Energy Beamforming for Wireless Information and Power Transfer in Backscatter Multiuser Networks
Wenyuan Ma Wei Wang Tao Jiang
cs.NI





Energy Beamforming for Wireless Information and Power Transfer in Backscatter Multiuser Networks
    Wenyuan Ma, Wei Wang^∗, Tao JiangSchool of Electronic Information and Communications, Huazhong University of Science and TechnologyEmail: {wenyuan_ma, weiwangw, taojiang}@hust.edu.cn
^∗The Corresponding author is Wei Wang (weiwangw@hust.edu.cn).
    December 30, 2023
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Wirelessly powered backscatter communication (WPBC) has been identified as a promising technology for low-power communication systems, which can reap the benefits of energy beamforming to improve energy transfer efficiency. Existing studies on energy beamforming fail to simultaneously take energy supply and information transfer in WPBC into account. This paper takes the first step to fill this gap, by considering the trade-off between the energy harvesting rate and achievable rate using estimated backscatter channel state information (BS-CSI). To ensure reliable communication and user fairness, we formulate the energy beamforming design as a max-min optimization problem by maximizing the minimum achievable rate for all tags subject to the energy constraint. We derive the closed-form expression of the energy harvesting rate, as well as the lower bound of the ergodic achievable rate. Our numerical results indicate that our scheme can significantly outperform state-of-the-art energy beamforming schemes. Additionally, the proposed scheme achieves performance comparable to that obtained via beamforming with perfect CSI.	        


Energy Beamforming for Wireless Information and Power Transfer in Backscatter Multiuser Networks
    Wenyuan Ma, Wei Wang^∗, Tao JiangSchool of Electronic Information and Communications, Huazhong University of Science and TechnologyEmail: {wenyuan_ma, weiwangw, taojiang}@hust.edu.cn
^∗The Corresponding author is Wei Wang (weiwangw@hust.edu.cn).
    December 30, 2023
=========================================================================================================================================================================================================================================================




§ INTRODUCTION

Wirelessly powered backscatter communication (WPBC), a kind of wireless transmissions by modulating and reflecting ambient wireless signals, emerges as a promising technology to achieve ultra-low-power communication for a broad range of the Internet of Things (IoT) applications, such as smart city, connected health, and smart farming <cit.>. Although achieving transmissions with power consumption of μWs via ambient wireless signals from television and WiFi communications, WPBC is limited to short ranges due to the small amount of energy harvested by a backscatter tag and low signal power which further attenuates significantly after reflection.

By forming sharp energy beams towards target users, energy beamforming is an attractive solution to improve communication ranges. Growing attempts have been devoted to exploring the merits of energy beamforming in WPBC <cit.>. Despite the fact that much has been understood through these studies, their beamforming designs fail to take the fundamental trade-off between energy supply and information transfer into account. These studies either only focus on optimizing the transmit beamforming to maximize the sum rate of a cooperative WPBC system<cit.>, or design energy beamforming just to maximize a total utility of the harvested energy <cit.>. Nevertheless, there are different optimal beams
for the achievable rate and energy harvesting rate. The reason lies in the fact that the energy harvesting rate at the tag only depends on the forward channel (i.e., transmitter-to-tag), while the achievable rate at the reader depends on the backscatter channel (i.e., transmitter-to-tag-to-receiver). As a consequence, energy beamforming that only maximizes harvested energy cannot guarantee the achievable rate to retain reliable communications. Likewise, energy beamforming that only maximizes data rate cannot ensure the energy supply.

 

To overcome the above limitation, we take the first step to investigate the energy beamforming design that considers the trade-off between energy supply and information transfer via the estimated backscatter channel state information (BS-CSI). As shown in Fig. <ref>, this paper considers a general WPBC network where a reader with massive antennas transmits energy to multiple single-antenna tags, and tags transmit information to the reader by backscatter modulation. To ensure reliable communication and user fairness, we formulate the beamforming design as a max-min optimization problem by maximizing the minimum achievable rate for all tags subject to tags' power consumption constraints. To solve this problem, we analyze the energy harvesting rate and achievable rate via beamforming using the estimated BS-CSI.  

The major contributions are summarized as follows.

	
  * We propose a beamforming scheme for energy supply and information transfer in WPBC using the estimated BS-CSI. To the best of our knowledge, this is the first work that considers the trade-off between the energy harvesting rate and the achievable data rate in WPBC.


	
  * We obtain the analytical expressions of the energy harvesting rate and lower bound on the ergodic achievable rate with the estimated BS-CSI, considering the effect of unknown forward and backward channels.


	
  * Our numerical results indicate that the minimum achievable rate of the proposed scheme is significantly higher than that of previous energy beamforming schemes, while slightly lower than that obtained by energy beamforming with perfect CSI. 


The rest of this paper is organized as follows. Section 2 establishes the system model. Section 3 elaborates on the energy beamforming design for energy supply and information transfer. Section 4 analyzes the harvested energy and achievable rate of the proposed scheme. Next, Section 5 shows the numerical results. Section 6 presents the state of the art. Finally, Section 7 concludes our paper.




§ SYSTEM MODEL

As shown in Fig. <ref>, we consider wireless energy transfer (WET) from a monostatic reader with M antennas to K single-antenna tags and backscatter communication between tags and the reader with another R antennas for a WPBC system. Each tag harvests energy for circuit consumption and transmits information to the reader by backscatter modulation. 

We assume transmission over time blocks. The duration of each time block is T symbol periods, each of which is shorter than the coherent interval. Each time block consists of a channel estimation (CE) slot, which lasts α symbol periods, followed by a simultaneous wireless energy and information transfer slot. First, the reader sends pilots and leverages backscattered signals to estimate the BS-CSI associated with each tag. Subsequently, the reader performs energy beamforming to transmit wireless energy and provide carrier signals for all tags. Simultaneously, tags transfer information to the reader. 



 §.§ Channel Model

Let h_k^f=[h_1k^f, h_2k^f, ... ,h_Mk^f]^T∈ℂ^M×1, h_k^b=[h_k1^b, h_k2^b, ... , h_kR^b]^T
∈ℂ^R×1 be the forward and backward channel vector, respectively. h_mk^f is the forward channel coefficient between the m-th transmit antenna of the reader and tag k, and h_kr^b is the backward channel coefficient between the k-th tag and the r-th receive antenna of the reader. We consider flat Rayleigh-fading channel model. Thus, we assume that h_mk^f, h_kr^b are independent and both distributed as 𝒞𝒩( 0, β_k ), where β_k denotes the path loss of the channel between the reader and tag k. β_k is constant over the coherent interval and known at the reader. Let H_k=h_k^bh_k^fT denote the R× M backscatter channel matrix of tag k, where [H_k]_mr=h_mkr=h_mk^fh_kr^b is the backscatter channel coefficient, which is a combination of the channel from the m-th transmit antenna to tag k with the channel from tag k to the r-th receive antenna of the reader.



 §.§ Backscatter Channel Estimation

In the backscatter CE slot, the reader estimates the channel matrix of each tag by controlling tags to reflect the pilot signals in sequence<cit.>. When estimating the backscatter channel of the k-th tag, the reader transmits orthogonal pilot sequences GB^1/2∈ℂ^D× M by M transmit antennas, where G∈ℂ^D × M satisfies G^HG=I_M, and B∈ℂ^M × M is a diagonal matrix with Dp_ce as element. p_ce and D represent the power of each antenna transmitting a pilot sequence and the length of pilot sequence (D≥ M), respectively. Then, K tags reflect the pilot signals with a reflection coefficient δ_k∈ℂ in sequence. 

The reader can eliminate the direct-link influence by employing self-interference cancellation techniques or frequency shifting. Thus, the received signal at the reader is given by

    Y_k^CE   =√(δ_k)H_k(GB^1/2)^T+N_k,
 
where N_k is an R× D matrix with independent and identically distribute (i.i.d.) elements and [ N_k]_mr = n_k,mr∼𝒞𝒩( 0, σ^2).

Given Y_k^CE, the least-square (LS) estimate of H_k can be expressed as

    H_k^LS   =Y_k^CEδ_k^-1/2G^*B^-1/2
       =H_k+N_kδ_k^-1/2G^*B^-1/2.
  
Denote the estimation error by E_k≜H_k^LS-H_k, where ẽ_k,mr, ĥ_mkr^LS, and h_mkr represent the m-th row , r-th column element of E_k, H_k^LS, and H_k, respectively. E_k is independent of H_k^LS. Thus, we can obtain ẽ_k,mr∼𝒞𝒩( 0, σ^2/Dp_ceδ_k).

However, energy beamforming needs to obtain forward channel state information (F-CSI). Therefore, we use the backward channel state information (B-CSI) as an unknown parameter to obtain the desired F-CSI through BS-CSI. Mathematically, we define random vector h_k^f≜h_kr/ h_kr^b, where h_k^f=[ĥ_1k^f, ĥ_2k^f, ..., ĥ_Mk^f]^T, h_kr=[ĥ_1kr, ĥ_2kr, ... ,ĥ_Mkr]^T, r∈{1, 2, ..., R}, and the error  e_k≜h_k^f-h_k^f. 
Then, we can obtain the element of error that follows complex Gaussian distribution with zero mean and variance

    σ_e,kr^2(h_kr^b)=σ^2/|h_kr^b|^2Dp_ceδ_k                                                             =Kσ^2/|h_kr^b|^2α p_ceδ_k.


Based on the above analysis, we can obtain the distributions with h_kr^b as a condition for subsequent analysis as follows:

    h_kr|h_kr^b   ∼𝒞𝒩(0_M,|h_kr^b|^2( β_k+σ_e,kr^2(h_kr^b)) I_M) 
    h_k^f|h_kr^b   ∼𝒞𝒩(0_M,( β_k+σ_e,kr^2(h_kr^b)) I_M) 
    h_k^f|h_kr,h_kr^b   ∼𝒞𝒩(β_kh_k^f/β_k+σ_e,kr^2(h_kr^b), β_kσ_e,kr^2(h_kr^b)/β_k+σ_e,kr^2(h_kr^b)I_M).
 



§ ENERGY BEAMFORMING FOR WIRELESS ENERGY AND INFORMATION TRANSFER



 §.§ Energy Transfer

The reader transmits a carrier signal to K tags concurrently. To improve the energy transmission efficiency, we use the estimated BS-CSI for beamforming. The reader transmits a signal u with 𝔼{| u |^2  }=p, and the average transmit power is w. Thus, we can obtain p=wT -α p_ce/T-α. The reader uses a weighted sum of the normalized estimated BS-CSI, since it is proved to be asymptotically optimal for WET<cit.>. The beamformer is denoted as  Φ=∑_k=1^K√(ζ_k)ĥ_kr^*/ĥ_kr, where ζ_k∈[0,1] ∀  k such that ∑_k=1^Kζ_k=1. So the signal emitted by the reader is given as uΦ, and the received signal at tag k is given by

    b_k =uΦ^Th_k^f + n_k,

where n_k denotes the noise and  n_k∼𝒞𝒩( 0, σ_k^2).

We assume that the noise energy cannot be harvested <cit.>. Then the incident signal power of tag k can be represented as 

    P_Ik=𝔼{| u |^2 |Φ^Th_k^f|^2  }
    =p𝔼{|Φ^Th_k^f|^2 }.

When the tag is activated to reflect the incoming signal, The rectifier converts the (1-δ_k) fraction of incident radio frequency signal into a direct current (DC)<cit.>. Otherwise, the whole incident signal power is rectified to DC.

To simplify the analysis, we assume that the harvested energy scales linearly with the input power <cit.>. Thus, the instantaneous energy harvesting rate when the tag is activated is given by 

    P_Ek=η(1-δ_k)P_Ik,

where η∈(0,1] is the rectifier efficiency. 



 §.§ Information Transfer

When the instantaneous energy harvesting rate exceeds the circuit power consumption rate, tags reflect and modulate the incident carrier signal. Then the received signal vector from tag k can be written as

    y_k    =√(δ_k)h_k^bb_ks_k+z_k
       =√(δ_k)us_kh_k^bh_k^fTΦ + v_k,

where δ_k and s_k are the reflection coefficient and the backscatter signal of tag k, respectively. We assume s_k is distributed as 𝒞𝒩( 0, 1 ), and the noise v_k is distributed as 𝒞𝒩( 0, Σ) <cit.>.

So the received signal vector at the reader from K tags can be represented as

    y = ∑_k=1^Ky_k   =H^b[(H^fTuΦ)∘(Λs)]+μ
       =H^bx+μ,

where ∘ represents Hadamard product, H^b=[h_1^b, h_2^b, ... , h_K^b], H^f=[h_1^f, h_2^f, ... , h_K^f], Λ=diag{√(δ_1), √(δ_2), ... , √(δ_K)}, s=[s_1, s_2, ... , s_K]^T, x=[√(p_1)s_1, √(p_2)s_2, ... , √(p_K)s_K]^T, the p_k = δ_k P_Ik denotes the reflect power, and μ is a vector of additive white Gaussian noise, μ has i.i.d. 𝒞𝒩( 0, σ̃^2 ) elements.









However, information detection needs to obtain the B-CSI. Therefore, we use F-CSI as an unknown parameter to obtain the desired B-CSI through BS-CSI. Mathematically, we define random vector h_k^b≜h_mk/ h_mk^f, where h_k^b=[ĥ_k1^b, ĥ_k2^b, ..., ĥ_kR^b]^T,  h_mk=[ĥ_mk1, ĥ_mk2, ... ,ĥ_mkR]^T, m∈{1, 2, ..., M}, and the error  ε≜H^b-H^b. Furthermore, we can obtain the distributions of h_mk|h_mk^f, h_k^b|h_mk^f, h_k^b|h_mk,h_mk^f according to the analytical results in Section 2-B.

The received vector after using a linear detector Q to detect the information from all tags is given by

    r   = Q^HH^bx + Q^Hμ = Q^H( H^bx - εx + μ).

Let r_k and x_k be the k-th elements of K × 1 vectors r and x, respectively. q_k, h_k^b, ε_k be the k-th column of detector Q, H^b, ε. Therefor, after using the linear detector, the received signal associated with the k-th tag can be written as 

    r_k = q_k^Hh_k^b x_k + ∑_i=1,i≠ k^Kq_k^Hh_i^b x_i - ∑_i=1^Kq_k^Hε_i x_i+ q_k^Hμ.


The ergodic achievable rate of the information transmission from tag k is given by

    R_k≜𝔼{log_2( 1 +  γ_k) },

where the signal-to-interference-plus-noise-ratio(SINR)

    γ_k    =    p_k |q_k^Hh_k^b|^2 /∑_i=1,i≠ k^K  p_i |q_k^Hh_i^b|^2  + |q_k^Hq_k|∑_i=1^K  p_iσ_ϵ,mi^2(h_mi^f)+ |q_k^Hq_k|σ̃^2.
  



 §.§ Energy Beamforming Design

We realize energy beamforming and information detection for energy supply and information transfer in WPBC using the estimated BS-CSI. We note that the energy harvesting rate at the tag relies on the forward channel while the achievable rate at the reader relies on the backscatter channel. Therefore, there are different optimal energy beams for achievable rate and harvested energy due to the difference between the forward and backscatter channels. We formulate the beamforming design as a max-min optimization problem to maximize the minimum rate for all tags by jointly optimizing the energy allocation weights ζ=[ζ_1, ζ_2,..., ζ_K] of the beamformer Φ, the CE time α and transmit power of pilot p_ce, subject to the power consumption constraint as follows.

    max_ζ,α, p_ce  min_1 ≤ k ≤ K R_k
       [           s.t.  P_Ek≥ρ,   ∀ k;                 ∑_k=1^Kζ_k=1,;                      0 ≤α≤ T,;                ζ_k≥ 0,   ∀ k, ]

where ρ denotes the circuit power consumption rate in backscattering. This problem maximizes the throughput with user-fairness assurance. However, a closed-form analytic solution to (<ref>) does not appear easy to find. To verify the performance improvement of the proposed scheme, we obtain the optimal resource allocation through numerical solver.

TempEqCnt                         
 
                           

 
       


§ ANALYSIS ON ENERGY HARVESTED RATE AND ACHIEVABLE RATE

To optimize beams in a multiuser system as formulated in Eq. (<ref>), we derive analytical expressions of the energy harvesting rate and achievable rate by using energy beamforming with the estimated BS-CSI. Additionally, we analyze the effect of the unknown channels on the energy harvesting rate and achievable rate. For analytical tractability, we obtain the bounds on the energy harvesting rate and achievable rate.



 §.§ Energy Harvesting Rate

Based on the energy beamforming, the energy harvesting rate (<ref>) can be derived as <cit.>

    P_Ek=η(1-δ_k)p β_k[ζ_k[ M-( M-1)ϕ(h_kr^b) - 1] +1],

where ϕ(h_kr^b)=𝔼_h_kr^b{1/β_k|h_kr^b|^2α p_ceδ_k/Kσ^2+1}. 
The result of (<ref>) is obtained by performing expectation over the unknown backward channel, which takes the fuzziness of h_kr^b into account. To simplify the calculation, we derive the upper and lower bounds on the incident signal power. 


	The boundary of the incident signal power of tag k, when energy bamforming is performed using the BS-CSI, is given by
	
    P_Ik^L   ≜ pβ_k[ζ_k (M-1)ℒ_1(α, p_ce) +1] < P_Ik(p_ce, p, α, ζ_k)
        < pβ_k[ζ_k (M-1) ℒ_2(α, p_ce)+1] ≜ P_Ik^U,

where

 ℒ_1(α, p_ce)=[1-Kσ^2/β_k^2 α p_ceδ_kln(1 + β_k^2 α p_ceδ_k/ Kσ^2 )],

ℒ_2(α, p_ce)=[1-Kσ^2/2 β_k^2 α p_ceδ_kln(1 +  2 β_k^2 α p_ceδ_k/ Kσ^2 )].	


  Please refer to Appendix A.




 §.§ Ergodic Achievable Rate

The exact expression for the rate R_k in (<ref>) is analytically intractable. Since f(x)=log(1+ 1/x) is a convex function, by utilizing Jensen's inequality, the lower bound on the ergodic achievable rate is given by (<ref>) shown at the top of this page. The result of (<ref>) is obtained by performing expectation over the unknown backward and forward channels, which takes the effect of unknown channels into account.


 With LS channel estimate, MRC detection, i.e. q_k = h_mk/h_mk, and R ≥ 2, we can obtain an achievable rate as follows
 
 
    R̃^MRC_k   = log_2(1 + γ̃^MRC_k),
 
 where p_k = δ_k P_Ik, and the SINR
 
    γ̃^MRC_k = p_k (R-1)β_k/( 1 - Kσ^2/β_k^2 α p_ceδ_kexp(Kσ^2/β_k^2 α p_ceδ_k) Γ(0,Kσ^2/β_k^2 α p_ceδ_k)) 
       1/( ∑_i=1,i≠ k^K  p_i(β_i + K σ^2/α p_ceδ_iΓ(0, τ/β_i) ) + ∑_i=1^K  p_iK σ^2/α p_ceδ_iΓ(0, τ/β_i) + σ̃^2 ).
 


	Please refer to Appendix B.


We also note that the SINR appears analytically intractable. For analytical tractability, we obtain the lower bound on the SINR as (<ref>) shown at the top of next page. We take the lower bound R̃^MRC_kL as the achievable rate, and P_Ek^L as instantaneous energy harvesting rate in Eq. (<ref>) to complete the beamforming design.


























































§ NUMERICAL RESULTS

In this section, we verify the performance improvement of the proposed scheme through numerical simulations. We consider two tags and assume the duration of each time block is T = 200 symbol periods. We set the noise power of the backward link as σ̃^2 = - 60 dBm and noise power of the whole backscatter link as σ^2 = - 90 dBm. We also use the long-term fading model β_k = 0.0086/4π d_k^2 <cit.>, where the distance d_1 = 4 m and d_2 =6 m. The energy conversion coefficient is η = 0.65 and the reflection coefficient of two tags is δ_1 = δ_2 = 0.3 + 0.4i <cit.>. We consider the circuit power consumption rates as ρ = 8.9 μ W <cit.>. 



To verify the performance gain of the proposed scheme, we give two benchmarks, i.e., the case of beamforming with perfect CSI and the case of omnidirectional transmission. In the first case, we assume the reader has perfect knowledge of F-CSI for energy beamforming and B-CSI for information detection. Thus, the step of the CE can be removed. In the second case, the reader performs omnidirectional transmission without energy beamforming. Channel estimation is only used for information detection. 

Figs. <ref> and <ref> show the efficiency of the proposed scheme. We first compare the achievable rate to the two benchmarks. As shown in Fig. <ref>, we can observe that the max-min rate of the proposed scheme is significantly higher than that of omnidirectional transmission, while slightly lower than that obtained by energy beamforming with perfect CSI. From Fig. <ref>, we can also observe that the energy harvesting rate of the proposed scheme has a large gain compared to that of omnidirectional transmission, only a slight loss compared to that obtained using perfect CSI. Furthermore, as M increases, the energy harvesting rate by using beamforming increases, while the energy harvesting rate by omnidirectional transmission remains as a small constant. Due to the extremely low energy harvesting rate of omnidirectional transmission, it is difficult to activate the tag for communication when the transmission power at the reader is low. However, beamforming can greatly increase the energy harvesting rate to activate tags and ensure uninterrupted backscattering of tags. 

Figs. <ref> and <ref> compare our energy beamforming scheme, which maximizes the minimum rate for all tags subject to the power consumption constraint, and the existing energy beamforming design that aims to maximize the minimum energy harvesting rate. Fig. <ref> indicates that the minimum achievable rate of the proposed scheme is significantly higher than that of previous energy beamforming scheme. Besides, it shows that the proposed beamforming scheme can effectively guarantee the reliable communications and fairness of all tags. Even though the energy harvesting rate of tag 1 is low, it can also reach the same achievable rate as tag 2. The beamforming scheme which maximizes the minimum energy can ensure the energy harvesting rate of the two tags is equivalent. However, due to the double near-far effect <cit.>, the rate of tag 2 is 80% lower than that of tag 1, when M=R=5. Thus, it is wasteful for the portion of the energy that exceeds the circuit power consumption of the tag.






















§ RELATED WORK

Significant progress has recently been made on the energy beamforming in wireless powered communication <cit.>. To harness the benefits of energy beamforming, there have been many efforts made to enable energy beamforming in WPBC. Long et al. <cit.> focus on optimizing the transmit beamforming to maximize the sum rate of a cooperative WPBC system. Gong et al. <cit.> investigate energy beamforming in a relay WPBC system. Both these studies assume that F-CSI can be known to achieve energy beamforming. However, the closed-loop propagation and power-limited tags make it hard to obtain F-CSI. Instead, Yang et al. <cit.> performs energy beamforming via the estimated BS-CSI to improve WET efficiency. Departure from these studies, our work considers the beamforming design for both energy supply and data transfer to ensure communication performance. 



§ CONCLUSION


In this paper, we investigate the energy beamforming via estimated BS-CSI to improve the performance of multiuser WPBC system. To ensure uninterrupted communication, achievable rate and user fairness, we propose a beamforming scheme for power and information transfer in WPBC using the estimated BS-CSI. Additionally, we obtain the analytical expressions of the energy harvesting rate and lower bound on the ergodic achievable rate to optimize resource allocation for maximizing the minimum rate for all tags. Our results indicate that the proposed beamforming scheme offers tremendous performance gains, compared to state-of-the-art energy beamforming schemes. We hope that our investigation on the proposed energy beamforming scheme can provide some implications for future designs.   

 
 



§ PROOF OF PROPOSITION 1
 

Based on (<ref>) and h_kr^b∼𝒞𝒩( 0, β_k ), define the random variable f=|h_kr^b|^2, thus, the f follows exponential distribution. According to the <cit.> (Lemma 3), we can obtain the expectation in (<ref>) as follows.


    ϕ(h_kr^b) = Kσ^2/β_k^2 α p_ceδ_kexp(Kσ^2/β_k^2 α p_ceδ_k) Γ(0, Kσ^2/β_k^2 α p_ceδ_k),
 
where Γ(0,t)≜∫_t^∞u^-1exp(-u)du and the bound of the Gamma function is <cit.>


    1/2 e^-xln( 1+ 2/x) < Γ(0,x) = E_1(x) < e^-xln( 1+ 1/x),    (x>0).

Substituting (<ref>) into (<ref>), Proposition 1 is proved. 



§ PROOF OF PROPOSITION 2
 

We use the MRC detection, i.e. q_k = h_mk/h_mk, and h_i^b= h_mi/h_mi^f.
Define ψ̃_i≜h_mk^Hh_mi/h_mk, and ψ̃_i∼𝒞𝒩(0,|h_mi^f|^2( β_i+σ_ϵ,mi^2(h_mi^f)) ). Then the expectation in (<ref>) can be rewritten as 


    𝔼   {∑_i=1,i≠ k^K  p_i1/|h_mi^f|^2 |ψ̃_i|^2  + ∑_i=1^K  p_iσ_ϵ,mi^2(h_mi^f)+ σ̃^2/ p_k1/|h_mi^f|^2h_mk^2 }
        = ( ∑_i=1,i≠ k^K  p_i𝔼_h_mi^f{β_i+σ_ϵ,mi^2(h_mi^f) }
          + ∑_i=1^K  p_i𝔼_h_mi^f{σ_ϵ,mi^2(h_mi^f) } + σ̃^2 )
    𝔼{1/ p_k1/|h_mi^f|^2h_mk^2}
        =  ( ∑_i=1,i≠ k^K  p_i(β_i + K σ^2/α p_ceδ_i𝔼_h_mi^f{1/|h_mi^f|^2}) 
          + ∑_i=1^K  p_iK σ^2/α p_ceδ_i𝔼_h_mi^f{1/|h_mi^f|^2} + σ̃^2 )
    𝔼{1/ p_k1/|h_mi^f|^2h_mk^2}.

calculate 𝔼_h_mi^f{1/|h_mi^f|^2}, let x̃_i = |h_mi^f|^2 ≥τ (τ>0), x̃_i∼exp( 1/β_i), then


    𝔼_h_mi^f{1/|h_mi^f|^2}  = ∫_τ^∞1/β_ix̃_i e^-x̃_i/β_i dx̃_i= Γ(0,τ/β_i)/β_i,


where Γ(0,τ/β_i) ≜∫_τ/β_i^∞ t^-1 e^-t dt.

Since h_mk|h_mk^f∼𝒞𝒩(0_R,|h_mk^f|^2( β_k+σ_ϵ,mk^2(h_mk^f)) I_R), Z_mk≜2/|h_mk^f|^2( β_k+σ_ϵ,mk^2(h_mk^f))h_mk^Hh_mk follows central chi-square distribution with 2R degree of freedom. Thus the 1/Z_mk∼ Inv-χ^2(2R), and we have 𝔼{1/Z_mk} = 1/2(R - 1). Then


    𝔼   {1/ p_k1/|h_mi^f|^2h_mk^2}  = 𝔼{2/ p_k( β_k +σ_ϵ,mk^2(h_mk^f) ) Z_mk}
        = 𝔼_h_mk^f{1/ p_k (R-1) ( β_k +σ_ϵ,mk^2(h_mk^f) ) }
        = 1/p_k (R-1)β_k𝔼_h_mk^f{1/1 + Kσ^2/|h_mk^f|^2 α p_ceδ_kβ_k}
        = 1/p_k (R-1)β_k( 1 - 𝔼_h_mk^f{1/|h_mk^f|^2α p_ceδ_kβ_k/Kσ^2+1}).
	
Then, we can utilize (<ref>), and substitute (<ref>) and (<ref>) into (<ref>). This completes the proof.

 






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