A Reconfigurable Subarray Architecture and Hybrid Beamforming for Millimeter-Wave Dual-Function-Radar-Communication Systems
Xin Jin Tiejun Lv Wei Ni Zhipeng Lin Qiuming Zhu Ekram Hossain H. Vincent Poor













A Reconfigurable Subarray Architecture and Hybrid Beamforming for 
Millimeter-Wave Dual-Function-Radar-Communication Systems

    
 Xin Jin, Tiejun Lv, Senior Member, IEEE, Wei Ni, Fellow, IEEE, Zhipeng Lin, Member, IEEE, Qiuming Zhu, Senior Member, IEEE, Ekram Hossain, Fellow, IEEE, and H. Vincent Poor, Life Fellow, IEEE

Manuscript received March 6, 2024; accepted April 22, 2024. This paper was supported in part by the National Natural Science Foundation of China under No. 62271068, the Beijing Natural Science Foundation under Grant No. L222046, the U.S National Science Foundation under Grants CNS-2128448 and ECCS-2335876. (corresponding author: Tiejun Lv.)

X. Jin and T. Lv are with the School of Communication and Information Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: {jxzoe, lvtiejun}@bupt.edu.cn).

W. Ni is with the Data61, Commonwealth Scientific and Industrial Research Organization, Sydney, Marsfield, NSW 2122, Australia.
(e-mail: wei.ni@ieee.org).

Z. Lin and Q. Zhu are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: {linlzp, zhuqiuming}@nuaa.edu.cn).

E. Hossain is with the Department of Electrical and Computer Engineering, University of Manitoba, Canada (e-mail: ekram.hossain@umanitoba.ca).

H. V. Poor is with the Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).   


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Dual-function-radar-communication (DFRC) is a promising candidate technology for next-generation networks. By integrating hybrid analog-digital (HAD) beamforming into a multi-user millimeter-wave (mmWave) DFRC system, we design a new reconfigurable subarray (RS) architecture and jointly optimize the HAD beamforming to maximize the communication sum-rate and ensure a prescribed signal-to-clutter-plus-noise ratio for radar sensing. Considering the non-convexity of this problem arising from multiplicative coupling of the analog and digital beamforming, we convert the sum-rate maximization into an equivalent weighted mean-square error minimization and apply penalty dual decomposition to decouple the analog and digital beamforming. Specifically, a second-order cone program is first constructed to optimize the fully digital counterpart of the HAD beamforming. Then, the sparsity of the RS architecture is exploited to obtain a low-complexity solution for the HAD beamforming. The convergence and complexity analyses of our algorithm are carried out under the RS architecture.  Simulations corroborate that, with the RS architecture, DFRC offers effective communication and sensing and improves energy efficiency by 83.4% and 114.2% with a moderate number of radio frequency chains and phase shifters, compared to the persistently- and fully-connected architectures, respectively.





	Reconfigurable subarray (RS) architecture, hybrid beamforming (HBF), dual-function-radar-communication (DFRC), penalty dual decomposition (PDD).  









§ INTRODUCTION


The sixth generation (6G) of wireless systems is anticipated to go beyond traditional mobile networks in continuous and ubiquitous sensing capabilities <cit.>. A promising concept for realizing simultaneous sensing and communication capabilities is dual-functional-radar-communication (DFRC). By sharing hardware, software, and information, DFRC enables these capabilities while coping with resource constraints <cit.>. Benefiting from advances in millimeter-wave (mmWave) technologies, mmWave DFRC can support simultaneous high-speed communications and high-resolution sensing in environment-aware scenarios <cit.>, e.g., smart homes and Internet of Things (IoT). For instance, the authors of <cit.> proposed a novel dual-function waveform design scheme that can estimate velocity and range at ultra-high resolutions in noisy environments.

To address severe path-loss of mmWave signals, massive multiple-input multiple-output (mMIMO) antenna arrays with significant beamforming gains have been studied for both communication and radar systems <cit.>. 
The authors of <cit.> provided a parameter design approach for high mobility scenarios to enhance the resilience to Doppler shift.
Unfortunately, a fully digital (FD) mMIMO architecture is of limited practical value due to its prohibitive hardware and energy costs. To this end, 
hybrid analog-digital (HAD) architectures, offering flexible connectivity between radio frequency (RF) chains and antennas, become promising to balance hardware complexity and system performance <cit.>. 

A similar concept in radar systems, i.e., a phased-MIMO radar, is used to join the high resolution of a MIMO radar and the coherent processing gain of a phased-array radar <cit.>. For instance, the authors of <cit.> developed a hybrid beamforming (HBF) phased-array architecture for active sensing applications, using a hybrid beamformer system for transmitting and receiving images. In <cit.>, an approach to synthesize probing beampatterns for mMIMO radars with fully-connected (FC) structures was presented based on learning-based RF chains and antenna selection.

As communication and sensing converge in mmWave mMIMO systems, it is imperative to design an effective transceiver architecture to avoid high hardware costs and enhance the integration gain of DFRC systems. To this end, we present a new DFRC system with a reconfigurable subarray (RS) architecture for improved spectral efficiency (SE), energy efficiency (EE), and sensing performance.






 §.§ Related Works


Many works have considered HAD DFRC systems, including FC  <cit.> and persistently-connected (PC) architectures <cit.>. In <cit.>, two-stage alternating minimization was proposed to optimize hybrid transmit beams and phase vectors for detecting several targets and communicating with several users, subject to the quality-of-service (QoS) requirements, e.g.,  signal-to-interference-plus-noise ratio (SINR), for communication. The authors of <cit.> conducted weighted optimization of optimal FD transmission and radar beampattern for an Internet of Vehicles (IoV) scenario to improve SE of vehicular communication as well as sensing accuracy. However, these studies focused on radar similarity constraints or beampattern approximation, and overlooked the sensing performance of the radar receiver. Considering that the radar detection probability directly depends on the signal-to-clutter-plus-noise ratio (SCNR) of the radar receiver, the authors of <cit.> optimized hybrid transmit beamforming to maximize the receive SCNR. Additionally, the authors of <cit.> opted for a communication-centric transmitter design under the Cramer-Rao bound (CRB). These studies were generally based on an FC architecture, where every RF chain is hardwired to all antennas via a phase shift (PS) network, which could be prohibitive in mMIMO. 

Another hybrid architecture is the PC architecture, where RF chains connect individual antenna subarrays. In <cit.>, an effective trade-off between sub-array MIMO radar and communication was achieved by minimizing a weighted sum of beamforming errors for both. The EE of the integrated system was maximized by activating a small number of RF chains <cit.>. In <cit.>, the communication rate was maximized under a sensing beampattern mismatch requirement in both single- and multi-user multi-carrier systems. More recently, the authors of <cit.> considered a more realistic clutter interference environment and used a double-phase-shifter (DPS) structure, providing a well-tolerated compromise between the number of PSs and the performance gain of the DPS HBF. Although the PC architecture significantly reduces hardware costs compared to the FC architecture, these HBF designs offer no flexibility. 


RS architecture-based HBF offers a promising solution to balance system performance and hardware costs in communication systems <cit.>. This is achieved by flexibly connecting an RF chain to a non-overlapping subarray through a switch network and PSs <cit.>. Hybrid subarrays were reconfigurable based on channel statistics in <cit.>, where a greedy algorithm was implemented to approach the SE achieved by exhaustive search. In multi-user scenarios, the authors of <cit.> designed HBF based on sub-channel differences. However, these discussions were limited to communication scenarios. A summary of these existing studies is provided in Table I.






 §.§ Challenges and Contributions

  
The incorporation of an RS architecture and HBF in a mmWave DFRC system is not straightforward. On the one hand, the tight multiplicative coupling of the analog and digital beamforming makes their joint optimization challenging. 
The need to selectively connect the RF chains and antennas in the RS architecture could result in an NP-hard mixed integer programming problem <cit.>. Also, a direct use of techniques, such as alternating optimization and block decomposition, is likely to get trapped in inferior solutions or even fail to converge. On the other hand, many existing DFRC designs focus on the transmit HBF, which can penalize the sensing performance in cluttered environments. To cope with clutter interference, the authors of <cit.> considered the processing of radar receive filters and their optimization on a DFRC system, which effectively utilizes the available degrees of freedom (DoFs) in radar receive arrays. But only small-scale FD beamforming was considered in <cit.>.  


This paper presents a new framework for HAD DFRC systems with hybrid transmit beamforming and radar receive beamforming, where the PSs and antennas are reconfigurably connected through adaptive switching networks. A dual-function base station (BS) delivers communication services to multiple users in the downlink, meanwhile detecting a potential target amidst signal-dependent interference sources, i.e., clutter. Unlike conventional approaches focusing on radar beampattern matching errors, we adopt the radar receive SCNR to quantify and constrain the sensing performance. We propose an effective algorithm that optimizes the digital and analog beamformers to maximize the achievable sum-rate of the HAD DFRC system under a prescribed SCNR requirement of its radar sensing performance. 




Following is the list of contributions of this paper:

	
  * We propose a new RS architecture for mmWave HAD DFRC systems in which every RF chain can be flexibly linked to a non-overlapping subset of antennas. This architecture can significantly reduce the hardware costs while maintaining the DFRC performance. 

    
  * We present a new problem formulation to maximize the non-convex sum-rate under the SCNR constraints of radar sensing in an HAD DFRC system with an RS architecture. 
 
	
  * Using penalty dual decomposition (PDD) and weighted minimum mean square error (WMMSE) as an interpretation of the non-convex sum-rate, we put forth a new WMMSE-PDD (WPDD) algorithm to solve the new problem through block coordination descent (BCD). As per iteration, a second-order cone program (SOCP) optimizes the FD transmit counterpart of the HBF. The sparsity of the RS architecture is exploited to obtain a low-complexity solution for the HBF.

    
  * The proposed WPDD algorithm is extended to DFRC systems with PC architectures. Its convergence and complexity are analyzed, indicating the benefit of HBF in DFRC systems in terms of EE. 


Extensive simulations are carried out to verify our analysis and assess the effectiveness of a DFRC system with an RS architecture between performance and cost by comparing it with the FC and PC architectures. RS-based HBF leads to a 83.4% improvement in EE compared to the PC-based HBF, and surpasses the EE achieved under the FC architectures by more than twofold.



The subsequent sections of this paper are arranged as follows. 
The studied problem is cast in Section II. The new WPDD algorithm is presented in Section III, followed by its convergence, complexity, and EE analyses in Section IV. A summary of simulation results is given in Section V. Lastly, the conclusions are presented in Section VI. 


Notation: 

Bold-face upper- and lower-cases indicate matrices and vectors, respectively. (·)^T, 
(·)^H, (·)^†, and tr( · ) denote transpose, 
conjugate transpose, pseudo-inverse, and trace, respectively, ℂ^M × N denotes all M × N complex matrices,

R{·} takes the real part of a complex value, · provides Euclidean or Frobenius norm for a vector or matrix, ·_0 is the 0-norm of a vector, 𝔼(·) takes the statistical expectation, CN (·,· ) represents the complex normal distribution, blkdiag(·) denotes block diagonal matrix, ⌈·⌉ stands for ceiling. The notation used is collated in Table <ref>.  










§ SYSTEM ARCHITECTURE AND PROBLEM STATEMENT





 
As depicted in Fig. <ref>, an mmWave DFRC BS and users both have uniform linear arrays (ULAs) installed.
The DFRC BS has M_T transmit antennas and N_RF^t RF chains to serve K users in the downlink. It also has M_R receive antennas and N_RF^r RF chains to detect a target of interest in cluttered environments. Each user has M_U receive antennas. 
The angle-of-departure (AoD) of the radar target is the same as the angle-of-arrival (AoA), as considered in the literature <cit.>. An RS-based HAD architecture is considered at both the receive and transmit sides of the DFRC BS, where every RF chain can be linked to a non-overlapping subset of transmit antennas via an adaptive switching network. 

Let s( t ) = [ s_1^T( t ),⋯,s_K^T( t )]^T∈ℂ^N_s (N_s ≤ N_RF^t≪M_T) denote the symbol sequence sent in the t-th slot for K users and the radar target. d_s is the number of data streams for user k (𝔼{s_k( t )s_k( t )^H} = I_d_s). The symbols are independent among the users. The transmit signal is obtained as

    x (t)  = T_RFT_Ds ( t ) = ∑_k = 1^K T_RFT_D,ks_k( t ),


where T_D,k∈ℂ^N_RF^t× d_s stands for the digital beamfomer for the k-th user, T_D = [ T_D,1,⋯,T_D,K], and 𝐓_RF∈ℂ^ M_T× N_RF^t stands for the analog beamformer. 






 §.§ Communication Model

Following the extended Saleh-Valenzuela model <cit.>, mmWave channel between the DFRC BS and user k is modeled as  

    H_k = √(M_TM_U/L_k)∑_l = 1^L_kα _k,la_r (ψ _k,l ) a_t^H ( ϕ _k,l),∀ k,

where L_k specifies the number of propagation paths to the user k, α_k,l∼CN ( 0,10^-0.1PL(d_k) ) is the complex gain of the l-th path, l= 1,⋯,L_k. PL(d_k) is the path loss over the distance, d_k, between the BS and user k. a_t ( ϕ _k,l ) and a_r ( ψ _k,l ) give the transmit and receive array response vectors, respectively:

    a_t (ϕ _k,l  )  =  1/√( M_T)[ 1, e^ - j2π/λ dsinϕ _k,l,⋯,e^ - j2π/λ d( M_T -  1) sinϕ _k,l]  ^T,


    a_r (ψ _k,l  )  =  1/√( M_U)[ 1, e^ - j2π/λ dsinψ _k,l,⋯,e^ - j2π/λ
        d( M_U -  1) sinψ _k,l]  ^T.

Here, ϕ_k,l and ψ_k,l are the AoD/AoA of the l-th path to the k-th user, respectively; λ indicates the signal wavelength; d = λ/2 specifies antenna spacing. Suppose H_k, ∀ k, is known at the DFRC BS. In practice, the channels can be estimated, e.g., using the algorithms developed in <cit.>.


The received signal of user k is

    y_k ( t ) = H_k ∑_k = 1^K T_RFT_D,ks_k( t )  + n_k( t ),

where n_k( t ) ∼ CN( 0,σ ^2I_M_U ) stands for the additive white Gaussian noise (AWGN) of the k-th user. n_k( t ) and s_k( t ) are independent of each other. For the brevity of notation, we suppress the time index t in the remainder of this paper. 

After FD receive beamforming, U_k ∈ℂ^ M_U× d_s, the detected signal of user k is 

    ŝ_k   =U_k^H y_k
       =U_k^H H_kT_RFT_D,ks_k_desired signal+ ∑_i  k^K U_k^H H_kT_RFT_D,is_i_multi-user interference+U_k^H n_k_noise.


Consequently, the overall achievable rate of the K users is

    R (  T_RF , T_D,k, U_k )   =    ∑_k = 1^K log ( I_d_s + U_k^H H_k T_RFT_D,k
       ×T_D,k^HT_RF^HH_k^H U_kR_k^-1),

where R_k =U_k^H ( σ ^2I_M_U+∑_i  k^K H_kT_RFT_D,iT_D,i^HT_RF^HH_k^H   ) U_k.



 §.§ Radar Model

Apart from sending communication symbols to the K users, the transmitted waveform is also exploited to accomplish a radar detection task. Considering an obstacle detection scenario, a target (e.g., a slowly moving pedestrian) is located at angle θ _0. There are also J stationary clutterers (e.g., trees and buildings) at angles θ _j, j = 1,⋯,J. In this case, the time delay and Doppler shift are relatively negligible, as discussed in <cit.>. The prior knowledge of θ _0 and θ _j is obtainable from an environmental dynamic database using a cognitive paradigm <cit.>. 

Suppose that the potential self-interference from the transmit array to the receive array of the MIMO radar is adequately addressed by implementing appropriate techniques, for instance, the one developed in <cit.>. According to (<ref>), the echo signal received by the BS can be written as


    y_R = β _0a_r( θ _0 )a_t^T( θ _0 )x+ ∑_j = 1^J β _ja_r( θ _j )a_t^T( θ _j )x + n_R,


where β _0 and β _j denote the round-trip attenuation and complex reflection coefficients concerning the target and the j-th clutter, respectively; 
𝔼( | β _0|^2 ) = σ _0^2; 𝔼( | β _j|^2 ) = σ _C^2; a_r ( θ ) gives the M_R× 1 receive steering vector; a_t ( θ ) specifies the M_T× 1 transmit steering vector; a_r ( θ ) and a_t(θ ) are obtained in the same way as (3); n_R∼ CN( 0,σ _R^2I_M_R ) is the AWGN. 

An RS-based HAD architecture is also considered at the DFRC BS receiver. The received echo signal after analog receive beamforming 𝐖_RF∈ℂ^ M_R× N_RF^r and digital receive beamforming W_D∈ℂ^N_RF^r× N_s, is given by

    y'_R   = W_D^H𝐖_RF^Hy_R
       = β _0W_D^H𝐖_RF^HA( θ _0 )x_target + W_D^H𝐖_RF^H∑_j = 1^J β _jA( θ _j )x_clutters
       + W_D^H𝐖_RF^Hn_R_noise,

where A(  θ )  = a_r( θ  )a_t^T( θ ). 



We adopt the SCNR as the metric to measure the radar's target detection and localization capabilities, as given by






    ( T_RF,T_D,W_RF,W_D ) 
       =𝔼( β _0W_D^H𝐖_RF^HA( θ _0 )x^2 ) /𝔼( W_D^H𝐖_RF^H∑_j = 1^J β _jA( θ _j)x^2)+ 𝔼( W_D^H𝐖_RF^Hn_R^2 )
    
          =tr (W_D^H𝐖_RF^HΣ_tW_RF𝐖_D) /tr(W_D^H𝐖_RF^HΣ_cnW_RF𝐖_D ) ,

where Σ_t =  σ _0^2A( θ _0 ) T_RF∑_k = 1^K (T_D,kT_D,k^H) T_RF^HA^H( θ _0), and Σ_cn =  ∑_j = 1^J σ _C^2A( θ _j ) T_RF∑_k = 1^K (T_D,kT_D,k^H) T_RF^HA^H( θ _j)  +  σ _R^2I. 


We note that SCNR is a measurable sensing performance indicator at the receiver, and can be reasonably adapted to different signal characteristics and application scenarios. Moreover, the radar detection probability is known to be a non-decreasing function of the SCNR[ The target detection probability (PD) increases monotonically with the received SCNR, i.e., P_D(γ) = Q (√(2γ),√(-2 ln P_FA)) <cit.>, where Q (·,·) is the Marcum Q function and P_FA gives the false-alarm probability.], indicating the relevance of the SCNR in DFRC systems. The measurability and relevance of SCNR offer excellent practicality, leading it to be broadly considered in radar and DFRC systems <cit.>.

 



 §.§ Problem Statement


We holistically design the hybrid transmit beamformers T_D,k and T_RF, and the hybrid receive beamformers W_D and W_RF under the RS-based HAD architecture, to elevate the downlink sum-rate while ensuring the radar detection performance. This problem is cast as follows:


    P_1: max_𝒳
            R (  T_RF , T_D,k, U_k )  
        
    s.t.     ( T_RF,T_D, W_RF,W_D ) ≥γ, 
       T_RFT_D_F^2 ≤ P_T, 
       T_RF( m,n )∈{0, F}, ∀ m,n, 
       T_RF( m,: )_0 = 1, ∀ m, 
       W_RF( m̃,ñ )∈{0,  F}, ∀m̃,ñ, 
       W_RF( m̃,: )_0 = 1, ∀m̃,


where 𝒳={T_RF, 
T_D,k,  U_k ,W_RF,W_D} collects all of the optimization variables. Constraint (<ref>) ensures the minimum SCNR γ. Constraint (<ref>) specifies the total transmit power budget. 
Constraints (<ref>)-(<ref>) specify the HBF mapping criteria. More specifically, if the n-th RF chain is associated with the m-th antenna via a PS, T_RF( m,n) ∈ F, m = 1,⋯,M_T,n = 1,⋯,N_RF^t, contains a nonzero phase, where F collects all possible values of a PS; otherwise, T_RF( m,n) = 0. Similarly, if the ñ-th RF chain is connected to the m̃-th antenna via a PS, W_RF( m̃,ñ) ∈ F, m̃ = 1,⋯,M_R,ñ = 1,⋯,N_RF^r; otherwise, W_RF( m̃,ñ) = 0. Since no subarrays overlap, the analog beamformer has only one nonzero element per row, i.e., T_RF( m,:)_0 = 1 for m = 1,⋯,M_T; W_RF( m̃,:)_0 = 1 for m̃ = 1,⋯,M_R. 


Problem P_1 poses significant challenges ascribed to non-convex objectives and constraints, and the strong coupling of digital and analog beamformers. The non-convexity makes it difficult to attain a globally optimal solution.
A direct attempt to tackle the coupling effect with block decomposition methods, e.g., block successive upper-bound minimization and inexact flexible parallel algorithm, can also lead to difficulty in ensuring convergence <cit.>. In what follows, we develop a tractable solution to this problem.



§ WPDD ALGORITHM FOR DFRC WITH RS ARCHITECTURE



 §.§ Reformulation of Problem P_1


To overcome the difficulty in solving the nonconvex sum-rate problem P_1 with coupled variables, we develop a tractable objective according to the WMMSE method to decrease the complexity of solving problem P_1. As revealed in Proposition <ref>, problem P_1 can be rewritten equivalently as a WMMSE minimization problem <cit.>.


   
Problem P_1 and the following problem of minimizing weighted sum mean square error (MSE) have the same global optimum:

 

    P_2: max_𝒳, {G_k },{E_k }   ∑_k = 1^K ( log (G_k) - tr (G_k E_k )  ) 
    s.t. 
           (<ref>)-(<ref>),


where G_k denotes the weighting factor for user k, U_k is the receive beamforming for user k, and E_k is the MSE of user k, as given by

    E_k   = 𝔼{(ŝ_k - s_k ) (ŝ_k - s_k )^H }
       = I_d_s -2 R(U_k^H H_k T_RFT_D,k) + σ ^2U_k^HU_k
         + ∑_i =1 ^K U_k^H H_kT_RFT_D,iT_D,i^HT_RF^HH_k^H U_k.
 




 Since G_k and U_k appear only in (<ref>), the equivalence between problems P_2 and P_1 can be established by substituting their optimal solutions (by first-order optimality condition) into (<ref>). For more details, please refer to <cit.>.




Although the WMMSE-based transformation makes the objective function easy to handle, the analog and digital beamforming are coupled in constraints (<ref>) and (<ref>).
Moreover, the non-convex SCNR constraint in (<ref>), and the L_0 constraints in (<ref>) and (<ref>) make the problem challenging. Jointly optimizing 
analog and digital beamforming is difficult. We introduce auxiliary variables, T_k = T_RFT_D,k, ∀ k, and W = W_RFW_D.
Problem P_2 is recast as


    P_3 : max_𝒳̃   ∑_k = 1^K ( log (G_k) - tr (G_k E_k )  ) 
    s.t.    tr (W^HΣ_tW )/tr (W^HΣ_cnW )≥γ, 
       ∑_k = 1^K T_k^2 ≤ P_T, 
       T_k = T_RFT_D,k,∀ k,   
       W = W_RFW_D,   
       (<ref>)-(<ref>),


where 𝒳̃={T_RF,T_D,k ,  U_k , W_RF,W_D,  G_k , T_k , W}. 




 §.§ PDD-Based Framework for Problem P_3


We resort to the PDD to solve P_3, as problem P_3 has the form of the problem (𝒫) in <cit.>. 

The PDD method integrates the penalty method, the augmented Lagrangian method, and the alternating direction method of multipliers (ADMM) and delivers a general algorithmic framework for minimizing a non-convex and non-smooth function while dealing with non-convex coupling constraints. The PDD comprises double-loop iterations, with the outer loops updating the dual variables and penalty parameters, and the inner loops executing the augmented Lagrangian method. The PDD suits problem P_3 with a differentiable objective function.  


Through the incorporation of two penalty terms for constraint (<ref>) and (<ref>) into (<ref>), the inner loop can be formulated as 


    P_4:min_𝒳̃   ∑_k = 1^K {tr ( G_k E_k ) -log ( G_k ) 
         + 1/2ρT_k - T_RFT_D,k + ρD_k^2}
         +  1/2ρW - W_RFW_D + ρD̃^2
    s.t.    (<ref>)- (<ref>), (<ref>)-(<ref>),


where ρ denotes the penalty parameter; D_k and D̃ are the dual variables of constraints (<ref>) and (<ref>), respectively. ρ, D_k, ∀ k, and D̃ are fixed throughout the inner-loop iterations.  


Next, we solve problem P_4 utilizing the PDD framework, where BCD is used to solve problem P_4.







1) Equivalent FD Sensing Receive Beamformer W: 

By letting t_k= vec (T_k), t = [t_1^T,⋯,t_K^T]^T, and w= vec (W), the SCNR in (<ref>) can be rewritten as

    ( T_RF,T_D,W_RF,W_D )  =σ _0^2| w^H Ã( θ _0) t|^2/w^HΣ̃_cnw,

where Ã(  θ )  = I⊗A(  θ ), Σ̃_cn =  ∑_j = 1^J σ _C^2Ã( θ _j ) tt^H  Ã^H( θ _j)+σ _R^2I.  

When the transmit beamforming is given and fixed, we employ the well-known minimum variance distortionless response (MVDR) beamformer <cit.> to obtain the maximum SCNR at the MIMO radar, as given by

    w = Σ̃_cn^-1Ã( θ _0 )t/t^HÃ^H( θ _0 )Σ̃_cn^-1Ã( θ _0 )t,

then W can be obtained by reshaping w. According to this optimal radar receive beamformer W, the SCNR in (<ref>) is further expressed as

    ( T_RF,T_D,k ) = ∑_k = 1^K tr(T_D,k^H T_RF^H ΦT_RFT_D,k),

where Φ = σ _0^2A^H( θ _0 )Σ_cn^-1A( θ _0 ).





2) Equivalent FD Transmit Beamformer
T_k: 

With the rest of variables fixed, the subproblem of problem P_4 for solving the auxiliary variable T_k is written as
  

    min_{T_k}   ∑_k = 1^K ( tr ( G_k E_k )  +1/2ρT_k - T_RFT_D,k + ρD_k^2 ) 
    s.t.    ∑_k = 1^K tr(T_k^HΦT_k) ≥γ,  
       ∑_k = 1^K T_k^2≤P_T,


where E_k is obtained by plugging (<ref>) into (<ref>), as given by

    E_k    =  I_d_s -2 R(U_k^H H_k T_k) + σ ^2U_k^HU_k
         + ∑_i =1 ^K U_k^H H_k T_iT_i^HH_k^H U_k.



Notice that (<ref>) provides a non-convex quadratic constraint quadratic programming (QCQP). Semi-definite relaxation (SDR) is often applied to resolve such a problem. 
However, SDR conducts a relaxation of rank-one constraints, resulting in an additional process like Gaussian randomization to restore the rank-one constraints on obtained solutions <cit.>. Extra computational complexity occurs.


We propose to transform  (<ref>) into a series of SOCPs <cit.>, using viable substitution and successive convex approximation (SCA) techniques and solve the SOCPs recursively to obtain a sub-optimal solution to (<ref>).  Let Z_k = T_k + ρD_k and  P_k = H_k^H U_k G_kU_k^HH_k. We introduce slack variables a_k,i > 0 and b_k > 0, i,k∈{1,⋯,K } such that t_i^H (I⊗P_k^H) t_i≤a_k,i and Z_k - T_RFT_D,k^2 ≤b_k, which can be further rewritten in the following second-order cone (SOC) constraints: 

    [ [ t_i^H (I⊗P_k^H)^1/2;         a_k,i - 1/2 ]]≤a_k,i + 1/2,  ∀ i,k;


    [ [ Z_k - T_RFT_D,k;       b_k - 1/2 ]]≤b_k + 1/2, ∀ k.

In the non-convex SCNR constraint (<ref>), we can rewrite tr(T_k^HΦT_k)=  t_k^H (I⊗Φ^H) t_k. Then, we resort to the SCA technique to replace this non-convex term with its linear lower bound (e.g., the first-order expansion), i.e., 

    _k =  2 R ( t̅_k^H Φ̅t_k ) - t̅_k^H Φ̅t̅_k,

where Φ̅= I⊗Φ^H, and t̅_k is the value of t_k in the previous SCA iteration.

Thus, the approximate convex problem of (<ref>) is 
 

    min_{t_k, a_k,i, b_k}   ∑_k = 1^K ( ∑_i = 1^K a_k,i-2 R( vec^H (G_k U_k^H H_k )^H t_k ) + 1/2ρb_k)
    s.t.    ∑_k = 1^K 2 R ( t̅_k^H Φ̅t_k ) - t̅_k^H Φ̅t̅_k≥γ ,   
    
           t≤√(P_T),
       (<ref>)-(<ref>).


We can use off-the-shelf toolboxes, e.g., CVX <cit.>, to efficiently solve (<ref>). By recursively solving (<ref>), we can obtain at least a local optimum of (<ref>).  


Through the use of the PDD framework, we can decouple hybrid transmit and receive beamforming for DFRC into the subproblems of optimizing the equivalent sensing receive beamformer W and the equivalent DFRC transmit beamformers T_k, ∀ k, which can be solved sequentially. The inter-user interference in the considered system leads to non-convexity in the objective function in (<ref>). The clutter interference leads to non-convexity in the SCNR constraint in (<ref>). To deal with this, variable substitution and SCA are conducted to convexify both the objective function and the SCNR constrain in the subproblem of T_k, ∀ k; see (<ref>), (<ref>) and (<ref>). As a result, the subproblem can be solved using off-the-peg CVX toolkits. 
 



Remark: 
PDD is a general algorithmic framework for solving problems with multiplicatively coupled variables. It has been applied to optimize hybrid transmit and receive beamforming for a single user in a MIMO channel, e.g., in <cit.>. By contrast, the studied problem of multi-user mMIMO can undergo non-negligible inter-user interference; see (<ref>). Moreover, our studied system suffers from interference from clutters to the radar sensing signals received by the BS; see (<ref>). This differs substantially from the work presented in <cit.>, where no such interference exists. In this sense, our studied problem is distinct from <cit.> in objectives and constraints. 




3) Digital Receive Beamformer U_k for User k: We optimize {U_k } by fixing all other variables, leading to an unconstrained problem:

    min_U_k ∑_k = 1^K tr ( G_k E_k ) .

By substituting (<ref>) into (<ref>) and then assessing the first-order derivative of (<ref>), the optimum receive combiner U_k for user k is obtained in closed form as follows:

    U_k =  (∑_i =1 ^K H_kT_iT_i^HH_k^H + σ^2 I_M_U )^-1H_k T_k, ∀ k.





4) WMMSE Weighting Matrix G_k: By substituting (<ref>) into (<ref>), the mean square error E_k is given by ( I - U_k^H H_k T_k ). Then, assessing the first-order optimality condition of problem P_4 concerning G_k, the closed-form solution of G_k is

    G_k = ( E_k )^ - 1 = ( I - U_k^H H_k T_k )^-1,∀ k.





5) Analog Transmit Beamformer T_RF: Recall that Z_k = T_k + ρD_k, and the analog transmit beamformer T_RF is only in the second term of (<ref>). Therefore, the subproblem concerning T_RF can be given by

    min_T_RF   ∑_k = 1^K Z_k - T_RFT_D,k^2
    s.t. 
           T_RF( m,n) ∈{0, F}, ∀ m,n,  
       T_RF( m,:)_0 = 1, ∀ m.


Let Z  = [ Z_1,⋯,Z_K]. Then, the objective ∑_k = 1^K Z_k - T_RFT_D,k^2 can be rewritten as Z - T_RFT_D^2. The constraints in (<ref>) indicate there is only one non-zero element with a constant magnitude in every row of T_RF, allowing (<ref>) to be solved row-by-row. 

Let S_n,n = 1,⋯,N_RF^t  denote the transmit antennas linked to the n-th RF chain. Because each of the antennas is linked to only an RF chain and the subarrays do not intersect each other, we can have

    ∪ _n = 1^N_RF^t S_n = {1,⋯,M_T}, ∀ n;


    S_p∩ S_q = ∅ ,∀ p  q.


Based on (<ref>) and (<ref>), we establish the following proposition to solve problem (<ref>).


Define Z( m,: )T_D( n,: )^H = | ζ _m,n|e^jϕ̃ _m,n for m ∈ S_n and n =  1,⋯,N_RF^t. Problem (<ref>) is rewritten as

    max_{φ _m,n}{ S_n}   ∑_n = 1^N_RF^t∑_∀ m∈ S_n| ζ _m,n| cos ( ϕ̃ _m,n - φ _m,n )
    s.t.    (<ref>),  (<ref>),

where φ _m,n stands for the phase of the ( m,n )-th element of T_RF, i.e., T_RF( m,n ) = e^jφ _m,n.   



See Appendix <ref>.



Clearly, the objective of problem (<ref>) is  maximized when φ _m,n = ϕ̃ _m,n. Therefore, for each antenna m, we map it to the n-th RF chain maximizing | ζ _m,n|cos ( ϕ̃ _m,n - φ _m,n ):

    n_m^⋆ = max_n | ζ _m,n|cos ( ϕ̃ _m,n - φ _m,n ),

and refresh the subset of transmit antennas for the n-th RF chain: S_n^⋆ =  S_n^⋆∪{ m }.The optimal subarray mapping scheme is obtained by performing (<ref>) for each antenna. Then the corresponding optimal analog beamformer is obtained as 

    T_RF( m,n )=e^jφ _m,n,    if m ∈ S_n^⋆,  n =  1,⋯,N_RF^t;  
    0,   otherwise.

Based on the sparsity in the reconfigurable connection constraint (<ref>), we guarantee the optimality of the analog beamformer T_RF since rows are enumerated one after another.


6) Analog Sensing Receive Beamformer W_RF: Define Q =W + ρD̃. The subproblem concerning W_RF is

    min_W_RF   Q - W_RFW_D^2
    s.t.    W_RF( m̃,ñ )∈{0,  F}, ∀m̃,ñ,  
       W_RF( m̃,: )_0 = 1, ∀m̃.

Since problem (<ref>) has the same form as problem (<ref>), we can solve the radar receive analog beamforming W_RF in the same way as described in Proposition <ref>, as given by 

    max_{φ̅ _m̃,ñ},{S̅_ñ}   ∑_ñ = 1^N_RF^r∑_∀m̃∈S̅_ñ| ζ̅ _m̃,ñ| cos ( ϕ̅ _m̃,ñ - φ̅ _m̃,ñ )
    s.t.    ∪ _ñ = 1^N_RF^rS̅_ñ = {1,⋯,M_R}, ∀ñ, 
       S̅_̅p̅∩S̅_q = ∅ ,∀ p  q,

where S̅_ñ, ñ= 1,⋯,N_RF^r stands for the set of receive antennas linked to the ñ-th RF chain; ζ̅ _m̃,ñ and ϕ̅ _m̃,ñ stand for the amplitude and phase of Q( m̃,: )W_D( ñ,: )^H, respectively; φ̅ _m̃,ñ is the phase of the ( m̃,ñ )-th element of W_RF. The corresponding optimal receive analog beamformer is given by

    W_RF( m̃,ñ )=e^jφ̅ _m̃,ñ,    if m̃∈S̅_ñ^⋆,  ñ=  1,⋯,N_RF^r;  
    0,   otherwise.

Here, S̅_ñ^⋆ is the set of optimal antennas connected to the ñ-th RF chain obtained from (<ref>).


7) Digital Transmit Beamformer T_D,k and Sensing Receive Beamformer W_D: 

The subproblem concerning the digital transmit beamforming T_D,k is an unconstrained least squares (LS) problem: 

    min_T_D,k∑_k = 1^K Z_k - T_RFT_D,k^2.

Subsequently, the optimal solution to (<ref>) is obtained as

    T_D,k = T_RF^†Z_k, ∀ k.


The subproblem concerning the radar digital receive beamforming W_D can be formulated in the same way as (<ref>). With reference to (<ref>), the optimal W_D can be given by:

    W_D = W_RF^†Q.
 








 §.§ Summary of the WPDD Algorithm


Algorithm <ref> summarizes the new WPDD algorithm for designing the HBF in the considered DFRC system by solving problem P_1. Let D= [ D_1,⋯,D_K ]. 
In the outer-loop iteration, we refresh the dual variable sets {D,D̃}  
and the penalty parameter ρ, based on the following constraint violation condition: 

    h(𝒳̃ ) =max(T_k- T_RFT_D,k, W- W_RFW_D).

If h(𝒳̃) 
≤η, we update the dual variables by

    D_k    ←D_k  +  1/ρ( T_k - T_RFT_D,k ), ∀ k, 
    
    
         D̃   ←D̃  +  1/ρ( W - W_RFW_D ).
 
If h(𝒳̃ ) > η, we reduce the penalty factor ρ by ρ← c ρ, with 0<c<1. 
Algorithm <ref> terminates if h(𝒳̃ ) 
< ε _2, where ε _2 is a prespecified threshold. The WPDD stabilizes at a stationary point of problem P_1 <cit.>, as discussed later in Section IV-A.








	












 §.§ Adaptation Under PC Architectures




A PC configuration represents a specific instance of RS connection, where every RF chain is persistently hardwired to a set of antennas. Specifically, the analog beamformer T_RF and W_RF are block diagonal matrices. For example, every block of transmit analog beamforming T_RF has an M-dimensional vector with a constant-modulus (CM) value, i.e.,

    T_RF =blkdiag{p_1,⋯,p_N_RF^t}∈ℂ^M_T×N_RF^t,

where p_i∈ℂ^M × 1 for i = 1,⋯,N_RF^t, M = M_T/ . -N_RF^t, and | p_i( ν )| = 1 for ν = 1,⋯,M.

By replacing the reconfigurable connection constraints (<ref>) and (<ref>) in problem P_1 with (<ref>), we can obtain the DFRC HBF design under the PC architecture. Fortunately, Algorithm <ref> is still applicable to this problem, except that the subproblems for updating the digital and analog beamformers, T_RF and T_D,k differ, that is, Step 3 in Algorithm <ref>, becomes different, as described below.


The subproblem of T_RF, i.e., (<ref>), can be rewritten as

    min_T_RFZ - T_RFT_D^2 
    
        =    min_φ _m,n∑_∀ n∑_∀ mZ( m,: ) - e^jφ _m,nT_D( n,: )^2.

Here, m =  1,⋯,M_T, and n = 1,⋯,⌈mN_RF^t/M_T⌉. 
Then, the phase of the (m,n)-th element of T_RF is given by

    φ _m,n = ( Z( m,:)T_D( n,: )^H).


On the other hand, the subproblem concerning T_D, i.e., (<ref>), can be rewritten as

    min_T_D   Z - T_RFT_D^2 ,
        
          s.t. 
        T_D^2 = N_RF^tP_T/M_T,

which is a non-convex 
QCQP. Here, we employ a computationally efficient algorithm to acquire the optimal 

T_D. 

By defining
Q̅ =T_RF^HT_RF, and
G̅ = T_RF^HZ, the objective of (<ref>) is reformulated as 

    tr(T_D^H Q̅T_D) - 2 R(tr(T_D^H G̅)).
 
By introducing the Lagrange multiplier λ_L, the Lagrangian function is formulated as

    ℒ (T_D,λ_L ) =     tr(T_D^H Q̅T_D) - 2 R(tr(T_D^HG̅)) 
       + λ_L (T_D^2 -  N_RF^tP_T/M_T) .

According to the optimality condition of (<ref>), we obtain

    T_D^opt = ( Q̅ + λ_L^optI )^ - 1G̅,
 
where λ_L ^opt can be obtained by bisection search <cit.>.

In general, the HBF for DFRC systems with PC architectures can be obtained by replacing the solution process of T_RF and T_D in Algorithm <ref>, i.e., replacing (<ref>) and (<ref>)
using (<ref>) and (<ref>) in Step 3, respectively. The description is omitted because the PC-based receive beamforming can be obtained similarly. 





§ SYSTEM PERFORMANCE ANALYSIS




 §.§ Convergence Analysis

 
Algorithm <ref> converges to an effective solution to problem P_1 by iteratively running the proposed double-loop process. Recall that 𝒳̃={T_RF, T_D,k,  U_k, W_RF,W_D,  G_k, T_k, W} collects all variables to be updated in the inner-loop, where W, U_k, and G_k depend uniquely and deterministically on T_k, followed by finding T_RFT_D,k and W_RFW_D to approximate T_k and W, respectively. In this sense, we can partition 𝒳̃ into three subsets: 𝒳̃_1 ={W, T_k, U_k, G_k }, 𝒳̃_2 ={T_RF, W_RF}, and 𝒳̃_3 = {T_D,k, W_D}, the three of which are updated alternately in each inner-loop iteration.

Given ρ, D, and D̃ specified at the beginning of the current outer-loop iteration, 𝒳̃_1 depends solely on the SCA process of T_k, ∀ k. The objective function (<ref>) is certainly non-increasing during the process. Given 𝒳̃_1, 𝒳̃_2 and 𝒳̃_3 are updated, which would further decrease (<ref>) due to the fact that (<ref>) and (<ref>) provide the optimal solutions to (<ref>) and (<ref>), respectively, and hence ensure the objective function of (<ref>) or (<ref>) is strictly non-increasing. Likewise, (<ref>) and (<ref>) can ensure the objective function of (<ref>) is strictly non-increasing. As a result, (<ref>) is strictly non-increasing during an inner-loop iteration and hence throughout the inner-loop iterations during an outer-loop iteration. On the other hand, (<ref>) is apparently lower bounded due to the finite and non-negative transmit power of the BS. 


In the outer loop, ρ, D, and D̃ are updated under the constraint violation condition, h(𝒳̃), until the PDD convergence. While (<ref>) may not be strictly non-increasing in the outer loop, the outer-loop iterations can increasingly penalize the violation of (<ref>) and (<ref>). When T_k stops changing in the inner loop, so do the rest of the variables in the inner loop, and the outer-loop iterations also terminate according to (<ref>) and (<ref>). With the optimal ρ^*, D^*, and D̃^* in the outer loop, 
Algorithm <ref> eventually converges.




 §.§ Discussion on Optimality

Due to the NP-hardness of problem P_1, existing techniques fall short of delivering global optima. Nevertheless, the FD counterpart of the considered system, i.e., problem P_1 with (<ref>)–(<ref>) dropped, can be solved using Algorithm 1 without (<ref>), (<ref>), (<ref>) and (<ref>) in Step 3, reach a Karush-Kuhn-Tucker (KKT) point, and serve as an upper bound for the proposed HBF algorithm. As evidenced numerically in Section V, the HBF design in Algorithm 1 significantly outperforms the existing alternative approaches, much closer to the FD counterpart. In the FD counterpart of the considered problem, only 𝒳̃_1 remains to be optimized. As discussed earlier, 𝒳̃_1 depends uniquely and deterministically on T_k, ∀ k, with the rest of the variables in 𝒳̃_1 all given in closed-form functions of T_k, ∀ k. To this end, the optimality of 𝒳̃_1 depends on that of T_k, ∀ k.

Under the FD architecture, the objective function of the T_k subproblem contains only the first term of (<ref>). The solution yields a_k,i = t_i^H (I⊗P_k^H) t_i, ∀ i,k. Or, one could reduce the auxiliary variable a_k,i to reduce (<ref>). 
After variable substitution and SCA, the T_k subproblem is convexified. 
The non-convex constraint tr ( T_k^HΦT_k ), ∀ k is replaced by the first-order Taylor approximation 2 R ( t̅_k^H Φ̅t_k ) - t̅_k^H Φ̅t̅_k, ∀ k. Here, tr ( T_k^HΦT_k ) and 2 R ( t̅_k^H Φ̅t_k ) - t̅_k^H Φ̅t̅_k yield the same value and the same gradient at the local point t̅_k. 
Moreover, the Slater’s condition is satisfied in every iteration of SCA; i.e., there is a feasible solution to the T_k subproblem such that all inequality constraints are strictly satisfied. According to <cit.>, this solution is a KKT stationary point, which is a local minimum if it is in the interior of the feasible set of the T_k subproblem. As a result, the WPDD can achieve a KKT point for the FD counterpart of the considered problem, which is a local minimum if it is inside the feasible region. 






 §.§ Computational Complexity

Since there are typically many more transmit antennas than receive antennas in DFRC systems, the complexity of the WPDD algorithm is predominantly contributed by the solution of T_k in Step 3. Solving for T_k in (<ref>) involves (KM_T + K^2 + K) variables and (K^2 + K +1) SOC constraints. The complexity of running the interior point method to solve for an ϵ-optimal solution for T_k is 𝒪 ( K^4M_T^3log ( 1/ ϵ ) ), ∀ k <cit.>. 
The worst-case complexity of WPDD under the RS architecture is 𝒪( I_OI_I(K^4M_T^3log ( 1/ ϵ ) )), where I_O and I_I indicate the iteration numbers in the outer and inner loops, respectively.




 §.§ Energy Efficiency (EE)




To evaluate the effect of the RS HAD DFRC system from the perspectives of hardware costs and power consumption, the average EE of each user is defined as 

    ηΔ =  1 /K R/P_tot,

where P_tot is the total energy consumption of the DFRC BS. 

The overall power consumption under various beamforming architectures with transmit power P_T is summarized in Table <ref>, where P_BB specifies the circuit power consumed by the baseband circuit; P_RF and P_SW are the powers consumed by an RF chain and a switch, respectively; P_PS gives the power consumed by a PS; N_PS is the number of PSs.
Clearly, the FD has the maximum power consumption because there are the same number of RF chains and transmit/receive antennas. Moreover, the number of PSs is proportional to that of the transmit/receive antennas and depends on the specific connection mode in the HAD architecture.





§ NUMERICAL RESULTS


In this section, the developed algorithm for DFRC systems with HBF architectures is numerically assessed by MATLAB simulations. The path loss PL(d_k) is modeled as 

    PL(d_k)  [dB] = α + 10 βlog_10(d_k) + ξ,

where ξ∼N (0,σ^2), α = 72.0, β = 2.92, and σ = 8.7 dB for a non-line-of-sight (NLoS) path <cit.>. The noise power of user k is σ _k^2 = -90 dBm. The AoDs ϕ_k,l, ∀ k,l, follow a uniform distribution within [- π/ 2, π/ 2 ]. The other simulation parameters are collated in Table <ref>. 

For the WPDD algorithm, the initial penalty factor is ρ=1, the control parameter is c=0.6, and the termination tolerances are ε _1 = ε _2 = 10^-4, where ε _1 controls the accuracy of inner iterations. 

Moreover, we set the maximum number of iterations I_max = 30 to prevent slow convergence in the inner loop. All of the simulations are conducted using the Matlab 2018b version using a standard PC with an i7-8700 3.20 GHz CPU, 16 GB RAM, and a 64-bit operating system.






The schemes used for evaluating and benchmarking the system performance are outlined as follows:

 
  * RS-WPDD: This is the proposed WPDD algorithm for HBF design in multi-user DFRC systems with the RS architectures (see Section III-C). 
 
  * PC-WPDD: This is the adaptation of the proposed WPDD algorithm for multi-user DFRC systems with the PC architectures (see Section III-D). 
 
  * DPC-THEREON: This is the joint hybrid waveform and radar receiver optimization algorithm in <cit.> based on the DPS architectures.
 
  * FC-TwoStage: This is the FC-based HBF algorithm developed in <cit.>, which first constructs an FD beamformer 𝐓 and then iteratively optimizes the analog and digital beamformers to approximate the achieved 𝐓. 
 
  * PC-TwoStage: This is the HBF algorithm developed under the PC architectures in <cit.>, which can be implemented similarly to FC-TwoStage algorithm.
 
  * FD/FA: This is the fully-digital/analog-based beamforming algorithm that provides an upper/lower bound for all HBF algorithms.















Fig. <ref> plots the convergence of the proposed WPDD algorithm for the HBF design with the increase of outer iterations, where the radar SCNR threshold γ is set to 10 dB and 15 dB. It is noticed that the communication sum-rate converges within about ten iterations under both the RS-WPDD and PC-WPDD algorithms. 

Moreover, RS-WPDD always obtains better communication performance than PC-WPDD.





 






Fig. <ref> plots the trade-off between the achievable sum-rate and the radar SCNR constraint under different beamforming design schemes, where γ ranges from 10 dB to 20 dB. The sum-rate declines as γ grows. The reason is that when the sensing demand is higher, fewer DoFs are used to improve communication performance. When γ≤ 12 dB, the multi-user communication sum-rate remains approximately constant since the system is saturated when the sensing performance is easily satisfied with small γ. When γ≥ 16 dB, the communication performance degrades rapidly. In addition, Fig. <ref> shows that the proposed RS-WPDD algorithm outperforms the PC-based algorithms. Even the proposed PC-WPDD has comparable performance with DPC-THEREON. Here, FD offers an upper bound for the performance of our proposed system, while FA gives a lower bound.

Fig. <ref> shows the trade-off between the communication and sensing of the WPDD framework under different DFRC radar receive architectures, including the hybrid receiver (HR) architecture and the fully digital receiver (FDR) architecture. The RS-based DFRC hybrid receive beamforming is closer to the FD receive beamforming than the PC architecture. Fig. <ref> plots the achievable sum-rate of the DFRC system with the increasing total transmit power P_T, where the number of transmit antennas is M_T=32, 64, and 128. It is shown that the proposed RS-WPDD is better than PD-WPDD, as observed in Fig. <ref>. The system performance improves as M_T increases, enhancing antenna diversity and better beamforming gain. Furthermore, the difference between RS-WPDD and PC-WPDD decreases as M_T increases.










We adopt the spatial beampattern, P( θ ) = | w^HA ( θ )x|^2 <cit.>, as a sensing performance metric. The optimized beampatterns of the proposed WPDD algorithm and the benchmarks with M_T = 32, M_R = 16, and γ = 18 dB are depicted in Fig. <ref>. Clearly, all schemes can be pointed at a 0^∘ target and achieve good suppression of clutters at -30^∘ and 30^∘. Clutters can be effectively suppressed through receive beamforming at the sensing receiver. Moreover, the RS-WPDD scheme has a peak-to-side-lobe ratio of about 30 dB, which is much better than the rest of the schemes.

We also plot the detection probability of the proposed RS-WPDD with the SCNR threshold γ, as shown in Fig. <ref>. When γ = 15 dB, the detection probability is 99.72% under a false alarm probability of 10^-6, and 99.99% under a false alarm probability of 10^-4. As shown in Fig. <ref>, RS-WPDD can obtain a satisfactory sensing performance with marginal or even negligible communication rate losses when γ = 15 dB.













We evaluate EE to show the effect of our proposed HBF on the system cost and performance, where P_BB = 200 mW, P_RF = 300 mW, P_PS = 50 mW, and P_SW = 5 mW <cit.>.
Fig. <ref> plots the average EE of the DRFC systems with the increase of γ. We see that the EE of the proposed RS-WPDD is better than that of PC-WPDD and significantly surpasses the other benchmarks. Specifically, when γ=15 dB, RS-WPDD improves the EE by 23.9% compared to PC-WPDD, 83.4% compared to PC-TwoStage, and 114.2% compared to FC-TwoStage. Fig. <ref> shows the change in EE with the number of users K   (K ≤N_RF^t), when each user has only one data stream, i.e., d_s=1, and N_RF^t = 8. The RS-WPDD still yields the best EE across the spectrum of the user number, as observed in Fig. <ref>. 
The system parameters can be selected to balance the hardware cost and system performance of DFRC with the RS architectures.



§ CONCLUSION

We have designed HBF for a multi-user mmWave DFRC system with a new RS HAD architecture. 

We have jointly optimized digital and analog beamformers to maximize the communication sum-rate under a prescribed SCNR of target detection. An efficient WPDD algorithm has been designed based on WMMSE and PDD methods, which can guarantee convergence to a stationary point. The effectiveness of WPDD has been confirmed in terms of convergence, communication SE, target beampattern, and system EE. Simulations have shown that with the RS architecture and HBF, DFRC systems excel in both SE and EE compared to the conventional PC architecture. The EE of the systems is dramatically improved compared to the FC architecture by 114.2%. 












































































§ PROOF OF PROPOSITION <REF>
 

Because the reconfigurable subarrays do not overlap and every transmit antenna is linked to only an RF chain, we can solve problem (<ref>) row by row, as follows:

    min_T_RFZ - T_RFT_D^2
      = ^ (a)   min_T_RF∑_n=1^N_RF^t∑_∀ m∈ S_nZ( m,:) - T_RF( m,n)T_D( n,:)^2
    
    =   min_T_RF∑_n = 1^N_RF^t∑_∀ m∈ S_n{Z( m,:)^2  + | T_RF( m,n)|^2T_D( n,:)^2 }
       - 2 R( Z( m,:)T_D( n,:)^HT_RF( m,n)^H),

where the equality (a) holds when the objective function is expanded in rows. 

Define φ _m,n as the phase of the (m,n)-th entry in T_RF, i.e., T_RF( m,n) = 1/√(M_T)e^jφ _m,n, and Z( m,:)T_D( n,:)^H = | ζ _m,n|e^jϕ _m,n, n =  1,⋯,N_RF^t , m ∈ S_n. By suppressing the constant term, (<ref>) is further equivalent to

    max_T_RF∑_n =1^N_RF^t∑_∀ m∈ S_n R( Z( m,:)T_D( n,:)^HT_RF( m,n)^H)
      = ^ (b)   max_{φ _m,n}∑_n =1^N_RF^t∑_∀ m∈ S_n R( e^ - jφ _m,nZ( m,:)T_D( n,:)^H)
    
    =   max_{φ _m,n}∑_n =1^N_RF^t∑_∀ m∈ S_n| ζ _m,n| cos( ϕ _m,n - φ _m,n).

Since the (m,n)-th entry of T_RF has a constant modulus value, the equality in (b) holds. The final establishment of the equivalence between problems (<ref>) and (<ref>) is achieved by integrating the objective (<ref>) with the reconfigurable connection constraints (<ref>) and (<ref>). 



























10
	9737357
	F. Liu, Y. Cui, C. Masouros, J. Xu, T. X. Han, Y. C. Eldar, and S. Buzzi,
	“Integrated sensing and communications: Toward dual-functional wireless
	networks for 6G and beyond,” IEEE J. Sel. Areas Commun., vol. 40,
	no. 6, pp. 1728–1767, Jun. 2022.
	
	10188491
	F. Liu, L. Zheng, Y. Cui, C. Masouros, A. P. Petropulu, H. Griffiths, and Y. C.
	Eldar, “Seventy years of radar and communications: The road from separation
	to integration,” IEEE Signal Process. Mag., vol. 40, no. 5, pp.
	106–121, Jul. 2023.
	
	9303435
	X. Yuan, Z. Feng, J. A. Zhang, W. Ni, R. P. Liu, Z. Wei, and C. Xu,
	“Spatio-temporal power optimization for MIMO joint communication and radio
	sensing systems with training overhead,” IEEE Trans. Veh. Technol.,
	vol. 70, no. 1, pp. 514–528, Jan. 2021.
	
	9851407
	T. Mao, J. Chen, Q. Wang, C. Han, Z. Wang, and G. K. Karagiannidis, “Waveform
	design for joint sensing and communications in millimeter-wave and low
	terahertz bands,” IEEE Trans Commun, vol. 70, no. 10, pp. 7023–7039,
	Oct. 2022.
	
	9606831
	Y. Cui, F. Liu, X. Jing, and J. Mu, “Integrating sensing and communications
	for ubiquitous IoT: Applications, trends, and challenges,” IEEE
		Netw., vol. 35, no. 5, pp. 158–167, Sep. 2021.
	
	8241348
	S. A. Busari, K. M. S. Huq, S. Mumtaz, L. Dai, and J. Rodriguez,
	“Millimeter-wave massive MIMO communication for future wireless systems: A
	survey,” IEEE Commun. Surv. Tutor., vol. 20, no. 2, pp. 836–869, 2nd
	Quart., 2018.
	
	8962251
	S. Fortunati, L. Sanguinetti, F. Gini, M. S. Greco, and B. Himed, “Massive
	MIMO radar for target detection,” IEEE Trans. Signal Process.,
	vol. 68, pp. 859–871, Jan. 2020.
	
	zhang2023doppler
	F. Zhang, T. Mao, and Z. Wang, “Doppler-resilient design of CAZAC sequences
	for mmwave/THz sensing applications,” arXiv:2305.07234, 2023.
	
	8371237
	I. Ahmed, H. Khammari, A. Shahid, A. Musa, K. S. Kim, E. De Poorter, and
	I. Moerman, “A survey on hybrid beamforming techniques in 5G: Architecture
	and system model perspectives,” IEEE Commun. Surv., vol. 20, no. 4,
	pp. 3060–3097, 4th Quart., 2018.
	
	7397861
	X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaief, “Alternating minimization
	algorithms for hybrid precoding in millimeter wave MIMO systems,”
	IEEE J. Sel. Top. Signal Process., vol. 10, no. 3, pp. 485–500, Apr.
	2016.
	
	7389996
	F. Sohrabi and W. Yu, “Hybrid digital and analog beamforming design for
	large-scale antenna arrays,” IEEE J. Sel. Top. Signal Process.,
	vol. 10, no. 3, pp. 501–513, Apr. 2016.
	
	5419124
	A. Hassanien and S. A. Vorobyov, “Phased-MIMO radar: A tradeoff between
	phased-array and MIMO radars,” IEEE Trans. Signal Process.,
	vol. 58, no. 6, pp. 3137–3151, Jun. 2010.
	
	9237135
	R. Rajamäki, S. P. Chepuri, and V. Koivunen, “Hybrid beamforming for active
	sensing using sparse arrays,” IEEE Trans. Signal Process., vol. 68,
	pp. 6402–6417, Oct. 2020.
	
	9729809
	C. Qi, W. Ci, J. Zhang, and X. You, “Hybrid beamforming for millimeter wave
	MIMO integrated sensing and communications,” IEEE Commun. Lett.,
	vol. 26, no. 5, pp. 1136–1140, May. 2022.
	
	10199478
	W.-C. Yang, H.-Y. Chang, R. Y. Chang, and W.-H. Chung, “Hybrid beamforming for
	dual-functional radar-communication systems,” in Proc. IEEE 97th Veh.
		Technol. Conf. (VTC2023-Spring), Jun. 2023, pp. 1–7.
	
	10049999
	X. Yu, L. Tu, Q. Yang, M. Yu, Z. Xiao, and Y. Zhu, “Hybrid beamforming in
	mmwave massive MIMO for IoV with dual-functional radar communication,”
	IEEE Trans. Veh. Technol., vol. 72, no. 7, pp. 9017–9030, Jul. 2023.
	
	9538857
	Y. Dai, K. Han, G. Wei, Y. Du, and Z. Wang, “Hybrid beamforming for DFRC
	system based on SINR performance metric,” in Proc. IEEE/CIC Int.
		Conf. Commun. China (ICCC Workshops), Jul. 2021, pp. 82–87.
	
	10279443
	J. Gong, W. Cheng, F. Chen, and J. Wang, “Hybrid beamforming for millimeter
	wave integrated sensing and communications,” in Proc. IEEE Int. Conf.
		Commun. (ICC), May. 2023, pp. 2958–2963.
	
	8683591
	F. Liu and C. Masouros, “Hybrid beamforming with sub-arrayed MIMO radar:
	Enabling joint sensing and communication at mmwave band,” in Proc.
		IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), May. 2019, pp.
	7770–7774.
	
	9500661
	A. Kaushik, C. Masouros, and F. Liu, “Hardware efficient joint
	radar-communications with hybrid precoding and RF chain optimization,” in
	Proc. IEEE Int. Conf. Commun. (ICC), Jun. 2021, pp. 1–6.
	
	9366836
	Z. Cheng, Z. He, and B. Liao, “Hybrid beamforming for multi-carrier
	dual-function radar-communication system,” IEEE Trans. Cogn. Commun.
		Netw., vol. 7, no. 3, pp. 1002–1015, Sep. 2021.
	
	9950549
	Z. Cheng, L. Wu, B. Wang, M. R. B. Shankar, and B. Ottersten,
	“Double-phase-shifter based hybrid beamforming for mmwave dfrc in the
	presence of extended target and clutters,” IEEE Trans. Wirel.
		Commun., vol. 22, no. 6, pp. 3671–3686, Jun. 2023.
	
	9110865
	H. Li, M. Li, Q. Liu, and A. L. Swindlehurst, “Dynamic hybrid beamforming with
	low-resolution PSs for wideband mmwave MIMO-OFDM systems,” IEEE
		J. Sel. Areas Commun., vol. 38, no. 9, pp. 2168–2181, Sep. 2020.
	
	7880698
	S. Park, A. Alkhateeb, and R. W. Heath, “Dynamic subarrays for hybrid
	precoding in wideband mmwave MIMO systems,” IEEE Trans. Wirel.
		Commun., vol. 16, no. 5, pp. 2907–2920, May. 2017.
	
	8502711
	X. Jing, L. Li, H. Liu, and S. Li, “Dynamically connected hybrid precoding
	scheme for millimeter-wave massive MIMO systems,” IEEE Commun.
		Lett., vol. 22, no. 12, pp. 2583–2586, Dec. 2018.
	
	9537599
	C. G. Tsinos, A. Arora, S. Chatzinotas, and B. Ottersten, “Joint transmit
	waveform and receive filter design for dual-function radar-communication
	systems,” IEEE J. Sel. Top. Signal Process., vol. 15, no. 6, pp.
	1378–1392, Nov. 2021.
	
	6649991
	G. Cui, H. Li, and M. Rangaswamy, “MIMO radar waveform design with constant
	modulus and similarity constraints,” IEEE Trans. Signal Process.,
	vol. 62, no. 2, pp. 343–353, Jan. 2014.
	
	6834753
	M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and
	E. Erkip, “Millimeter wave channel modeling and cellular capacity
	evaluation,” IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp.
	1164–1179, Jun. 2014.
	
	8462337
	J. Sung, J. Choi, and B. L. Evans, “Narrowband channel estimation for hybrid
	beamforming millimeter wave communication systems with one-bit
	quantization,” in Proc. IEEE Int. Conf. Acoust., Speech Signal
		Process. (ICASSP), Apr. 2018, pp. 3914–3918.
	
	9207745
	X. Wei, C. Hu, and L. Dai, “Deep learning for beamspace channel estimation in
	millimeter-wave massive MIMO systems,” IEEE Trans. Commun.,
	vol. 69, no. 1, pp. 182–193, Jan. 2021.
	
	8286860
	K. Wu, W. Ni, T. Su et al., “Fast and accurate estimation of
	angle-of-arrival for satellite-borne wideband communication system,”
	IEEE J. Sel. Areas Commun., vol. 36, no. 2, pp. 314–326, Feb. 2018.
	
	9724259
	R. Liu, M. Li, Q. Liu, and A. L. Swindlehurst, “Joint waveform and filter
	designs for STAP-SLP-based MIMO-DFRC systems,” IEEE J. Sel.
		Areas Commun., vol. 40, no. 6, pp. 1918–1931, Jun. 2022.
	
	6404093
	A. Aubry, A. DeMaio, A. Farina, and M. Wicks, “Knowledge-aided (potentially
	cognitive) transmit signal and receive filter design in signal-dependent
	clutter,” IEEE Trans Aerosp Electron Syst, vol. 49, no. 1, pp.
	93–117, Jan. 2013.
	
	6832464
	A. Sabharwal, P. Schniter, D. Guo et al., “In-band full-duplex
	wireless: Challenges and opportunities,” IEEE J. Sel. Areas Commun.,
	vol. 32, no. 9, pp. 1637–1652, Sep. 2014.
	
	10007634
	M. Ashraf, B. Tan, D. Moltchanov, J. S. Thompson, and M. Valkama, “Joint
	optimization of radar and communications performance in 6G cellular
	systems,” IEEE Trans. Green Commun. Netw., vol. 7, no. 1, pp.
	522–536, Mar. 2023.
	
	9120361
	Q. Shi and M. Hong, “Penalty dual decomposition method for nonsmooth nonconvex
	optimization—part I: Algorithms and convergence analysis,” IEEE
		Trans. Signal Process., vol. 68, pp. 4108–4122, Jun. 2020.
	
	5756489
	Q. Shi, M. Razaviyayn, Z. Luo, and C. He, “An iteratively weighted MMSE
	approach to distributed sum-utility maximization for a MIMO interfering
	broadcast channel,” IEEE Trans. Signal Process., vol. 59, no. 9, pp.
	4331–4340, Sep. 2011.
	
	8239836
	L. Wu, P. Babu, and D. P. Palomar, “Transmit waveform/receive filter design
	for MIMO radar with multiple waveform constraints,” IEEE Trans.
		Signal Process., vol. 66, no. 6, pp. 1526–1540, Mar. 2018.
	
	5447068
	Z. Luo, W. Ma, A. M. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of
	quadratic optimization problems,” IEEE Signal Process. Mag., vol. 27,
	no. 3, pp. 20–34, May. 2010.
	
	9772613
	X. Zhang, Y. Ren, J. Wang, and T. Lv, “Joint design of multicast transmission
	and in-network caching for green internet of things,” IEEE Sens. J.,
	vol. 22, no. 12, pp. 12 404–12 414, Jun. 2022.
	
	grant2014cvx
	M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex
	programming, version 2.1,” 2014.
	
	8332507
	Q. Shi and M. Hong, “Spectral efficiency optimization for millimeter wave
	multiuser MIMO systems,” IEEE J. Sel. Top. Signal Process.,
	vol. 12, no. 3, pp. 455–468, Jun. 2018.
	
	boyd2004convex
	S. P. Boyd and L. Vandenberghe, Convex Optimization.1em plus
	0.5em minus 0.4emCambridge Univ. Press, 2004.
	
	marks1978general
	B. R. Marks and G. P. Wright, “A general inner approximation algorithm for
	nonconvex mathematical programs,” Oper. Res., vol. 26, no. 4, pp.
	681–683, Aug. 1978.
	
	ben2001lectures
	A. BenTal and A. Nemirovski, Lectures on Modern Convex Optimization:
		Analysis, Algorithms, and Engineering Applications.1em plus 0.5em
	minus 0.4emSIAM, 2001.
	














[
    < g r a p h i c s >
]Xin Jin received the B.S. degree in communication engineering from Sichuan Normal University (SICNU), Chengdu, China, in 2020. She is currently pursuing the Ph.D. degree with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT), Beijing, China. Her research interests include integrated sensing and communications, massive MIMO, hybrid beamforming, and array signal processing.






[
    < g r a p h i c s >
]Tiejun Lv(Senior Member, IEEE) received the M.S. and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1997 and 2000, respectively. From January 2001 to January 2003, he was a Postdoctoral Fellow at Tsinghua University, Beijing, China. In 2005, he was promoted to Full Professor at the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT). From September 2008 to March 2009, he was a Visiting Professor with the Department of Electrical Engineering at Stanford University, Stanford, CA, USA. He is the author of four books, more than 150 published journal papers and 200 conference papers on the physical layer of wireless mobile communications. His current research interests include signal processing, communications theory and networking. He was the recipient of the Program for New Century Excellent Talents in University Award from the Ministry of Education, China, in 2006. He received the Nature Science Award from the Ministry of Education of China for the hierarchical cooperative communication theory and technologies in 2015. He has won best paper award at CSPS 2022. 





[
    < g r a p h i c s >
]Wei Ni (Fellow, IEEE) received the B.E. and Ph.D. degrees in Electronic Engineering from Fudan University, Shanghai, China, in 2000 and 2005, respectively. He is a Principal Research Scientist at CSIRO, Sydney, Australia, and a Conjoint Professor at the University of New South Wales. He is also an Adjunct
Professor at the University of Technology Sydney, and an Honorary Professor at Macquarie University. He serves as a Technical Expert at Standards Australia in support of the ISO standardization of AI and Big Data. He was a Postdoctoral Research Fellow at Shanghai Jiaotong University from 2005 to 2008; Deputy Project Manager at Bell Labs, Alcatel/Alcatel-Lucent from 2005 to 2008; and Senior Researcher at Devices R&D, Nokia from 2008 to 2009. He has co-authored one book, ten book chapters, more than 300 journal papers, more than 100 conference papers, 26 patents, ten standard proposals accepted by IEEE, and three technical contributions accepted by ISO. His research interests include 6G security and privacy, machine learning, stochastic optimization, and their applications to system efficiency and integrity.

Dr. Ni has been an Editor for IEEE Transactions on Wireless Communications since 2018, an Editor for IEEE Transactions on Vehicular Technology since 2022, and an Editor for IEEE Transactions on Information Forensics
and Security and IEEE Communications Surveys and Tutorials since 2024. He served first as the Secretary, then the Vice-Chair and Chair of the IEEE VTS NSW Chapter from 2015 to 2022, Track Chair for VTC-Spring 2017, Track Co-chair for IEEE VTC-Spring 2016, Publication Chair for BodyNet 2015, and Student Travel Grant Chair for WPMC 2014.




[
    < g r a p h i c s >
]Zhipeng Lin(Member, IEEE) received the Ph.D. degrees from the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, China, and the School of Electrical and Data Engineering, University of Technology of Sydney, NSW, Australia, in 2021. Currently, He is an Associate Researcher in the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His current research interests include signal processing, massive MIMO, spectrum sensing, and UAV communications.





[
    < g r a p h i c s >
]Qiuming Zhu(Senior Member, IEEE) received his B.S. in electronic engineering from Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing, China, in 2002 and his M.S. and Ph.D. in communication and information system from NUAA in 2005 and 2012, respectively. Since 2021, he has been a professor in the Department of Electronic Information Engineering, NUAA. From Oct. 2016 to Oct. 2017, June 2018 to Aug. 2018 and June 2018 to Aug. 2018, he was also an academic visitor at Heriot Watt University,
Edinburgh, U. K. He has authored or coauthored more than 120 articles in refereed journals and conference proceedings and holds over 40 patents. His current research interests include channel sounding, modeling, and emulation for the fifth/sixth generation (5G/6G) mobile communication, vehicle-to-vehicle (V2V) communication and unmanned aerial vehicles (UAV) communication systems.






[
    < g r a p h i c s >
]Ekram Hossain(Fellow, IEEE) is a Professor and the Associate Head (Graduate Studies) of the Department of Electrical and Computer Engineering, University of Manitoba, Canada. He is a Member (Class of 2016) of the College of the Royal Society of Canada. He is also a Fellow of the Canadian Academy of Engineering and the Engineering Institute of Canada. He has won several research awards, including the 2017 IEEE Communications Society Best Survey Paper Award and the 2011 IEEE Communications Society Fred Ellersick Prize Paper Award. He was listed as a Clarivate Analytics Highly Cited Researcher in Computer Science in 2017-2023. Previously, he served as the Editor-in-Chief (EiC) for the IEEE Press (2018–2021) and the IEEE Communications Surveys and Tutorials (2012–2016). He was a Distinguished Lecturer of the IEEE Communications Society and the IEEE Vehicular Technology Society. He served as the Director of Magazines (2020-2021) and the Director of Online Content (2022-2023) for the IEEE Communications Society.




[
    < g r a p h i c s >
]H. Vincent Poor(Life Fellow, IEEE) received the Ph.D. degree in EECS from Princeton University in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the faculty at Princeton, where he is currently the Michael Henry Strater University Professor. During 2006 to 2016, he served as the dean of Princeton’s School of Engineering and Applied Science. He has also held visiting appointments at several other universities, including most recently at Berkeley and Cambridge. His research interests are in the areas of information theory, machine learning and network science, and their applications in wireless networks, energy systems and related fields. Among his publications in these areas is the recent book Machine Learning and Wireless Communications. (Cambridge University Press, 2022). Dr. Poor is a member of the National Academy of Engineering and the National Academy of Sciences and is a foreign member of the Chinese Academy of Sciences, the Royal Society, and other national and international academies. He received the IEEE Alexander Graham Bell Medal in 2017.


