Transmit Beamforming Design for ISAC with Stacked Intelligent Metasurfaces
Shunyu Li Fan Zhang Tianqi Mao Rui Na Zhaocheng Wang George K. Karagiannidis













Transmit Beamforming Design for ISAC with Stacked Intelligent Metasurfaces
     
Shunyu Li, Student Member, IEEE, 
Fan Zhang, Student Member, IEEE,
Tianqi Mao, Member, IEEE, Rui Na, Zhaocheng Wang, Fellow, IEEE, and George K. Karagiannidis, Fellow, IEEE


        
This work was supported by National Natural Science Foundation of China under Grant No. 62088101. (Corresponding authors: Tianqi Mao, Rui Na.)
S. Li, T. Mao and R. Na are with State Key Laboratory of CNS/ATM, Beijing Institute of Technology, Beijing 100081, China. T. Mao is also with Beijing Institute of Technology (Zhuhai), Zhuhai 519088, China. R. Na is also with Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing), Jiaxing 314019, China (e-mails: li.shunyu@bit.edu.cn, maotq@bit.edu.cn, narui@bit.edu.cn).
F. Zhang and Z. Wang are with Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mails: zf22@mails.tsinghua.edu.cn, zcwang@tsinghua.edu.cn).
G. K. Karagiannidis is with Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Greece and also with Artificial Intelligence & Cyber Systems Research Center, Lebanese American University (LAU), Lebanon (geokarag@auth.gr).

    September 9, 2024
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§ ABSTRACT 
This paper proposes a transmit beamforming strategy for the integrated sensing and communication (ISAC) systems enabled by the novel stacked intelligent metasurface (SIM) architecture, where the base station (BS) simultaneously performs downlink communication and radar target detection via different beams. 
To ensure superior dual-function performance simultaneously, we design the multi-layer cascading beamformer by maximizing the sum rate of the users while optimally shaping the normalized beam pattern for detection.
A dual-normalized differential gradient descent (D^3) algorithm is further proposed to solve the resulting non-convex multi-objective problem (MOP), where gradient differences and dual normalization are employed to ensure a fair trade-off between communication and sensing objectives.
Numerical results demonstrate the superiority of the proposed beamforming design in terms of balancing communication and sensing performance.



Stacked intelligent metasurfaces (SIM), reconfigurable intelligent surface (RIS), integrated sensing and communication (ISAC), beamforming.




§ INTRODUCTION

Integrated Sensing and Communication (ISAC) is considered one of the most promising technologies for next-generation wireless networks <cit.>. Such philosophy aims at realizing the convergence of communication and sensing systems by sharing hardware platforms, spectrum resources, and even waveforms, which can alleviate spectrum congestion and hardware costs <cit.>. Therefore, this integrated implementation can support many emerging applications such as augmented reality, autonomous driving, and the Industrial Internet of Everything (IoE), where communication and sensing devices with compact deployment are highly demanded <cit.>. 

To further enhance the communication and sensing capabilities, the ISAC framework tends to incorporate the multi-antenna technology for additional spatial degrees of freedom (DoF) achieved by large-scale antenna array <cit.>.
Despite the fascinating capabilities of digital/hybrid beamforming, classical array-based ISAC systems inevitably suffer from excessive power consumption and hardware cost, resulting from the numerous radio frequency (RF) chains or complex feeding networks with phase shifters and microstrip lines <cit.>.
To solve this problem, the programmable metasurface, also known as reconfigurable intelligent surface (RIS), can be implemented at the transceiver by replacing the classical phased array antenna, where the RF chains and feeding networks are no longer required <cit.>. 

In <cit.>, a reconfigurable distributed antenna and reflecting surface aided ISAC system was proposed, combining the distributed gain of the distributed antenna system with the passive beamforming gain of RIS to significantly improve system performance while reducing hardware cost.
Besides, <cit.> developed a RIS-enabled integrated sensing, communication, and computing system where RF chain-free transmissions were realized by using RIS as passive information carriers and modulators. 

The aforementioned literature mainly focused on the single-layer metasurface structure, whose ability to control electromagnetic waves may be limited. 
To this end, the stacked intelligent metasurface (SIM) has recently been proposed as a promising approach which cascades multiple layers of transmissive metasurfaces. This SIM technology can further enhance the degrees of freedom (DoF) in manipulating the electromagnetic environment, which exhibits desirable communication performance in terms of multi-user beamforming <cit.>.
Despite <cit.> that deployed SIM between the base station and users/targets for ISAC performance enhancement, research on ISAC system design with the transmit SIM architecture is still in its infancy.

In this context, we propose a transmit beamforming design for ISAC systems equipped with SIM at the BS to perform downlink communication and sensing simultaneously.
Specifically, we consider dedicated streams for communication and sensing, respectively. To achieve the desired dual-function performance tradeoff, we optimize the reflection coefficients of SIM elements to form the corresponding sensing beam patterns toward the desired target in the 2D angular domain, while maximizing the total communication data rate.
Furthermore, we propose the Dual-normalized Differential Gradient Descent (D^3) algorithm to tackle this non-convex multi-objective problem (MOP) with coupled variables induced by the cascaded structure of SIM. 
The proposed D^3 algorithm uses gradient differences to balance the communication and sensing objectives, while incorporating two levels of normalization. The first normalization equalizes the magnitudes of the communication and sensing gradients, ensuring effective gradient differences, while the second normalization balances the phase shifts between SIM elements for convergence.  
Simulation results validate the superiority of our proposed transmit beamforming design and elucidate the impact of SIM parameters on dual-function performance, providing useful guidelines for SIM-enabled ISAC implementations.





§ SYSTEM MODEL FOR SIM-ENABLED ISAC

As shown in Fig. <ref>, we consider a SIM-enabled ISAC system serving N_C single-antenna users and N_S sensing targets.
The transmitter (BS) consists of a SIM planar array illuminated by a uniform linear array (ULA) of N_BS antennas. 
For simplicity and to focus on the SIM-enabled beamforming capabilities, we assume uniform power distribution over the ULA at the base station.
Apart from the N_C parallel communication data streams, we assume N_S independent streams for sensing to extend the degrees of freedom (DoF) for the transmit beamforming design, i.e,  N_BS= N_C + N_S, where the number of communication and sensing signal streams is equal to the number of users and targets, respectively <cit.>.

For clarity, the SIM is assumed to consist of L layers of metasurfaces of size M = M_r× M_c, where M_r and M_c denote the number of meta-atoms in the rows and columns of each metasurface layer, respectively.

According to Rayleigh-Sommerfeld diffraction theory, which is widely employed in SIM-related literature <cit.>, the inter-layer channel coefficient w_m, m^'^l from the m^'-th meta-atom on the (l-1)-th metasurface layer, i.e., Layer (l-1), to the m-th meta-atom on Layer l, can be expressed by

    w_m, m^'^l=A_t cosχ_m, m^'^l/r_m, m^'^l(1/2 π r_m, m^l-j 1/λ) e^j 2 πr_m, m^'^l/λ,

where λ is the wavelength, A_t represents the size of each meta-atom, r_m, m^'^l denotes the transmission distance, while χ_m, m^'^l specifies the angle between the propagation direction and the normal direction of the Layer (l-1).
By applying (<ref>) to each meta-atom, we obtain the inter-layer diffraction matrix, denoted as 𝐖^l ∈ℂ^M × M for l =2,3,⋯,L.
Besides, we have 𝐖^1 ∈ℂ^M × N_BS for l=1, which represents the diffraction matrix from the feeding antenna array to the Layer 1 of the SIM.
Additionally, the diagonal phase shift matrix Φ^l of the Layer l of the SIM can be formulated as

    Φ^l=diag(e^j θ_1^l, e^j θ_2^l, ⋯,e^j θ_m^l,⋯, e^j θ_M^l),

where e^j θ_m^l denotes the phase shift applied by the m-th meta-atom on Layer l with θ_m^l ∈[0,2 π), for l =1,2,⋯,L and m =1,2,⋯,M.
Afterwards, the SIM-enabled beamforming matrix 𝐅_SIM∈ℂ^M × N_BS can be expressed as

    𝐅_SIM=Φ^L 𝐖^L Φ^L-1𝐖^L-1⋯Φ^1 𝐖^1 .


By adopting the Saleh-Valenzuela channel model for MIMO systems <cit.>, the channel between the SIM and the n-th communication user, i.e., user n, for n=1,2,…,N_C can be characterized as

    𝐡_n=√(M/Q_n)∑_q=1^Q_n g_q^(n)α^H( θ_q^(n),φ_q^(n)),

where Q_n is the number of resolvable channel paths, while g_q^(n) represents the channel gain for q=1,2,⋯,Q_n. 
Specifically, for the LOS path, the channel gain is distributed as g_q=1^(n)∼𝒞𝒩(0, β), while for NLOS paths, the channel gain is distributed as g_q>1^(n)∼𝒞𝒩(0, 0.01β), for n = 1, …, N_C. Here, β denotes the distance-dependent path loss modeled as β = C_0 d_n^-α, where C_0 is the free space path loss, α is the path loss exponent, and d_n is the distance from the SIM to the user n.
Moreover, θ_q^(n) and φ_q^(n) represent the elevation and azimuth angles of departure (AoD) of the q-th channel path relative to those of the LoS path component. 
α( θ, φ) ∈ℂ^M × 1 denotes the channel steering vector, which is a function of elevation and azimuth angles expressed by

    α(θ, φ)=[1, ⋯, e^-j 2 π(M_r-1) sin (θ) cos (φ) d_x/λ] ⊗[1, ⋯, e^-j 2 π(M_c-1) sin (θ) sin (φ) d_y/λ],

where ⊗ stands for the Kronecker product, and d_x and d_y represent the horizontal and perpendicular spacings between adjacent meta-atoms, respectively <cit.>.

The transmit signal at the BS before passing through the SIM is denoted as x∈ℂ^N_BS, where E{x}=0 and E{xx^H}=I_N_BS. The first N_C elements of x, i.e., x_1, x_2, ⋯, x_N_C, represent the information symbols for the N_C communication users, while the remaining N_S elements, i.e., x_N_C+1, ⋯, x_N_BS, correspond to the sensing waveforms for the N_S sensing targets.
Hence, the received signals by different users, denoted as y∈ℂ^N_C× 1, can be expressed as 

    y=𝐇𝐅_SIMx+n,

where 𝐇=[𝐡_1, 𝐡_2, ⋯, 𝐡_N_C]^T ∈ℂ^N_C× M denotes the total channel matrix, and n∈ℂ^N_C× 1 is the additive white Gaussian noise (AWGN) vector with n∼𝒞𝒩(0, σ^2 𝐈_N_C). Here, σ^2 is the noise power at the receivers, and 𝐈_N_C is the N_C-by-N_C identity matrix.
For user n, both the signals from the other users and sensing targets are regarded as interference. Therefore, the signal-to-interference-plus-noise ratio (SINR) of user n can be expressed as

    γ_n=|[𝐇 𝐅_SIM]_n, n|^2/∑_i=1, i ≠ n^N_BS|[𝐇 𝐅_SIM]_n, i|^2+σ^2,

where [𝐇 𝐅_SIM]_n, i denotes the element in the n-th row and i-th column of the 𝐇 𝐅_SIM. Then, the sum rate of the N_C users, according to the Shannon-Hartly theorem, can be written as

    R_sum = ∑_n=1^N_Clog_2 (1 + γ_n).


In terms of sensing, communication signals are regarded as supplementary to enhance the sensing signals. Let {ψ_1, ψ_2, ⋯, ψ_N_D} and {ϕ_1, ϕ_2, ⋯, ϕ_N_D} denote the sampling points evenly distributed in the elevation and azimuth angle domains, respectively. Then the beam pattern gain 𝐏_S∈ℝ^N_D× N_D of the SIM in the direction  {ψ_j, ϕ_k} for j, k = 1,2, ⋯, N_D can be expressed as

    [𝐏_S]_j, k=α^H(ψ_j, ϕ_k) 𝐅_SIM𝐅_SIM^H α(ψ_j, ϕ_k),

where α(ψ_j, ϕ_k) is the channel steering vector obtained by (<ref>). Then the normalized beam pattern  𝐏_S is calculated as

    𝐏_S = 𝐏_S/𝐏_S_1.

Given the desired beam pattern 𝐏_D∈ℝ^N_D× N_D, we define the mean square error (MSE) between 𝐏_D and 𝐏_S as the beam-matching error <cit.>, calculated as

    J_MSE = 𝐏_S - 𝐏_D_2^2.

Here, ·_1 and ·_2 denote the ℓ_1 and ℓ_2 norms, respectively.


§ SIM-BASED TRANSMIT BEAMFORMING DESIGN



 §.§ Problem Fomulation

In order to facilitate downlink communication and sensing with desirable performance trade-off, an MOP is formulated to maximize the sum rate of users and concurrently minimize the beam-matching error under uniform transmit power allocation. These can be realized by properly configuring the phase shifts imposed by each meta-atom in the SIM.
Denoting ϑ={θ^1, θ^2, ⋯, θ^L} as the set of optimization variables with θ^l =[θ_1^l, θ_2^l, ⋯, θ_M^l]^T, the MOP is shown as



P1: 
    max_ϑ          R_sum  

    min_ϑ          J_MSE  

    s.t.          [t]
     θ_m^l ∈[0, 2π), 

     ∀l = 1,2,⋯,L, 

     ∀m = 1,2,⋯,M.
 


Following <cit.>, we apply the weighted-sum method to problem (P1), transforming the MOP into a single objective problem (SOP) written as



P2: 
    min_ϑ          J_MSE - R_sum  

    s.t.          [t]
     θ_m^l ∈[0, 2π), 

     ∀l = 1,2,⋯,L, 

     ∀m = 1,2,⋯,M.
 


The problem (P2) is inherently non-convex due to the form of the objective function in (<ref>). Furthermore, the strong coupling among the multiple phase shift matrices across the L layers of the SIM makes the problem even more intractable. 


 §.§ D^3 Algorithm


To address the aforementioned problem, we propose an efficient Dual-Normalized Differential Gradient Descent (D^3) algorithm to achieve a quasi-optimal solution. More specifically, our approach addresses a MOP that includes both communication and sensing tasks. It includes gradient differences and additional normalization steps that adjust the gradient components to similar scales across different tasks. This dual-normalization ensures that the optimization process achieves a balanced trade-off between communication and sensing tasks, and avoids over-promoting any single objective.

First, the phase shifts θ_m^l ∈ [0, 2π) for l = 1, 2, ⋯, L and m = 1, 2, ⋯, M are initialized as, e.g., a uniform random distribution. 
Next, the partial derivatives of J_MSE in (<ref>) with respect to θ_m^l are derived as 

    ∂ J_MSE/∂θ_m^l = 1/N_D^2∑_j=1^N_D∑_k=1^N_D 2([𝐏_S]_j, k-[𝐏_D]_j, k) · [𝐄]_j, k,

where 𝐄 represents the partial derivatives of 𝐏_S in terms of phase shifts θ_m^l, expressed as

    [𝐄]_j, k = 𝐏_𝐒_1 [𝐄]_j, k - [𝐏_S]_j, k𝐄_1/𝐏_S_1^2.

Furthermore, 𝐄 denotes the partial derivatives of 𝐏_S with respect to phase shifts θ_m^l, which can be calculated as

    [𝐄]_j, k = 2Im{ e^j θ_m^lα(ψ_j, ϕ_k)^H 𝐕_:, m^l 
    𝐔_m,:^l 𝐅_SIM^H α(ψ_j, ϕ_k) }.

Im{·} denotes the imaginary part, and 𝐕_:, m^l and 𝐔_m,:^l denote the m-th column of 𝐕^l and the m-th row of 𝐔^l, respectively, which are defined by

    𝐔^l    =𝐖^l Φ^l-1𝐖^l-1⋯Φ^2 𝐖^2 Φ^1 𝐖^1,          if  l ≠ 1, 
    𝐖^1,          if  l=1,
    
    𝐕^l    =Φ^L 𝐖^L Φ^L-1𝐖^L-1⋯Φ^l+1𝐖^l+1,     if  l ≠ L, 
    𝐈_M,     if  l=L.


Besides, the partial derivatives of R_sum in (<ref>) with respect to θ_m^l are derived as 

    ∂ R_sum/∂θ_m^l = 2 log_2 e ∑_p=1^N_Cδ_p(η_p, p-γ_p ∑_q=1, q ≠ p^N_Cη_p, q),

where δ_p and η_p, q in (<ref>) are given by <cit.>

    δ_p    = 1/∑_q=1^N_C|[𝐇𝐅_SIM]_p, q|^2 + σ^2,
    η_p, q   = Im{[𝐇 𝐕^l]_p, m[𝐔^l]_m, q[𝐇 𝐅_SIM]_p, q^* e^j θ_m^l}.

























Inspired by the gradient updating strategy in <cit.>, the partial derivatives obtained from (<ref>) and (<ref>) are then normalized in an element-wise manner. Afterwards, the corresponding differential gradient 𝐆∈ℂ^M × L, can be expressed as

    [𝐆]_m, l=w_1∂ J_MSE/∂θ_m^l/√((∂ J_MSE/∂θ_m^l)^2+ϵ)-w_2∂ R_sum/∂θ_m^l/√((∂ R_sum/∂θ_m^l)^2+ϵ),

where w_1 and w_2 denote the weights of the communication and sensing metrics, respectively. ϵ is the smoothing term as a small constant parameter, typically set to 10^-8 <cit.>.

Additionally, in order to mitigate gradient explosion and vanishing issues during optimization <cit.>, global normalization is applied to the differential gradient. Thus, the dual-normalized differential gradient 𝐆 can be expressed as

    [𝐆]_m, l = π/max (𝐆)· [𝐆]_m, l,

where m = 1, 2, ⋯, M, l = 1, 2, ⋯, L, and max (·) denotes the operation of extracting the largest element. 

Ultimately, the phase shifts θ_m^l ∈ϑ across each meta-atom in SIM can be updated via 𝐆 at each iteration, expressed as

    θ_m^l ←θ_m^l-μ·[𝐆]_m, l,

where μ is the step size. Specifically, we adopt an exponentially decreasing learning rate schedule as the iteration proceeds, which is updated by <cit.>

    μ←μβ,

where β is a hyperparameter determining the decay rate, satisfying 0 < β < 1.


By iteratively applying (<ref>)-(<ref>), the objective function (<ref>) will reach convergence when its decrease is smaller than a preset threshold or reaching the maximum iteration number.




§ NUMERICAL RESULTS

This section validates the effectiveness of the transmit beamforming design with the proposed D^3 algorithm. The simulation parameters are summarized in Table <ref>.
The desired beam pattern for sensing is assumed as

    [𝐏_D]_j,k =
    
    1    if  (j,k) ∈{(9, 27), (27, 9)}, 
    
    0    otherwise,

where j, k = 1,2, ⋯, N_D correspond to specific elevation and azimuth angles. Specifically, indices 9 and 27 correspond to the angle ranges [-45^∘, -40^∘] and [45^∘, 50^∘], respectively.


 

To prevent premature convergence to a local optimum due to inappropriate initialization, we generate five sets of initial phase shifts and run the D^3 algorithm in parallel for each set. The algorithm iterates until either the relative change in the objective function falls below a threshold of 10^-6 or the maximum number of 60 iterations is reached. Unless otherwise specified, we set the initial learning rate to η = 1 and the decay parameter to β = 0.5. After parallel execution, we choose the solution that minimizes J_MSE - R_sum as the quasi-optimal solution for the problems (P1) and (P2) <cit.>.

Figure <ref> shows the beam pattern obtained by the proposed beamforming design using the D^3 algorithm, where M = 100, L = 7, and w_1 = w_2 = 1.
It can be observed that the beam pattern gain reaches its maximum (about 2.5 × 10^-6) in the directions towards the sensing targets within the regions of [-45^∘, -40^∘] in elevation angle space and [45^∘, 50^∘] in azimuth angle space, as well as [45^∘, 50^∘] in elevation angle space and [-45^∘, -40^∘] in azimuth angle space, where the beam-matching error is calculated as J_MSE = 0.0526. Meanwhile, the beam peaks for the four communication users are located at  (-60^∘, -45^∘), and (-60^∘, -35^∘). Although the average strength of the communication beams is not comparable to the sensing beams, i.e., below the gain of 0.75 × 10^-6, superior data throughput with R_sum = 15 bit/s/Hz can be achieved with the proposed D^3 algorithm. Therefore, the proposed beamforming design successfully realizes a desirable balance between sensing and communication functions.


Figure <ref> illustrates the performance comparison of different weighting coefficients w_1 and w_2 with varying numbers of meta-atoms M. The weighting coefficients w_1 and w_2 dictate the trade-off between sensing and communication performance. When w_1=1 and w_2=0, the D^3 algorithm functions as a sensing-only scheme, while w_1=0 and w_2=1 conversely represents a communication-only scheme. For cases where both w_1 and w_2 are non-zero, the system functions as an ISAC scheme, with their ratio determining the priority of sensing or communication in the system. It can be observed that the sensing-only scheme (w_1=1, w_2=0) exhibits poor communication performance, with sum rates that are almost nil. In contrast, the ISAC scheme with w_1=w_2=1 achieves a sum rate approaching that of the communication-only scheme (w_1=0, w_2=1), while still maintaining satisfactory sensing performance. This illustrates that our suggested D^3 method grants the SIM-enabled ISAC system to achieve both high-performance communication and sensing capabilities concurrently.
Furthermore, Fig. <ref> shows that the sum rate of users increases with the number of meta-atoms M and the number of layers L. This improvement is due to the additional flexibility in shaping the electromagnetic wavefront provided by the multi-layer massive metasurface array, which can enable more sophisticated beamforming patterns to better accommodate the dual-functional requirements.





Figure <ref> depicts the convergence of the proposed D^3 algorithm. The solid lines represent beam-matching error (J_MSE, left y-axis), while the dashed lines show the sum rate of users (R_sum, right y-axis) over 60 iterations. Each color corresponds to a different channel realization. Notably, the D^3 algorithm converges rapidly in all cases, achieving its optimal value after approximately 15 iterations. This demonstrates that the proposed algorithm possesses the capacity to effectively adapt to different channel conditions, hence confirming its efficacy and reliability in practical ISAC systems.





§ CONCLUSION

This paper proposed a novel beamforming design for ISAC applications under the framework of transmit SIM-based antenna array. To realize a desirable dual-function performance trade-off, a non-convex MOP was established by simultaneously maximizing the communication sum rate and optimally shaping the sensing beam pattern by adjusting the reflection coefficients of the meta-atoms. To solve this problem, the D^3 algorithm was proposed, which effectively balances the trade-off between communication and sensing objectives through gradient differences and dual normalization. Numerical results confirmed the superiority of our proposed ISAC beamforming design under various channel conditions.




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