RIS-Aided Wideband Holographic DFRC

To enable non-line-of-sight (NLoS) sensing and communications, dual-function radar-communications (DFRC) systems have recently proposed employing reconfigurable intelligent surface (RIS) as a reflector in wireless media. However, in the dense environment and higher frequencies, severe propagation and attenuation losses are a hindrance for RIS-aided DFRC systems to utilize wideband processing. To this end, we propose equipping the transceivers with the reconfigurable holographic surface (RHS) that, different from RIS, is a metasurface with an embedded connected feed deployed at the transceiver for greater control of the radiation amplitude. This surface is crucial for designing compact low-cost wideband wireless systems, wherein ultra-massive antenna arrays are required to compensate for the losses incurred by severe attenuation and diffraction. We consider a novel wideband DFRC system equipped with an RHS at the transceiver and a RIS reflector in the channel. We jointly design the digital, holographic, and passive beamformers to maximize the radar signal-to-interference-plus-noise ratio (SINR) while ensuring the communications SINR among all users. The resulting nonconvex optimization problem involves maximin objective, constant modulus, and difference of convex constraints. We develop an alternating maximization method to decouple and iteratively solve these subproblems. Numerical experiments demonstrate that the proposed method achieves better radar performance than non-RIS, random-RHS, and randomly configured RIS-aided DFRC systems.


I. INTRODUCTION
Reconfigurable intelligent surfaces (RISs) have recently emerged as an enabling technology for future wireless systems.A RIS consists of several passive or near-passive sub-wavelength metasurface elements [1].Conventionally, wireless systems assume that the fading channel is uncontrollable and is a significant factor that limits the performance because of random signal reflections, diffraction, and scattering in the wireless environment [2].RIS overcomes these fading channel limitations through the ability of metasurfaces to manipulate electromagnetic waves for applications such as arbitrary aperture beamforming [3], frequency selective and highimpedance surfaces [4], polarization conversion [5], leaky-wave antenna [6], beam focusing [7], and holographic imaging [8].
In general, RIS is deployed as a reflector in wireless media.By exploiting the non-line-ofsight (NLoS) paths, the RIS-aided sensing [9] and communications [10,11] systems extend their coverage [12,13], suppress interference [11], and secure the information transfer [14].There is a rich heritage of research on non-RIS-based NLoS radars (see, e.g., [15] and the references therein); but these techniques generally require prior and accurate knowledge of the geometry of propagation environment.In contrast, RIS-aided sensing exploits the NLoS echoes to compensate for the LoS path loss [9,16].In wireless communications, RIS has been shown to enhance the coverage area by reflecting the impinging signals and hence overcome the severe line-of-sight (LoS) attenuation or blockage between the base station (BS) and multiple users (MU) [13].
For example, in [17], RIS was employed to minimize the total transmit power while ensuring the signal-to-interference-plus-noise ratio (SINR) among all users.Further, both the active and largely passive beamformers which employ at BS and RIS, respectively, are able to improve the overall quality-of-service (QoS) [18].
Recently, the investigations of RIS focus on enhancing the performance of dual-function radar-communications (DFRC) [19,20], wherein sensing and communications jointly utilize the spectral and hardware resources [21,22].A single-RIS-aided DFRC was proposed in [23] to maximize the radar SNR while utilizing the reflecting surface to simultaneously facilitate the target detection and single-user communications which is considered as the radar-centric design.This set-up was extended to wideband DFRC with multiple RISs in [22].In practice, the RIS phase shifts are not continuous but quantized.This issue has been analyzed for DFRC in [24,25].
A few other recent studies on communications-centric DFRC design, where the RIS facilitates in maximizing secrecy rates [14].
Early investigations on RIS-aided DFRC focused on narrowband sub-6 GHz frequencies.
Lately, rapid developments have taken place at millimeter-wave (mmWave) communications and sensing [19] to develop short-range technologies that exploit the large operational bandwidth at mmWave.This band is characterized by severe attenuation during signal propagation.To compensate for these losses, extremely dense antenna arrays comprising a massive number of antenna elements are employed.To this end, as one of the representative metamaterial antennas, reconfigurable holographic surface (RHS) [26] has been proposed to realize such large arrays.
Different from RIS, the RHS is embedded in a large number of metamaterial radiation elements which are connected with a radio-frequency (RF) chain and are generally integrated with the transceivers.The RHS radiation elements exploit the holographic interference principle [27] to control the radiation amplitude of the incident electromagnetic waves while also leading to compact and lightweight transceiver hardware [28,29].This low-cost amplitude-control beamforming design was first proposed for the conventional antenna under orthogonal frequency-divisionmultiplexing (OFDM) transmissions [30].Then, it was extended to the holographic scenario for swift radiation beam control [31].
Initial RHS investigations were limited to wireless communications applications for flexible beam steering [32,33].Recently, it has been demonstrated for DFRC, wherein the holographic beam is aligned toward the target to ensure the communications signal-to-interference-plus-noise ratio (SINR) meets the requirements [34].However, even with improved beam control, the RHSaided systems yield poor performance in the absence of a stable line-of-sight (LoS) link thereby precipitating the need to also employ an RIS [35,36].Previous research largely focused on RISassisted wireless solutions for narrowband signaling thereby leading to frequency-independent passive beamformers.However, future wireless systems are expected to scale up in the spectrum and, therefore, exploit wide bandwidths available at the higher frequencies [19].Narrowband beamforming techniques are not usable for such wideband systems, where the resulting beamsquint effect [37] could no longer be ignored.
To overcome the above-mentioned limitations, in this paper, we jointly exploit the advantages of both RIS and RHS in a wideband DFRC system.We deploy the passive RIS [38] in the channel as a reflector while equipping the RHS at BS as the transceiver.Then, we jointly design the digital, holographic, and passive beamformers for the digital DFRC, RHS transceiver, and RIS, respectively.Our objective is to maximize the worst-case radar SINR over all the target, while also ensuring a certain minimum SINR for the different communication users.The resulting optimization problem involves nonconvex quadratic constraint quartic programming (QCQP) with coupled variables.We solve this challenging problem by first decoupling it into several subproblems which are solved via an alternating optimization (AO) algorithm.
We summarize our main contributions in this paper as follows.
1) Wideband RHS model with beam-squint: Different from previous works focused on narrowband RHS-assisted DFRC systems [29,34], in this paper, we propose a more comprehensive wideband RHS model with OFDM signaling.Our proposed model allows for varying the digital beamforming on different subcarriers, thereby offsetting the beam squint effect.Meanwhile, RHS can adjust the beam by controlling the radiation amplitude of the input signal.
2) RIS-aided DFRC with RHS: Contrary to prior works, we consider simultaneously harnessing the benefits of both RIS and RHS for DFRC applications.This joint deployment is especially helpful at higher frequencies, where the channel is LoS-dominant and NLoS-assisted [19,22,37].Here, the RIS beamforming in the NLoS paths boosts the indirect echoes.Further, RHS transceivers have a small form-factor and are able to quickly shape the radiation beampattern to overcome the fast fading channel.However, this deployment scheme imposes a new challenge for jointly designing the passive and holographic beamformers.
3) Joint digital, holographic, and passive beamformer design: We design the digital, holographic, and passive beamforming, and the receive filter, simultaneously, to maximize the worstcase radar SINR accounting for all the targets while guaranteeing the communications SINR.
To this end, we develop an alternating optimization (AO) framework to tackle the resulting nonconvex maximin problem.We first utilize the generalized Rayleigh quotient (GRQ) method to obtain the closed-form solution for the receive filter design.Then, we combine the Dinkelbach and majorization-maximization (MM) algorithm to tackle the digital and holographic beamforming design.Finally, the consensus alternating direction of multipliers (C-ADMM) [39] and Riemannian steepest decent (RSD) [40] approaches are jointly utilized to solve the phase-shift design problem approximately.The remainder of this paper is organized as follows.In the next section, we introduce the signal model and problem formulation for a RIS-aided wideband DFRC system with RHS.In ) along the xand y-axes, respectively.The resulting wideband space-frequency steering vectors of DFBS, RIS, and user are, respectively, given by where ]) are azimuth (elevation) angles; λ k = f k /c and f k = f c + fk are, respectively, wavelength and frequency of the k subcarrier; f c and fk = k△f are the carrier frequency and k-th subcarrier offset frequency, respectively; △f is the subcarrier spacing and )) denote the steering vectors along the x-(y-) axis as where µ x = d x cos θ cos ψ, and µ y = d y sin θ cos ψ are the direction cosines [41].To simply the notation, hereafter, we denote x ×N U y as the total number of array elements in the antennas of the DFBS, RIS and communication user.

A. RHS-Based Wideband Transmit Signal Model
The RHS-based transceivers generate the emitted signal following the holographic interference principle.We refer the interested readers to [32] (and references therein) for the details of the operational principle of holographic systems.Here, we follow the same model but adapt it for wideband DFRC.
DRAFT May 9, 2023 a) OFDM Precoding: We denote the transmit symbol vector at the k-th subcarrier by Let F k denote the frequency-dependent beamformer to enable multiuser (MU) communications and mitigate the beam-squint effect [42].
After the digital beamforming, the frequency-domain signal at k-th subcarrier is Further, N RF RF chains on the DFBS are connected to a RHS having N B discrete antenna elements 1 .Applying N RF K-point inverse discrete Fourier transform (IDFT) to (4) yields the baseband signal where t ∈ (0, T s ] and T s denotes the OFDM duration excluding the cyclic prefix (CP).
b) RHS Beamforming: Following the model presented in [32], the electromagnetic response of the RHS at the k-th subcarrier takes the form where the matrix amplitude-control beamformer of (x, y)-th RHS element.Further, V k (p, q) = e −2jπγDp,q/λ k , where D p,q denotes the distance between the p-th RHS element and q-th feed, p = 1, and γ is the refractive index of the RHS material.Consequently, the matrix c) DFBS transmit signal: After the holographic beamforming to (5), the transmitted signal in passband (excluding CP) takes the form This signal is utilized to detect the targets and enable MU communications, simultaneously.
Meanwhile, for wideband DFRC, the transmit power should meet the system requirement.In order to fully utilize the bandwidth, herein, we assume the transmit power satisfies where P k is the maximum power assigned to the k-th subcarrier.

B. Communications Receiver
Denote the direct DFBS-user, RIS-user, and DFBS-RIS (in which only the LoS component is considered) channels at the k-th subcarrier frequency by H dir Cu,k , H RIS Cu,k , and G k , respectively.Following these prevalant channel models summarized in [37] (and references therein), the aforementioned wideband channel components are given, respectively, by where g * ,k = K 0 ( r 0 r ) ǫ is the distance-dependent path loss, K 0 is the path loss at the r 0 reference distance, r is the distance of the corresponding path, L d and L r denote the number of NLoS path for DFBS-user link and RIS-user link, ǫ is the path loss exponent (ranging from 2−4), and Cu denote the Rician factor for the corresponding path such that To simplify the design procedure, the aforementioned channels are estimated a priori [43,44].
Subsequent to baseband conversion, CP removal and N U K-point discrete Fourier transform (DFT), the received signal of u-th user on the k-th subcarrier is where H Cu,k denotes the composite channel between BS and u-th user, Φ denotes the phase-shift matrix which is common across all subcarriers, n Cu [f k ] denotes the zero mean white Gaussian noise with covariance σ 2 r I at the receiver front end for k-th subcarrier.subcarriers [45][46][47].Hence, the beam squint effect which degrades the performance of wideband system is also inevitable at RIS.While there exist some works aiming to design subcarrier specific phase-shifts to overcome the squint [38], the complexity of their hardware implementation is beyond the envisaged DFRC system [48].Hence, in this paper, we consider the fully passive RIS-assisted wideband DFRC with RHS.The active wideband RIS scenario will be explored in future work.
At the k-th subcarrier of u-th user, a digital combiner w k,u is utilized to filter the received signal and estimate the transmitted symbol as The quality of the estimate, s k,u , is determined by the SINR, the typical metric for benchmarking the link performance of communication.According to (14), the SINR of u-th user on k-th subcarrier is Then, the average SINR over all subcarriers for u-th user is Note that ( 16) is composed by the summation of a set of quartic fractional function in terms of receive filter and passive and holographic beamforming which is difficult to tackle.Hence, we reformulate ( 16) as the sum-average SINR of u-th user [49]: where Λ u denotes the selection matrix with u-th diagonal element is one and the others are zero and Λ u = I U −Λ u , the numerator and denominator of the right hand side of (17) denote the average desired signal power and total multi-user interference (MUI) plus noise power, respectively, for u-th user over all subcarriers.
Lemma 1.For any communication user, the sum-average SINR in (17) is a lower bound on the average SINR in (16), i.e.SINR Cu ≤ SINR Cu .

May 9, 2023 DRAFT
Proof: Let us first simplify the SINR in ( 17) and ( 16) as where Based on Sedrakyan's inequality, we have Further, leveraging on Cauchy-Schwarz inequality, we have Substituting ( 19) and ( 20) into ( 18), we can conclude that SINR C ≤ SINR C and the equality holds if and only if This completes the proof.

C. Radar Receiver
The signal at the radar receiver follows a model similar to that of communications with the addition of the two way propagation.Similar to Section II-B, after sampling, CP removal and applying N B K-point FFT, the echo signal on the k-th subcarrier at the radar receiver is where the composite channels across DFBS-t-th target-receiver and DFBS-q-th clutter-receiver are, respectively, where α t and α q denote the RCS for the t-th target and q-th clutter, respectively, denotes the path from RIS to target.
For the radar system, the performance of target detection is largely determined by the output SINR 2 and the detection performance for a given false-alarm improves with SINR.Thus, the 2 Here, interference includes clutter response in addition to response from other targets.
DRAFT May 9, 2023 maximization of SINR is widely used as the optimization criterion [50,51].As in the communications system, in case of multiple targets, the SINR for each of the targets need to be improved, particularly, improving the echo signal while suppressing interference.In this context, based on (21), we define the radar SINR for the t-th target as where w k,t denotes the radar receive filter at the k-subcarrier [52].
Remark 2. Transmit beampattern matching is another approach widely employed as the optimal design criterion for radar sensing in the conventional and RHS-aided DFRC system without RIS [34,[53][54][55].However, this may not be directly applicable for RIS-assisted radar-only or DFRC system, especially, in dense environments [12,56] because it is difficult to focus the beam towards the target and RIS directions without pathloss information.For example, if the LoS path is totally blocked, then the allocation of power to the direct link is not needed.Further, in such environments with a weak or no line-of-sight channel, aggregated interference caused by multiple reflections from other objects in the environment could result in performance loss, despite the beampattern design.Hence, the output SINR, which includes these artefacts and the design of received filter, is recognized as the proper radar metric for RIS-aided DFRC [57,58].

D. Problem Formulation
Our goal in a radar-centric DFRC is to maximize the worst-case radar SINR while guaranteeing the communications SINR over all users.We formulate this optimization problem as maximize where |Φ| = 1 indicates unit magnitude for each diagonal entry of the phase matrix Φ, m nx,ny = 0 denotes the (n x , n y )-th RHS element is disabled and m nx,ny = 1 denotes the (n x , n y )-th RHS element is unit gain, and η denotes the threshold of communications user.Note that the above optimization problem involves the maximin objective function, difference of convex (DC).and unimodular constraints.Meanwhile, it is a fractional quadratically constrained quartic program (QCQP) problem in multiple variables and thus difficult to solve directly.Despite the existence of several approaches for non-linear optimization addressing fractional QCQP for single variable, the problem in (25) poses unique challenges that prevent an adaptation of using existing methods.
These challenges include, highly coupled variables, maximin objective and several nonconvex constraints as well as presence of discrete variables.As a consequence, we develop the AO algorithm in the sequel.

III. ALTERNATING OPTIMIZATION
We first decouple the nonconvex fractional QCQP into four subproblems of designing the receive filter along with digital, holographic, and passive beamformers.Then, we resort to AO procedure to solve these problems.
A. Sub-problem 1: Update of receive filter w k,t and w k,u We first define the communications filter T for all subcarriers.Then, for fixed Φ, F k and M, the subproblem with respect to w k,t and w k,u is where η = Kη and the block diagonal matrix used in P 1 is given by and Note that the objective function of problem ( 26) is separable in terms of the variables w t and w u .
Hence, we obtain an optimal solution for the maximin problem (26) by solving the following disjoint problems where Based on Proposition 1, the close-form solution of P 1.1 is given by

B. Sub-problem 2: Update of digital beamforming F k
For the fixed w k,t , w k,u , M, and Φ , the subproblem with respect to F k is where and the block diagonal matrices Σ P 2 t , Σ P 2 t , Σ P 2 u and Σ P 2 u are, respectively, defined as and It is worth noting that P 2 is highly nonconvex due to the fractional quadratic objective function and difference of convex (DC) constraint.Inspired by the minorization-maximization (MM) algorithm [59], we can linearize the corresponding convex function.Specifically, for the function f (x) = x H Hx, the following inequality is always satisfied where H is positive semidefinite (PSD) matrix, x (l) denotes the current point (at the l-th iteration), and the equality holds if and only if x = x (l) ; See [60].
Based on (34), we simplify P 2 as where f (l) denotes the value of f at l-th outer AM iteration.This is a standard fractional maximin problem that can be solved using the generalized Dinkelbach-based method [61].Thus, we can solve problem (35) by reformulating it as and solving using Algorithm 1. Noted that for the simplified two variable quadratic programming with convex constraints (36), Dinkelbach algorithm can be convergent to the global optimal solution [61].

C. Sub-problem 3: Update of holographic beamforming M
With w k,t , w k,u , F k , and Φ fixed, the subproblem with respect to M is where m = M T 1 NxNy and the matrices Σ P 3 t , Σ P 3 t , Σ P 3 u and Σ P 3 u are similarly defined as Similar to P 2 , we reformulate P 3 as This subproblem is similar to the previous P 2. 1 and hence it can be also solved by Algorithm 1 with variables appropriately substituted.Notice that, based on the inequality (34), the objective value in ( 37) is always equal or great than the simplified problem (39) which guarantees the monotonic increasing of radar SINR in the MM iteration.

D. Sub-problem 4: Update of passive beamforming Φ
With w k,t , w k,u , F k , M fixed, the subproblem with respect to phase-shift design is where φ = Φ T 1 N R denotes the phase-shift vector, ϕ denotes the auxiliary variable, k,u and Σ P 4 k,u are defined as following Accordingly, P 4 is reformulated as where where φ = diag(ϕ), and the variables related to (43b) can be similarly defined as (43a) and hence omit it herein.
Rewriting the problem P 4.1 as where z = [z 1 , • • • , z T ] denotes the weight vector, ft (φ, ϕ) is the right side of inequality (34) and thus a lower bound function of f t (φ, ϕ), and p P 4 u is the reconstructed vector to linearize the SINR constraint.Then, the augmented Lagrangian function of ( 46) is where ρ is the penalty parameter, u and w denote the auxiliary variables, and Then, as the previous work problem [39], P 4.3 is solved by the C-ADMM algorithm which is summarized in Algorithm 2. Noted that in each ADMM iteration, the nonconvex unit-sphere programming ( 48) and ( 49) can be directly solved by RSD algorithm, see details in [40].Output: φ (l+1) = φ l 4 .

5:
Update φ l 4 via solving Update ψ l 4 via solving Update the dual variable u and w; 8: Based on above, the proposed AM algorithm for the optimization problem ( 25) is summarized in Algorithm 3.

E. Computational complexity
The overall computational burden of Algorithm 3 is linear with the number of outer iterations.
Meanwhile, at each outer iteration, the closed-form solution of radar filter . Then, for the update of the transmit beamforming matrix F k , k = 1, • • • , K, the computational cost of Algorithm 1 is linear with the number of inner iterations l 2 .At each inner iteration of the Dinkelbach-based method, the problem is solved by the CVX [62] with the complexity of O(K 3 N 3 RF U 3 ).Similarly, for the update of the holographic beamforming matrix M, the complexity is O(N 3 B ) In order to update the phase-shift matrix Φ in Algorithm 2, the C-ADMM and RSD algorithm are combined with the total complexity O(l 4 (2l , where l 3 and l 4 denote the maximum iteration number and DFBS-RIS (g BR,k ) are obtained by appropriate substitution for ǫ and r in the expression K 0 ( r 0 r ) ǫ .Without loss the generality, we set the RCS of targets and clutters as α t = 1 and α q = 1, respectively.The transmit power at each subcarrier is set to P k = 5 dBw, ∀k and the noise variances are set to σ 2 R = −45 dBm and σ 2 C = −55 dBm for radar and communication, respectively.We set the SINR threshold η = 9 dB for all users.The step size for RSD algorithm is set to 3.98 and the termination threshold for AM algorithm is set as ζ 3 = 10 −4 .The initialization of Φ (0) is randomly generated diagonal matrix, whose entries are assumed be constant modulus with random phase-shifts.The maximum iteration for RSD, C-ADMM and AM are set as 100, 24 and 30.

A. Convergence of the Proposed Algorithm
Fig. 2(a) illustrates the convergence of C-ADMM algorithm for solving the subproblem related to transmit beamformer design in the first AM iteration.We compare two different scenarios: (1).Random RHS, which keep M fixed before C-ADMM; (2).Optimal RHS, which update M by Algorithm 1 before C-ADMM.Note that even though the RSD algorithm can not obtain a close-form solution for the nonconvex manifold optimization problem in each ADMM iteration, the objective value of the proposed C-ADMM is still monotonically increasing in the first AM iteration which indicates the phase-shift update provides additional enhancement of minimum radar SINR.Meanwhile, the optimal RHS case reaps the better radar SINR performance.Fig. 2(b) demonstrates the overall convergence of proposed AM algorithm in different scenarios: (1).Random RHS, w/o RIS involves optimal receive filter, digital beamforming, random RHS, and non-RIS, which is set as the benchmark; (2).Random RHS, random RIS with optimal receive filter, digital beamforming, random RHS, and random RIS; (3).Random RHS, optimal RIS comprises optimal receive filter, digital beamforming, random RHS, and optimal RIS; (4).
Optimal RHS, w/o RIS involves optimal receive filter, digital beamforming, RHS, and non-RIS; (5).Optimal RHS, random RIS with optimal receive filter, digital beamforming, RHS, and random RIS; (6).Optimal RHS, optimal RIS comprises optimal receive filter, digital beamforming, RHS, and RIS.Despite multiple linearization steps being utilized for both objective and constraint functions in the subproblems, the proposed AM algorithm is convergent in about 10 iterations.The minimum radar SINR objective is monotonically increasing with the iterations since the objective value of the original problem is always greater than the linearized problem and Dinkelbach method can also guarantee the monotonic objective value.The cases with the optimal RHS achieve the higher worst-case radar SINR compared with the random RHS.Meanwhile, with random phase-shifts for RIS, the worst-case radar SINR is lower than that the optimal phase-shifts but higher than the non-RIS case.Optimizing the phase-shifts leads to at least 2.8 and 3.7 dB radar SINR gain for random RIS and non-RIS deployment, respectively.

B. Effect of Transmit Power
We illustrate the variation in the minimum achievable radar SINR with respect to different parameters to demonstrate the flexibility of our approach.Fig. 3 shows the achievable worstcase radar SINR versus the P k for each subcarrier.Clearly, if we simultaneously optimize the receive filter, digital beamformer, holographic beamformer and passive beamformer, the highest radar SINR is achieved compared with random RHS, non-RIS and random RIS.Furthermore, for the RIS-assisted DFRC, the optimization of the holographic beamforming by RHS can provide around 1.3 dB SINR gain owing to its flexibility on radiation amplitude controlling.
It is also observed that for all six cases (i.e., random RHS, optimized RHS, w/o RIS, random RIS and optimized RIS), increasing of 1 dB transmit power can bring around 0.4 dB radar SINR enhancement.Meanwhile, the proposed algorithm with both optimal RHS and RIS achieves the DRAFT May 9, 2023  highest radar SINR which indicates the advantage of our proposed RIS-assisted holographic DFRC system.

C. Effect of radar LoS Pathloss Exponent
Fig. 4 shows the achievable worst-case radar SINR versus radar LoS pathloss exponent ǫ dir (i.e., direct pathloss exponent between RHS and targets/clutters).The radar SINR is gradually reduced with the increase in the LoS pathloss exponent.Meanwhile, the proposed methods with optimal RIS (random or optimal RHS) provide better radar SINR with the weak direct path, i.e., ǫ dir ≥ 3.6.This demonstrates RIS is more effective when a stable LoS path is missing.x (or N R y ).It is observed that random RIS can not provide a stable performance enhancement compared with the optimal RIS, even with the larger element number.Different from that, the proposed method with the optimal RHS and RIS is able to enhance the system continuously.Meanwhile, it is seen that each passive RIS element enhancement w.r.t x-(or y-) axis can offer around 0.4 dB SINR improvement under the optimal receive filter, digital beamforming, holographic beamforming, and passive beamforming, see red curve.This kind of enhancement is crucial for DFRC system, especially in the dense environment, where the LoS components are weak.Fig. 5(b) shows the achievable worst-case radar SINR versus RHS element number w.r.t x-(or y-) axis, N B

D. Effect of Surface Element Number
x (or N B y ).With the increase in RHS elements, random RHS does not provide a stable radar SINR improvement.On the contrary, the optimal RHS is able to enhance the system continuously which highlights the importance of proposed joint design scheme for RIS-assisted holographic DFRC.

E. Effect of Communication SINR
In Fig. 6, the results of the achievable worst-case radar SINR versus η are presented.We vary the communications SINR threshold from 0 to 18 dB which covers the demands in most existing wireless applications.We observe that, in the DFRC system, the higher requirement on the communications SINR leads to the performance loss for radar SINR.Hence, even with   the assistance of RIS, the proposed holographic DFRC system still demonstrates a performance trade-off between radar and communications, albeit, an enhanced one.Note that the threshold of communications SINR can be flexibly selected based on the system requirement, e.g., decoding performance or outage probability, and it can be also varied for different users.

F. Effect of Number of Users
Fig. 7 depicts the achievable worst-case radar SINR as a function of U. It again indicates that the proposed method with the optimal RIS and RHS achieves the best radar SINR compared with non-RIS, random-RIS and random-RHS cases in terms of different user numbers.On the other side, the increase of one user can lead to only around 0.35 dB radar SINR loss which expresses the robustness of the proposed DFRC system.

V. SUMMARY
We considered the joint deployment of the RHS and RIS to assist a wideband DFRC system with OFDM signaling.Our design of digital, holographic, and passive beamformers shows improvement in the performance of the DFRC system when compared with non-RIS and non-RHS systems.The key challenge to the design problem arises from the coupling of various parameters and nonconvexity.We showed that our alternating optimization approach facilitates not only decoupling but also a tractable design.
For highly dynamic wideband channels, machine learning methods may be employed to estimate the channel state [37].This may also be incorporated with the RHS DFRC systems.A particularly complicated procedure in the RHS is estimation of angle-of-arrivals because, unlike phased arrays whose feeds directly receive the signals, RHS feeds receive the signals after modulation by holographic patterns.This requires an additional maximum likelihood estimation step for AoA estimation.
Holographic DFRC is currently at an early stage of research.As a result, substantial challenges in prototyping, channel modeling, and optimized design remain.Graphene-based arrays and leaky-wave antennas are other alternatives for realizing these antenna structures [63].Further, exploiting other non-OFDM multiple access technologies for signaling also offers a promising research avenue for holographic DFRC in the near future [34].Note that problem ( 52) is the complex-valued homogeneous QCQP, which can not be directly solved due to the convex objective function and quadratic constraint.However, we can write the Lagrangian function of problem (52) as where λ denotes the corresponding Lagrange multiplier.Setting the derivative of the Lagrangian in (53) with respect to w to zero, i.e., ∂L(w, λ) ∂w = 2Aw − 2λBw = 0.
Based on above, the extreme value of equation ( 53) and should satisfy According to (55), we found that λ and w are the eigenvalue and corresponding eigenvector of B −1 A, respectively.Meanwhile, from ( 52) and ( 54), we have Hence, we conclude that the maximum values λ * of w H Aw and the generalized Rayleigh quotient (51) is λ * = λ max (B −1 A), and the corresponding vector is w * = ρ max (B −1 A).This completes the proof.

Remark 1 .
Due to the passive nature of RIS which precludes baseband signal processing unit, RIS can only work in the resonance frequency and thus the phase-shift is common for all DRAFT May 9, 2023

t = 1 ,Proposition 1 .
• • • , T and u = 1, • • • , U. Note that P 1.1 is composed by a set of generalized Rayleigh quotient programming which can be solved by the following Proposition 1.For a fixed Hermitian matrix A, and a fixed positive definite matrix B, the maximum values λ * of the generalized Rayleigh quotient w H Aw w H Bw , where w = 0 and the corresponding vector w * satisfy: λ * = λ max (B −1 A), w * = ρ max (B −1 A), where λ max (•) and ρ max (•) denotes the operation of largest eigenvalue and principal eigenvector.Proof: See Appendix A.

4 :
Update f (l) via Algorithm 1 and reconstruct F M (l) via Algorithm 1 by changing the input variables; 6:

Remark 3 .
and Φ (l) ; The optimization framework is general and can handle different architectures by appropriate constraints on M and choice of V.In particular, letting N RF = N B , and M = V = I N RF , the problem reduces to classical digital beamforming based DFRC.Further, letting M = I and different choice of V leads to different hybrid analog digital beamforming architectures, including partial and fully connected.

Fig. 5 (
Fig.5(a), shows the achievable worst-case radar SINR versus RIS element number w.r.t x-(or y-) axis, N R x (or N R y ).It is observed that random RIS can not provide a stable performance enhancement compared with the optimal RIS, even with the larger element number.Different
be equivalently reformulated asmaximize w =0w H Aw, subject to w H Bw = 1.
Section III, we develop our AO-based algorithm to tackle the formulated nonconvex maximin problem, in which the corresponding subproblems are solved iteratively.We evaluate our methods in Section IV through extensive numerical examples.Finally, we conclude in Section V.