ|/π^N∫_𝐱∈ℂ^Nexp[
            𝐱^ HΣ𝐱 
            + 2 ( 𝐱^ H𝐛)
            + j 2 ( 𝐱^ H𝐜)
        ]  d𝐱
    
    
    
    
    
           
        =   exp[ 
            ‖Σ^-1/2𝐛‖^2
            - ‖Σ^-1/2𝐜‖^2
            + j (𝐛^ HΣ^-1𝐜)
        ],


and since ω = ω_1 + jω_2, we have 


    Φ_Z|μ_, μ_(jω)
        =   | Σβ_|^-1/ρ̅_^2Nexp[
        -   ρ_^2/ρ̅_^2 ‖ h_‖^2 
    
         
        +   ρ_^2/ρ̅_^4‖Σ^-1/2β_^1/2 h_‖^2
            -   |ω|^2/4ρ_^2
            ‖Σ^-1/2β_^1/2 h_‖^2
    
         
        + j ρ_ρ_/ρ̅_^2(
                μ_^ HΣ^-1β_^-1μ_ω)
        ], ω∈ℂ.



After some mathematical manipulations, we obtain (<ref>). This completes the proof of Lemma <ref>.





§ PROOF OF THEOREM <REF>

 

The PDF of |Z|^2 is derived as f_|Z|^2(z) = [ 1/σ_Z^2 f__ R( z/σ_Z^2) ]. 
    By adopting the identity [g(X)] = ∫_x g(x) f_X(x) d x for the RV X with PDF f_X(x), we obtain


    f_|Z|^2(z)  
        =  ∫_0^∞ f__ R( z/x)/x f_σ_Z^2(x) d x.



We note that a noncentral χ^2 RV can be characterized as a central χ^2 RV with Poisson-distributed d.o.f. as χ_k^2(λ) =^d χ_ K+k^2, where K follows a Poisson distribution with mean λ. 
    Applying this property, we obtain σ_Z^2 =^d _Zχ_ K_1 + N k_ Z^2, where K_1 follows a Poisson distribution with mean N λ_ Z. 
Hence, the PDF of σ_Z^2 is rewritten as 
    f_σ_Z^2(x)
    =   ∑_k_1 = A_1^B_1p_ K_1(k_1)/_Z 
    f_χ^2( N k_ Z + k_1; x/_Z) for x > 0. Similarly, we obtain _ R=^d _ Rχ_ K_2 + k_ R^2, where K_2 follows Poisson distribution with mean λ_ R.
Moreover, the PDF of _ R can be rewritten as f__ R(x)
    =   ∑_k_2 = A_2^B_2p_ K_2(k_2)/_ R
    f_χ^2( k_ R + k_2; x/_ R) for x > 0. 

Plugging the foregoing results into (<ref>), then applying <cit.> and the scaling property f__(z) = _^-1 f_|Z|^2( _^-1 z ), we obtain (<ref>), which concludes the proof of Theorem <ref>.



IEEEtran




 Paper Title: TinyBird-ML: An ultra-low Power Smart Sensor Node for Bird Vocalization Analysis and Syllable Classification.











TinyBird-ML: An ultra-low Power Smart Sensor Node for Bird Vocalization Analysis and Syllable Classification 


     Lukas Schulthess1,
Steven Marty1,
Matilde Dirodi2,
Mariana D. Rocha2,



Linus Rüttimann2,
Richard H. R. Hahnloser2,
Michele Magno1


1Dept. of Information Technology and Electrical Engineering, ETH Zürich, Switzerland
2Institute for Neuroinformatics,
ETH Zürich, Switzerland

    August 5, 2024
=========================================================================================================================================================================================================================================================================================












§ ABSTRACT 
Animal vocalisations serve a wide range of vital functions. Although it is possible to record animal vocalisations with external microphones, more insights are gained from miniature sensors mounted directly on animals' backs. We  present TinyBird-ML; a wearable sensor node weighing only 1.4 g for acquiring, processing, and wirelessly transmitting acoustic signals to a host system using Bluetooth Low Energy. TinyBird-ML embeds low-latency tiny machine learning algorithms for song syllable classification.

To optimize battery lifetime of TinyBird-ML during fault-tolerant continuous recordings, we present an efficient firmware and hardware design. We make use of standard lossy compression schemes to reduce the amount of data sent over the Bluetooth antenna, which increases battery lifetime by 70% without negative impact on offline sound analysis. Furthermore, by not transmitting signals during silent periods, we further increase battery lifetime. 

One advantage of our sensor is that it allows for closed-loop experiments in the microsecond range by processing sounds directly on the device instead of streaming them to a computer. 
We demonstrate this capability by detecting and classifying song syllables with minimal latency and a syllable error rate  of 7%, using a light-weight neural network that runs directly on the sensor node itself. Thanks to our power-saving hardware and software design, during continuous operation at a sampling rate of 16 kHz, the sensor node achieves a lifetime of 25 hours on a single size 13 zinc-air battery.




Wireless sensor node, low power, audio compression, machine learning, bird monitoring, bluetooth low energy.





§ INTRODUCTION

Among all animal forms of communication, vocal expression contains much information about an individual's physiological and behavioral states <cit.>. Thus, to better understand animals' behaviours and their group dynamics, it is of utmost importance to study vocal patterns and their meanings <cit.>. Unfortunately, the unambiguous assignment of a sound to the individual that generated it can be a major challenge, especially when animals rapidly move and irregularly vocalize in the midst of background sounds. A possible solution to meet this challenge is to record animals with wireless animal-borne sensors to directly capture vocalizations at their source <cit.>.

Such animal-borne bio-loggers have been successfully used on large and mid-sized animals <cit.>.
For small animals such as songbirds, in particular zebra finches (Taeniopygia guttata), the design of animal-borne sensor nodes is inflicted by many constraints <cit.>. 

 
Design imperatives are avoidance of stress from the weight or shape of the sensor node, because otherwise the behavioral readout may be of no value <cit.>. For this reason, bio-loggers need to be comfortable, very small and light-weight <cit.>.


At the same time, to ensure an undisturbed observation period, a long battery lifetime is desired for chronic recordings. A day-long recording typically lasts 14 hours, which provides sufficient time for a battery change at night when birds do not sing and move <cit.>.


Traditionally, animal-borne sensors are based on frequency modulation (FM) to transmit data <cit.>. Although this approach can achieve minimal power consumption <cit.>, there are well-known issues in terms of signal quality and interference when more than a single transmitter is in the same area <cit.>. Moreover, FM transmitters allow only one-way communication from the sensor to the receiver, and there is no way to send information back to the sensor node on the bird <cit.>.

Recent technological advances in the fields of sensors, wireless communication, and power-efficient power MCUs Microcontrollers (MCUs) have paved the way for new approaches <cit.>. Switching from fully analog to digital solutions such as Bluetooth Low Energy (BLE) offers several advantages: communication is bidirectional and it allows the host station to send messages and commands to a sensor node, which is not possible with FM transmitters. Moreover, BLE allows simultaneous communication with multiple nodes and the on-board MCU is capable of performing complex mathematical operations such as running small neural networks to detect specific events in the data stream <cit.>. Such events in the field of bird vocalization analysis are the occurrence of syllables, which involves segmenting the syllables in time and identifying the syllable types <cit.>. Moving from post-processing to real-time on-device syllable detection enables new research opportunities.

Since longitudinal observations are preferable and frequent battery changes can cause stress to an animal, animal-borne devices must be power-optimized on both hardware and firmware levels. Energy-efficient compression algorithms as described in <cit.> enable compression with tolerable data loss. On-board compression therefore can contribute to improving the energy efficiency by reducing the transmitted data volume <cit.>. 

The paper presents TinyBird-ML, a miniaturized and power-optimized digital animal-borne sensor node that features a zero-power digital micro electromechanical system (MEMS) microphone and an inertial measurement unit (IMU). The node also hosts non-invasive electrodes in combination with a state-of-the-art electrostatic sensor to enable the acquisition of signals such as heart rate, respiration rate and muscle contractions which provides additional information for assessing the birds behaviour <cit.>.
The paper demonstrates the importance of on-board processing using tiny machine learning, proposing a light-weight neural network that fully runs on the TLE microcontroller and allows real-time signal analysis. We present a dual-stage classifier for detection of syllables and classification of their types. Due to the low-power design, including a "zero-power listening" acoustic sensor and on-board processing, TinyBird-ML achieves a lifetime of up to 25 h on a single 0.8 gram zinc-air battery. 




















§ SYSTEM ARCHITECTURE

TinyBird-ML has been designed with low power and energy efficiency in mind to maximize operating time whilst keeping the total weight and therefore the impact on birds' natural behaviors at a minimum. Furthermore, the sensor nodes' elongated shape is designed to not disturb a bird during flight.
<ref> shows a simplified block diagram of the proposed light-weight animal-borne sensor node for vocal monitoring. The system can be divided into three sub-parts:

<ref>; the core of the sensor node is the ANNA-B112 BLE, a module from ublox, <ref>; the sensors for data acquisition and <ref>; the power management based on a power-efficient boost-converter. Finally, the hardware design is described in <ref>.





 §.§ Bluetooth Low Energy Module

The ANNA-B112 BLE module from ublox is an ultra compact System in Package (SiP) based on the nRF52832 System on Chip (SoC) from Nordic Semiconductor. It integrates Radio Frequency (RF) matching as well as an internal antenna. The ARM Cortex-M4F microcontroller with 64 RAM and 512 flash memory can run at a clock frequency up to 64 and thus offers enough computational power and memory to perform on-board audio processing and to run small neural networks. To increase the computational efficiency, the integrated Digital Signal Processor (DSP) can be exploited for compression algorithms. With a current consumption of [per-mode = symbol]58
while running from flash memory, together with a Tx peak current of 5.3 at 0, the module offers outstanding performance in a small package. Its excellent trade-off between light weight, size, and low power consumption is the main reasons for selecting the ANNA-B112 BLE module.



 §.§ Sensors

For sound acquisition, we used the high performance digital piezoelectric MEMS microphone VM3011 from Vesper MEMS.
It features adaptive zero-power listening, allowing it to reduce the microphone current consumption down to 10 while still being capable of detecting sounds louder than the background noise level. This functionality allows TinyBird-ML to further minimize the power consumption while the bird is not vocalizing, for example after the stressful sensor node attachment.
As soon as the bird has recovered and the desired behavior (e.g. song) is recognized, the microphone generates a wake-up signal and the microcontroller leaves its power-saving idle state. 

The LSM6DSV16X, an IMU with integrated Qvar (Q = charge var = variation) electrostatic sensor from ST Microelectronics for non-invasive vital signs acquiring completes the sensor node. 
Both, MEMS microphone and IMU-Qvar sensor communicate over individual I2C peripherals and, depending on the application, can be power-gated using load switches.



 §.§ Power Management

TinyBird-ML is powered by a non-rechargeable A13 standard-size 1.45 zinc-air battery, which is commonly used in hearing aids. Such batteries benefit from a very high theoretical energy density of [per-mode = symbol]1086 
<cit.>, which is crucial for our application. To overcome the low cell voltage, the nano-power synchronous boost converter MAX17220 from Maxim is
integrated. It converts the low input voltage to the sensor node's operating voltage of 3. 
The boost converter's minimal input voltage of 400 allows to completely deplete the battery and, together with a high conversion efficiency of over 90, ensures an optimal power usage during operation.



 §.§ Hardware Design

The entire sensor node is realized on a rigid 4-layer PCB with a total thickness of 0.4. The zinc-air battery is held by a weight-optimized battery clip, allowing for a reliable contact, facilitating battery replacement and achieving device reusability. Two contacts on the sensor node's top side can be used to attach two Qvar electrodes for cutaneous measurements of electrocardiograms (ref. <ref>). 
For bird-borne sensor nodes, the overall weight is very crucial. The total weight of TinyBird-ML, including battery, amounts to 1.4.
<ref> summarizes the weight and size of each part.













§ FIRMWARE



Once TinyBird-ML is powered up, it starts to send advertising packets. After having set up a connection, the central node subscribes to the TinyBird-ML's BLE characteristic responsible for the audio data. The recording does not start immediately because the bird may need time to acclimate itself to the backpack. To save power during this period, the BLE Connection Interval (CI) is set to a high value to save transmission power.



 §.§ Smart Compression




In applications of vocal communication research, the main emphasis is on calls and song syllables. Therefore, to save energy, we tried to avoid transmission of sounds due to the movement of the bird and during silent periods altogether. Nevertheless, to allow for reconstruction of vocal output, we transmitted the temporal information about non-vocal periods. 
A simple approach is an amplitude threshold: acoustic signals below the threshold are classified as silent and lead to incrementation of a silence counter. Silent phases are thereby encoded as the counter magnitude. When the amplitude exceeds the threshold, the recording is started, compressed and send to the central, the silence counter is prepended to the audio block to allow correct reconstruction of the audio signal in the time domain.



 §.§ Syllable Classification using TinyML

Zebra finch calls and song syllables can span over half a second, and the gaps between them can be as short as a few milliseconds <cit.>. This indicates that no practical analysis window is small enough to always contain parts of just a single syllable and large enough to always include an entire syllable. To detect and analyze syllables, the data is windowed into non-overlapping blocks of a power-of-two samples, such that the Fast Fourier Transform (FFT) could be computed efficiently.
The signal blocks are processed by using two hierarchical classifiers. The first classifier analyses each block and decides, based on a single-layer perceptron with 16 Mel-frequency Cepstral Coefficients (MFCCs) as input, whether it contains parts of a syllable. This keeps the computational overhead minimal. 
The second classifier recognizes syllable types and is based on a mix of convolutional and fully connected layers, as illustrated in <ref>. By waiting until the end of the syllable before classifying its type, more information about the syllable can be gathered. Three evenly distributed blocks are selected between syllable on- and offset. Each block is then transformed into sixteen MFCCs and used as input to the syllable classifier network. After nonlinear activation functions, the activations are flattened and fed through a fully connected layer with a softmax layer at the end.








§ EXPERIMENTAL RESULTS




 §.§ Smart Compression

<ref> shows spectrograms of raw and of compressed audio signals. During reception of the latter, the current consumption only rises above the baseline level when vocalizations are detected and data is streamed over BLE. 
Comparing the two spectrograms reveals high-fidelity reconstruction from the compressed data.





 §.§ Compression Algorithms

Compression algorithms reduce the number of bytes required to represent data at the cost of a computational overhead. To be effective, a compression scheme must achieve a reasonable quality wherein the energy savings from data reduction outweigh the power needed for compression. A comparison between different compression algorithms with respect to current consumption and memory usage is presented in <ref>. Algorithms with a compression rate of 16 (Bitrate of [per-mode = symbol]2) represent the signals insufficiently for vocal analysis.
Among the remaining algorithms, the Adaptive Differential Pulse Code Modulation (ADPCM) shows the best performance with a minimal computational overhead of only 50 and a low average BLE current consumption of 190 







 §.§ Syllable Classification


The proposed 8-bit quantized two-stage classification network achieved a syllable error rate <cit.> of 7 on eight different syllable classes with an overall inference time of 5.4.
<ref> summarizes the classification networks performances and memory requirements.



For syllable detection, the microphone's buffer is fed through the detection network every 16. The classification network only runs if a syllable has been detected (ref. <ref>). Detected syllables sent over BLE require little data, including only the time between each syllable and the syllable type.
An average power consumption of 5.73 has been achieved. It consists of the microphone's constant power consumption of 5.25 and the classifier's computational overhead of 0.48.



The total power consumed is less than during audio streaming with any other standard compression algorithm, which makes on-board syllable detection an attractive power-saving solution.





§ CONCLUSION

This paper presented TinyBird-ML a miniaturized and power-optimized digital animal-borne sensor node that can be placed on small birds such as zebra finches. Besides a zero-power digital MEMS microphone, TinyBird-ML features an IMU with an integrated electrostatic sensor to enable monitoring a bird's vital signs such as its heart beats or muscle contractions. A light-weight neural network running on a microcontroller detects and classifies up to eight syllable types in real-time with a syllable error rate of 7%. Weighing only 1.4 grams, TinyBird-ML minimizes weight whilst achieving an operating time of up to 25 hours during continuous data acquisition. 





§ ACKNOWLEDGMENT

This work has been partially found by: the Swiss National Science Foundation (Agreement No 31003A-182638 and the NCCR Evolving Language Agreement no. 51NF40-180888), the BRAIN Initiative (Targeted BRAIN Circuits Project on ‘Neural sequences for planning and production of learned Vocalizations’), and the European Union’s Horizon 2020 Marie Sklodowska-Curie (grant 101025762 to M.D.R).


IEEEtran





 Paper Title: Cooperative Orbital Angular Momentum Wireless Communications.






















IEEE Transactions on Vehicular Technology, VOL. 73, NO. 1, JANUARY 2024
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals

Cooperative Orbital Angular Momentum Wireless Communications
    Ruirui Chen, Wenchi Cheng, Senior Member, IEEE, Jinyang Lin, and Liping Liang

This work is supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20200650, in part by National Natural Science Foundation of China under Grant 62071472, in part by China Postdoctoral Science Foundation under Grant 2019M660133, in part by Program for “Industrial IoT and Emergency Collaboration” Innovative Research Team in CUMT under Grant 2020ZY002, in part by Fundamental Research Funds for the Central Universities under Grant 2019QNB01 and 2020ZDPY0304. (Corresponding author: Ruirui Chen.)

Ruirui Chen and Jinyang Lin are with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China (emails: rrchen, linjinyang@cumt.edu.cn). Wenchi Cheng and Liping Liang are with State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, 710071, China (e-mail: wccheng, liangliping@xidian.edu.cn).




    August 5, 2024
======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
Orbital angular momentum (OAM) mode multiplexing has the potential to achieve high spectrum-efficiency communications at the same time and frequency by using orthogonal mode resource.
However, the vortex wave hollow divergence characteristic results in the requirement of the large-scale receive antenna, which makes users hardly receive the OAM signal by size-limited equipment. To promote the OAM application in the next 6G communications, this paper proposes the cooperative OAM wireless (COW) communication scheme, which can select the cooperative users (CUs) to form the aligned antennas by size-limited user equipment. First, we derive the feasible radial radius and selective waist radius to choose the CUs in the same circle with the origin at the base station. Then, based on the locations of CUs, the waist radius is adjusted to form the receive antennas and ensure the maximum intensity for the CUs. Finally, the cooperative formation probability is derived in the closed-form solution, which can depict the feasibility of the proposed COW communication scheme.
Furthermore, OAM beam steering is used to expand the feasible CU region, thus achieving higher cooperative formation probability.
Simulation results demonstrate that the derived cooperative formation probability in mathematical analysis is very close to the statistical probability of cooperative formation, and the proposed COW communication scheme can obtain higher spectrum efficiency than the traditional scheme due to the effective reception of the OAM signal.



Orbital angular momentum, spectrum efficiency, user, signal reception, cooperative formation probability.






§ INTRODUCTION

Plane electromagnetic wave based wireless communications have almost made full use of traditional orthogonal resources (such as time, frequency, etc) <cit.>. Virtual reality, high-definition video, etc, which are emerging applications, bring about the rapidly increasing wireless traffic demand for the next 6G communications <cit.>. To further improve the spectrum efficiency, orbital angular momentum (OAM) is expected to achieve a breakthrough in the information transmission approach <cit.>. OAM describes the characteristic of electromagnetic wave rotating around the propagation axis, which produces the phase wavefront of electromagnetic wave in a vortex shape <cit.>. The electromagnetic wave with OAM is called vortex wave, and different OAM modes are orthogonal with each other <cit.>. Therefore, as a new orthogonal resource, OAM mode can be utilized to enhance the access capability and improve the reliability of anti-jamming by using mode division multiple access and the mode hopping technique, respectively <cit.>. Particularly, with orthogonality, different OAM modes can be used to transmit multiple data streams simultaneously for the significant increase of spectrum efficiency in the next 6G communications <cit.>.

 Since the light beam with OAM was discovered in 1992, it attracts great interest in wireless communications due to its high spectrum efficiency <cit.>.
The authors in <cit.> proposed the adaptive optic technique to mitigate the turbulence effect for OAM based underwater optical communications.
To improve the effective coverage, the authors of <cit.> investigated the beam and probabilistic shaping scheme for OAM based indoor optical communications.
Besides OAM based optical communications, OAM mode has been applied to wireless communications in the radio frequency (RF) domain <cit.>.
The optimal uniform circular array (UCA) design was proposed to select the OAM mode and radius of receive UCA for OAM wireless backhaul communications <cit.>.
To improve the spectrum efficiency, the authors of <cit.> proposed an OAM mode allocation method, which provided adjacent cells with different carriers of different OAM modes.
To date, the research on OAM mode multiplexing has made great progress in wireless communications <cit.>.
It is worth noting that in 2013, 4 Gbps successful transmission of uncompressed video over the 60 GHz OAM wireless channel was demonstrated in <cit.>.
In <cit.>, two wireless signals were successfully multiplexed at the same frequency in OAM mode multiplexing based wireless communications for the first time.
The authors of <cit.> analyzed the microwave based OAM mode multiplexing communications, which demonstrated that the high spectrum efficiency can be achieved by using the low complexity receiver for future wireless communications.
In OAM multiplexing communications, a coaxial separation and convergence scheme for multiple OAM modes was proposed, and the crosstalk between multiple OAM channels can be effectively reduced <cit.>.
To achieve high spectrum efficiency in sparse multipath environment, the authors of <cit.> proposed a scheme that combines OAM mode multiplexing with orthogonal frequency division multiplexing (OFDM).
For further increase of the multiplexing gain, concentric UCAs were utilized to achieve the 100 Gbps data rate of OAM wireless communications <cit.>. By using plane spiral OAM mode-group, a partial arc sampling receive scheme was proposed to realize OAM wireless communications, which can obtain robust performance in the multipath environment <cit.>. The authors in <cit.> proposed a shared aperture patch antenna, which had the simple circuit design and easy integration, to realize OAM wireless communications.

However, the transceiver antenna alignment, which requires the perfect alignment between the transmit and receive antenna arrays, imposes a crucial challenge for the high spectrum efficiency achievement of OAM wireless communications <cit.>. The authors in <cit.> drew a conclusion that if the transceiver antenna is not aligned, the performance of OAM wireless communications deteriorates quickly.
For OAM wireless communications, the misalignment between the transmitter and receiver would cause interference among different OAM modes <cit.>.
To overcome this problem, the authors of <cit.> studied the effect of misalignment on the channel crosstalk of OAM wireless communications, which can be relieved by selecting the proper aperture size and mode spacing of the transmitted OAM beams.

In <cit.>, the authors proposed the beam steering method, which used switches connected to array elements, to solve the aperture alignment of the transmitter and receiver in OAM wireless communications.
To avoid the performance deterioration caused by antenna misalignment, the authors of <cit.> compared the analog beam steering scheme and digital beam steering scheme, and concluded that the baseband digital OAM transceiver with digital beam steering has a better performance for misaligned multi-mode OAM broadband wireless communications.
Considering the size and cost of UCA, the authors in <cit.> developed an efficient technique for beam steering of arbitrary UCA sizes based on OAM modes. A hybrid mechanical and electronic beam steering approach was proposed in <cit.> to maximize the channel capacity of OAM wireless communications.

Besides the transceiver antenna alignment, another challenge of the OAM application is that the size-limited user equipment is hard to receive the OAM signal at long distance transmission due to the hollow divergence.
Specifically, the hollow divergence of the vortex wave beam is the zero intensity circle around the beam propagation axis <cit.>. The intensity distribution of the vortex wave beam is a ring, which is proportional to the OAM mode and increases as the transmission distance increases. In other words, the large OAM mode will result in serious hollow divergence, and the energy ring of the vortex wave beam will diverge with the increase of the transmission distance <cit.>. To reduce the divergence of the OAM beam, the authors of <cit.> proposed the spherical Luneberg lens, which improved the transmission performance and reliability of OAM wireless communications.
Note that the great majority of research on OAM wireless communications is based on the centralized UCA <cit.>. The authors in <cit.> showed that the spectrum efficiency of UCA-based OAM wireless communications is affected by the transmission distance and the size of UCA. However, the energy ring of the vortex wave beam will diverge as the transmission distance increases, thus making the receive UCA size big at long distance transmission <cit.>. The hollow divergence leads to difficulty in receiving and utilizing the OAM signal.
Furthermore, the authors in <cit.> proved that the hollow divergence of the vortex wave beam greatly increases as the carrier frequency decreases. Therefore, the easily realized communication scheme, which can efficiently receive the OAM signal, is urgently demanded for  size-limited user equipment to realize the OAM mode multiplexing, where the hollow divergence is serious in the RF domain.

Motivated by the OAM signal reception of size-limited user equipment, the cooperative OAM wireless (COW) communication scheme, which can select the cooperative users (CUs) to form aligned receive antennas, is proposed in this paper to overcome the vortex wave hollow divergence and achieve the high spectrum-efficiency OAM point-to-point transmission (especially for the long distance).
The main contributions of the paper are summarized as follows:


  
  * According to the intensity distribution of the vortex wave beam, the feasible radial radius and selective waist radius are derived to select the CUs in the same circle with the origin at the base station (BS). Then, we propose the COW communication scheme to adjust the waist radius to form the aligned receive antennas and ensure the maximum intensity for the CUs.
  
  * To depict the feasibility of the proposed COW communication scheme, we derive the closed-form expression of cooperative formation probability, which is the probability that at least two CUs can be selected as one cooperative pair (CP).
  
  * To expand the feasible CU region, OAM beam steering is used to compensate the phase distortion based on oblique angles, which can achieve higher cooperative formation probability.
  
  * The proposed COW communication scheme achieves higher spectrum efficiency than the traditional scheme due to the effective signal reception by size-limited user equipment, and the derived cooperative formation probability in mathematical analysis is very close to the statistical probability of cooperative formation.


The rest of this paper is organized as follows. Section II presents the system model of OAM wireless communications. Based on the intensity distribution of the vortex wave beam, we derive the feasible radial radius and selective waist radius in Section III. In Section IV, we propose a COW communication scheme to realize the COW communications.
Section V presents the cooperative formation probability analysis, which can validate the feasibility of the proposed COW communication scheme.
The signal compensation and spectrum efficiency performance are analyzed for our proposed COW communication scheme in Section VI.
Section VII gives numerical simulation results to validate the theoretical analysis of the COW communication scheme.
The conclusion is drawn in Section VIII.



§ SYSTEM MODEL

This paper assumes that there exists a BS and K users in hotspot region, which is caused by the concert, festivals, and stadium games. The user set is represented by 𝒦={1,2⋯,K}. The BS is equipped with the UCA, which has M antenna elements denoted by the set ℳ={1,2⋯,M}. The users are all equipped with one antenna and randomly scattered on the ground.

The radius and central angle of UCA are R and φ=2π/M, respectively. Furthermore, the user location information is known to the BS, which is easy for intelligent user equipment at present <cit.>.
As shown in Fig. <ref>, this paper considers the downlink COW communications, where there exists the Line-of-Sight (LoS) path and the channel experiences free-space path loss.

We define the two CUs as the k_1-th and k_2-th users. As shown in the bottom of Fig. <ref>, CU k1 and CU k2 are two users with size-limited equipment that can form a CP. A COW communication unit is defined as a unit that is composed of the BS and a CP. Thus, the COW communications can be divided into a number of COW communication units. This implies that in the COW communication model, each user uniquely belongs to a CP and also uniquely belongs to one COW communication unit. Based on the above analysis, we can focus on the COW communication unit to investigate the COW communications. Note that one user may carry multiple equipment that can cooperate with each other to form a CP. Furthermore, in hotspot region, there are many users clustered together, and incentive mechanism can be used to encourage user cooperation <cit.>, which can guarantee the COW communications.



In the 3D cylindrical coordinate system, the BS with height H is at the origin (0, 0, 0), and the z-axis is perpendicular to the ground through BS. The UCA center of the BS and k-th (k∈𝒦) user are at the locations (0, 0, H) and (r_k, θ_k, 0), respectively. The complex amplitude distribution of the Laguerre-Gaussian (LG) beam propagating along the z-axis is expressed as follows:

    U_p,ℓ(r,θ,z)=   √(2p!/π w^2(z)(p+|ℓ |)!)  L_p^|ℓ | (2r^2/w^2(z))  [r√(2)/w(z)]^|ℓ |
       exp (iℓθ)exp [iβ r^2z/2(z^2+z_R^2)]exp [-r^2/w^2(z)]
    
       exp [-i(2p+|ℓ |+1)tan ^-1 (z/z_R)],

where θ is the azimuth angle, β represents the wave number, w_0 is the waist radius at z=0, w(z) = w_0√(1 + (z/z_R)^2) denotes the radius of the beam at z, z_R=π w_0^2/λ represents the Rayleigh range, L_p^|ℓ |(·) denotes the associated Laguerre polynomial, and (2p+|ℓ |+1)tan ^-1(z/z_R) is the Gouy phase. We can see that the LG beam is related with two parameters: the azimuthal index ℓ and the radial index p. The azimuthal index ℓ is the topological charge, which is also called OAM mode or vortex wave mode. The radial index p is the number of radial nodes on the cross section of the LG beam intensity.



§ FEASIBLE RADIAL RADIUS AND SELECTIVE WAIST RADIUS

The OAM beam is the LG beam with p=0. The intensity distribution is expressed as

    U_ℓ(r,z)=2/π w^2(z)|ℓ |! [√(2)r/w(z)]^2|ℓ |exp [-2r^2/w^2(z)].

 Since U_ℓ(r,z) is a strictly convex function with respect to the radial radius r, we can obtain r corresponding to the maximum U_ℓ(r,z) as follows <cit.>:

    r_(z)=√(|ℓ |/2)w(z).

The maximum intensity is given as

    U_(z)=2|ℓ |^|ℓ |exp (-|ℓ |)/π w^2(z)|ℓ |!.


According to (<ref>), we derive the waist radius w_0 corresponding to the optimal radial radius r_(z) as

    w_0=√(r_^2(z)/|ℓ |±√(r_^4(z)/|ℓ |^2-z^2λ^2/π^2)).

The following feasible radial radius should be satisfied to make the waist radius w_0 exist

    r_(z) ≥ R_(z) ≜√(zλ |ℓ |/π).

Considering the size limitation of the BS, the smaller w_0 is the selective waist radius w_0, i.e.,

    w_0=√(r_^2(z)/|ℓ |- √(r_^4(z)/|ℓ |^2-z^2λ^2/π^2)).


To realize the L OAM mode multiplexing communications, L antennas uniformly locate at the 2π/a arc, and different OAM modes l_n  (n=1, 2, 3...N) satisfy l_n-l_0=na, where l_0 is a constant integer and n is an arbitrary integer <cit.>. Note that the user receives the OAM signal by using the size-limited equipment with only one antenna in general. We can see that L CUs are required to achieve L OAM mode multiplexing communications. Obviously, it is meaningless to realize the 1 OAM mode multiplexing communications. Furthermore, due to the random distribution of users in hotspot region, the more OAM modes are, the more difficult the realization of OAM multiplexing communications is. Therefore, we choose two CUs to achieve OAM wireless communications, which is the most practical case and can achieve 2 OAM mode multiplexing communications.



§  COOPERATIVE OAM WIRELESS COMMUNICATION SCHEME

In the 3D cylindrical coordinate system, the UCA center of the BS and k-th (k∈𝒦) user are at the locations (0, 0, H) and (r_k, θ_k, 0), respectively.
The users are all uniformly scattered on the ground in hotspot region (i.e., horizontal plane z=0).
The location of the m-th (m∈ℳ) antenna element is at the location (R, mφ, H).
The distance between the two CUs is denoted by d_k_1,k_2.

Based on (<ref>), the two CUs should be close to the radial radius with maximum intensity. To this end, the two CUs should have the same distance to the BS and be located at the diameter of the circle with radius r_(z^c), i.e., r_k_1 =r_k_2 and d_k_1,k_2=2r_(z^c), where z^c denotes the distance from the UCA center of BS to the midpoint of the connection between the two CUs. Furthermore, the maximum communication distance is defined as D_ for the CUs.

Note that the propagation direction of z^c is the propagation axis of the LG beam, and the BS adjusts the waist radius w_0 to guarantee that the two CUs are located at the diameter of the circle with radius r_(z^c). In other words, the waist radius w_0 is adjusted based on the radial radius r_(z^c)=d_k_1, k_2/2, and the transmission direction of the LG beam is from the UCA center of BS to the midpoint of the connection between the two CUs. Based on (<ref>), the condition d_k_1,k_2≥ 2R_(z^c) should be satisfied to guarantee the existence of the waist radius.

The CU set and potential user (PU) set are defined as ℛ_c and ℛ_p, respectively. The cardinalities of sets ℛ_c and ℛ_p are |ℛ_c| and |ℛ_p|, which are respectively the numbers of users that belong to the CU and PU sets.
The ring with inner radius r_s-ϵ and outer radius r_s+ϵ is defined as the r_s search ring, where r_s denotes the search radius and ϵ is the extremely small value.
The search radius r_s belongs to (ϵ,R_-ϵ), where R_ is the maximum search radius (i.e., the BS coverage region).
Then, we propose the COW communication scheme, which can be summarized as follows.

[0.25]8.8cm0.5pt
Cooperative OAM wireless communication scheme
Initialization:
 1) CU and PU sets ℛ_c=ℛ_p=∅ and given user location
  (r_k, θ_k, 0);
 2) Set arbitrarily small value ϵ and search radius r_s = ϵ;
Iteration:
 3) while (ℛ_c=∅) do
 4)   while (r_s⩽ R_-ϵ) do
 5)     For r_s search ring, find PU set ℛ_p;
 6)     if  (|ℛ_p|≥ 2)
 7)        In PU set ℛ_p, select two CUs k_1 and k_2, which have
        the minimum d_k_1,k_2∈ [ 2R_(z^c), D_];
 8)       Add CUs k_1 and k_2 to the CU set ℛ_c;
 9)     end if
10)     Set r_s = r_s + 2ϵ;
11)   end while
12)   Set ϵ=2ϵ;
13) end while
[0.25]8.8cm1pt



Figure <ref> gives the two selected CUs to illustrate the proposed COW communication scheme. The two CUs are within the r_s search ring, which is the green circle with radius r_s=r_k_1=r_k_2. At the same time, the two CUs are located at the orange circle with diameter d_k_1, k_2 marked by the dotted line.
The distance from the UCA center of BS to the midpoint of the connection between the two CUs, which is denoted by z^c, can be expressed as follows:

    z^c = √(r_s^2 - d_k_1, k_2^2/4 + H^2) .

The BS with height H transmits OAM signals to the two CUs through the propagation direction of z^c (i.e., the propagation axis of the LG beam) marked by the orange solid line.

The PU set ℛ_p can be found in the r_s search ring according to the updated search radius r_s (step 5).
If the number of the PUs is not smaller than two, the two CUs k_1 and k_2 that have the minimum d_k_1,k_2∈ [ 2R_(z^c), D_] are selected from the PU set ℛ_p and added to the CU set ℛ_c (step 4-11).
The above process should be repeated until the CUs are selected by the iteration of the r_s search ring, i.e., execute the inner and outer while loop (steps 3-13).

For the complexity of proposed COW communication scheme, the maximum iteration numbers of the inner and outer loops are R_/2ϵ and _2(1+R_/ϵ), respectively. The maximum iteration number of steps 5-10 is bounded by R_/2ϵ_2(1+R_/ϵ). For the sequential statements in steps 5-10, the complexity is determined by the highest complexity among the sequential statements, which can be expressed as a constant variable W. Therefore, the complexity of the proposed COW communication scheme is O(R_/2ϵ_2(1+R_/ϵ)W). Note that the complexity of the proposed COW communication scheme is proportional to O(1/ϵ_2(1+R_/ϵ)). Based on the above analysis, the complexity of the proposed COW communication scheme can be bounded by O(1/ϵ_2(1+R_/ϵ)), which is the tolerable complexity. Note that the BS has the ability to be equipped with multiple UCAs for covering all directions. Furthermore, the beam steering can be employed to expand the BS coverage and adjust the propagation direction of the LG beam <cit.>.



§ COOPERATIVE FORMATION PROBABILITY ANALYSIS

To validate the feasibility of the proposed COW communication scheme, the cooperative formation probability P_ is defined as the probability that BS can select at least two CUs as one CP. Furthermore, the cooperative formation probability P_ is derived under the assumption that the users are randomly distributed in the coverage of BS. To this end, we first give the probability P_(r_s) that two CUs locate at the same r_s search ring. Then, based on the cosine theorem, we derive the probability P_(r_s) that two CUs satisfy the communication distance requirement. Finally, the cooperative formation probability P_ can be obtained by integrating the feasible search radius region.

The probability that one CU locates at the r_s search ring can be derived as

    P_(r_s)   =S_/S_ = ( r_s + ϵ)^2 - ( r_s - ϵ)^2/R_^2,

where S_ is the area of the r_s search ring, S_ represents the area of BS coverage, and R_ is the BS coverage radius. Since two CUs should have the same distance to the BS, two CUs should locate at the same r_s search ring. Furthermore, users are randomly distributed within the BS coverage, and the probability of two CUs located at the same r_s search ring can be written as

    P_(r_s)=P^2_(r_s).


In the proposed COW communication scheme, device-to-device (D2D) communications are used to realize the COW communications. The distance between two CUs that locate at the same r_s search ring should satisfy the communication distance condition d_k_1,k_2∈ [ D_, D_]. D_ is the maximum D2D communication distance that can guarantee the efficient information transmission.   Based on (<ref>) and (<ref>), the minimum communication distance d_k_1,k_2 (i.e., D_) can be given as

    D_min    =2R_(z^c)
        =2√(√(H^2 + r_s^2 - ( D_min/2)^2)×λ×| l |/π).

Then, (<ref>) can be transformed into the following expression

    D_(r_s) =  √(2λ| l |/π×( √(λ ^2l^2/π ^2 + 4( H^2 + r_s^2))  - λ| l |/π)).

To obtain the probability that two CUs satisfy the communication distance condition d_k_1,k_2∈ [ D_, D_], we calculate the minimum communication angle θ_(r_s) and maximum communication angle θ_(r_s) corresponding to the minimum communication distance d_k_1,k_2=D_ and maximum communication distance d_k_1,k_2=D_, respectively.


The minimum communication angle θ_(r_s) and maximum communication angle θ_(r_s) are shown in Figs. <ref> and <ref>, respectively. Based on the cosine theorem, the expressions of θ_(r_s) and θ_(r_s) can be respectively derived as

    θ _(r_s)≜arccos2r_s^2 - D_^2/2r_s^2

and

    θ _(r_s)≜arccos2r_s^2 - D_^2/2r_s^2.


We can observe that θ _(r_s) is equal to π for r_s ∈ [R_,D_/2],
where R_ can be expressed as follows:

    R_ = R_(H) = √(H λ| l |/π).

In general, the condition R_≤D_/2 holds due to the small value of wavelength λ. Therefore, the probability P_(r_s) that two CUs satisfy the communication distance condition can be derived as

    P_(r_s) = π  - θ _/π,

which is also the proportion of the angle range that satisfies the communication distance condition.

Since D_ belongs to [D_, 2R_], the maximum search radius is R_ = min{R_,   R_}
, where

    R_ = √(( π ^2D_^4/16λ ^2l ^2) + ( D_/2)^2 - H^2).

R_ is obtained by solving the equation D_ = D_.
For D_/2 < r_s ≤R_, the probability P_(r_s) can be given as

    P_(r_s) = θ _  - θ _/π.

Based on the above analysis, the probability P_(r_s) that two CUs satisfy the communication distance requirement can be summarized as follows:

    P_(r_s) =
    {[     π  - θ _/π,  R_≤ r_s ≤D_/2;;    θ _ - θ _/π, D_/2 < r_s ≤R_;;              0,      otherwise. ].


Then, we obtain the probability that two CUs locate at the same r_s search ring and meet the communication distance condition, which can be expressed as

    P_ = ∫_0^R_P_(r_s)×P_(r_s)dr_s.

 C_K^0 represents the number of the combination that 0 CU is chosen from K users covered by the BS, which can be expressed as

    C_K^0 = 0!/0!(K - 0)! = 1/K!.

Therefore, the cooperative formation probability is written as

    P_ = 1 - C_K^0 × P_^0 ×( 1 - P_)^K,

where P_c^0 denotes the zero power of P_c.




§ PERFORMANCE ANALYSIS

In the COW communications, there exists the BS, which can generate and transmit OAM signals. The two selected CUs, which are uniformly located at the π angular arc, can sample OAM signals. For each CP, the phase-shift component is used to achieve 2 OAM mode multiplexing communications. Thus, the receive OAM signal can be given as

    y = GQ^Hx + n,

where y is the receive vector, G represents the channel matrix, Q^H denotes the multiplexing matrix, x is the transmission information, and n denotes the noise. Q^HQ is the identity matrix. Based on (<ref>), the (ℓ,k)-th element of the channel matrix G can be expressed as

    g(ℓ,k)=U_ℓ(r_k,z^c)=2/π w^2(z^c)|ℓ |! [√(2)r_k/w(z^c)]^2|ℓ |exp [-2r_k^2/w^2(z^c)].

For the COW communications, the channel matrix G can be further written as

    G = Q^HΛQ,

where Λ  = diag{β _1,β _2} is a diagonal matrix with the eigenvalues of G on its diagonal. The ℓ-th diagonal element β_ℓ is the transmission gain corresponding to the ℓ-th OAM channel. Note that we only consider the OAM mode ℓ=1,2, which can obtain the highest spectrum efficiency. For the ℓ-mode OAM transmission, the demultiplexing OAM signal can be written as

    y_de(ℓ ) = q(ℓ )(Gq^H(ℓ )x + n) = g_de(ℓ )x + ñ,

where q(ℓ ) ∈Q, g_de(ℓ ) = q(ℓ )Gq^H(ℓ ) is the ℓ-mode effective OAM channel and ñ = q(ℓ ) n. For the COW communications, the spectrum efficiency of ℓ-mode OAM communication can be given as follows:

    C(ℓ)= _2 (1+P_ℓ| g_de(ℓ )|^2/σ^2 ),

where P_ℓ denotes the power allocated to the ℓ-th OAM channel based on the water-filling method and σ^2 is the noise power.

For the proposed COW communication scheme, selected CUs are required to locate at the feasible CU region, which is the limited angular range on the same circumference. To expand the feasible CU region,  OAM beam steering can be used to compensate the phase distortion based on oblique angles ϕ and ψ. The oblique angle ϕ is the included angle between the vertical line of the connection between the two CUs and the connection between the UCA center of BS and the midpoint of the two CUs. The oblique angle ψ is the included angle between the vertical line of the UCA and the connection between the UCA center of BS and the midpoint of the two CUs.

Based on <cit.>, the transmit beam steering vector and receive beam steering vector are written as b  =  [1, e^-jV_2,..., e^-jV_M] and a  =  [1, e^-jT_2,..., e^-jT_K_c], respectively. Furthermore,

    V_m = 2π R/λcos (2π(m-1) /M - π/2)sinψ


    T_k = 2πr_max(z^c)/λcos (2π(k-1) /K_c + π/2)sinϕ ,

where m denotes the m-th transmit antenna element, k is the k-th CU, and K_c represents the number of selected CUs.
By utilizing the beam steering, the ℓ-mode effective OAM channel can be rewritten as follows:

    g'_de(ℓ ) = (q(ℓ ) ⊙a)G(b^T⊙q^H(ℓ )).

For the COW communications with beam steering, the spectrum efficiency of ℓ-mode OAM communication can be expressed as

    C'(ℓ)=_2 (1+P_ℓ| g'_de(ℓ )|^2/σ^2 ).


Based on the above analysis, the expansion of the feasible CU region can be achieved through the compensation of the phase distortion based on oblique
angles by OAM beam steering, which can relax the limited angular range on the same circumference for selected CUs.
This enlarges the ϵ in (<ref>), which can be expressed as

    ϵ = D_maxsin (ϕ+ψ) /2.

As ϵ becomes larger, the cooperative formation probability P_ will become larger, which makes our proposed COW communication scheme more feasible.



§ SIMULATION RESULTS


In this section, simulation results are provided to evaluate the performance of our proposed COW communication scheme. The users are randomly distributed in the BS coverage. We set the operating frequency, BS transmit power, and noise power as 1, P=30, and σ^2=-90, respectively. We compare the proposed COW communication scheme with the traditional OAM wireless communications with the fixed UCA scheme.



Figure <ref> is given to demonstrate the selected CUs by using our proposed COW communication scheme. 50 users marked by “∘" are uniformly distributed in 200 ×200 area, and the green dashed rings are the determined r_s search rings. 6 users in the green dashed rings are the PUs, and the “73" are the selected CUs, which can achieve the COW communications.



For the proposed COW communication scheme, Fig. <ref> plots the cooperative formation probability versus BS coverage radius with different user numbers 1000, 2000, 3000, and 4000. It can be observed that the cooperative formation probability is a decreasing function of the BS coverage radius. The proposed COW communication scheme with user number 4000 obtains the largest cooperative formation probability. This is due to the fact that the increase of user density (i.e., user number in the BS coverage) can decrease the distance between users, which increases the probability to select at least two CUs as one CP. Furthermore, we can see that the theoretical cooperative formation probability is consistent with the statistical cooperative formation probability, which validates the correctness of our derived closed-form theoretical cooperative formation probability.



 Under various BS coverage radii, the cooperative formation probabilities with different maximum transmission distances 10 m, 15 m, 20 m, and 25 m are illustrated in Fig. <ref> for the validation of the proposed COW communication scheme. For all the maximum transmission distances between two CUs, the cooperative formation probabilities monotonically decrease as the BS coverage radius increases. In addition, we can see that the larger the maximum transmission distance between two CUs is, the larger the cooperative formation probability is. This is because the increase of the maximum transmission distance between two CUs can increase the number of PUs that can satisfy the communication distance condition of the proposed COW communication scheme. In this paper, we set the maximum transmission distance between two CUs as 10 m, which can guarantee the LoS path between two CUs and achieve efficient cooperative communications for two CUs.



Figure <ref> shows the spectrum efficiency of the proposed COW communication scheme with different user numbers.
For the fixed UCA scheme, there are two antennas to receive the OAM signal.
It can be seen that the proposed COW communication scheme obtains higher spectrum efficiency than the fixed UCA scheme.
We can observe that the spectrum efficiency is a monotonic decreasing function of the BS coverage radius.
Furthermore, the larger the user density is, the larger the spectrum efficiency of the proposed COW communication scheme is. This is due to the fact that with the increase of the user density for the fixed BS coverage radius, the proposed COW communication scheme has the larger probability to select the CUs that can efficiently receive the OAM signal. Therefore, the proposed COW communication scheme can support the high spectrum efficiency over the long transmission distance. The more CUs are, the higher the spectrum efficiency of the proposed COW communication scheme is.



 In Fig. <ref>, the number of CUs selected by the proposed COW communication scheme is depicted under various BS coverage radii. It can be observed that the number of selected CUs monotonically decreases as the BS coverage radius increases. Moreover, as the user density increases, the proposed COW communication scheme can select more CUs to achieve COW communications for spectrum efficiency improvement. This is because the proposed COW communication scheme has the large probability to select the CUs for the large user density.



 Figure <ref> presents the spectrum efficiency performance under various BS heights.
There are two antennas to receive the OAM signal for the fixed UCA scheme.
The proposed COW communication scheme achieves better spectrum efficiency performance than the fixed UCA scheme.
From Fig. <ref>, we can see that the proposed COW communication scheme with user number 4000 can obtain higher spectrum efficiency than those with user numbers 1000, 2000, and 3000. Furthermore, the spectrum efficiency of the proposed COW communication scheme is a monotonic decreasing function of BS height. This is due to the fact that as the BS height increases, the transmission distance of the OAM signal will increase, thus making the minimum communication distance between two CUs larger. Based on the proposed COW communication scheme, it can be observed that as the minimum communication distance between the two CUs increases, the proposed COW communication scheme has the small probability to select the CUs.



Figure <ref> demonstrates the number of selected CUs versus BS height. We can observe that the number of CUs selected by the proposed COW communication scheme monotonically decreases as the BS height increases. Moreover, with the increase of the user density, the proposed COW communication scheme can select more CUs. This is because as the user density increases, the proposed COW communication scheme obtains the large probability to realize long distance information transmission. The larger the user density is, the more the number of selected CUs is.



For the proposed COW communication scheme, Fig. <ref> plots the cooperative formation probability with different ϵ. The maximum transmission distance is set as D_max=10m, and the oblique angle is 6^∘ based on <cit.>. According to (<ref>), we can obtain ϵ≈ 0.52. Therefore, the cooperative formation probability is plotted with ϵ=0.05, 0.25, and 0.5. It can be observed that the larger the ϵ is, the larger the cooperative formation probability is. This is due to the fact that the increase of ϵ can expand the feasible CU region, which increases the number of PUs and makes our proposed COW communication scheme more feasible. The cooperative formation probability monotonically decreases as the BS coverage radius increases.




§ CONCLUSIONS


In this paper, a COW communication scheme, which realizes the effective reception of the OAM signal, is proposed to select the CUs with size-limited equipment for COW communications at the long distance. To this end, the feasible radial radius and selective waist radius were obtained by deriving the intensity distribution with respect to the radial radius. Then, the two CUs were chosen by using the same circle with the origin at the BS. Finally, to guarantee the reception of the OAM signal, we adjusted the waist radius to make the two CUs at the maximum intensity. Simulation results showed that compared with the traditional OAM wireless communications with the fixed UCA scheme, the proposed COW communication scheme can achieve higher spectrum efficiency over the long transmission distance. Furthermore, the theoretical cooperative formation probability was very close to the statistical cooperative formation probability, which validated the feasibility of the proposed COW communication scheme.



99
1
R. Lyu, W. Cheng, B. Shen, et al., “OAM-SWIPT for IoE-Driven 6G," IEEE Communications Magazine, vol. 60, no. 3, pp. 19-25, Mar. 2022.

2
R. Chen, H. Zhou, M. Moretti, et al., “Orbital angular momentum waves: generation, detection, and emerging applications," IEEE Communications Surveys and Tutorials, vol. 22, no. 2, pp. 840-868, 2020.

3
S. M. Mohammadi, L. K. S. Daldorff, J. E. S. Bergman, et al., “Orbital angular momentum in radio–a system study," IEEE Transactions on Antennas and Propagation, vol. 58, no. 2, pp. 565-572, Feb. 2010.

4
R. Chen, F. Cheng, J. Lin, et al., “Performance analysis of rate splitting multiple access based vortex wave communications," IEEE Wireless Communications Letters, vol. 11, no. 8, pp. 1570-1574, Aug. 2022.

5
R. Lyu, W. Cheng and W. Zhang, “Modeling and performance analysis of OAM-NFC systems," IEEE Transactions on Communications, vol. 69, no. 12, pp. 7986-8001, Dec. 2021.

7
W. Cheng, W. Zhang, H. Jing, et al., “Orbital angular momentum for wireless communications," IEEE Wireless Communications, vol. 26, no. 1, pp. 100-107, Feb. 2019.

7a
L. Liang, W. Cheng, W. Zhang, et al., “Mode hopping for anti-jamming in radio vortex wireless communications," IEEE Transactions on Vehicular Technology, vol. 67, no. 8, pp. 7018-7032, Aug. 2018.

6
T. Hu, Y. Wang, X. Liao, et al., “OAM-based beam selection for indoor millimeter wave MU-MIMO systems," IEEE Communications Letters, vol. 25, no. 5, pp. 1702-1706, May 2021.

8
A. A. Amin and S. Y. Shin, “Capacity analysis of cooperative NOMA-OAM-MIMO based full-duplex relaying for 6G," IEEE Wireless Communications Letters, vol. 10, no. 7, pp. 1395-1399, Jul. 2021.




10
L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, et al., “Orbital angular momentum of light and the transformation of LaguerreGaussian laser modes," Physical Review A, vol. 45, no. 11, pp. 8185-8189, Jun. 1992.

10A
L. Zhu, H. Yao, H. Chang, et al.,“Adaptive optics for orbital angular momentum-based internet of underwater things applications," IEEE Internet of Things Journal, vol. 9, no. 23, pp. 24281-24299, Dec. 2022.

10A1
J. Li, Q. Yang, X. Dai, et al., “Joint beam-and-probabilistic shaping scheme based on orbital angular momentum mode for indoor optical wireless communications," Journal of Lightwave Technology, 2023. Available: https://ieeexplore.ieee.org/stamp/stamp.jsp?t p=&arnumber=10158487.

10A2
Y. Ren, L. Li, G. Xie, et al., “Line-of-sight millimeter-wave communications using orbital angular momentum multiplexing combined with conventional spatial multiplexing," IEEE Transactions on Wireless Communications, vol. 16, no. 5, pp. 3151-3161, May 2017.

 10B
H. Jing, W. Cheng, W. Zhang, et al., “Optimal UCA design for OAM based wireless backhaul transmission," IEEE International Conference on Communications, pp. 1-6, 2020.

10C
Z. Zhu, C. Zhang, L. Wang, et al., “Research on orbital angular momentum different modes networking method in wireless communication," IEEE Wireless Communications Letters, vol. 11, no. 5, pp. 1007-1011, May 2022.



10a
K. A. Opare, Y. Kuang, and J. J. Kponyo, “Mode combination in an ideal wireless OAM-MIMO multiplexing system," IEEE Wireless Communications Letters, vol. 4, no. 4, pp. 449-452, Aug. 2015.

10b
W. Cheng, H. Zhang, L. Liang, et al., “Orbital angular-momentum embedded massive MIMO: achieving multiplicative spectrum-efficiency for mmWave communications," IEEE Access, vol. 6, pp. 2732-2745, 2018.




11a
F. E. Mahmouli and S. D. Walker, “4-Gbps uncompressed video transmission over a 60-GHz orbital angular momentum wireless channel," IEEE Wireless Communications Letters, vol. 2, no. 2, pp. 223-226, 2013.

11
F. Tamburini, E. Mari, A. Sponselli, et al., “Encoding many channels in the same frequency through radio vorticity: first experimental test," New Journal of Physics, vol. 14, no. 3, Art. 2012.

12
W. Zhang, S. Zheng, X. Hui, et al., “Mode division multiplexing communication using microwave orbital angular momentum: an experimental study," IEEE Transactions on Wireless Communications, vol. 16, no. 2, pp. 1308-1318, Feb. 2017.

12a
Y. Zhao and C. Zhang, “Compound angular lens for radio orbital angular momentum coaxial separation and convergence," IEEE Antennas and Wireless Propagation Letters, vol. 18, no. 10, pp. 2160-2164, 2019.

13
L. Liang, W. Cheng, W. Zhang, et al., “Joint OAM multiplexing and OFDM in sparse multipath environments," IEEE Transactions on Vehicular Technology, vol. 69, no. 4, pp. 3864-3878, Apr. 2020.

14
D. Lee, H. Sasaki, H. Fukumoto, et al., “An experimental demonstration of 28 GHz band wireless OAM-MIMO (orbital angular momentum multi-input and multi-output) multiplexing," 2018 IEEE 87th Vehicular Technology Conference (VTC Spring), pp. 1-5, 2018.
14a
X. Xiong, S. Zheng, Y. Chen, et al., “Plane spiral OAM mode-group orthogonal multiplexing communication using partial arc sampling receiving scheme," IEEE Transactions on Antennas and Propagation, vol. 70, no. 11, pp. 10998-11008, Nov. 2022.
14b
X. He, L. Deng, B. Feng, et al., “Angular momentum multiplexing via a shared-aperture patch antenna," IEEE Transactions on Antennas and Propagation, 2023. Available: https://ieeexpl-ore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10024147.

15
Z. Zhang, Y. Xiao, Z. Ma, et al., “6G wireless networks: vision, requirements, architecture, and key technologies," IEEE Vehicular Technology Magazine, vol. 14, no. 3, pp. 28-41, Sept. 2019.

15a
R. Chen, H. Xu, M. Moretti, et al., “Beam steering for the misalignment in UCA-based OAM communication systems," IEEE Wireless Communications Letters, vol. 7, no. 4, pp. 582-585, 2018.

15b
H. Jing, W. Cheng and X. Xia, “A simple channel independent beamforming scheme with parallel uniform circular array," IEEE Communications Letters, vol. 23, no. 3, pp. 414-417, Mar. 2019.

15c
K. Liu, Y. Cheng, Z. Yang, et al., “Orbital angular-momentum-based electromagnetic vortex imaging," IEEE Antennas and Wireless Propagation Letters, vol. 14, pp. 711-714, 2015.




16
P. Yang, Y. Xiao, M. Xiao, et al., “6G wireless communications: vision and potential techniques," IEEE Network, vol. 33, no. 4, pp. 70-75, Aug. 2019.

16a
Y. Zhang, W. Feng, and N. Ge, “On the restriction of utilizing orbital angular momentum in radio communications," 2013 8th International Conference on Communications and Networking in China (CHINACOM), pp. 271-275, Aug. 2013.

17
G. Xie, L. Li, Y. Ren, et al., “Performance metrics and design considerations for a free-space optical orbital-angular-momentum-multiplexed communication link," Optica, vol. 2, no. 4, pp. 357-365, 2015.

17a
Q. Song, Y. Wang, K. Liu, et al., “Beam steering for OAM beams using time modulated circular arrays," Electronics Letters, vol. 54, no. 17, pp. 1017-1018, 2018.

18
Z. Tian, R. Chen, W. Long, et al., “Broadband beam steering for misaligned multi-mode OAM communication systems," Journal of Systems Engineering and Electronics, vol. 32, no. 4, pp. 779-788, Aug. 2021.

19
R. Chen, W. Long, X. Wang, et al., “Multi-mode OAM radio waves: generation, angle of arrival estimation and reception with UCAs," IEEE Transactions on Wireless Communications, vol. 19, no. 10, pp. 6932-6947, Oct. 2020.

19.1
R. Chen, Z. Tian, W. Long, et al., “Hybrid mechanical and electronic beam steering for maximizing OAM channel capacity,"  IEEE Transactions on Wireless Communications, vol. 22, no. 1, pp. 534-549, Jan. 2023.



 19A1
M. J Padgett, F. M Miatto, M. P J Lavery, et al.,“Divergence of an orbital-angular-momentum-carrying beam upon propagation,"  New Journal of Physics, vol.17, no.2, 2015.

19A2
M. Oldoni, F. Spinello, E. Mari, et al.,“Space-division demultiplexing in orbital-angular-momentum-based MIMO radio systems,"  IEEE Transactions on Antennas and Propagation, vol. 63, no. 10, pp. 4582-4587, Oct. 2015.

19B
G. Xie, Y. Ren, H. Huang, et al., “Analysis of aperture size for partially receiving and de-multiplexing 100-gbit/s optical orbital angular momentum channels over free-space link," 2013 IEEE Globecom Workshops (GC Wkshps), pp. 1116-1120, Dec. 2013.
19.2
R. Chen, J. Zhou, W. Long, et al., “Hybrid circular array and luneberg lens for long-distance OAM wireless communications," IEEE Transactions on Communications, vol. 71, no. 1, pp. 485-497, Jan. 2023.

19a
H. Jing, W. Cheng, W. Zhang, et al., “OAM based wireless communications with non-coaxial UCA transceiver," 2019 IEEE 30th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 1-6, 2019.

19b
G. Gil, J. Lee, H. Kim, et al., “Comparison of UCA-OAM and UCA-MIMO systems for sub-THz band line-of-sight spatial multiplexing transmission," Journal of Communications and NetworksView article, vol. 23, no. 2, pp. 83-90, Apr. 2021.

19c
W. Yu, B. Zhou, Z. Bu, et al., “UCA based OAM beam steering with high mode isolation," IEEE Wireless Communications Letters, vol. 11, no. 5, pp. 977-981, May 2022.

19d
Z. Guo and G. Yang, “Radial uniform circular antenna array for dual-mode OAM communication," IEEE Antennas and Wireless Propagation Letters, vol. 16, pp. 404-407, 2017.

19e
F. Qin, L. Li, Y. Liu, et al., “A four-mode OAM antenna array with equal divergence angle," IEEE Antennas and Wireless Propagation Letters, vol. 18, no. 9, pp. 1941-1945, Sept. 2019.

20
W. Yu, B. Zhou, Z. Bu, et al., “Analyze UCA based OAM communication from spatial correlation," IEEE Access, vol. 8, pp. 194590-194600, 2020.

21
Y. Yan, G. Xie, M. Lavery, et al., “High-capacity millimetre-wave communications with orbital angular momentum multiplexing," Nature Communications, vol. 5, no. 1, 2014.
22
Y. Yuan, Y. Zhao, B. Zong, et al., “Potential key technologies for 6G mobile communications," Science China Information Sciences, vol. 5, no. 1, pp. 213-231, 2020.
22a
W. Long, R. Chen, M. Moretti, et al., “Joint spatial division and coaxial multiplexing for downlink multi-user OAM wireless backhaul," IEEE Transactions on Broadcasting, vol. 67, no. 4, pp. 879-893, Dec. 2021.
22b
T. Zhang, H. Wang, X. Chu, et al., “A signaling-based incentive mechanism for device-to-device content sharing in cellular networks," IEEE Communications Letters, vol. 21, no. 6, pp. 1377-1380, June 2017.
22c
Y. Chen, S. He, F. Hou, et al., “An efficient incentive mechanism for device-to-device multicast communication in cellular networks," IEEE Transactions on Wireless Communications, vol. 17, no. 12, pp. 7922-7935, Dec. 2018.
23
W. Zhang, S. Zheng, Y. Chen, et al., “Orbital angular momentum-based communications with partial arc sampling receiving," IEEE Communications Letters, vol. 20, no. 7, pp. 1381-1384, Jul. 2016.







[
    < g r a p h i c s >
]Ruirui Chen
received the Ph.D. degree in military communication from Xidian University, Xi'an, China, in 2018.

He is currently a lecturer with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. He is also a master supervisor of electronic information in China University of Mining and Technology. His research interests focus on OAM wireless communications, UAV communications and B5G/6G wireless communications.


[
    < g r a p h i c s >
]Wenchi Cheng
(Senior Member, IEEE) received the B.S. and Ph.D. degrees in telecommunication engineering from Xidian University, Xi'an, China, in 2008 and 2013, respectively.
He was a Visiting Scholar with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA, from 2010 to 2011. He is currently a Full Professor with Xidian University. He has published more than 100 international journal and conference papers in IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, IEEE Magazines, IEEE TRANSACTIONS, IEEE INFOCOM, IEEE GLOBECOM, and IEEE ICC. His current research interests include B5G/6G wireless networks, emergency wireless communications, and OAM wireless communications.

Prof. Cheng received the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award in 2021, the URSI Young Scientist Award in 2019, the Young Elite Scientist Award of CAST, and four IEEE journal/conference best papers award. He has served or is serving as an Associate Editor for IEEE SYSTEMS JOURNAL, IEEE COMMUNICATIONS LETTERS, and IEEE WIRELESS COMMUNICATIONS LETTERS; the Wireless Communications Symposium Co-Chair for IEEE ICC 2022 and IEEE GLOBECOM 2020; the Publicity Chair for IEEE ICC 2019; the Next Generation Networks Symposium Chair for IEEE ICCC 2019; and the Workshop Chair for IEEE ICC 2019/IEEE GLOBECOM 2019/INFOCOM 2020 Workshop on Intelligent Wireless Emergency Communications Networks.



[
    < g r a p h i c s >
]Jinyang Lin
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2021. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in OAM wireless communications. Email: linjinyang@cumt.edu.cn.



[
    < g r a p h i c s >
]Liping Liang
received the Ph.D. degree in communication and information system with Xidian University, Xi'an, China. She is a prospective associate professor with Xidian University. Her research interests focus on 5G wireless communications with emphasis on radio OAM wireless communications and anti-jamming communications. Email: liangliping@xidian.edu.cn.





 Paper Title: Blind and robust estimation of adaptive optics point spread function and diffuse halo with sharp-edged objects.

 








Application to asteroid deconvolution and moon enhancementBased on observations made with ESO Telescopes at the Para­nal Observatory under programmes ID 60.A-9362(A) (Yang et al. 2014), ID 296.C-5038(A) (Yang et al. 2016), ID 199.C-0074 (Vernazza et al. 2021) and reduced data publicly available at https://observations.lam.fr/astero/https://observa-tions.lam.fr/astero/ and observations obtained at Keck Observatory under programme ID U58N2 (de Pater et al. 2005).




	LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 5 place Jules Janssen, 92195 Meudon, France
	National Astronomical Research Institute of Thailand, Center for Optics and Photonics, 260 Moo 4, T. Donkaew, A. Maerim, Chiang Mai 50180, Thailand
	

	anthony.berdeu@obspm.fr






 



Initially designed to detect and characterise exoplanets, extreme adaptive optics (AO) systems open a new window onto the Solar System by resolving its small bodies. Nonetheless, their study remains limited by the accuracy of the knowledge of the AO-corrected point spread function (AO-PSF) that degrades their image and produces a bright halo, potentially hiding faint moons in their close vicinity.



To overcome the random nature of AO-PSFs, I aim to develop a method that blindly recovers the PSF and its faint structured extensions directly into the data of interest, without any prior on the instrument or the object's shape. The objectives are both to deconvolve the object and to properly estimate and remove its surrounding halo to highlight potential faint companions.



My method first estimated the PSF core via a parametric model fit, under the assumption of a sharp-edged flat object. Then, the resolved object and the PSF extensions were alternatively deconvolved with a robust method, insensitive to model outliers, such as cosmic rays or unresolved moons. Finally, the complex halo produced by the AO system was modelled and removed from the data.



The method is validated on realistic simulations with an on-sky AO-PSF from the SPHERE/ZIMPOL instrument. On real data, the proposed blind deconvolution algorithm strongly improves the image sharpness and retrieves details on the surface of asteroids. In addition, their moons are visible in all tested epochs despite important variability in turbulence conditions.



My method shows the feasibility of retrieving the complex features of AO-PSFs directly from the data of interest. It paves the way towards more precise studies of asteroid surfaces and the discovery and characterisation of Solar System moons in archival data or with future instruments on extremely large telescopes with ever more complex AO-PSFs.




Blind and robust estimation of adaptive optics point spread function and diffuse halo with sharp-edged objects
    
	Anthony Berdeu1,2
	
    Received 2 August 2023 / Accepted 7 May 2024
==============================================================================================================
















§ INTRODUCTION



The performance of ground-based instruments is limited by the atmospheric turbulence that corrugates the incident wavefront of the observed target <cit.>. For short exposures, the diffraction-limited point spread function (PSF) of the telescope breaks down to a random speckle field. For long exposures, their average is equivalent to the PSF of a 10 to 20 telescope <cit.>, strongly degrading the sensitivity and the resolution.


Introduced in the 1990s, adaptive optics (AO) systems are now commonly deployed in instruments of ground-based observatories to mitigate the effects of the turbulence <cit.>. The coupling of (i) PSF prediction to optimise the instrument design and performance <cit.>, with (ii) PSF modelling <cit.> to improve the data post-processing via model-fitting or deconvolution <cit.>, has resulted in a leap forward in numerous fields such as photometry and astrometry of stellar populations <cit.>, spectroscopy and kinematics of distant galaxies <cit.>, or the study of extended object surface topology <cit.>.

In the last decade, the arrival of extreme AO systems pushed the performance in high-contrast and high-resolution imaging even further  <cit.>. In this context, PSF prediction and modelling reach their limits, unable to faithfully reproduce the complexity of observed AO-corrected PSFs (AO-PSFs). Point spread function estimation or reconstruction techniques must then be considered <cit.>. But direct estimation of the PSF parameters from the AO telemetry <cit.> or reference PSFs obtained on calibration sources (internal or natural stars) before or after the observation <cit.> are not always sufficient, and can for example lead to strong deconvolution artefacts <cit.>.

Among others, the challenge arises from the evolving nature of AO-PSFs. By essence, the turbulence is random and the AO loops are not perfect, suffering from measurement noise, or temporal and aliasing errors <cit.>. Finally, flexions or thermal dilations in the instruments induce quasi-static speckles that slowly drift with time, changing the PSF's small and faint structures <cit.>. As a consequence, a given observation represents one realisation of an AO-PSF that is barely reproducible, or even impossible to reproduce.

To overcome this random nature of the AO-PSF, the only solution is to extract and reconstruct it directly from the data of interest, a problem known as blind deconvolution <cit.>. The challenge is to overcome the degeneracy between the estimation of the PSF and the object <cit.> and to solve an ill-posed problem with potentially twice more unknowns than measurements if both the images of the PSF and the object are to be retrieved from a single data image. As a consequence, no method is strictly `blind', and there is a need to make some assumptions about the PSF or the deconvolved object (or both) to avoid local minima of the problem and converge on the best `PSF and object' pair.

One solution is to implement marginal approaches <cit.>. The problem is rewritten to split the PSF and the object contributions, in order to integrate over the space of the object parameters to marginalise them out of the problem. The deconvolution of the object is only performed as a second step. Marginalisation has proved to be efficient when combined with parametric PSF models that strongly limit the number of unknowns to fit <cit.>. These methods nonetheless imply a need to know some prior on the object, such as the structure of its power spectrum density  <cit.> or its overall shape <cit.>. In addition, such PSF models, relying on a limited number of parameters, cannot fully grasp the complexity of real and potentially broadband AO-PSFs of high-contrast and high-resolution imagers. The challenge will be even greater with the incoming extremely large telescopes (ELTs). Their segmented primary mirror and large spiders holding the secondary mirror will produce intricate and structured PSFs that are very sensitive to AO residuals <cit.>. More versatile methods than parametric approaches are consequently needed.

Finally, in the context of asteroid study, beyond the deconvolution of the images of asteroids' surfaces <cit.>, the approximate knowledge of the PSF also limits the study of asteroids' close vicinity and the detection of faint moons. Similar constraints apply when observing the moon systems around the giant planets of our Solar System <cit.>. To get rid of the bright halo induced by the PSF extensions around the main object, techniques were adapted from exoplanet detection algorithms, based on local averaging or median filters <cit.>. But the problem to solve is quite different, the halo being produced not by a coherent source (an unresolved star) but by an extended object (the resolved asteroid or planet). These techniques consequently do not account for the physics at the origin of the halo that is poorly estimated. They thus inherently suffer from self-subtraction problems and can bias the moon photometry or even prevent its detection <cit.>.


It is thus essential to properly recover the AO-PSF extensions. But compared to the PSF core, their intensity is several orders of magnitude fainter. This echoes high-contrast imaging challenges where outliers must be carefully handled to avoid any corruption of the model. Some outliers, such as dead pixels, can be identified through proper calibration <cit.>, handled, and removed from the problem. But others may be random or specific to a given acquisition, such as hot pixels, cosmic rays, or model errors (signals not predicted by the forward model of the problem). This second kind of outlier must be identified on the fly, directly in the data of interest.

To reduce the incidence of such outliers, robust penalisation approaches have been developed <cit.>. They consist of replacing the conventional quadratic penalisation of the problem by a robust estimator that is approximately quadratic around zero but that grows sub-quadratically for large deviations to reduce their impact <cit.>.

One solution is to directly change the cost function of the problem <cit.>. Another solution, so-called iterative reweighted least squares <cit.>, is to stay in the least squares framework but to solve a sequence of least squares problems whose weights are iteratively updated with a robust estimator. The mathematical formalism associated with such robust penalisations will be further detailed in sec:wing.


This paper is a continuation of a previous pragmatic method that was initially developed to detect and characterise the third moon of  <cit.>. In this previous work, the performance of an AO system being better in the infrared, a simple parametric model was sufficient to describe the PSF, to deconvolve the object, and to properly model and subtract its bright halo. In addition, the PSF was directly estimated on a bright moon in the data of interest, acting as a reference point source.

In this work, I intend to provide a more general and blind approach for more complex AO-PSFs with limited priors on the object and the PSF, while removing the need for a point source to be present in the field of view (FoV). There are three main objectives: (i) to reconstruct the AO-PSF in its complexity, reproducing its faint structured extensions, so-called wings, directly from the data; (ii) to deconvolve the object image with this blindly estimated AO-PSF; and (iii) to carefully model the AO residual halo in order to remove it from the data and enhance the signal of potential moons.

The proposed method, which alternates between object and PSF deconvolution, is described in sec:method. It is then validated on realistic simulated data in sec:res_sim and applied to real data in sec:res_real. Finally, in 
sec:conclu, I discuss several ways of further improving the proposed method. For readers interested in knowing more about the algorithms and their implementation, the pseudo-codes of the different steps are detailed in app:algo.





§ PROPOSED METHOD






In the isoplanetic domain (the PSF does not depend on the position in the FoV), the forward model of the problem  canonically states that the image data,  (discretised onto the sensor pixel grid), is the (discrete) convolution  of the extended object, , with the long exposure PSF, , that combines the telescope and instrument response and the AO residuals <cit.>,

    =  + n ,

where n is a nuisance term. This equation is pictured in fig:deconv_model. As is seen in fig:deconv_modeld, the nuisance term encompasses the classical detector readout noise and photon shot noise, but also outliers such as defective pixels (`salt & pepper' noise) and cosmic rays that may hit the sensor during the acquisition (orange arrows). I emphasise here that the way the problem is posed – the convolution of an extended object with a PSF – implies that the unresolved moons (point sources), are also considered to be outliers, and thus belong to the nuisance term (coloured circles). As was stated above, the objective is to split  and  only from the knowledge of  without being corrupted by n.



The PSF presented in fig:deconv_modelc was obtained on a star, as part of the European Southern Observatory (ESO) Large Programme ID 199.C-0074 <cit.> and using the imaging mode of the Zurich IMaging POLarimeter instrument <cit.>. This figure emphasises all the complexity of an AO-PSF. The AO cut-off frequency (orange annulus) delimits the classical two regimes of AO-PSFs: (i) the outer region (red), dominated by the atmospheric turbulence halo and left uncorrected by the AO system, and (ii) the inner region dominated by the AO residuals. The roughly symmetric core of the PSF (white and grey) is surrounded by a wind-driven halo (green) elongated in the wind direction (temporal error of the AO system, so called servo-lag) and a speckle field produced by the non-common path aberrations <cit.> between the science part of the instrument and the AO system (yellow and orange). The diffraction spikes produced by the spiders holding the secondary mirror are also visible (yellow cross along the diagonals). All these features are responsible for the extended and structured bright halo visible in fig:deconv_modela that hides the neighbouring moons.

I propose to solve eq:forward_model step by step, using minimal assumptions on the object and the PSF, summarised in tab:deconv_assumtions. (I) Firstly, as is pictured in fig:deconv_core, the problem can be approximated, `from a distance', as a sharp-edged flat object, fig:deconv_coreb, convolved with a simple PSF core, fig:deconv_corec. In the following, such an object will qualify as `binary' in the sense of morphological operations <cit.>. This strongly simplifies the problem in order to get a first estimated separation between the object, , and the PSF, . (II) Secondly, as is pictured in fig:deconv_obj, this PSF core was used to deconvolve the main extended object, fig:deconv_objb. (III) Thirdly, as is pictured in fig:deconv_wing, the deconvolution paradigm is reversed and the object was used to deconvolve the faint PSF extensions, so-called wings, fig:deconv_wingc. Finally, after the removal of the halo model, fig:deconv_winga, the moon can be seen in the residuals, fig:deconv_wingd. A detection algorithm could then be applied to find the signal of potential faint moons further hiding in the noise. This is nonetheless beyond the scope of this paper. All of these steps are further detailed in the following.







 §.§ Step 1. Estimation of the PSF core



The main objective of this step is to estimate a first separation between the object, , and the PSF, , as is shown in fig:deconv_core. As a consequence, it must depend on a very limited number of parameters to be robustly constrained by the data.




Concerning the PSF, the method assumes that its core can be approximated by a parametric description. Such a simple model does not include the physics of an AO system and cannot properly model the turbulent halo. It thus dilutes the energy and is consequently not quantitative <cit.>. Parametric model-based profiles can partially solve this issue <cit.> but they imply some knowledge of the instrument (such as, for example, its pupil shape, the number of actuators of its AO system, or the wavelength). As a consequence, they are harder to tune. But such precision is not needed for this given step. Indeed, <cit.> showed that a Moffat profile <cit.> is a general but sufficient approximation to qualitatively retrieve the morphology of the observed objects, while depending only on a limited number of parameters. A two-dimensional (2D) Moffat pattern is defined as

    , α, β, θ = 
        	1 + 
            r_1^2/α_1^2
            + r_2^2/α_2^2^-β ,

where β is the power parameter of the pattern, θ is its orientation, r_1 = _1cosθ + _2sinθ and r_2 = -_1sinθ + _2cosθ are the 2D coordinates in its rotated frame, and α = α_1, α_2 are its elongations along its two axes.


As we shall see later in sec:res_simsec:res_real, this Moffat approximation worked well with all the tested datasets. Nonetheless, in the hypothetical case that this model is not sufficient for a given dataset, eq:Moffat can be replaced by another more adapted parametric model if needed. Doing so does not change the presented methodology as long as the number of parameters remains limited.


The PSF core is thus simply proportional to this 2D pattern:

    _0, α, β, θ, γ≜γ-_0, α, β, θ .

Its parameters mainly control how the object is blurred by the PSF. Most of the signal is then given by the object's edges rather than its surface texture. If the shape of the object is overall smooth with sharp edges, applying a threshold, d̅, on the data, , defined as

    thr, d̅ = 
        
    		1  if ≥d̅
    
            0  otherwise ,

is a sufficient approximation of its support to extract the information needed to estimate the parameters of eq:PSF_core. Indeed, in a parametric approach with a limited number of arguments, namely the parameters of the core and the threshold on the data, a crude description of the object is sufficient because there are far more measurements than unknowns to recover.

The values of the different parameters of eq:PSF_coreeq:thres are obtained by minimising the following cost function,

    core, wd̅, γ, _0, α, β, θ≜   
       wls,
    			_0, α, β, θ, γthr, d̅,
    			 ,

which is a pure data fidelity term between the data, , and the model, _0, α, β, θ, γthr, d̅, weighted by . 𝒟^wls is defined as the weighted least square difference (wls)

    wlsφ_1,φ_2,≜   
       1/2∑_, ^', ^'φ_1 -φ_2φ_1^' -φ_2^' ,

where  is the inverse of the data covariance matrix. Its role is to weigh the measurements in the data fidelity term in terms of confidence.

The readout noise is assumed to be of a Gaussian statistic of variance, . On the other hand, for fluxes higher than a few photons, the Poisson statistic of photon shot noise can be approximated by a Gaussian statistic whose variance is proportional to the incident flux (and thus approximately to the data after proper calibration and pre-reduction) by a factor, η, that accounts for the conversion between photons and analogue to digital unit (ADU) and for the quantum efficiency of the pixels <cit.>. In the general case, these noise terms are independent from one pixel of the sensor to another. As a consequence, the inverse of the data covariance is diagonal,

    , ^' = 
         if  = ^'
    
            0  otherwise ,

and is given by <cit.>

    = 1/η+ .

The sum in eq:WLS_full is thus a sum on  only

    wlsφ_1,φ_2,≜1/2∑_φ_1 -φ_2^2 .

I emphasise that η and   are technically vectors as their values may depend on the position in the FoV, for example because of inhomogeneities in the pixel response or detector gains. Thus, contrary to other methods that approximate the noise level as being homogeneous across the pixels, to allow the marginalisation of the object and the PSF in the Fourier domain <cit.>, the proposed method includes a physical model of the noise statistics that depends on the pixels and that is a function of the data intensities (measured, see eq:weight_diag, or expected by the model, see eq:weight_mod below). This subtlety is critical: for the other methods, the unknown is the deconvolved image of the bright asteroid for which a homogeneous photon noise is a good enough approximation. But the objectives of my method are to additionally retrieve the PSF wings and remove the diffuse halo, while discarding faint outliers, which implies having a fine model of the noise according the intensity level.

As was discussed above, most of the signal is given by the blurring of the object's edges, and is condensed in the bright core of the halo. To avoid any corruption by the faint extensions of the halo that are not modelled under the parametric PSF core assumption, a threshold is applied to the data to reweight, , as is detailed in the alg:core of app:algo_core. Only the pixels not shaded in red in fig:deconv_corea are considered in the fit.

Finally, I mention that this first step can be replaced by more classical approaches, for example using reference PSFs obtained on a star or myopic approaches <cit.>. Indeed, its main objective is to get a good enough PSF core to initialise the object deconvolution of sec:deconv. The global PSF will be further refined later, in sec:wing.




 §.§ Step 2. Deconvolution of the main object





As is shown in fig:deconv_obj, the objective of this step is to deconvolve the image of the object, , assuming that the PSF, , is known, for example from the fit of its core done in the previous section sec:core. It is a classical deconvolution problem solved by minimising the following cost function <cit.>:

    obj,,≜wls, , 
    	+ 
    	^objobj ,

under the physical constraint that ≥0. This hard constraint also limits oscillating artefacts close to sharp edges, commonly seen in deconvolution problems <cit.>. ^obj is a hyperparameter to balance the regularisation, obj, compared to the data fidelity term. obj was chosen to favour smooth objects with sharp edges, by encouraging the sparsity of spatial gradients <cit.>

    objφ≜   
       ∑_√(∇_1φ^2+∇_2φ^2+ϵ^obj^2)-ϵ^obj ,

where ∇_1 and ∇_2 correspond to finite difference operators along the first and second spatial dimensions of the image. ϵ^obj>0 is a threshold controlling the transition between a ℓ^1-norm (edge-preserving) and a ℓ^2-norm (smoothness). ϵ^obj also ensures that eq:reg_obj is differentiable at zero. Such an edge-preserving regularisation or equivalents are common in asteroid deconvolution <cit.>. In practice, these hyperparameters, ^obj and ϵ^obj, were manually tuned. ^obj mainly depends on the object's characteristics. Since all asteroids share similar properties, it is not expected that this parameter changes a lot for resolved asteroids. ϵ^obj varies with the awaited contrast on the asteroid surface: (i) for a well-resolved asteroid, values of a few percent of the object's dynamics provide good results, (ii) for poorly resolved targets where surface details are hardly expected, values of around a thousandth or less better favour sharp-edged binary reconstructions.



At this stage, there is still no constraint on potential moons. They are thus in the deconvolved model and disappear from the residuals, as is shown in fig:deconv_objd. Once the object is deconvolved, it can be segmented with a threshold to remove the moons or other artefacts from the image model, as is described in app:algo_deconv, alg:obj_wing.



 §.§ Step 3. Deconvolution of the extended PSF wings









Now that a de-blurred image of the main object, , has been correctly estimated, it is possible to revert the problem and use this image as the convolution kernel to deconvolve the AO-PSF wings, as is shown in fig:deconv_wing. In this approach, the moons, which are a priori unknown, are not explainable by a model of an extended object convolved with a PSF, and should thus appear in the residuals; so should defective or hot pixels or random cosmic rays impacting the sensor, which cannot be calibrated in advance. These outliers should not corrupt the reconstruction of the PSF wings. As was introduced in sec:intro, the mean square error of eq:WLS is consequently not adapted. I propose to tackle this problem with robust penalisation <cit.>. This consists of replacing the quadratic norm of the residuals in eq:WLS with a function, , that limits the influence of large (and aberrant) values in the minimisation:

    φ_1,φ_2,w≜∑_√(w)φ_1 -φ_2 .


Of the different robust estimators, , available in the literature <cit.>, I chose the Cauchy function:

    r≜γ^2/2ln1+ r^2 / γ^2 .

If the confidence model,  in eq:robdata, is properly scaled and calibrated so that the argument

    r = √(w)φ_1 -φ_2
 follows a normalised Gaussian noise statistics (centred on zero and of a standard deviation equal to one) in the absence of outliers, setting γ = 2.385 ensures that minimising eq:WLS or eq:robdata gives similar results. In doing so, the Cauchy function has the advantages of having no tuning parameter and of being differentiable.

In more detail, 

    r=1/r∂r/∂ r=1+r^2 / γ^2^-1 ,

the so-called robust weight in the following, is the correction factor to apply to the weights in the least square error of eq:WLS to make it equivalent to the robust penalisation of eq:robdata. This is typically used in the context of IRLS, as was introduced in sec:intro:

    ^irls = r .


fig:weighta gives an example of a robust weight map, obtained by applying  of eq:Cauchy_weight to a weighted halo residual map when fitting the full PSF, . The outliers can directly be identified in black, with equivalent weights tending towards zero. It is then possible to discard them on the fly, using a conservative threshold,  (typically 50, as is discussed in app:algo_deconv), and as is shown in fig:weightb. This is done by updating eq:weight, with the additional remark that this equation is in fact only an approximation. Indeed, the noisy data, , is not the real flux. A better estimate of the weighting confidence term, , can be obtained by replacing  with the forward model

    ^mod =

in eq:weight. In total, as is detailed in alg:obj_wing of app:algo_deconv, the updated weight map now reads as

    =
    	
    		0  if ≤
    
    		1/η^mod+ otherwise .

It is iteratively updated in an approach similar to IRLS, eq:weight_IRLS, but with (i) a binary update when it comes to the robust estimator, and (ii) a refinement of the expected noise level with the intensity predicted by the forward model. Doing so prevents any further corruption of the deconvolved PSF wings by strong outliers.




The cost function that is iteratively minimised to reconstruct the PSF, , is

    psf, , ≜wls,,
    	+ ^psfpsf ,

under the physical constraint that ≥ 0. ^psf is a hyperparameter to balance psf. I define this regularisation term as

    psfφ≜∑_∇_1lnφ^2+∇_2lnφ^2 ,

which consists of the classical ℓ^2-norm on the gradient but applied to the logarithm of the PSF. The former ensures a smooth reconstruction, while the latter acts as a `dynamic whitening' term. Indeed, the challenge in regularising the PSF wings is that  spans multiple orders of magnitude between the PSF core vicinity and the external turbulence halo, as is seen in fig:deconv_modelc. psf must act through this high dynamic range. Similar to  in eq:WLSeq:robdata, which scales the residuals according to the admissible awaited noise level (noise whitening), the regularisation must scale the PSF wings. Since

    ∇_ilnφ = ∇_iφ / φ ,

I let the opportunity for the gradients in eq:reg_psf to evolve proportionally to the local intensity of the total PSF, despite the large dynamics of the unknowns, both refining the PSF core and recovering its wings. As for ^obj in sec:deconv, the hyperparameter ^psf is a function of the noise level of the data, , and was manually tuned in this study.

Technically, one could stop here if the main objective were to retrieve the faint moon hidden in the bright halo. Nonetheless, now that a better model of the AO-PSF is known, it is possible to loop back on the object deconvolution step and iterate the steps of sec:deconvsec:wing to further refine the reconstruction while robustly updating the noise model, , via eq:weight_mod in eq:cost_objeq:cost_wing. This implementation is further detailed in alg:obj_wing of app:algo_deconv. This proved to be efficient for complex PSF cores, badly described by a 2D Moffat pattern, for example in the presence of motion blur or for bad seeing conditions. On top of a better object deconvolution, this also improves the residual quality close to the object edges, potentially unveiling moons very close to the main body.





§ VALIDATION ON SIMULATIONS



The method was tested on simulated data representative of real conditions. The simulated object, shown in fig:deconv_modelb, is an image of 67P/Churyumov–Gerasimenko obtained by Rosetta (European Space Agency), downscaled to the resolution and size (between 100 and 200 milliarcseconds) that are typical of the targets observed within the ESO Large Programme described in <ref>. As was mentioned in sec:method, the PSF used to blur the object image is a real AO-PSF obtained with ZIMPOL on a star. Three moons with different contrasts and distances were added to the main body (coloured circles in fig:deconv_modeld). The photon shot noise and the readout noise parameters, η and , are assumed to be identical for all the pixels in the FoV:

    ∀∈FOV, η = η and  =  .

On top of these noises, random pixels are modelled as hot or dead (`salt & pepper' noise) and five cosmic rays are randomly simulated (orange arrows).



The results of the different steps of the method are shown in Figs. <ref>, <ref>, and <ref>. fig:sim_analysis focuses on the main body deconvolution, providing zooms on the deconvolved images, ^mod, the model residuals, - ^mod, and the object residuals, - κ^mod.  stands for the shifted true object, . Indeed, due to the PSF core deformation, the PSF in fig:deconv_modelc may not be properly centred. By construction (see app:algo), my method reconstructs a centred PSF. As a consequence, the deconvolved object may be translated compared to the truth. This shift is estimated by fitting a perfect 2D Moffat model in the true PSF.  is the cubic interpolation of  translated by the position of this fitted 2D Moffat pattern. fig:sim_analysisi shows that this shift is vertical, producing the bright arcs at the top and the bottom of the object residuals. To emphasise the surface texture retrieval rather than a bias in the global intensity, the deconvolved ^mod is further normalised with

    κ = κ̃∑_^mod-κ̃ ,

using the absolute value in the sum to be robust to outlier regions, badly deconvolved or smoothed by the deconvolution such as edges or very bright or dark areas.

fig:deconv_core shows that the oversimplified model of the first step efficiently fits the bright inner part of the halo that is correctly removed from the residuals of fig:deconv_cored and fig:sim_analysisf. By design, the outer part of the structured halo, mainly driven by the AO cut-off frequency, is still present in the residuals, as it cannot be explained by this simple model. As for the object, the threshold applied to the data produces a binary object (see fig:deconv_coreb and fig:sim_analysisb, whose edges are consistent with the blurred object, as is shown by the orange contour in fig:deconv_corea). This is further confirmed by the slices of fig:sim_analysism, the edges of the binary approximation (dark grey) being within a few pixels of the true slice of the object (light grey). As was expected, the overall object intensity is underestimated by more than 10 because the energy diluted by the PSF wings is not integrated and of course none of the surface structure is retrieved, fig:sim_analysisj. As for the PSF core, the radially averaged profile of the blindly estimated Moffat (blue curve in fig:sim_analysisn) shows a very good agreement with the theoretical radial profile of the 2D Moffat (red curve) fitted directly in the true PSF.

fig:deconv_obj shows that using the fitted PSF core is indeed sufficient to retrieve the morphology of the object, as is discussed in sec:core. Structures at the surface of the deconvolved object, fig:deconv_objb and fig:sim_analysisc, are consistent with the simulated truth, fig:deconv_modelb and fig:sim_analysisa, as is shown by the good residuals of fig:sim_analysisk. Nonetheless, the fact that the Moffat model cannot properly grasp the deformations of the PSF core produces two kinds of artefacts: (i) a bright edge on the left side of the object in fig:sim_analysisc,k that suggests that this edge is `over-deconvolved', and (ii) non-homogeneous structures in the residuals at the edge's location, shown in fig:sim_analysisg. This can also be seen in the blue slice in fig:sim_analysism, which shows unnatural deconvolution peaks at the edges of the reconstructed object. And as was already discussed, the fact that the PSF core does not account for the PSF wings induces an underestimation of the object's intensity, here by more than  6. I also emphasise that, by construction, the halo is found in this deconvolved image of fig:deconv_objb, as well as the moons and other noise artefacts. They consequently appear in the modelled data, fig:deconv_obja, and the moons are not visible in the residuals, fig:deconv_objd.

All of these problems are solved after a few iterations of the alternate algorithm, once the PSF wings are properly deconvolved, as is shown by fig:deconv_wing. Concerning the object, the zooms of fig:sim_analysisd,l and the green slice of fig:sim_analysism show that, despite being slightly smoothed by the regularisation of eq:cost_obj, the details are quantitatively retrieved with the correct intensity within 100.2-97.3 ≤ 3. The sharp edges are also well defined at the correct location (see the green and light grey curves). Compared with fig:sim_analysisc,k, the `over-deconvolution' problem on the left side of the object is greatly improved. Nonetheless, the combination of (i) the threshold to segment the main object, (ii) the regularisation in eq:cost_obj, and (iii) the approximate knowledge of the PSF core (or equivalently the cut-off of the optical transfer function of the system) within a blind approach, limits the precision of the object's edge recovery, as is visible in the data residuals of fig:sim_analysish. In addition, the spatial regularisation erases the finest details on the surface of the object, as is seen in the model residuals of fig:sim_analysisl. These are known limits of the edge-preserving regularisation <cit.> but are a necessary trade-off with the PSF reconstruction to control the alternate blind deconvolution convergence. Otherwise, the residuals reach the noise level, below the percent of the data dynamic (see colour bars of fig:deconv_wingd for fig:sim_analysish). There is no noticeable feature at the location of the object's surface that would imply an over-regularisation in the deconvolution, indicating that the details smoothed by the regularisation are indeed lost in the noise level.

Concerning the PSF, all the features mentioned in sec:method are quantitatively retrieved in fig:deconv_wingc. The wind-driven halo is well defined. The brightest structures on the AO cut-off frequency ring and the diffraction spikes are clearly visible and deconvolved. Only the resolution on the speckle pattern is smoothed by the regularisation but its global structure has the correct dynamics. In fig:sim_analysisn, the radial profile of the deconvolved PSF perfectly matches the radial profile obtained on the true PSF. As is explained in sec:wing, the moons and other outliers are robustly penalised in the equivalent weights in fig:weighta and the brightest ones are successfully rejected from the fit, fig:weightb. Thus, they do not impact the model of the halo, fig:deconv_winga, and are consequently nicely visible in the residuals, fig:deconv_wingd. Compared with the injected signals in fig:deconv_modeld, the moons are slightly dimmed and erased from the data, showing that the method suffers from slight self-subtraction. This is nonetheless a cost worth paying to efficiently retrieve the halo and constitutes a trade-off in the regularisation of eq:cost_wing between the reconstruction of the PSF structures and the moon rejection from their deconvolution.






§ EXAMPLES IN REAL DATA





 §.§ The multiple asteroids and 



The ESO Large Programme `Asteroids as tracers of Solar System formation: Probing the interior of primordial main belt asteroids' <cit.> targeted several asteroids of the Solar System's main belt in order to characterise the albedo, surface, shape, and mass of the ones accompanied by moons. The data were obtained with ZIMPOL, mounted on the Spectro Polarimetric High-contrast Exoplanet REsearcher <cit.> of the Very  Large Telescope (VLT) observatory, equipped with the SPHERE Adaptive eXtreme Optics system <cit.>. Mainly designed for polarimetry acquisitions, ZIMPOL has an imaging mode. Working in visible wavelengths, it offers an unprecedented resolution of small Solar System bodies at the cost of a more complex AO-PSF. I applied my method to two targets of this Large Programme: and . 

Among the 100 class asteroids, is quite original. It has a dumbbell shape and is orbited by two known moons, which constrain its gravitational field, and thus its mass and density <cit.>. Its elongated shape and the availability of state-of-the-art deconvolutions and 3D reconstructions make it a good case with which to test the deconvolution performance of the proposed method.


is also an asteroid of the main belt, surrounded by three moons <cit.>. Its faintest and closest companion was missed until recently due to the lack of proper data reduction and data analysis tools <cit.>. It was detected in archival data obtained in 2014 with the Integral Field Spectrograph of SPHERE <cit.>. The complexity of ZIMPOL AO-PSFs and the variability of the brightness and distance of the different moons relative to the main body represent a challenging case with which to test the proposed method. was observed at different epochs in 2018 and 2019 by <cit.>, each epoch gathering multiple frames. In order to assess the consistency of the method results, both in terms of deconvolution and moon enhancement, two different frames of four epochs were selected. To show the versatility of the method and the large acceptability of its assumptions, I also selected additional datasets from other instruments: the IFS (ID 60.A-9362(A), PI: Yang et al. 2014, YJH filter, λ from 0.95 to 1.65), the InfraRed Dual-band Imager and Spectrograph <cit.> of SPHERE, and the Near Infrared Camera 2 (NIRC2, ID U58N2, PI: de Pater et al. 2005, Kp filter, λ=1.248, Δλ=163) of the Keck II telescope.




 §.§ Main body deconvolution and surface detail recovery




fig:Elektra_profilesfig:Kleopatra gather the results obtained on and for different epochs with different seeing conditions. The noise model parameters, η,, of eq:weight, critical to scale the robust penalisation, are empirically adjusted in the data, as is described in app:noise_model.









fig:Elektra_profiles focuses on the final deconvolutions of , obtained after the alternate algorithm. The reconstructed edges are sharp without any `corona artefact' (see below discussion on fig:Kleopatra) or motion blur residual, even for 2019-07-30 (2), which is a challenging case, as is highlighted by the reference deconvolution or the important deformation of the PSF core in fig:comp_instru. Nonetheless, for the 2019-08-05 frames, the overshoot on the left side of the asteroid also indicates a slight over-deconvolution, as was already seen in the simulations, sec:res_sim. Features at the asteroid's surface are resolved, as is seen in the zooms of fig:Elektra_profiles. The evolution of the projected shadows on the surface is visible between the frames of the best epochs, 2019-08-05 and 2019-08-06, and the corresponding slices of fig:Elektra_profiles. This supports the facts that the method is consistent and that these are not deconvolution artefacts. For 2019-08-06, the slices show that the contrast in the structures decreases between the two frames, as the land forms face the Sun more. Except for the main shadow of 2019-08-05, none of these details can be seen in the reference deconvolution obtained by <cit.>.





fig:Kleopatra details the different steps of the blind deconvolution algorithm on . For each epoch, the first columns show a zoom of the data, the fitted binary model, and the deconvolved images based on the knowledge of the PSF core or the PSF wings (and the corresponding residuals) as well as intensity slices along the coloured lines. The deconvolutions obtained by <cit.> are also given to serve as references, b. c show that the binary threshold of the data provides a good approximation of the object support, both in terms of size and shape. Then, it clearly appears that the mere knowledge of the PSF core is sufficient to retrieve the object body, but the deconvolved images of f suffer from what is sometimes called `corona artefact', a stepwise artefact surrounding the object edges that originates from a coupling between the edge-preserving regularisation and an incorrect description of the PSF core <cit.>. These images are similar to the reference reconstructions. For some epochs, such as 2017-07-27, 2017-08-10, and 2017-08-22, the artefacts surrounding the object strongly suggest hiccups of the AO loop or even motion blur. This could be explained by a shift in the telescope pointing direction in between the four Detector Integration Times (DITs), which are averaged in one ZIMPOL-reduced frame by the data reduction pipeline <cit.>. This is particularly visible in the blue slices of 2017-07-27 and 2017-08-10 that cut along this motion blur.


Deconvolving the full PSF solves these issues. In the final step of the proposed method, the edges of are de-blurred and sharp, and the duplicated images are collapsed into a single image in the cases of strong motion blur. This is also emphasised by the green slices in fig:Kleopatra. These deconvolved images obtained via a blind deconvolution algorithm with limited priors on the object and the PSF compare well with state-of-the-art marginal deconvolutions and are consistent with the previously derived 3D model of the asteroid <cit.>. The orange slices of fig:Kleopatra show that the reference method underestimates the object's intensity and can only retrieve smooth and large-scale structures. Due to these limitations, only a difference in albedo between the two lobes of the asteroid was studied by <cit.>. The small-scale structures in the green slices of fig:Kleopatra suggest that finer details from 's surface can be retrieved with my method.



Interestingly, the deconvolution residuals in fig:Kleopatra show a line-by-line horizontal stripe pattern. This can be directly linked to the ZIMPOL imaging mode where only one out of two lines of the sensor is exposed. To produce `high-resolution' images, the reduction pipeline interpolates the missing lines <cit.>. This implies that the proposed deconvolution algorithm reaches the level of the artefacts introduced by the data reduction. To limit the impact of this `zebra' in the deconvolved image, I had to increase the regularisation parameter, ^obj, compared to sec:res_sim, limiting the performance of the method of edge and detail recovery. This explains why spatial features can still be seen in the residuals. These reduction features that correlate neighbouring pixels also mean that the assumption of independent noise previously mentioned in sec:core for eq:weight_diag is not strictly correct and that the diagonal weighting term, , in eq:cost_wing should actually be the inverse of a covariance matrix, as in eq:WLS_full. This effect was nonetheless neglected in this study. 






 §.§ Point spread function reconstruction and halo removal


A quick look at the halo brightness in the data column of fig:Kleopatra shows that the turbulence conditions change between the different epochs. This directly impacts the deconvolved AO-PSF wings and their contrast. For the two epochs 2017-07-14 and 2017-08-10, the PSF is even mainly dominated by the wind-driven halo. Nonetheless, the blind PSF deconvolution successfully retrieves the AO cut-off limit at the correct location as well as its brightest speckles, despite the absence of an instrumental prior in the PSF model. For good seeing conditions, in spite of the elongated shape of , the reconstructed speckles appear nicely resolved and split. In addition, the orientation and elongation of the wind-driven halo do not appear to correlate with 's orientation (see especially 2017-07-14 and 2017-08-22), as one could have feared from a potential cross-talk during the blind deconvolution. This overall consistency favours the robustness of the method in connection with the shape of the object and with the turbulence conditions.

Looking at fig:comp_instrufig:Elektra_data, the conclusions are similar with , with the additional supporting fact that for each epoch the two reconstructed PSFs are extremely consistent in terms of turbulence features (contrast and wind-driven halo). Furthermore, the rotation of the spider diffraction spikes is clearly visible (see 2019-07-30 and 2019-08-06 in particular).




After halo removal, the two moons of are clearly visible at all epochs in the halo residuals, as is shown in the last column of fig:Kleopatra (coloured circles). The transition between a readout noise-limited regime to a photon shot noise-limited regime is also nicely visible: the noise level increases in the object's vicinity where the halo is very bright in the data, especially for bad seeing conditions (2017-07-14, 2017-08-10, and 2017-08-22). Looking at the halo residuals in fig:comp_instru, this transition is also visible, especially for the bad seeing cases of 2019-07-30 and 2019-08-04 with an increased noise level in the object's vicinity. The three moons of are visible in all the presented ZIMPOL frames (coloured circles), even when the deconvolution is not perfect (see 2019-07-30). Indeed, by construction, even if the object and the PSF are not perfectly split, moons (point sources) remain outliers of the model and should appear in the residuals. This is an encouraging result that shows the robustness of the approach. The counterparts of the moons in the robust weight maps are also nicely visible. The orange arrows point towards discarded outliers, such as defective pixels or random cosmic rays as well as camera defects impacting some columns of the detector.

The last lines of fig:comp_instru present results on data obtained with other instruments and reference PSFs measured on a star. They show that the method is robust for a large variety of AO-PSFs.

Looking at the IFS line, my method is able to reproduce the results of my previous pragmatic approach <cit.>, extracting the three moons from the diffuse halo. The poor AO performance of this dataset explains why the diffraction rings of the reconstructed PSF differ from the IFS reference. Nonetheless, the diffraction spikes are visible and the reconstructed multi-lobed core nicely matches the PSF produced by the brightest moon (red circles): it presents a strong secondary lobe, at the top left of the main core. A similar complex PSF core is retrieved with ZIMPOL for the 2019-07-30 (2) frame (secondary lobe on the right). This shows that the method is, to some extent, robust for multi-lobed PSF cores diverging from a simple Moffat pattern, slightly relaxing this potential limitation of the method. The robust weights obtained with the IFS appear quite low. This comes from the wrong noise model fitted in the reduced data via the method presented in app:noise_model. Indeed, the reduction performed by PIC of IFS data <cit.> is based on a spatial regularisation that biases and correlates the noise in the reconstruction. Such an incorrect noise model can be handled by relaxing the constraint on  to lower values for automatic outlier rejection.

The IRDIS data are very noisy, with a dense pollution with defective pixels (common with infrared detectors), while only spans a few pixels. Despite this unfavourable situation, the first Airy rings of the PSF are reconstructed with the correct dynamics and with negligible signal in the residuals. The first ring is fragmented due to poor seeing conditions. Unfortunately, only the brightest moon is visible because of the high noise level and high density of unusable pixels.

Finally, with NIRC2, is barely resolved, with an angular size similar to that of the first diffraction ring of the instrument PSF. Despite this challenging situation, the characteristic hexagonal shape of this ring, due to the fragmentation of the Keck II mirror with hexagonal segments, is nicely recovered. So are the secondary diffraction peaks (two in green at the top right of the core for the first order and a few in yellow for the second order) that produce replicated images of in the saturated data. Concerning the moons, only the brightest is visible in the residuals, highlighting the need for more frames and epochs and dedicated detection algorithms to increase the signal-to-noise ratio.




§ CONCLUSIONS AND PERSPECTIVES



In this paper, I have presented a novel blind deconvolution technique that can retrieve both the object and the faint wings of AO-PSFs. The only main priors on the object are that it be contrasted with a smooth shape and sharp edges. Contrary to most of the existing blind deconvolution algorithms, it is not based on a parametric model of the PSF, allowing one to reconstruct a large variety of shapes and structures, as featured by PSFs after an AO correction. The proposed inverse problem approach is an alternate deconvolution with an edge-preserving regularisation for the object and a regularisation on the smoothness of the gradients for the PSF with intensity whitening. Tested on simulated and real data, my method outperforms state-of-the-art techniques <cit.>, by both efficiently reconstructing details at the surface of the objects and faithfully retrieving the features of the AO-PSFs. It produces a physical and realistic model of the bright halo surrounding the object, which can then be carefully removed. This enhances faint moons in its close vicinity with a better contrast and at closer distances than previous halo-subtraction methods <cit.>. 

Implemented in a general framework and with a limited number of assumptions, my method can be applied to different instruments mounted on different telescopes equipped with different AO systems. It paves the way for the study of resolved asteroids or moon systems around the giant planets of our Solar System via archival data of different instruments for which previous methods could not reveal the companions. It also prepares the arrival of the future generation of ELTs, whose complex and sensitive PSFs and AO systems will require robust and versatile processing techniques. Nonetheless, as is discussed below, there are several ways to further improve this method.

Solving the dual deconvolution problem implies carefully tuning the method. This is mainly done through the hyperparameters of the regularisations on the object and the PSF, namely ^obj, ϵ^obj, ^psf, and . In particular, one could decide to first keep a strong regularisation on the object to prioritise the PSF reconstruction, and to relax it the second time, once the PSF has been correctly estimated, to better retrieve the object's details. Providing an automatic tuning of such parameters to get a fully unsupervised algorithm is an active field of research <cit.> and implementing such methods was beyond the scope of this paper. In this work, these parameters were manually tuned and fixed once and for all for each instrument. The fact that they worked for all the tested datasets is encouraging and shows their good admissible range in terms of noise level, object shape, and seeing conditions. This thus limits the need for further manual tuning by the user.

It was emphasised in sec:res_real that lots of datasets suffer from poor AO-PSF quality or even motion blur. Robustness is obtained via a higher regularisation level, but this can degrade the image quality or produce over-deconvolution. To increase the resolution and the deconvolution quality, and since the targets are very bright, one should work on the individual ZIMPOL DITs rather than the stacked image. The results also show that the method reaches the limits of the ZIMPOL reduction pipeline, whose artefacts dominate the residuals in some datasets. Implementing such changes implies modifying the reduction pipeline and was beyond the scope of this paper.

As is shown in sec:res_real, most of the outliers, such as defective pixels or cosmic rays, are successfully and robustly identified and discarded on the fly by the proposed algorithm. Nonetheless, some of them are missed, and so are the faintest moons. As is discussed in sec:res_sim, the proposed method consequently suffers from slight self-subtraction, which could dim the signal of potential moons. Even if it is less pronounced than with other methods <cit.>, some precautions must be taken when fitting the photometry of a moon that can be underestimated, or when announcing the absence of moons below a given noise-limited contrast. It is possible to assess these biases by injecting false moons as is done in the standard exoplanet detection algorithm to quantitatively estimate the contrast performance <cit.>. This was beyond the scope of the paper, but should be kept in mind for further studies.

Finally, the natural next step of this method is to use to the reconstructed PSF within matching filter algorithms to detect even fainter moons in the residuals. Combined with the refined noise model and with the robust weights to limit false positive detections by discarding outliers, this would push the detection limits even further. Echoing the challenges of exoplanet detection, it comes nonetheless with the additional difficulty of the rapid motion of the moons around the main object <cit.> and is an open research field.











	The author would like to warmly thank the Referee Laurent Mugnier for his pertinent feedback and numerous remarks and suggestions that greatly contributed to improving the quality and clarity of the article.
	

	The author would like to thank Éric Thiébaut and Ferréol Soulez for the fruitful discussions on robust detection algorithms and adequate prior for high dynamic unknowns as well as Benoît Carry for his careful proofreading.
	

	All the ZIMPOL reductions are based on public data acquired for the ESO Large Programme ID 199.C-0074 (Vernazza et al. 2021, `Asteroids as tracers of Solar System formation: Probing the interior of primordial main belt asteroids') and available at https://observations.lam.fr/astero/https://observations.lam.fr/astero/
.
	

	The IFS and IRDIS reductions are based on public data provided by the ESO Science Archive Facility. The author would like to thank Antoine Kaszczyc and Maud Langlois who provided the PSFs and reduced the IRDIS raw data.
	

	The NIRC2 reductions are based data provided by the Keck Observatory Archive (KOA), which is operated by the W. M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration. The author would like to thank Kate Minker who reduced the raw data and provided the PSF.
	

	This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101004719.






aa













§ PSEUDO-CODES OF THE METHOD



This appendix gathers the pseudo-codes of the different steps of the proposed method, described in sec:method. In practice, all the code is implemented in MATLAB within the open-source  framework <cit.>. I also detail in this appendix how the different cost functions are minimised.



 §.§ Estimation of the point spread function core



The different parameters of the first step of sec:core were alternately fitted, as was described in alg:core. Indeed, applying a threshold to the data is not a differentiable operation, complicating the minimisation with iterative gradient descent methods. A first set of parameters of the PSF core was guessed, corefirst_guess, on a coarse grid of threshold values, corerough_set. These values were then refined by alternating five fits on (i) the threshold on the data, along with the amplitude parameter, coreobj_thr, and (ii) the PSF core parameters, corePSF_fit. The optimisation problems were solved with the simplex search method of <cit.>. For each problem, 200 iterations were performed. The parameters of the Moffat profiles were initialised with α = 3 and β=1.6, which are standard values <cit.>. To reweight, , corereweight, a general threshold of 2.5 was applied to the data. From the perspective of the dynamic of the PSF, this value roughly corresponds to the transition between its wind-driven core and its wings (see fig:deconv_modelc). Only the meaningful pixels in the brightest part of the halo, not shaded in red,  fig:deconv_corea, are thus kept in the data fidelity term.






 §.§ Alternate object and point spread function deconvolution




alg:obj_wing details the implementation of the dual deconvolution algorithm of the object and the PSF wings of sec:deconvsec:wing. The role of obj_wingth_obj is to define the main object support. I empirically chose a threshold of d̅^sup between 15 and 25 of the maximal value of the median-filtered deconvolved object, med (kernel of 5×5 pixels, as in alg:noise). Such a threshold is low enough to account for partially illuminated pixels on the object's edge, while being conservative enough to correctly constrain and tackle the `corona artefacts' mentioned in sec:res_real. In practice, this threshold was applied only in the vicinity of the main body. To further  avoid crenellated edges on the segmented image, this support was dilated by one pixel. The rest of the FoV was automatically excluded to avoid bright moons or artefacts, such as cosmic rays in the deconvolved image, exceeding this threshold.

Conversely, for obj_wingw_rob_obj and obj_wingw_rob_wing, the  threshold was applied only in the halo to exclude the moons or artefacts, such as cosmic rays, but not in the main body vicinity, to keep this area (see fig:weightb). Indeed, this area is critical to fit the PSF core features but hard to fit due to the artificial threshold on the deconvolved image and the edge-preserving regularisation, as is seen in the robust weights of fig:comp_instru. I typically used the conservative value of 50, as it is important to remove any moon suspicion while the deconvolution of the smoothly varying PSF is robust for a high rate of missing data. In the presence of strong outliers (dead pixels) on the object support or its vicinity, as in fig:weightb, it is still possible to apply a less conservative threshold to this region, typically below 10.

The two minimisation problems of obj_wingth_obj and obj_wingwing were solved with 1000 iterations of the variable metric with limited memory-bounded algorithm <cit.>, a limited-memory quasi-Newton method with Broyden-Fletcher-Goldfarb-Shanno updates <cit.> that handles bound constraints. In the case of strong motion blur, typically alt=30 alternate iterations are looped.

In obj_wingw_rob_obj and obj_wingw_rob_wing, the weights were robustly updated only after a few alternate loops,  wgt=5. I would like to remark here that if a proper calibration of the reduced data has been made, the robust weights, obj_wingw_rob_0, can be initialised with a map of already known aberrant pixels. This was done for the IRDIS dataset.







§ EMPIRICAL NOISE MODEL



The attentive reader may notice that the proposed method implies knowledge of the noise model parameters, η,, in eq:weight. If they are known in the simulated case of sec:res_sim, they are a priori unknown for the real data of sec:res_real. In principle, the noise model can be obtained through proper calibration and analysis of both calibration and science data <cit.>. Nonetheless, most of the data used in this paper are ZIMPOL data from the ESO Large Programme described in sec:res_data that were already reduced <cit.>. It was beyond the scope of this paper to fully recharacterise the ZIMPOL sensor noise and rerun a reduction procedure. As a consequence, I chose to implement a pragmatic approach that could empirically fit these parameters directly in the data of interest and that could be easily applied to reduced data from other instruments. The principle of the method is pictured in fig:noise_model.



The noise model of eq:weight is basically an affine law that links the data intensity and its associated variance. With the assumption that the noise parameters are constant over the FoV, eq:weight_const, the two noise parameters, η,, can be fitted on the full data image using the diversity of information, as is implemented in alg:noise.


First, as is shown in fig:noise_modelb, the noise map was approximated by removing from the data, a smoothed image obtained after applying a median filter of 5×5 pixels, noisemed. The pixels were then split among arc arcs (with a typical width of 5 pixels and length of 20 pixels for the ZIMPOL data) centred on the main body, shown in fig:noise_modela. The intensities (resp. variances) of the model were empirically estimated by averaging the data (resp. by computing the variance of the noise map) on each arc, noiseavg (resp. noisevar). To be robust with regard to the presence of outliers among this point cloud, fig:noise_modelc, the affine law was robustly fitted via an approach based on the median absolute deviation of the model from the empirical points, noisemad, a robust estimator of the standard deviation <cit.>. 



This method was tested on the simulated data of sec:res_sim, for which the true noise parameters are known. The results of the fit, given in fig:noise_modelc, show that the noise model was successfully estimated.

I point out here that these noise parameters could be iteratively refined during the iterations of the alternate algorithm once the model of the convolution between the object and the PSF, namely the theoretical intensity map, and the model of the halo residuals, namely the theoretical noise map, are better constrained.[See for example the documentation of the Epifluorescence DEconvolution MICroscopy plug-in <cit.> at https://icy.bioimageanalysis.org/plugin/epidemic/https://icy.bioimageanalysis.org/plugin/epidemic/.] This was nonetheless beyond the scope of this paper and is kept for a future work. Indeed, the homogeneity of the weight maps in fig:comp_instrufig:Elektra_data shows that the proposed method is a good enough approximation, correctly whitening both the readout noise and the photon shot noise.








§ ADDITIONAL FRAMES OF 



fig:Elektra_data comes in addition to fig:comp_instru and presents the second frame of the couples introduced in fig:Elektra_profiles. This figure supports the consistency of my method that produces similar results within each epoch, in terms of PSF structures and dynamics as well as moon enhancement.







 Paper Title: Doppler Ambiguity Elimination Using 5G Signals in Integrated Sensing and Communication.







Doppler Ambiguity Elimination Using 5G Signals in Integrated Sensing and Communication
 

This work was funded by the European Commission through the H2020 MSCA 5GSmartFact project under grant agreement number 956670. Also, this work was supported by the project 6G-DISAC under the European Union's Horizon research and innovation program with grant agreement number 101139130.

    Keivan Khosroshahi12, Philippe Sehier2, and Sami Mekki2,

1
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, Gif-sur-Yvette, France

2 Nokia Standards, Massy, France

 keivan.khosroshahi@centralesupelec.fr, {philippe.sehier, sami.mekki}@nokia.com

    August 5, 2024
==========================================================================================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
The industrial point of view towards integrated sensing and communication (ISAC), the preference is to leverage existing resources and fifth-generation (5G) infrastructure to minimize deployment costs and complexity. In this context, we explore the utilization of current 5G new radio (NR) signals aligned with 3rd generation partnership project (3GPP) standards. Positioning reference signals (PRS) for sensing and physical downlink shared channel (PDSCH) for communication have been chosen to form an ISAC framework. However, PRS-based sensing suffers from Doppler ambiguity when the Doppler frequency shift is severe. To address this challenge, we introduce a novel method within the ISAC system that leverages the demodulation reference signal (DMRS) present in PDSCH to eliminate Doppler ambiguity. Furthermore, we formulate a resource allocation problem between PRS and PDSCH to achieve a Pareto optimal point between communication and sensing without Doppler ambiguity. Through simulations and analysis, we demonstrate the effectiveness of our proposed method on joint DMRS-PRS exploitation in mitigating Doppler ambiguity and the efficiency of the resource allocation scheme in achieving Pareto optimality for ISAC within a 5G NR framework.


PRS – PDSCH – DMRS – ISAC – Doppler ambiguity 




§ INTRODUCTION

The adoption of integrated sensing and communication (ISAC) technology in upcoming mobile networks is increasingly gaining consensus due to the emergence of future applications in the domains of industry 4.0, automotive, internet of things (IoT) in smart cities and health care, which impose new requirements on wireless communication networks <cit.>.
Integrating sensing services into the mobile infrastructure enables new services to be deployed at low cost, reusing the mobile coverage base already deployed and without requiring additional spectrum resources, while minimizing the impact on existing equipment.

Several reference signals have been defined in fifth-generation (5G) new radio (NR), addressing different objectives. They have been specified to provide excellent passive detection performance, robust anti-noise capabilities, and favorable auto-correlation characteristics <cit.>. Moreover, using pilots in ISAC systems leads to low hardware costs, easy implementation, and low computational complexity <cit.>. Consequently, significant interest has emerged in designing ISAC signals based on pilots. Among the pilots available in 5G networks, the use of the positioning reference signal (PRS) for sensing purposes holds significant promise, given its rich time-frequency resources and flexible configuration. PRS, introduced in 3rd generation partnership project (3GPP) Release 16 of the 5G specification, aims to improve the positioning accuracy of connected user equipments (UEs) due to its high resource element (RE) density and superior correlation properties compared to existing reference signals due to the diagonal or staggered PRS RE pattern.  In sensing context, PRS is used in <cit.> to estimate the range and Doppler, and its performance is compared with other pilots available in 5G standard, namely demodulation reference signal (DMRS), channel state information reference signal (CSI-RS) and synchronization signal (SS). A two-stage joint pilot optimization, target detection, and channel estimation scheme has been introduced in <cit.> for ISAC. In <cit.>, DMRS and CSI-RS are separately used for velocity and range estimation. In <cit.>, a multi-range joint automotive radar and communication system based on pilot-based orthogonal frequency-division multiplexing (OFDM) waveform is investigated. 

However, exploiting the existing reference signals of the 5G-NR raises new issues. This is because the sensed targets might be far from the gNB, or their speed may be high. Consequently, the reflected signals may be received with a large time delay, above the cyclic prefix length, and large Doppler shifts. 
Periodogram-based estimation for delay and Doppler exhibit spurious peaks in these situations, creating ambiguities in distance or velocity estimates. Such fake peaks can be interpreted as real targets. While the proposed solution in <cit.> has focused on multiple reference signals to address the ambiguity, they are not directly applicable in ISAC scenarios due to potential interference between physical downlink shared channel (PDSCH) for data transmission and other reference signals such as PRS and CSI-RS. Additionally, the method described in <cit.> leads to a significant reduction in the maximum detection range, especially when employing large PRS comb sizes. Hence, to the best of our knowledge, a 3GPP-compliant solution to address range ambiguity within the ISAC framework has not been proposed.


In this work, we introduce a novel approach leveraging the current reference signals in 5G NR to enable sensing for unconnected targets with enhanced maximum detectable Doppler frequency shift without Doppler ambiguity. To provide sensing services alongside data transmission, we propose an ISAC OFDM resource grid comprising PDSCH for communication and PRS for sensing. To remove the Doppler ambiguity, we propose a method to reuse DMRS within PDSCH for sensing in addition to communication. Furthermore, we propose time and frequency resource allocation between PRS and PDSCH since sensing and communication symbols cannot share the same REs in an OFDM resource grid. Building upon this, we formulate a resource allocation problem between PDSCH and PRS to achieve Pareto optimality between communication and sensing without Doppler ambiguity. Ultimately, via extensive simulations, we prove the efficacy of our proposed method in addressing Doppler ambiguity and finding the Pareto optimal point between communication and sensing.





§ PRS AND DMRS DESCRIPTION

The 5G-NR waveform is built on a time-frequency resource grid as described in technical specifications (TS) 38.214 <cit.> and the smallest time-frequency resource is a physical resource
block (PRB), consisting of 12 contiguous sub-carriers and 14 consecutive symbols. In the following two sections, we explain about PRS and DMRS in 5G NR.



 §.§ PRS

The generation of PRS as explained in TS 38.211 <cit.> as follows



    r(m) = 1/√(2)(1 - 2c(2m)) + j1/√(2)(1 - 2c(2m + 1)),


where c(i) is denoted as Gold sequence of length-31 and the starting value of c(i) for the PRS is provided in <cit.>. PRS is generated from the Gold sequence and it has favorable anti-noise and auto-correlation characteristics. The PRS is allocated a minimum of 24 PRBs and a maximum of 272 PRBs, depending on the carrier size, showing the flexible transmission parameters supported in 5G NR. This flexibility allows for a versatile configuration of time-frequency resources to fulfill sensing accuracy requirements across diverse applications and scenarios. As outlined in TS 38.211 <cit.>, PRS supports four comb structures, i.e., Comb {2,4,6,12}, in the frequency domain and five symbol number configurations, i.e., Symbol {1,2,4,6,12}, in the time domain. The structure of PRS with 12 symbols and two comb sizes is illustrated in Fig. <ref>.




 §.§ DMRS

Another reference signal 
that can potentially be used in the ISAC framework is DMRS, available in the PDSCH. DMRS replaces the cell-specific reference signal (CRS) in long-term evolution (LTE) for channel estimation, and PDSCH decoding <cit.>. DMRS generation is also based on (<ref>), and the initial value of c(i) for DMRS can be found in <cit.>. DMRS generation occurs within the designated PDSCH allocation, as detailed in TS 38.211 <cit.>. The allocated resources for PDSCH are available in the bandwidth part (BWP) of the carrier as explained in TS 38.214 <cit.>. DMRS is used for channel estimation and is confined to the resource blocks (RBs) assigned for PDSCH. Its structure is adaptable to various deployment scenarios and use cases. The positioning of DMRS symbols depends on the mapping type, which can be slot-wise (Type A) or non-slot-wise (Type B). Additional DMRS symbol positions are determined by tables outlined in TS 38.211 <cit.>. Furthermore, front-load DMRS may occupy 1 to 4 OFDM symbols in the time domain. DMRS is within physical channels allocated for communication receivers and is transmitted using the same beam selected for communication receivers. 








§ ISAC SYSTEM MODEL

The existing 5G NR signals were not initially designed for radar sensing applications. This section outlines a methodology for efficiently leveraging both communication and sensing signals within the ISAC framework. Our vision involves implementing a downlink ISAC system where a gNB directs signals toward a sensing area where K point-like targets exist. The receiver captures echoes from these targets to estimate their range and velocity. We assume that the targets are within the line-of-sight (LOS) of both the transmitter and the receiver. Fig. <ref> depicts an example of the considered scenario. If we consider an OFDM resource grid with N symbols in the time domain and M sub-carriers in the frequency domain, the transmitted signal in the time domain can be written in the following form: <cit.>




    s(t) = ∑_n = 0^N-1rect(t - nT_0/T_0)∑_m=0^M-1 v(m,n) e^j2π mΔ f(t - nT_0),

where rect(t/T_0) denotes the rectangular pulse, Δ f represents the subcarrier spacing. T_0 = T_CP + T_s is the total duration of the OFDM symbol, where T_s = 1/Δ f represents the symbol duration, and T_CP denotes the duration of the cyclic prefix (CP). v(m,n) is the complex transmitted symbol at OFDM symbol n and sub-carrier M, where n=0,...,N-1 and m=0,...,M-1 within an M × N OFDM resource grid. If there are k targets present in the sensing area, the echo signals from targets received by the receiver can be written as <cit.>



    r(t)=∑_k=1^Kβ_k s(t-τ_k)e^j2π f_d,kt + u(t),

where β_k denotes the attenuation factor of the k-th target, f_d,k represents the Doppler frequency shift of the k-th target, τ_k signifies the delay of the k-th target and u(t) ∈ℂ refers to complex additive white Gaussian noise (AWGN) with zero mean and variance of 2σ^2.


Conventional OFDM receivers proceed by sampling the signal at each symbol time, followed by performing fast fourier transform (FFT) to extract the modulated symbols. Under normal circumstances, delay and Doppler effects are compensated, leading to mitigation of intersymbol interference (ISI). The cyclic prefix is typically leveraged to address receive timing errors and multipath spread. However, in our scenario, the path delay and Doppler may not fall within the usual range due to significant uncertainties regarding the target's position and speed. As a result, it becomes necessary to consider ISI in both the time and frequency domains. Consequently, the received modulation symbols can be expressed as:



    y(m,n) =   ∑_k=1^Kβ_k e^j2π n T_0 f_d,k e^-j2π m Δ f τ_k v(m,n) + ISI
       + q(m,n).


The term q(m,n) ∈ℂ represents the complex AWGN on the m-th subcarrier and the n-th OFDM symbol. This noise has a zero mean and a variance of 2σ^2, and it is obtained from the sampling and FFT process over r(t).



§ ESTIMATION OF THE RANGE AND VELOCITY 

Doppler and delay would cause a rotation on the time and frequency axis, respectively over the received samples. Determining range and Doppler necessitates a two-dimensional search. Multiple techniques are outlined in <cit.>; however, we focus here on a computationally efficient method operating in two stages: initially estimating the range and subsequently the Doppler, after compensating for the delay. The bistatic distance, i.e., from the transmitter to the target and then from the target to the receiver, can be derived from range assessment using the periodogram <cit.>. To achieve this, we eliminate the transmitted sensing symbols from the received echoes through point-wise division as described below. 



    g(m,n) =    
        y(m,n)/v(m,n),        If   v(m,n) ≠ 0,
    
          0,        If   v(m,n) = 0.


Afterward, we compute M-point inverse fast fourier transform (IFFT) of the PRS on the n-th column of the g(m,n)



    r_n(l) =    |IFFT(g(m,n))| = |∑_k=1^K (β_k e^j2π n T_0 f_d,k∑_m=0^M-1
        e^-j2π m Δ f  R^tot_k/c e^j2πml/M+ q'(m,n)/v(m,n)e^j2πml/M)|,

where l= 0,...,M-1, |.| is the absolute value, q'(m,n) = q(m,n) + ISI, and we replaced τ_k = R^tot_k/c while R^tot_k is the bistatic range, c is the light speed. The absolute value is taken to reduce Doppler sensitivity. When the arguments of e^-j2π m Δ f  R^tot_k/c e^j2πml/M in (<ref>) cancel each other, the maximum value occurs. The next step is to perform IFFT over all columns of the g(m,n) and average over them to increase the signal-to-noise ratio (SNR):



    r(l) = 1/N∑_n=0^N-1 r_n(l).


Afterward, we need to find the index of the maximum value named l̂_k in (<ref>). Then, the bistatic distance of each target can be calculated as <cit.> 



    R̂^tot_k = l̂_kc/Δ f M.


The range resolution is



    Δ R = c/Δ f M,

and the expression for the maximum detection range can be written as



    R_max =  c/Δ f.


According to <cit.>, enhancing accuracy performance is achievable by increasing the number of IFFT points m_a as shown below



    r_n(l) =    |IFFT(g(m,n))| = |∑_k=1^K (β_k e^j2π n T_0 f_d,k∑_m=0^m_aM-1
        e^-j2π m Δ f  R^tot_k/c e^j2πml/m_aM + q'(m,n)/v(m,n)e^j2πml/m_aM)|.


In this case, (<ref>) will change into

    R̂^tot_k = l̂_kc/m_aΔ f M.


As highlighted in <cit.>, increasing m_a results in range accuracy enhancement, albeit at the expense of heightened computational complexity.

After compensating the delay caused by the targets, for Doppler estimation, we perform N-points FFT on the m-th row of the g(m,n)



    v_m(d) =    |FFT(g(m,n))| = |∑_k=1^K (β_k ∑_n=0^N-1 e^j2π n T_0 f_d,k e^-j2πnd/N
        + q'(m,n)/v(m,n)e^-j2πnd/N)|,

where d= 0,...,N-1. We perform FFT across all rows of the g(m,n) and then average the results as shown below:



    v(d) = 1/M∑_m=0^M-1 v_m(d).


Afterward, we determine the index of the maximum value, denoted as d̂_k for target k in (<ref>), and the Doppler frequency can be estimated as <cit.> 



    f̂_d,k = d̂_k/T_s N.


The Doppler frequency resolution is



    Δf̂ = 1/T_s N.

and the maximum detectable Doppler frequency estimation is 



    f̂_max = 1/T_s.


We can improve the accuracy of Doppler frequency estimation by increasing the FFT point similar to range estimation.



 §.§ Doppler Ambiguity

PRS was initially designed for positioning in 5G NR and not for sensing. In positioning, time-frequency uncertainties are mitigated by the time-frequency servo loop, which is not the case for sensing. PRS features a periodic structure that can be adjusted by the comb size, as demonstrated in Fig. <ref>, where an example with a comb size, i.e., K_comb = 6 is presented. 
The presence of empty REs within the PRS structure leads to the emergence of fake peaks in Doppler estimation using the method described in section <ref>. These peaks introduce ambiguity in distinguishing between the real and fake Doppler frequency shift resulting in Doppler ambiguity as shown in Figs. <ref> and <ref>. 
The cost of the approach proposed in <cit.> to circumvent this ambiguity is the reduction of the maximum detectable range and Doppler frequency shift by a factor of K_comb. In scenarios where precise Doppler resolution is crucial, reducing the subcarrier spacing, as defined in (<ref>), becomes necessary. However, this adjustment with larger comb sizes leads to a notable reduction in the maximum detectable Doppler frequency shift. Hence, an alternative method is necessary to alleviate Doppler frequency ambiguity while ensuring a high maximum detectable Doppler frequency.





 §.§ Proposed Method

The key idea to eliminate ambiguity in Doppler estimation is to leverage DMRS as a reference signal within the PDSCH for ISAC scenarios where the OFDM frame contains PRS for sensing and PDSCH for communication. To this end, we generate an OFDM frame, and based on the sensing and communication requirements of each application, we perform the time and frequency resource allocation. Fig. <ref> depicts an example of such resource allocation in the frequency domain between PRS and PDSCH. In this scenario, DMRS serves not only for communication tasks like channel estimation but also for enhancing sensing capabilities by accurately distinguishing between real and fake Doppler frequency shifts. Additionally, this improvement is achieved without compromising accuracy or the maximum detectable Doppler frequency. Additionally, it does not require any modifications to the gNB configuration, thus fully aligning with 3GPP standards. 

Initially, we set double symbol for DMRS configuration and one additional DMRS symbol. Based on DMRS configuration type, the total number of subcarriers per PRB that can be allocated to DMRS, i.e., M', is 4 or 6. We define g_DMRS(m,n) and g_PRS(m,n) as the result of point-wise division similar to (<ref>) over DMRS and PRS symbols, respectively. Then, we form the following



    g_PRS,DMRS^m,m'(n) = g_PRS^m(n) + g_DMRS^m'(n),

where g_PRS^m(n) is the m-th row of g_PRS(m,n) and g_DMRS^m'(n) is the m'-th row of g_DMRS(m',n). Based on TS 38.211 <cit.>, m' = {0,2,4,6,8,10} for DMRS configuration type 1 and m' = {0,1,6,7} for DMRS configuration type 2. Then we calculate FFT over m-th row of g_PRS,DMRS^m,m'(n) as follows




    v_m^m'   (d) = |FFT(g_PRS,DMRS^m,m'(n))| = |∑_k=1^K (2β_k ∑_n=0^N-1 e^j2π n T_0 f_d,k
       .e^-j2πnd/N + (q'(m,n)/v_PRS(m,n) + q'(m',n)/v_DMRS(m',n))e^-j2πnd/N)|,

where v_PRS and v_DMRS are transmitted complex symbols of PRS and DMRS, respectively. Afterward, we perform averaging over m as follows



    v^m'(d) = 1/M∑_m=0^M-1 v_m^m'(d).

Eventually, we obtain the element wise multiplication of v^m'(d) over m' as follows

    v(d) = ∏_m'=0^M'-1 v^m'(d).


At the end, we use (<ref>) to find the index of the maximum value and estimate Doppler. The Doppler frequency shift resolution and the maximum detectable Doppler frequency can be obtained using (<ref>) and (<ref>), respectively. This approach disrupts the periodic pattern of PRS, effectively reducing the magnitude of spurious peaks. Consequently, distinguishing between real and false Doppler frequencies becomes straightforward. 
If PRS and PDSCH have different resources allocated in the time domain, i.e., slots, we pad zeros to the reference signal with the smaller slots so g_PRS^m(n) and g_DMRS^m'(n) can have similar vector sizes.





§ PARETO OPTIMALITY IN ISAC RESOURCE ALLOCATION

In this section, we introduce an optimization problem to reach Pareto optimal point in time and frequency resource allocation between communication and sensing without Doppler ambiguity through the proposed algorithm. Towards this goal, we define the range resolution metric as c/Δ f, R_0 as the maximum throughput of the received PDSCH per one slot and one PRB, and 1/T_s as the Doppler resolution metric. We present the following resource allocation problem to identify the Pareto optimal point.




    m_0,n_0,...,m_K,n_K max   α_0 m_0 n_0/R_max R_0 -  ∑_k=1 ^K( γ_k,1/d_max  m_kc/Δ f
       + γ_k,2/ν_max  n_k1/T_s ),  
    s.t.    m_i ∈{1,...,M_max-1},  i = 0,...,K, 
         n_i ∈{1,...,N_max-1}, i= 0,...,K,  
       ∑_i=0^K m_i = M_max, 
       ∑_i=0^K n_i = N_max,


where α_0, γ_k,1, and γ_k,2 represent the communication weight, range resolution weight, and Doppler resolution weight of target k, respectively. These values can be selected based on the priorities of the application, ensuring that α_0 + ∑_k=1^K (γ_k,1 +γ_k,2) = 1. M_max and N_max are available resources in frequency and time domain, respectively. R_max, d_max, and ν_max denote the maximum achievable throughput, range resolution, and Doppler resolution, respectively, when all available resources in the time i.e., N_max, and frequency i.e., M_max, domains are allocated for communication data transmission or range and Doppler estimation. Additionally, m_0 and n_0 represent the number of PRBs and slots, respectively, allocated for communication, while m_k and n_k denote the number of PRBs and slots allocated for range and Doppler estimation of the target k.
By solving this optimization problem, we can identify the Pareto optimal point for resource allocation between communication and sensing.






§ SIMULATION RESULTS 

To validate our approach, we exploited MATLAB 5G toolboxes, ensuring compatibility with 3GPP standards. We conducted simulations for a scenario where two targets caused the Doppler frequency shift of 4kHz and 9kHz. The central frequency was set to 4GHz with a subcarrier spacing (SCS) of 30kHz, and the SNR is 0dB. PRS number of symbols is 12, DMRS configuration type is 2, the number of front-loaded DMRS symbols is 2 with 1 additional DMRS symbol position which make 4 symbols in total, PDSCH mapping type is "A", Code rate used to calculate transport block sizes is 490/1024 based on <cit.>. Virtual resource block (VRB) bundle size is 4, VRB to PRB interleaving is disabled, and modulation is 16 quadrature amplitude modulation (16-QAM) for PDSCH and β_1=β_2 =1. In Fig. <ref>, we can observe the Doppler ambiguity using PRS only, with the real and false peaks of Doppler frequency shifts. However, in Fig. <ref>, the magnitudes of the fake Doppler frequencies are significantly suppressed by using the proposed method, and we can easily select the two highest peaks originating from two targets. It is worth mentioning that the number of the targets can be estimated
by model order selection based on information-theoretic criteria <cit.>. In Fig. <ref>, we can see an extreme Doppler ambiguity when K_comb = 12 while
Fig. <ref> shows the performance of the proposed method with extreme ambiguity in Doppler estimation. In this case, we can still easily select the highest two peaks and estimate the Doppler frequency shifts of the targets correctly. Once the Doppler ambiguity is mitigated, the Pareto optimal point, as illustrated in Fig. <ref> in time and frequency resource allocation can be found. F in Fig. <ref> is the objective function of the optimization problem in (<ref>). In this simulation, we set M_max = 48, N_max = 20, K=1, α_0 = 2, and γ_1,1 = γ_1,2 = 1. As illustrated in Fig. <ref>, the Pareto optimal point is attained by allocating 3 slots and 5 PRBs for PRS, with the remaining resources allocated to PDSCH.















§ CONCLUSION

In this work, we addressed the Doppler ambiguity appeared using PRS  by exploiting DMRS within PDSCH for unconnected targets in ISAC framework. We proposed a novel method to alleviate Doppler ambiguity, which is fully compliant with 3GPP standards and can be integrated into existing 5G networks. Furthermore, our approach enables sensing without imposing additional overhead, requiring modifications to gNB configurations, or compromising Doppler estimation accuracy, while also enhancing the maximum detectable Doppler shift. Finally, we introduced a resource allocation problem between PDSCH and PRS to determine the Pareto optimal point between communication and sensing while effectively eliminating Doppler ambiguity.
 






IEEEtran




 Paper Title: Flexible Beam Coverage Optimization for Movable-Antenna Array.





Flexible Beam Coverage Optimization for Movable-Antenna Array


    Dong Wang, Weidong Mei, Boyu Ning, and Zhi Chen
National Key Laboratory of Wireless Communications 

University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China
 

Emails: DongwangUESTC@outlook.com; wmei@uestc.edu.cn; boydning@outlook.com; chenzhi@uestc.edu.cn


    August, 2024
======================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
Fluid antennas (FAs) and movable antennas (MAs) have attracted increasing attention in wireless communications recently. As compared to the conventional fixed-position antennas (FPAs), their geometry can be dynamically reconfigured, such that more flexible beamforming can be achieved for signal coverage and/or interference nulling. In this paper, we investigate the use of MAs to achieve uniform coverage for multiple regions with arbitrary number and width in the spatial domain. In particular, we aim to jointly optimize the MAs' weights and positions within a linear array to maximize the minimum beam gain over the desired spatial regions. However, the resulting problem is non-convex and difficult to be optimally solved. To tackle this difficulty, we propose an alternating optimization (AO) algorithm to obtain a high-quality suboptimal solution, where the MAs' weights and positions are alternately optimized by applying successive convex approximation (SCA) technique. Numerical results show that our proposed MA-based beam coverage scheme can achieve much better performance than conventional FPAs.




§ INTRODUCTION

	Beamforming is a pivotal signal-processing technology for multiple-antenna systems, which is able to enhance/reduce the signal strength over desired/undesired directions by optimizing the amplitude and/or phase of the antenna weight vector (AWV)<cit.>.  However, due to the fixed geometry of conventional fixed-position antenna (FPA) arrays, their steering vectors have inherent correlation/orthogonality among different steering angles, thus resulting in difficulty in achieving flexible beamforming for interference nulling and/or beam coverage. 
	
	Recently, fluid antennas (FAs) and movable antennas (MAs) have emerged as a new technology that enables local antenna movement within a confined region at the transmitter/receiver, thereby providing additional spatial degrees of freedom (DoFs) to enhance the wireless communication performance<cit.>. Inspired by its compelling benefits, there have been some prior studies focusing on its use for point-to-point communications, multi-user systems, secure communicationes, among others<cit.>. In particular, the authors in <cit.> and <cit.> have shown the capability of FAs/MAs to achieve flexible interference nulling, which is able to amplify signals over desired directions and mitigate interference over undesired directions at the same time.
			
	Different from the above papers, we investigate in this paper the use of MAs for flexible beam coverage, aiming to achieve uniform beam gain over a single or multiple regions with different width in the spatial domain, as shown in Fig. <ref>. Note that such a beam coverage problem can find broad applications in wireless communications. For example, due to the mobility of users, the beamwidth can be dynamically adjusted to cover an area based on the distribution of the users within it<cit.>. Additionally, the number of covered areas may fluctuate over time due to varying network traffic demands. Consequently, more efficient beam coverage approaches are required compared to beamforming design with conventional FPAs only to accommodate diverse requirements for beamwidth and beam number, which motivates this work.
	In particular, we aim to jointly optimize the MAs' weights and positions within a linear array to maximize the minimum beam gain over the desired spatial regions. However, the resulting problem is non-convex and difficult to be optimally solved. To tackle this challenging problem, we propose an alternating optimization (AO) algorithm to obtain a high-quality suboptimal solution to it, where the MAs' weights and positions are alternately updated by applying the successive convex approximation (SCA) technique. Numerical results show that our proposed MA-based beam coverage scheme can achieve much better coverage performance than conventional FPAs.
	
	    
 Notations: ℝ and ℂ represent the sets of real and complex number, respectively. For a complex number a, Re{a} and | a| denote its real part and modulus, respectively. For a complex-valued vector x, x^T, x^H, ‖x‖_2 and x(n) denote its transpose, conjugate transpose, l_2-norm, n-th element, respectively. For a matrix A, Tr(A) represents its trace. x∼𝒞𝒩(μ,R) represents that x follows circularly symmetric complex Gaussian (CSCG) distribution with mean μ and covariance matrix R.



§ SYSTEM MODEL

As depicted in Fig. 1, we consider a transmitter equipped with a linear array with N MAs, whose positions can be flexibly adjusted within a one-dimensional (1D) region denoted as 𝒞_t. The length of the MA array is set to D. For convenience, we assume a reference origin point x_0=0 in the transmit region 𝒞_t, as shown in Fig. 1. Let x_n denote the coordinate of the n-th MA's position, n∈𝒩≜{1,2,⋯,N}. Accordingly, the antenna position vector (APV) of all N MAs is denoted by x≜[x_1,x_2,⋯,x_N]∈ℝ^N×1. Let f_c and λ denote the operating carrier frequency of the considered system and carrier wavelength, respectively. For a given steering angle θ, the array response vector of the MA array can be determined as a function of the APV x and θ, i.e.,

    a(x,θ) = [e^j2π/λx_1cosθ,⋯,e^j2π/λx_Ncosθ]^T∈ℂ^N× 1.


To ease the practical implementation, we assume analog beamforming with a constant magnitude at the transmitter, which is given by

    ω = 1/√(N)[e^jϕ_1,e^jϕ_2,⋯,e^jϕ_N]^T∈ℂ^N×1,

where ϕ_n denotes the phase shift at the n-th MA.
 
Based on the above, the beam gain over the steering angle θ can be expressed as

    G(ω,x,θ)=|ω^Ha(x,θ)|^2.

	
As shown in Fig. 1, in this paper, we aim to achieve uniform beam coverage in the angular domain by jointly optimizing the APV x and the analog beamformer ω. In particular, we consider that the desired coverage region is composed of K disjoint subregions, i.e.,  ℛ=ℛ_1∪ℛ_2∪⋯∪ℛ_K, with ℛ_k={θ∈[θ_min^(k),θ_max^(k)]}, where θ_min^(k) and θ_max^(k) denote the boundary angles of the k-th subregion, with θ_min^(k) < θ_max^(k). It follows that by changing the number of the subregions (i.e., K) and the width of each subregion k (i.e., θ_max^(k) - θ_min^(k)), flexible beam coverage can be achieved.

To achieve uniform beam coverage over the K subregions, we aim to maximize the minimum beam gain over them, i.e.,

    G_min(ω,x)=min_θ∈ℛG(ω,x,θ),

by jointly optimizing the APV x and the analog beamforming vector ω. The associated optimization problem can be formulated as


    (P1)      max_ω,xG_min(ω,x)
          0≤ x_n≤ D,   n∈𝒩, 
          x_n-x_n-1≥ d_min,   n∈𝒩\{1},
         |ω(n)|=1/√(N),   n∈𝒩,


where constraint (<ref>) ensures that the MAs are located within the transmit region 𝒞_t; (<ref>) ensures the minimum distance between adjacent MAs (denoted as d_min) to avoid antenna coupling; constraint (<ref>) is with respect to (w.r.t.) the non-convex modulus constraints on the analog beamforming vector. 

However, (P1) is generally difficult to be optimally solved due to the continuous coverage region, the coupling of the analog beamformer ω and APV x in objective function, as well as the unit-modulus constraints. Moreover, the objective function appears to be non-convex in both ω and x. Next, we propose an AO algorithm to solve (P1) iteratively.




§ PROPOSED ALGORITHM

First, due to the continuous nature of the angle within each subregion ℛ_k, we first discretize it into L_k discrete values, i.e.,

    θ_l^(k) = θ_min^(k)+l^(k)-1/L^(k)-1Z_k,l^(k)=1,⋯,L^(k),

where L^(k) represent the number of the sampling points in ℛ_k, and Z_k=θ_max^(k)-θ_min^(k) represents the width of ℛ_k.

Next, we introduce an auxiliary variable t to convert (P1) into its epigraph form, i.e.,


    (P2)      max_ω,x,t t 
          
    	G(ω,x,θ_l^(k))≥ t, ∀ k,l
    	
          (<ref>),(<ref>),(<ref>).



However, (P2) is still difficult to solve due to its non-convex constraints w.r.t. both w and x. Next we propose an AO algorithm to optimize them alternately with the SCA technique.



 §.§ Optimization of ω for Given x

First, we optimize the analog beamforming vector ω in (P2) with a given APV x. Then, (P1) can be simplified as

    (P2.1)      max_ω,t t 
          (<ref>),(<ref>).

To deal with the non-convex constraint (<ref>) , we first rewrite the beam gain at θ_l^(k) as 

    G(ω,x,θ_l^(k))   =|ω^Ha(x,θ_l^(k))a^H(x,θ_l^(k)) ω|^2
    	
       =Tr(R_l^(k)V),

where R_l^(k)=a(x,θ_l^(k))a^H(x,θ_k^(l)) and V=ωω^H, with rank(V)=1. If we apply the semidefinite relaxation (SDR) to relax the rank-one constraint, (P2.1) can be recast as


    (P2.2)     max_t,V t
    	
         Tr(R_l^(k)V)≥ t,  ∀ l, k
       V(n,n)=1/N,    n∈{1,⋯,N}.
    	
       V≽0,


Note that as (<ref>), (<ref>) and (<ref>) are all affine functions, (P2.2) is a classical semi-definite program (SDP) problem, which thus can be optimally solved via the interior-point algorithm<cit.>. However, the optimized matrix V may not be rank-one. Although Gaussian randomization procedures may be applied to yield a rank-one solution, the resulting performance gap with the globally optimal solution may be large. To tackle this issue, we propose a penalty-based algorithm in this paper, which moves the rank-one constraint to the objective function. Specifically, note that the rank-one constraint can be equivalently written as

    (V)=1 ⇔ f(V)≜Tr(V)-σ(V)=0,

 where σ(V) represents the maximum singular value of V. As such, we can consider the following objective function by adding a penalty term, i.e.,

    max_t,Vt-ρ f(V),

where ρ≥ 0 represents a penalty parameter ensuring that the objective function is small enough if Tr(V)-σ_max≠ 0. However, such an objective function is still non-convex as the maximum singular value of V, σ(V), is a convex function (instead of being concave) in V. Nonetheless, it enables us to apply the SCA algorithm to obtain a locally optimal solution.  In particular, 
	for a given local point V^(i) in the i-th SCA iteration, we replace f(V) as its first-order Taylor expansion, i.e., 
	
    f(V)≥f̃(V|V^(i))≜Tr   (V)-σ(V^(i))+
       Re{Tr(ss^H)(V-V^(i))},

where s represents the singular vector corresponding to the maximum singular value of V^(i). where the equality holds only when V=V^(i). Based on the above, the optimization of V in the i-th SCA iteration is given by

    (P2.3)     max_t,V  t-ρf̃(V|V^(i))
    	
         (<ref>),(<ref>),(<ref>).

It can be seen that the objective function of (P2.3) is currently a linear function of t and V. As such, (P2.3) is an SDP problem which can be optimally solved<cit.>. Based on our simulation results, by properly setting the value of ρ, the converged solution of V is always rank-one. Next, we proceed to the (i+1)-th SCA for V by updating V^(i+1)=V^(i). Finally, to extract ω based on the converged V, we can perform the singular value decomposition (SVD) on V as V=U_1^HΛU_2, where U_1, Λ and U_2 are the left eigenvector matrix, the diagonal matrix of the singular values and the right eigenvector matrix of V, respectively. Then we can obtain the analog beamforming solution to (P2.1) as ω=1/√(N)e^j(U_1^HΛ^1/2r), where r is the left singular vector corresponding to the largest singular value of V.



 §.§ Optimization of x for Given ω

Next, we optimize the APV x in (P2) with a given analog beamformer 
  ω. To tackle the non-convex constraint (<ref>), we first expand G(ω,x,θ_l^(k)) into the following form
  
    G   (ω,x,θ_l^(k))=|ω^Ha(x,θ_l^(k))|^2
       =∑_p=1^N∑_q=1^N1/Ncos(2π/λcos(θ_l^(k))(x_p-x_q)-(ϕ_p-ϕ_q))
       ≜∑_p=1^N∑_q=1^N1/Ncos(u_l^(k)(x_p,x_q)).

where cos(u_l^(k)(x_p,x_q))≜ (α_l^(k))^2(x_p-x_q)-(ϕ_p-ϕ_q) and α_l^(k)=2π/λcos(θ_l^(k)). The difficulty in dealing with (14) lies in the cosine function therein. To tackle this challenge, we relax (<ref>) by using the SCA again. For a given z_0∈ℝ, its second-order Taylor expansion can be expressed as

    cos(z_0)-sin(z_0)(z-z_0)-1/2cos(z_0)(z-z_0)^2.

As cos(z_0)≤1 and (z-z_0)^2≥0, we can construct a quadratic function g(z|z_0) to replace cos(z) as

    cos(z)≥ g(z|z_0)≜cos(z_0)-sin(z_0)(z-z_0)-1/2(z-z_0)^2.

Based on the above, for any given x^(i) in the ith iteration of SCA for the APV x, by substituting (<ref>) into (<ref>), we can construct a surrogate function of G(ω,x,θ_k) as

    G(ω,x,θ_k^(l))   ≥∑_p=1^N∑_q=1^N1/Ng(u_l^(k)(x_p,x_q)|u_l^(k)(x_p^(i),x_q^(i)))
       ≜x^TA_l^(k)x+[b_l^(k)]^Tx+c_l^(k),

where A_l^(k)∈ℝ^N× N, b_l^(k)∈ℝ^N× 1, and c_l^(k)∈ℝ are given by

    A_l^(k)=-(α_l^(k))^2(I_N-1/N1_N)≜-(α_l^(k))^2W,


    b_l^(k)(p)=2α_l^(k)/N∑_q=1^N[α_l^(k)x_p,q^(i)-sin(u_l^(k)(x_p^(i),x_q^(i)))],


    c_l^(k)=   1/N∑_p=1^N∑_q=1^N[cos(u_l^(k)(x_p^(i),x_q^(i)))+
       α_l^(k)x_p,q^(i)sin(u_l^(k)(x_p^(i),x_q^(i)))-1/2(α_l^(k)x_p,q^(i))^2],

where I_N and 1_N represent the identity matrix and all-ones matrix of size N, respectively, x_p,q^(i)≜ x_p^(i)-x_q^(i), and W≜I-1/N1_N. As such, constraint (<ref>) can be relaxed as

    x^TA_l^(k)x+[b_k^(l)]^Tx+c_l^(k)≥ t, ∀ k,l.

Next, we show that (21) is a quadratic constraint (QC). To this end, it is noted that both A_l^(k) and W are hermitian matrices for any given k and l and have the same eigenvectors. Thus, A_l^(k) can be diagonalized as

    A_l^(k)=[U_l^(k)]^HD_l^(k)U_l^(k)≜-[U_l^(k)]^H(α_l^(k))^2DU_l^(k)

where U_l^(k) is an unitary matrix composed of the eigenvectors of A_l^(k), D_l^(k) and D are diagonal matrices composed of the eigenvalues of A_l^(k) and W, respectively, with D_l^(k)=-(α_l^(k))^2D. Furthermore, any eigenvalue of D is non-negative according to Gerschgorin disk theorem. Hence, the eigenvalue of D_l^(k) is non-positive and A_l^(k) is a negative semi-definite matrix. It follows that constraint (<ref>) is a convex QC w.r.t. x. 



Therefore, in the i-th iteration of the SCA for x, x can be optimized by solving the following optimization problem


    (P2.4)    max_x,tt
        (<ref>),(<ref>),(<ref>).


Since constraints (<ref>) and (<ref>) are linear and (<ref>) is a QC w.r.t x, (P2.2) is a classic quadratically constrained quadratic program (QCQP) problem which can be effectively solved via the interior-point algorithm. Next, we proceed to the (i+1)-th SCA for x by updating x^(i+1)=x^(i).



 §.§ Overall Algorithm and Complexity Analysis

The overall AO algorithm can be executed as follows. Let 
	ω(j-1) and x(j-1) denote the values of ω and x at the beginning of the j-th AO iteration. Then, in this AO iteration, we first optimize V via SCA with fixing x=x(j-1) and V^(0)=ω(j-1)ω^H (j-1) and obtain ω(j) by performing SVD on the converged V. 
	Next, we optimize x via SCA with fixing ω=ω(j) and x^(0)=x(j-1) and obtain x(j) as the converged solution, and the (j+1)-th AO iteration follows. The overall procedures of the AO algorithm are summarized in Algorithm 1.


Next, we prove that the proposed AO algorithm with SCA is ensured to converge. Specifically, regarding the SCA for V, let v^(i)≜ t-ρ f(V^(i)) denote the objective value of (11) in the i-th SCA iteration for V. Then, the following inequalities hold, i.e.,

    v^(i)   ≜ t-ρ f(V^(i))(a)=t-ρf̃(V^(i)|V^(i))
    	
       (b)≤ t-ρf̃(V^(i+1)|V^(i))(c)≤ t-ρ f(V^(i+1))≜ v^(i+1),

where the equality (a) holds since the first-order Taylor expansion in (<ref>) is tight at V^(i); the inequality (b) holds as V^(i+1) is the optimal solution to (P2.3) and thus maximizes the function t-ρf̃(V|V^(i)); the inequality (c) holds due to (12). Based on the above, the sequence {v^(i)} is non-decreasing and thus ensured to converge. Similarly, it can be shown that the SCA for optimizing the APV x also converges, for which the details are omitted for brevity. It follows that the proposed AO algorithm in Algorithm 1 must converge.


Finally, we analyze the complexity Algorithm 1. It can be shown that the complexity of optimizing the analog beamformer ω and the APV x are both in the order of 𝒪(√(L)N(N^2+L))<cit.>, where L=∑_k=1^KL^(k) denotes the total number of sampling points.




 §.§ Initialization

The converged performance of the AO algorithm depends critically on the initialization. In this subsection, we propose an efficient initialization method to detemine x(0) and ω(0) in Algorithm 1. For the initial MA positions, we consider that all MAs are uniformly deployed within the linear array with equal spacing. As such, we set x(0)=[D/N+1,2D/N+1,⋯,ND/N+1]^T.

Next, to determine ω(0), we consider solving the SDR problem (P2.2) directly with fixing x=x(0) via the interior-point algorithm. Denote by V(0) the associated optimal solution to (P2.2). Then, we obtain ω(0) by performing Gaussian randomization based on V(0), i.e., 


    ω(0)=1/√(N)e^j(U_0^HΛ_0^1/2r_0),

where U_0 and Λ_0 are unitary matrix and diagonal matrix composed by eigenvectors and eigenvalues of the matrix V(0), respectively, and vector r_0∈ℂ^N× 1 is a random vector following 𝒞𝒩(0,I_N). By comparing the objective value of (P2.2) under different realizations of r_0, we set ω(0) as that corresponding to the one yielding the maximum objective value of (P2.2).





§ NUMERICAL RESULTS

In this section, we provide numerical results to evaluate the performance of our proposed MA-enabled beam coverage. Unless otherwise stated, the simulation parameters are set as follows. The penalty parameter for updating AWV is set to ρ=20. The carrier frequency is f_c=1 GHz. The length of the MA linear array is set to D=8λ. Meanwhile, we set the convergence threshold for the AO algorithm, the SCA for V, and the SCA for x as 10^-5, 0.01, and 0.01, respectively. For performance comparison, we consider the following two benchmarks:

1) FPA: N FPAs are deployed with half-wavelength spacing and the analog beamfroming is optimized based on the SCA algorithm as in Section III-A.

2) MAs with fixed analog beamforming (MA-FAB): In this benchmark, we set the analog beamforming ω=ω(0) as presented in Section III-D and optimize MA positions based on the SCA algorithm as in Section III-B.
 
 First, by setting K=1, θ_min^(1)=0, and θ_max^(1)=180^∘, i.e., ℛ=[0,180^∘], Figs. <ref> and <ref> plot the optimized beam gains over R by different schemes for N=6 and N=8, respectively. Note that we have introduced a common shift to the MA positions such that x_1=0 without affecting the coverage performance. The optimized MA positions by different schemes are also shown in Figs. <ref> and <ref> for N=6 and N=8, respectively. It is observed from Figs. <ref> and <ref> that our proposed AO algorithm can achieve a considerably higher max-min beam gain over ℛ as compared to the two benchmarks, thereby resulting in flatter beam gain for both N=6 and N=8. 
 It is also observed that the MA-FAB benchmark outperforms the FPA benchmark, which indicates that antenna positions may play a more significant role than antenna weights in terms of beam coverage. 
 In addition, it is observed from Figs. <ref> and <ref> that the optimized MA positions are different in different schemes. In particular, for both N=6 and N=8, the proposed scheme and the MA-FAB benchmark result in a larger antenna aperture, i.e., x_N-x_1, than the FPA benchmark, so as to fully exploit the spatial DoFs available. Moreover, the inter-antenna spacing is observed to be non-uniform in the former two schemes employing MAs, unlike the FPA benchmark with an equal antenna spacing. In addition, by comparing Figs. 2(c) and 2(d), it is observed that the antenna aperture under M=8 is larger than that under M=6 in the proposed scheme (1.2 versus 1.14), so as to accommodate the movement of more MAs.   
      
Next, Fig. <ref> plots the optimized beam gain over ℛ by setting K=3, ℛ_1=[0,30^∘], ℛ_2=[70^∘,110^∘] and ℛ_3=[160^∘,170^∘]. The number of MAs is set to N=8. The optimized MA positions by the considered schemes are shown in Fig. <ref>. It is observed that in the case of multi-region coverage, the proposed scheme still achieves much higher max-min beam gain than the other two benchmarks. In particular, the minimum beam gain by our proposed scheme is observed to be even larger than the maximum beam gain by the FPA benchmark in ℛ_1 and ℛ_3. While in ℛ_2, although the FPA can achieve a maximum beam gain of 6 dB, its minimum beam gain is considerably lower (-19 dB) than that by the proposed scheme (-1 dB). Moreover, the MA-FAB benchmark is observed to achieve a slightly higher minimum beam gain than the proposed scheme in ℛ_2. This implies that as far as each subregion concerned, the proposed scheme may not achieve the highest minimum beam gain. It is also observed from Fig. <ref> that similar to Figs. <ref> and <ref>, the proposed scheme and the MA-FAB benchmark result in a larger antenna aperture than the FPA benchmark. Particularly, x_8 increases from 1.2 in Fig. <ref> to 1.65 in Fig. <ref>, which suggests that multi-region coverage may demand a larger antenna aperture than its single-region counterpart due to the discontinuous subregions. Interestingly, it is observed that the optimized MA positions by the proposed scheme can be viewed as being composed of three subarrays, with the first and second MAs in the first subarray, the third to fifth MAs in the second subarray, and the sixth to eighth MAs in the third subarray, which helps cover the three subregions.  
 
	
Finally, by setting K=1 and θ_min^(1)=0, we plot the max-min beam gain versus θ_max^(1) in Fig. <ref>. The number of MAs is set to N = 8. It is observed that the max-min beam gains by all considered schemes monotonically decrease with θ_max^(1). This is expected as the total beam gain over the entire spatial domain is a constant. Thus, when θ_max^(1) or the coverage width increases, the max-min beam gain should reduce accordingly. It is also observed that in the case with a narrow-to-moderate coverage width, i.e., θ_max^(1)≤50^∘, the performance gap between the proposed scheme and the MA-FAB benchmark is negligible, while both of them significantly outperform the FPA benchmark. This indicates that adjusting the MA positions suffices to achieve uniform coverage for a narrow region given the proposed initial analog beamforming design. However, with the increase in θ_max^(1), the performance gap between the proposed scheme and the MA-FAB benchmark becomes larger. This indicates that the analog beamforming design will play a more significant role for the more challenging wide-region coverage.





§ CONCLUSION

In this paper, we investigate the beam coverage design with multiple MAs by jointly optimizing their weights and positions to maximize the minimum beam gain over the desired spatial regions. An AO algorithm was proposed to obtain a high-quality suboptimal solution to this problem. Numerical results show that our proposed MA-enabled beam coverage design significantly outperforms the conventional FPAs by more flexibly catering to the geometry of the subregions. It was also shown that the antenna weights may play a more significant role as the width of the region increases.


0
b1
 J. Mietzner, R. Schober, L. Lampe, W. H. Gerstacker and P. A. Hoeher, “Multiple-antenna techniques for wireless communications - a comprehensive literature survey,” IEEE Commun. Surveys Tuts., vol. 11, no. 2, pp. 87-105, Second Quarter 2009.
 
    
b2
K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” IEEE Trans. Wireless Commun., vol. 20, no. 3, pp. 1950-1962, Mar. 2020.

b3
K.-K. Wong, K.-F. Tong, and C.-B. Chae, “Fluid antenna system-part II: Research opportunities,” IEEE Commun. Lett., vol. 27, no. 8, pp. 1924-1928, Aug. 2023.

b4
L. Zhu, W. Ma, and R. Zhang, “Modeling and performance analysis for
	movable antenna enabled wireless communications,” IEEE Trans. Wireless Commun., early access, 2023.

b5
L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless communication: Opportunities and challenges,”  IEEE Commun. Mag., early access, 2023.

b7
Y. Wu, D. Xu, D. W. K. Ng, W. Gerstacker, and R. Schober, “Movable antenna-enhanced multiuser communication: Jointly optimal discrete antenna positioning and beamforming,” in Proc. IEEE Global Commun. Conf., Kuala Lumpur, Malaysia, Dec. 2023, pp. 7508-7513.

b8
L. Zhu, W. Ma, B. Ning, and R. Zhang, “Movable-antenna enhanced multiuser communication via antenna position optimization,” IEEE Trans. Wireless Commun., 2023, early access.

b9
W. Mei, X. Wei, B. Ning, Z. Chen and R. Zhang, “Movable-antenna position optimization: A graph-based approach,” IEEE Wireless Commun. Lett., early access.

b10 
 G. Hu, Q. Wu, K. Xu, J. Si and N. Al-Dhahir, “Secure wireless communication via movable-antenna array,” IEEE Signal Processing Letters, vol. 31, pp. 516-520, 2024.

b11
Z. Cheng, N. Li, J. Zhu, X. She, C. Ouyang and P. Chen, “Enabling secure wireless communications via movable antennas,” in Proc. ICASSP, Seoul, Korea, 2024, pp. 9186-9190. 

b12
W. Ma, L. Zhu, and R. Zhang, “MIMO capacity characterization for movable antenna systems,” IEEE Trans. Wireless Commun., vol. 23, no. 4, pp. 3392-3407, April 2024. 

b13   
W. Ma, L. Zhu and R. Zhang, “Multi-beam forming with movable-antenna array,” IEEE Commun. Lett., vol. 28, no. 3, pp. 697-701, Mar. 2024.

b14
L. Zhu, W. Ma and R. Zhang, “Movable-antenna array enhanced beamforming: Achieving full array gain with null steering,” IEEE Commun. Lett., vol. 27, no. 12, pp. 3340-3344, Dec. 2023


 ning
 B. Ning, T. Wang, C. Huang, Y. Zhang, and Z. Chen, “Widebeam designs for terahertz massive MIMO: SCA-ATP and S-SARV,” IEEE Internet Things J., no. 12, pp. 10857-10869, June 2023.



b16
S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2009.

o
K. Wang, et al., “Outage constrained robust transmit optimization for multiuser MISO downlinks: Tractable approximations by conic optimization,” IEEE Trans. Signal Process., vol. 62, no. 21, pp. 5690-5705, Nov. 2014.











 Paper Title: Wireless-Powered Mobile Crowdsensing Enhanced by UAV-Mounted RIS: Joint Transmission, Compression, and Trajectory Design.















Wireless-Powered Mobile Crowdsensing Enhanced by UAV-Mounted RIS: Joint Transmission, Compression, and Trajectory Design
    Yongqing Xu, Graduated Student Member, IEEE, Haoqing Qi, Zhiqin Wang, Xiang Zhang, 
Yong Li, Member, IEEE, and Tony Q.S. Quek, Fellow, IEEE
        
The work of Y. Xu, H. Qi, and Y. Li was supported by the National Research and Development Program “Distributed Large Dimensional Wireless Cooperative Transmission Technology Research and System Verification” under Grant 2022YFB2902400. The work of Tony Q. S. Quek was supported by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Future Communications Research & Development Programme. (Corresponding Author: Yong Li.)
	
	Y. Xu, H. Qi, and Y. Li are with the Key Laboratory of Universal Wireless Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China. Email: {xuyongqing; qihaoqing; liyong}@bupt.edu.cn.
	
	Z. Wang and X. Zhang are with China Academy of Information and Communications Technology, Beijing 100191, China. Email: {zhiqin.wang; zhangxiang1}@caict.ac.cn.
	
	T. Q. S. Quek is with the Singapore University of Technology and Design, Singapore 487372 (e-mail: tonyquek@sutd.edu.sg).
    August 5, 2024
======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
Mobile crowdsensing (MCS) enables data collection from massive devices to achieve a wide sensing range. Wireless power transfer (WPT) is a promising paradigm for prolonging the operation time of MCS systems by sustainably transferring power to distributed devices. However, the efficiency of WPT significantly deteriorates when the channel conditions are poor. Unmanned aerial vehicles (UAVs) and reconfigurable intelligent surfaces (RISs) can serve as active or passive relays to enhance the efficiency of WPT in unfavourable propagation environments. Therefore, to explore the potential of jointly deploying UAVs and RISs to enhance transmission efficiency, we propose a novel transmission framework for the WPT-assisted MCS systems, which is enhanced by a UAV-mounted RIS. Subsequently, under different compression schemes, two optimization problems are formulated to maximize the weighted sum of the data uploaded by the user equipments (UEs) by jointly designing the WPT and uploading time, the beamforming matrics, the CPU cycles, and the UAV trajectory. A block coordinate descent (BCD) algorithm based on the closed-form beamforming designs and the successive convex approximation (SCA) algorithm is proposed to solve the formulated problems. Furthermore, to highlight the insight of the gains brought by the compression schemes, we analyze the energy efficiencies of compression schemes and confirm that the gains gradually reduce with the increasing power used for compression. Simulation results demonstrate that the amount of collected data can be effectively increased in wireless-powered MCS systems.



Wireless power transfer, mobile crowdsensing, compression, unmanned aerial vehicle, reconfigurable intelligent surfaces.




§ INTRODUCTION

With the development of wireless networks, numerous application scenarios have emerged, such as smart cities, smart homes, smart transportation, smart grids, and smart healthcare <cit.>. By connecting and organizing numerous terminals within these scenarios, Internet-of-Things (IoT), ambient IoT (A-IoT), and artificial intelligence of things (AIoT) networks are established, where terminals have the abilities to sense, compute, and exchange information to make intelligent decisions <cit.>. Mobile crowdsensing (MCS) is a crucial technology for IoT, A-IoT, and AIoT, enabling mobile phones and wearable smart devices to upload sensing data, aggregating and fusing the sensed data in the cloud to achieve comprehensive area sensing and intelligent decision-making <cit.>. Research on MCS is currently underway to investigate network architectures, resource allocation schemes, and frame structures of MCS systems. For example, the authors of <cit.> proposed a joint sensing, communication, and computation framework for MCS systems, where the transmission parameters and CPU cycles are jointly optimized. The authors of <cit.> considered an energy-efficient transmission scheme, which can minimize the energy consumption for data sensing and transmission while satisfying the minimal sensing and transmission requirements. Despite the numerous benefits of MCS, the energy constraints of many devices with a small or no battery result in short and unstable MCS operation times.



 §.§ Wireless-Powered Mobile Crowdsensing

Wireless power transfer (WPT) is an emerging paradigm that enables delivering energy wirelessly <cit.>. WPT can prolong the operational time of MCS systems by charging distributed devices via electromagnetic waves <cit.>. The core of WPT-powered MCS lies in using energy to exchange sensing information, thereby motivating more device involvement. Specifically, the authors of <cit.> developed an energy-data trading mechanism for WPT-powered MCS systems and offered a corresponding network architecture. In <cit.>, the authors jointly controlled the wireless-power allocation at the base station (BS), the sensing and compression parameters of each user equipment (UE) to maximize data utility and minimize energy consumption simultaneously. Moreover, in <cit.>, a WPT-powered MCS-assisted sustainable federated learning architecture is proposed, where an optimization model is established to minimize the total task completion time. However, the efficiency of WPT depends on the hardware circuitry and channel conditions, and it significantly deteriorates in scenarios with poor channel conditions, such as emergency rescue, traffic inspection, and forest protection scenarios <cit.>.



 §.§ UAV-Assisted Wireless-Powered Mobile Crowdsensing

Unmanned aerial vehicle (UAV) is recognized as a promising technology for WPT systems because the efficiency of WPT can be significantly enhanced by the high mobility, flexibility, and high probability of the line-of-sight (LoS) path associated with UAVs <cit.>. Specifically, the authors of <cit.> derived optimal UAV locations to maximize the sum of harvested energy. The authors of <cit.> jointly designed the UAV trajectory and the orientation of the UAV antenna to maximize the harvested energy. Moreover, in <cit.>, the authors modelled the UAV's power consumption and proposed two types of UAV trajectories, which can maximize the power transferred by the UAV and guarantee that the UAV safely flies to the BS. More importantly, MCS systems can also benefit from the advantages of UAVs. For example, a joint WPT, compression, and data uploading framework for UAV-assisted MCS systems was proposed in <cit.>. In <cit.>, the authors formulated an uploading rate maximization problem in a UAV-enabled WPT system and proposed an alternative algorithm to solve the formulated non-convex problem. In <cit.>, the authors considered a two-stage transmission framework for a UAV-enabled WPT system, where the time durations of two stages and transmit parameters are optimized. Furthermore, the authors of <cit.> considered a more practical UAV-assisted MCS scenario, where the outage performance of the WPT and data rate are analyzed, and an outage probability minimization problem under specific constraints was formulated and solved. All the aforementioned works assume that UAVs have sufficient energy to perform WPT and data reception. However, UAVs are powered by onboard batteries, and the manoeuvring actions consume most of the UAVs' energy <cit.>, which limits the UAV capacity for WPT and data reception.



 §.§ RIS and UAV-Assisted Wireless-Powered Mobile Crowdsensing

Reconfigurable intelligent surface (RIS) is emerging as a critical technology for next-generation wireless networks due to its cost-effective and energy-effective characteristics <cit.>. By integrating UAV with RIS, preferable radio environments for WPT systems can be created <cit.>. The efficiency of WPT has the potential to be further enhanced <cit.>, e.g., the WPT protocols for RIS-assisted UAV systems proposed by the authors of <cit.> and <cit.> can enhance the WPT efficiency comparing with the schemes without RIS. Furthermore, the integration of RIS and UAV in MCS systems has garnered significant interest because of the high transmission efficiency <cit.>. Research on RIS and UAV-assisted MCS can be categorized into two types. The first type is the MCS assisted by separate deployment of RISs and UAVs, such as <cit.>, where RISs are installed on the wall of tall buildings. The authors focused on the joint optimization of UAV transmit power, UAV trajectory, UE time allocation, and RIS beamforming matrix. The second type is the MCS assisted by UAV-mounted RISs, such as <cit.>, where RISs are mounted on UAVs and the authors shifted their research focus to the joint design of the transmission parameters of BSs and UEs, the beamforming matrics of RISs, and the UAV trajectories. The design complexity of MCS systems assisted by separate deployment of RIS and UAV is low. However, the onboard batteries of UAVs limit their application as active aerial relays. In UAV-mounted RIS-assisted MCS systems, the UAVs act as passive aerial relays, which can enhance transmission efficiency. Research on UAV-mounted RIS-assisted MCS systems is still in its infancy. There remains significant potential to further explore and harness the capabilities of UAV-mounted RIS-assisted MCS technology.



 §.§ The Contributions of This Work

Motivated by the above-mentioned works, we first introduce the computational capabilities of UEs into UAV-mounted RIS-assisted MCS systems and establish a transmission framework for wireless-powered MCS systems to maximize the amount of data uploaded by UEs. The main contributions of  our work are listed as follows:

	
  * We design a transmission model based on the developed time-division multiple access (TDMA) protocol for UAV-mounted RIS-assisted MCS systems. We also establish the WPT, compression, and communication models for the considered wireless-powered MCS system enhanced by a UAV-mounted RIS.
	
  * We establish two optimization problems to maximize the amount of uploading data under two compression schemes. Both problems require to jointly optimize the WPT time of the BS, the transmit and receive beamforming vectors of the BS, the RIS beamforming matrices, the CPU cycles of UEs, the transmit power of UEs, the uploading times of UEs, and the UAV trajectory. Furthermore, we propose a block coordinate descent (BCD) algorithm to solve the formulated problems, which can iteratively converge to a near-optimal solution.
	
  * To highlight the gains brought by lossless compression and the performance losses caused by lossy compression, we analyze the energy efficiencies of both methods by comparing them with the direct uploading scheme. Moreover, we analyze the computational complexity for solving the formulated optimization problems and the convergence performance of the BCD algorithm.
	
  * We also perform numerous simulations from different perspectives to verify the effectiveness of the proposed transmission model and algorithms.




 §.§ Paper Organization

The remainder of this paper is organized as follows. Section 2 introduces the system model and performance metrics. Section 3 proposes a BCD-based algorithm to design the parameters of BS, RIS, UEs, and UAV under the lossless compression model. Section 4 formulates a problem for maximizing the amount of uploading data under the lossy compression model. Section 5 analyzes the energy efficiencies. Section 6 analyzes the complexity and convergence performance of the proposed algorithms. Section 7 presents the simulation results. Section 8 concludes the paper.



 §.§ Notations

Throughout this paper, matrices are denoted by bold uppercase letters, e.g., X, vectors are denoted by bold lowercase letters, e.g., x, and scalars are denoted by normal fonts, e.g., x. |·| represents the absolute value operation. ‖·‖ represents Euclidean norm of a vector. (·)^T, (·)^H, and (·)^* represent the transpose, the conjugate transpose, and the conjugate, respectively. ℂ^(·) denotes the complex space. ℝ^(·) denotes the real space. diag(x) represents using x to formulate a diagonal matrix. arg(·) represents the angle of a complex number.



§ SYSTEM MODEL


The considered scenario is shown in Fig. 1, where a BS equipping M antennas transfers power to K single-antenna UEs and receives data from the UEs. A UAV mounting a N-element RIS assists the power transfer of BS and the data uploading of UEs during its flying duration (i.e., T time slots). The BS and the RIS antenna arrays are configured as uniform linear arrays (ULAs). The inter-distance of the BS antennas and the RIS elements are half-wavelength. Under a three-dimensional (3-D) Cartesian coordinate system, the coordinates of the BS and the k-th UE are denoted as q_0=[x_B,y_B,z_B]^T∈ℝ^3×1 and q_k=[x_k,y_k,z_k]^T∈ℝ^3×1, respectively. The coordinate of the UAV at the t-th slot is q[t]=[x[t],y[t],z]^T∈ℝ^3×1. The heights of the BS, the UEs, and the UAV are all assumed to remain fixed during the flying duration of the UAV. The sets of UEs and time slots are represented by K and T.

The UEs harvest the transferred energy by the BS via their energy harvesting modules. Then, the UEs compress their collected sensing data and upload the data to the BS using the harvested energy via the data compression modules and the data uploading modules[For energy efficiency, only a part of the sensing data is compressed by the UEs. The details are discussed in Section 5.]. To avoid self-interference, the wireless power transfer of the BS and the data uploading of the UEs are scheduled in a time-division multiplexing (TDM) manner. The proposed TDMA protocol is shown in Fig. <ref>.





 §.§ Wireless Power Transfer Model

The harvested energy of the k-th UE at the t-th time slot can be expressed as

    E_k[t] = η _0δ _t| ( h_k^H + g_k^H[t]Θ_k[t]G_UR^H[t])w_k[t]|^2a_k[t],∀ k∈𝒦,

where η_0 denotes the energy conservation efficiency; δ_t denotes the length of one time slot; h_k=h_ka(ϕ_k)∈ℂ^M×1 denotes the channel vector between the BS and the k-th UE; g_k=g_kb(φ_k)∈ℂ^N×1 and  G_UR[t]=G_0a(ϕ_0[t])b^T(φ_0[t])∈ℂ^M× N denote the channels between the RIS and the k-th UE and between the BS and the RIS at the t-th time slot; h_k=√(h_0/d_B2U,k^α_BU), g_k[t]=√(h_0/d_R2U,k^α_RU[t]), and G_0[t]=√(h_0/d_B2R^α_BR[t]) denote the large scale channel gains, where h_0 represents the channel gain at the reference distance of one meter,  α_BU, α_RU, and α_BR are the pathloss exponents, d_B2U,k=‖q_k-q_0‖, d_R2U,k[t]=‖q[t]-q_k‖, and d_B2R[t]=‖q[t]-q_0‖; a(ϕ)=[1,⋯,e^-jπ(N-1)sin(ϕ)]^T∈ℂ^M×1 and b(φ)=[1,⋯,e^-jπ(M-1)sin(φ)]^T∈ℂ^N×1 denote the array steering vectors of the BS antenna array and the RIS; ϕ_k and φ_k[t] represent the angles of departure (AODs) of the signals transmitted and reflected by the BS to the k-th UE and by the RIS to the k-th UE at the t-th time slot; ϕ_0[t] and φ_0[t] denote the AODs between the BS and the RIS and between the RIS and the BS at the t-th time slot; the sine of the AODs can be written as sin(ϕ_k)=x_k/d_B2U,k, sin(φ_k[t])=√(d_R2U,k^2[t]-(z-z_k)^2)/d_R2U,k[t], sin(ϕ_0[t])=z-z_B/d_B2R[t], sin(φ_0[t])=√(d_B2R^2[t]-(z-z_B)^2)/d_B2R[t]. Moreover, w_k[t]∈ℂ^M×1 denotes the transmitting beamforming vector of the k-th UE at the t-th time slot; Θ_k[t]=diag(θ_k[t])∈ℂ^N× N denotes the RIS beamforming matrix used to serve the WPT for the k-th UE at the t-th time slot; 0≤ a_k[t]≤1 denotes the time ratio of the t-th time slot occupied by the k-th UE for harvesting energy.



 §.§ Compression Model

In this subsection, we introduce two types of the compression model. The first one is the lossless compression model, which aims to evaluate the performance upper bound obtained by the compression scheme. The second one is the lossy compression model, which aims to investigate the performance loss caused by imperfect compression.



  §.§.§ Lossless compression model

Lossless compression refers to the fact that the compressed data can perfectly reconstruct the raw sensing data. Many compression methods, such as Huffman, run-length, and Lempel-Ziv encoding, can achieve lossless compression. We assume that all UEs adopt the same compression method and that their compression ratios are equal. The energy used for compression of the k-th UE at the t-th time slot is defined as <cit.>

    E_C,k[t] ≜γδ _tγ _cf_k^3[t],∀ k∈𝒦,

where γ denotes the time ratio of the compression at each time slot, γ _c denotes a constant depending on the hardware, f_k[t] denotes the CPU-cycle frequency of the k-th UE at the t-th slot. The total sensing data being compressed is defined as

    S_k[t] ≜γδ _tf_k[t]/e^ε /κ _k - e^ε,∀ k∈𝒦,

where e^ε /κ _k-e^ε denotes the total CPU cycles that needs to compress one bit sensing data with ε denoting a constant depending on the compression method, κ_k∈(0,1] denotes the ratio of the compressed data to the raw data before compression. It is noted that the compression energy consumption is proportional to the cube of the compressed data volume. Therefore, the compression scheme can increase the energy efficiency when the compressed data volume is small. On the contrary, it decreases the energy efficiency when the compressed data volume is relatively large. We will quantitatively analyze the energy efficiency in  Section 5. 



  §.§.§ Lossy compression model

Lossy compression refers to the fact that the compressed data imperfectly reconstruct the raw sensing data due to the imperfections of hardware and compression algorithms. We first define the information loss caused by the lossy compression as <cit.>

    l_k≜1-√(κ̅_k),∀ k∈𝒦,

where κ̅_k∈(0,1] denotes the ratio of the compressed data to the raw data before the lossy compression. Specifically, l_k=0 when κ̅_k=1, which represents no sensing data has been compressed. l_k→1 when κ̅_k→0, which represents all sensing data has been lost due to the compression. Then, the sensing data can be effectively compressed is expressed as

    S̅_k[t] ≜γδ _tf̅_k[t]√(κ̅_k)/e^ε /κ̅ _k - e^ε,∀ k∈𝒦,

where f̅_k[t] denotes the CPU-cycle frequency under the lossy compression scheme.



 §.§ Communication Model

The data uploaded by the k-th UE at the t-th slot is

    R_k[t] = Bδ _tb_k[t]
       ·log _2( 1 + | ( h_k^H + g_k^H[t]Θ̅_k[t]G_UR^H[t])u_k[t]|^2P_k[t]/σ _B^2),∀ k∈𝒦,

where B denotes the bandwidth of uplink; 0≤ b_k[t]≤1 denotes the time ratio occupied by the k-th UE at the t-th time slot for uploading data; Θ̅_k[t]=diag(θ̅_k[t])∈ℂ^N× N denotes the RIS beamforming matrix used to serve the uploading of the k-th UE at the t-th slot; P_k[t] denotes the transmit power of the k-th UE at the t-th slot; σ _B^2 is the noise variance at the BS; u_k[t]∈ℂ^M×1 denotes the receive beamforming vector of the k-th UE.

Remark 1: The uploading data consists of the compressed data and the data without compression. Specifically, the amount of the raw sensing data uploaded by the k-th UE under the lossless compression scheme is R_k[t]+(1-κ_k)S_k[t], where (1-κ_k)S_k[t] represents the amount of the raw sensing data saved due to compression. The amount of the raw sensing data uploaded by the k-th UE under the lossy compression scheme is R_k[t]+(1-κ̅_k)S̅_k[t].

Our goal is to maximize the amount of the raw uploading data of all UEs by joint designing the BS transmit and receive beamforming vectors, the RIS beamforming matrics, the UAV trajectory, the CPU cycles, transmit power, energy harvesting time, and uploading time of all UEs.



§ UPLOADING DATA MAXIMIZATION BASED ON LOSSLESS COMPRESSION



 §.§ Problem Formulation

The problem of maximizing the amount of the total uploading data based on lossless compression can be formulated as

	
	
    (P1)max_X,Y,Q ∑_t = 1^T ∑_k = 1^K λ_k{R_k[t] + ( 1 - κ _k)S_k[t]}7a
             s.t.  f_k[t],P_k[t],a_k[t],b_k[t] ≥ 0,∀ k∈ K,∀ t ∈ T,7b
                 w_k[t]^2 ≤P_T,∀ t ∈ T,∀ k ∈ K,7c
                 | θ _k,n,n[t]| = 1,∀ t ∈ T,∀ k ∈ K,7d
                 ∑_i = 1^t [ γδ _tγ _cf_k^3[i] + b_k[i]δ _tP_k[i]]
                   ≤∑_i = 1^tE_k[i],∀ k ∈ K,∀ t ∈ T,7e
                 ∑_i = 1^t γδ _tf_k[i]κ _k/e^ε /κ _k - e^ε≤∑_i = 1^t R_k[i],∀ k ∈ K,∀ t ∈ T,7f
                 ∑_k = 1^K a_k[t]  + ∑_k = 1^K b_k[t]≤ 1,∀ t ∈ T,7g
                 ∑_k = 1^K b_k[t]≤ 1 - γ ,∀ t ∈ T,7h
                 q[ t + 1] - q[ t ]≤V_maxδ _t,∀ t ∈ T,7i
                 q[ 1 ] = q_1,q[ T ] = q_T7j,


where X≜{f_k[t],P_k[t],a_k[t],b_k[t],∀ k ∈ K,∀ t ∈ T}, Y≜{w_k[t],u_k[t],Θ_k[t],Θ̅_k[t],∀ k ∈ K,t ∈ T}, Q≜{q[ t ],∀ t ∈ T}, and λ_k is the weight of the k-th UE. The objective function in (<ref>) aimes to maximize the weighted sum of the amount of the raw uploading data of all UEs. The constraints in (<ref>) guarantee that the transmit power of the BS at each time slot is less than the transmit power budget P_T. The constraints in (<ref>) denote that the reflection magnitude of each RIS element at each time slot should be less than one. The constraints in (<ref>) guarantee that the accumulated energy at each UE is more than its consumed energy for compressing and uploading. The constraints in (<ref>) constrain the amount of compressed data of each UE to be less than its uploading rate. Moreover, to guarantee the WPT, the uploading, and the compression are scheduled in a TDM manner, we introduce the constraints in (<ref>) and (<ref>). The constraints in (<ref>) guarantee that the WPT and uploading of all UEs are scheduled in a TDM manner. The constraints in (<ref>) guarantee the uploading and compression of all UEs are scheduled in a TDM manner. The constraints in (<ref>) constrain the UAV speed is less than V_max. The two constraints in (<ref>) constrain the UAV initial and final positions. Due to the coupled variables and the non-convex constraints, (P1) is a non-convex problem. A BCD algorithm is developed to facilitate solving (P1).



 §.§ Subproblem of Designing CPU-cycle, Uploading Power, and Time Allocation

To design X, we first keep Y and Q fixed in this subsection. The problem of designing X can be reformulated as

	
	
    (P2)max_X ∑_t = 1^T ∑_k = 1^K λ_k{R_k[t] + ( 1 - κ _k)S_k[t]}8a
            s.t.  (<ref>), (<ref>), (<ref>), (<ref>), (<ref>).8b


It is observed that (P2) is still non-convex due to the coupled b_k[t] and P_k[i]. We define p_k[t]≜ b_k[t]P_k[t]. Then, the uploading data of the k-th UE can be rewritten as

    R̅_k[t] = Bδ _tb_k[t]
       ·log _2( 1 + | ( h_k^H + g_k^H[t]Θ̅_k[t]G_UR^H[t])u_k[t]|^2p_k[t]/b_k[t]σ _B^2),∀ k∈𝒦.

(P2) can be formulated as

	
	
    (P3)max_X/P_k[t],{p_k[t]} ∑_t = 1^T ∑_k = 1^K λ_k{R̅_k[t]+( 1 -κ _k)S_k[t] }10a
            s.t.  f_k[t],p_k[t],a_k[t],b_k[t] ≥ 0,∀ k∈ K,∀ t ∈ T,10b
                ∑_i = 1^t [ γδ _tγ _cf_k^3[i] + δ _tp_k[i]]
                   ≤∑_i = 1^tE_k[i],∀ k ∈ K,∀ t ∈ T,10c
                ∑_i = 1^t γδ _tf_k[i]κ _k/e^ε /κ _k - e^ε≤∑_i = 1^t R̅_k[i],∀ k ∈ K,∀ t ∈ T,10d
                (<ref>), (<ref>).10e


(P3) is a convex problem that can be solved by convex programming tools, such as CVX <cit.>. The uploading power P_k[t] can be obtained by P_k[t]=p_k[t]/b_k[t].



 §.§ Subproblem of Beamforming Designing

We keep X and Q fixed in this subsection. The problem of designing Y can be formulated as

	
	
    (P4)max_Y ∑_t = 1^T ∑_k = 1^K λ_kR_k[t]11a
            s.t.  (<ref>), (<ref>), (<ref>), (<ref>),11b


where the terms that are not related to Y in (<ref>) are omitted. At each time instant, the BS and RIS only serve one UE due to the fact of that the downlink WPT and uplink uploading are scheduled in a TDM manner. A low-complexity alternating optimization (AO) algorithm is developed to design Y.

Proposition 1: With the fixed RIS beamforming matrics, according to the maximum-ratio transmission (MRT) <cit.>, the optimal transmit and receive beamforming vectors are

    w_k^*[t]={[ √(P_T)( h_k^H + g_k^H[t]Θ_k[t]G_UR^H[t])^H/‖h_k^H + g_k^H[t]Θ_k[t]G_UR^H[t]‖,                                                                     a_k[t]>0,;                                                                        0_M×1,                                                                     a_k[t]=0, ].

and

    u_k^*[t]={[ ( h_k^H + g_k^H[t]Θ̅_k[t]G_UR^H[t])^H/‖h_k^H + g_k^H[t]Θ̅_k[t]G_UR^H[t]‖,                                                                 b_k[t]>0,;                                                                    0_M×1,                                                                 b_k[t]=0. ].

Proof: Please refer to Section 3 of <cit.>.▪

Proposition 2: With the fixed BS beamforming vectors, the optimal value of the n-th RIS element serving the downlink WPT at the t-th time slot is

    e^jθ_k,n^*[t]=
       {[ e^{jθ_k,0[t]-j(g_UR,n^H[t]w_k[t])-j(g_k,n^H[t])},                                         a_k[t]>0,;                                                0,                                         a_k[t]=0, ].

where θ_k,0[t]=(h_k^Hw_k[t]), g_UR,n[t] denotes the n-th column of G_UR[t], and g_k,n[t] denotes the n-th element of g_k[t]. The optimal value of the n-th RIS element serving the uploading at the t-th time slot is

    e^jθ̅_k,n^*[t]=
       {[ e^{jθ̅_k,0[t]-j(g_UR,n^H[t]u_k[t])-j(g_k,n^H[t])},                                          b_k[t]>0,;                                                 0,                                          b_k[t]=0, ].

where θ̅_k,0[t]=(h_k^Hu_k[t]).

Proof: Please refer to Appendix A.▪

Using Proposition 1 and 2, we can obtain the optimal beamforming vectors and matrics by iteratively updating the BS beamforming vectors and the RIS beamforming matrics.



 §.§ Subproblem of UAV Trajectory Designing

With the fixed X and Y, the problem of designing the UAV trajectory can be written as

	
	
    (P5)max_Q ∑_t = 1^T ∑_k = 1^K λ_kR_k[t]16a
            s.t.  (<ref>), (<ref>), (<ref>), (<ref>),16b


where the terms that are not related to Q in (<ref>) are omitted. The phase shifts of BS-UEs links are aligned with those of BS-RIS-UEs links when the subproblem of designing Y is convergent. Hence, the amount of uploading data and the harvested energy of the k-th UE at the t-th time slot can be reformulated as

    R̃_k[t]    = Bδ _tb_k[t]
       ·log _2{ 1 + P_k[t]/σ _B^2(|A_k[t]^2|+2|B_k[t]|h_0/d_R2U,k^α_RU/2[t]d_B2R^α_BR/2[t]
             +|C_k[t]|^2h_0^2/d_R2U,k^α_RU[t]d_B2R^α_BR[t])},∀ k∈𝒦,

and

    Ẽ_k[t] = η _0δ _t(   |D_k[t]|^2+2|F_k[t]|h_0/d_R2U,k^α_RU/2[t]d_B2R^α_BR/2[t]
       +|G_k[t]|^2h_0^2/d_R2U,k^α_RU[t]d_B2R^α_BR[t])a_k[t],∀ k∈𝒦,

where A_k[t]=h_k^Hu_k[t], D_k[t]=h_k^Hw_k[t],

    B_k[t]=h_k^Hu_k[t]u_k^H[t]a( ϕ _0[t])b^T( φ _0[t])Θ̅ _k^H[t]b( φ _k[t]),


    C_k[t]=b^H( φ _k[t])Θ̅ _k[t]b^*( φ _0[t])a^H( ϕ _0[t])u_k[t],


    F_k[t]=h_k^Hw_k[t]w_k^H[t]a( ϕ _0[t])b^T( φ _0[t])Θ _k^H[t]b( φ _k[t]),


    G_k[t]=b^H( φ _k[t])Θ _k[t]b^*( φ _0[t])a^H( ϕ _0[t])w_k[t].

Furthermore, (P5) can be reformulated as

	
	
    (P6)max_Q,{x_k[t]},{y[t]} ∑_t = 1^T ∑_k = 1^K λ_kBδ _tb_k[t]log_2{1 + P_k[t]/σ _B^2
              ( | A_k[t]|^2 + 2| B_k[t]|h_0/x_k^1/2[t]y^1/2[t] + | C_k[t]|^2h_0^2/x_k[t]y[t]) }23a
        s.t. ∑_i = 1^t [ γδ _tγ _cf_k^3[i] + b_k[i]δ _tP_k[i]]≤η_0δ_t∑_i = 1^t(|D_k[i]|^2+ 
             2|F_k[i]h_0|/x_k^1/2[i]y^1/2[i]+|G_k[i]|^2h_0^2/x_k[t]y[t]),∀ k ∈ K,∀ t ∈ T,23b
           ∑_i = 1^t γδ _tf_k[i]κ _k/e^ε /κ _k - e^ε≤∑_i = 1^tBδ_tb_k[i]log_2{1 + P_k[t]/σ _B^2( | A_k[i]|^2
             +2| B_k[i]|h_0/x_k^1/2[i]y^1/2[i] + | C_k[i]|^2h_0^2/x_k[i]y[i])},∀ k ∈ K,∀ t ∈ T,23c
           x_k^1/α_RU≥‖q[t]-q_k‖,∀ k ∈ K,∀ t ∈ T,23d
           y^α_BR≥‖q[t]-q_0‖,∀ k ∈ K,∀ t ∈ T,23e
           (<ref>), (<ref>),23f


where x_k[t],y[t],∀ k ∈ K,∀ t ∈ T are slack variables. Due to the non-convex functions in (<ref>), (<ref>), and (<ref>), (P6) is a non-convex problem. We propose a successive convex approximation (SCA) algorithm to solve (P6). The lower bounds of the uploading data rate and the harvested energy can be obtained using the first order Taylor expansion, which is given by (<ref>) and (<ref>) at the top of the next page. In (<ref>) and (<ref>), we have



    H_k^(l)[t]= 
       ( 1 + P_k[t]/σ _B^2( | A_k[t]|^2 + 2| B_k[t]|h_0/√(x_k^(l)[t]y^(l)[t]) + | C_k[t]|^2h_0^2/x_k^(l)[t]y^(l)[t])),

and

    U_k^(l)[i] = | D_k[i]|^2 + 2| F_k^(l)[i]|h_0/√(x_k^(l)[i]y^(l)[i]) + | G_k^(l)[i]|^2h_0^2/x_k^(l)[i]y^(l)[i],

where l denotes the number of SCA iteration. Finally, the problem of designing the UAV trajectory is formulated as

	
	
    (P7)max_Q,{x_k[t]},{y[t]} ∑_t = 1^T ∑_k = 1^K λ_kR_lb,k[t]28a
               s.t.  ∑_i = 1^t [ γδ _tγ _cf_k^3[i] + b_k[i]δ _tP_k[i]]
                     ≤∑_i = 1^tE_lb,k[i],∀ k ∈ K,∀ t ∈ T,28b
                   ∑_i = 1^t γδ _tf_k[i]κ _k/e^ε /κ _k - e^ε
                     ≤∑_i = 1^t R_lb,k[i],∀ k ∈ K,∀ t ∈ T,28c
                   (<ref>), (<ref>), (<ref>).28d


(P7) is a standard convex problem that can be solved by CVX tools. The SCA algorithm for designing the UAV trajectory is presented in Algorithm 1. The overall BCD-based algorithm for designing X, Y, and Q is presented in Algorithm 2. In Algorithms 1 and 2, O^(i) and O̅^(j) denote the value of objective function in (<ref>).





§ UPLOADING DATA MAXIMIZATION BASED ON LOSSY COMPRESSION

The problem of the uploading data maximization based on lossy compression can be formulated as

	
	
    (P8)max_X̅,Y,Q ∑_t = 1^T ∑_k = 1^K λ_k{R_k[t] + ( 1 - κ _k)S̅_k[t]}29a
             s.t.  f̅_k[t],P_k[t],a_k[t],b_k[t] ≥ 0,∀ k∈ K,∀ t ∈ T,29b
                 ∑_i = 1^t [ γδ _tγ _cf̅_k^3[i] + b_k[i]δ _tP_k[i]]
                   ≤∑_i = 1^tE_k[i],∀ k ∈ K,∀ t ∈ T,29c
                 ∑_i = 1^t γδ _tf̅_k[i]κ _k/e^ε /κ _k - e^ε≤∑_i = 1^t R_k[i],∀ k ∈ K,∀ t ∈ T,29d
                 (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>),29e


where X̅≜{f̅_k[t],P_k[t],a_k[t],b_k[t],∀ k ∈ K,∀ t ∈ T}. Similar with (P1), (P8) can also be solved by the BCD algorithm. Therfore, we omit the details of solving (P8).



§ ANALYSIS OF ENERGY EFFICIENCY

The essence of the fact that compression can improve system performance lies in its enhancement of energy efficiency. Therefore, the energy efficiencies of the compression and uploading are analyzed in this section. The compression schemes can improve system performance when the power used for compression is relatively small. However, the benefits of the compression schemes gradually reduce when the power used for compression becomes relatively large.

The energy efficiency of directly uploading data is defined as

    η_U≜R_k[t]/P_k[t]=Bδ _tb_k[t]
       ·log _2( 1 + | ( h_k^H + g_k^H[t]Θ̅_k[t]G_UR^H[t])u_k[t]|^2P_k[t]/σ _B^2)/P_k[t].

The energy efficiencies of the lossless and lossy compressions are defined as

    η _C≜( 1 - κ _k[t])S_k[t]/E_C,k[t]   = ( 1 - κ _k)f_k[t]/( e^ε /κ _k - e^ε)P_k[t]
       = ( 1 - κ _k)/γ _c^1/3( e^ε /κ _k - e^ε)P_k^2/3[t],

and

    η̅_C = ( 1 - κ_k)S̅_k[t]/γδ _tγ _cf̅_k^3[t]   = √(κ_k)( 1 - κ_k)f̅_k[t]/( e^ε /κ_k - e^ε)P_k[t]
       = √(κ_k)( 1 - κ_k)/γ _c^1/3( e^ε /κ_k - e^ε)P_k^2/3[t].

Proposition 3: There exists P^* that makes η_U<η_C when P_k[t]<P^*, η_U≥η_C when P_k[t]≥ P^*

Proof: Please refer to Appendix B.▪

Proposition 4: There exists P̅^* that makes η_U<η̅_C when P_k[t]<P̅^*, η_U≥η̅_C when P_k[t]≥P̅^*

Proof: The proof is omitted since it is similar to that of Proposition 3.▪


To illustrate the insight of Proposition 3 and 4, we observe the energy efficiency versus the power used for compression and uploading in Fig. <ref>, where the energy efficiency of the direct uploading scheme gradually becomes bigger than the lossless and lossy compression schemes. Moreover, P̅^* < P^* is because the energy efficiency of the lossless compression scheme is bigger than that of the lossy scheme.



§ ANALYSIS OF COMPLEXITY AND CONVERGENCE PERFORMANCE



 §.§ Analysis of Computational Complexity

We mainly focus on the complexity analysis of solving the convex optimization problem, i.e., the complexity of solving (P3) and (P7). (P3) and (P7) can both be solved by the interior-point method, whose complexity is 𝒪(n^3.5) with n denoting the dimension of the optimization variables <cit.>. Therefore, the computational complexity of solving (P3) and (P7) are 𝒪((4KT)^3.5) and 𝒪((2T)^3.5), respectively. The complexity of Algorithm 1 is 𝒪(m(2T)^3.5) with m denoting the number of SCA iteration. The complexity of Algorithm 2 is 𝒪(n((4KT)^3.5+KT(24MN+16N^2+4M)+m(2T)^3.5)) with n denoting the number of BCD iteration and KT(24MN+16N^2+4M) denoting the complexity to obtain the optimal beamforming vectors and matrics. 



 §.§ Analysis of Convergence Performance

Denoting the sum of uploading data at the i-th BCD iteration by R(X[i],Y[i],Q[i]). We have

    R_max   (a)≥R(X[i+1],Y[i+1],Q[i+1])
       (b)≥R(X[i+1],Y[i+1],Q[i])
       (c)≥R(X[i+1],Y[i],Q[i])(d)≥R(X[i],Y[i],Q[i]),

where R_max denotes the upper bound of the sum of uploading data, (b) holds since the Taylor expansion in (<ref>) is tight and (P7) can be optimally solved, (c) holds since Proposition 1 and 2 guarantee the uploading data non-decreasing, and (d) holds since the optimal X[i+1] can be obtained by solving (P3). Therefore, the convergence of the proposed BCD algorithm is guaranteed.



§ SIMULATION RESULTS


We present simulation results in this section to demonstrate the effectiveness of the proposed schemes. If not otherwise specified, the parameter values used in our simulations are listed in Table 1.


Fig. <ref> shows the UAV trajectories obtained by the lossless compression scheme, the lossy compression scheme, and the scheme without compression, which are respectively labelled as “Lossless”, “Lossy”, and “Wo”. First, under the lossless scheme with equal UE weights, the UAV flies towards the BS and then hovers around the BS. The trajectory is nearly symmetrical about the straight line x=0. This is intuitive because a symmetric trajectory provides equal WPT and uploading opportunities for UEs with equal weights. Then, the trajectory under the lossless scheme with different weights tends to be close to the UEs with higher weights since the UAV can transfer more energy to these UEs and collect more data from them under such a trajectory. Moreover, the trajectories under the lossy scheme and the without compression scheme with different weights are closer to the UEs with higher weights than that of the lossless scheme. This is because the UAV needs to fly close to these UEs to provide the same level of transmission efficiency as the lossless scheme.



Fig. <ref> shows the time allocation of UEs with equal and different weights. The WPT time of each UE is longer than their uploading time. This is because WPT can benefit simultaneously from both BS and RIS beamforming gains, whereas uploading can only leverage the RIS beamforming gain. Therefore, more energy needs to be transmitted to the UEs for them to upload data. Moreover, the WPT and uploading times for UE2 and UE3 are longer than those for UE1 and UE4. This is because the distances between UE2 and the BS and between UE3 and the BS are shorter than those distances of UE1 and UE4, and the path losses of the channels associated with UE2 and UE3 are smaller than those of UE1 and UE4. Additionally, when the UE weights are different, the time slots are almost entirely allocated to the UEs with higher weights to maximize the amount of data uploaded.



Fig. <ref> shows the amount of uploading data of all schemes versus the transmit power of the BS when N=16 and N=128. The amount of uploading data of all schemes increases with the transmit power, as the UEs can harvest more energy and upload more data with higher transmit power. The slope of the lossless scheme is steeper than the other two schemes because of its higher energy efficiency. Moreover, the amount of uploading data of N=128 is greater than that of N=16 because more RIS elements can provide higher beamforming gain. Furthermore, with a fixed number of RIS elements, the uploading data of the lossless scheme is greater than that of the lossy scheme, which is greater than that of the without compression scheme. Additionally, the amount of uploading data of the without compression scheme gradually approaches that of the lossy scheme as the transmit power increases. This is because, with the increase in transmit power, the energy efficiency of the lossy scheme gradually becomes lower than that of the without compression scheme, leading to a convergence in the amount of uploaded data. This is consistent with Proposition 3.



Fig. <ref> shows the amount of uploading data of all schemes versus the number of RIS elements when M=4 and M=32. The amount of uploading data of all schemes increase with the number of RIS elements, as the increasing RIS elements can provide higher beamforming gain. Moreover, the amount of uploading data of M=32 is greater than that of M=4 because more BS antennas can provide higher degrees of freedom (DoF). Additionally, with a fixed number of BS antennas, the uploading data of the lossless scheme is greater than that of the lossy scheme, which is greater than that of the without compression scheme.



Fig. <ref> shows the proportion of compressed data in uploading data versus the channel power gain when the UAV hovers at (0m, 0m, 8m). The proportions of compressed data of both the lossless scheme and lossy scheme decrease as the channel power gain increases. This is because higher proportions of compressed data can improve the transmission efficiency and overcome the effects of poor channel conditions. Additionally, the lossy scheme has lower proportions of compressed data than the lossless scheme since some compressed data are lost due to the compression error. Furthermore, in each compression scheme, the proportions of UE1 and UE4 are the same, and the proportions of UE2 and UE3 are also the same, since the path losses of UE1 and UE4 are the same, and the path losses of UE2 and UE3 are also the same. The proportions of compressed data of UE1 and UE4 are higher than those of UE2 and UE3 since the path losses of UE1 and UE4 are more serious. UE1 and UE4 need to enhance their transmission efficiencies by compressing more data.



Fig. <ref> demonstrates that the proposed BCD algorithm can achieve convergence with ten BCD iterations under different parameter configurations. The fast convergence is because we can obtain the globally optimal X by solving (P3) and the nearly globally optimal Y by Proposition 1 and 2, and the Taylor expansions used for dealing with (P6) construct tight lower bounds for the uploading data and the harvest energy.



§ CONCLUSION

In this paper, we have proposed a novel transmission framework for the WPT-assisted MCS systems enhanced by UAV-mounted RISs to enhance transmission efficiency. Two optimization problems are established to maximize the weighted sum of uploading data under different compression schemes. A BCD algorithm has been proposed to decompose the optimization problem into three subproblems, which are solved alternately. The subproblem for designing the transmission parameters can be easily converted into a convex problem. The closed-form optimal beamforming vectors and matrices are obtained when solving the beamforming subproblem. We utilize the SCA algorithm to approximate the lower bounds of the harvested energy by UEs and the uploading data for designing the UAV trajectory. The energy efficiencies of the compression schemes have been analyzed to further evaluate the gains obtained by compression. We have also analyzed the complexity and the convergence performance of the proposed algorithms. Numerous simulation results demonstrate that the proposed transmission framework and algorithms can effectively increase the amount of the collected data.



§ APPENDIX A
PROOF OF PROPOSITION 2

In the expression of the harvested energy (i.e., (<ref>)), the term that is related to the RIS beamforming matrix can be expressed as

    | ( h_k^H + g_k^H[t]Θ_k[t]G_UR^H[t])w_k[t]|
       =| h_k^Hw_k[t] + g_k^H[t]Θ_k[t]G_UR^H[t]w_k[t]|
       (1)≤| h_k^Hw_k[t] | +| g_k^H[t]Θ_k[t]G_UR^H[t] w_k[t] |,

where (1) holds due to the triangle inequality. When (h_k^Hw_k[t])=(g_k^H[t]Θ_k[t]G_UR^H[t] w_k[t]), the equality holds. The RIS beamforming matrix is used to align the phase shift of the reflected link with the phase shift of the direct link. The phase shift difference between the two links is (h_k^Hw_k[t])-(g_UR,n^H[t]w_k[t])-(g_k,n^H[t]). Then, we can obtain the optimal values of the RIS elements in (<ref>). We omit the derivation of (<ref>) since it is similar to that of (<ref>). This completes the proof of Proposition 2.



§ APPENDIX B
PROOF OF PROPOSITION 3

We first check the first-order partial derivatives of η_U and η_C, which are expressed in (<ref>) and (<ref>). (<ref>) is shown at the top of this page.


    ∂η_C/∂ P_k[t]=-2/3·( 1 - κ _k)/γ _c^1/3( e^ε /κ _k - e^ε)P_k^5/3[t].

∂η_U/∂ P_k[t] < 0 when P_k[t]→0^+. ∂η_U/∂ P_k[t] < 0 when P_k[t]>0. Therefore, the first-order partial derivatives of η_U and η_C are all less than 0 when P_k[t] >0. η_U and η_C are monotonically decreasing when P_k[t] >0. Furthermore, we check the value of the energy efficiencies at the end of the define fields, i.e., at 0 and +∞. We have lim_P_k[t]→0^+η_C/η_U→+∞, i.e., η_C>η_U, and lim_P_k[t]→+∞η_U/η_C→+∞, i.e., η_U>η_C. Therefore, there must have exactly one P^* that makes η_U<η_C when P_k[t]<P^*, η_U≥η_C when P_k[t]≥ P^*. This completes the proof of Proposition 3.

IEEEtran




