Downlink Beamforming for Dynamic Metasurface Antennas

Dynamic metasurface antennas (DMAs) have great potential to be used in the radiator/receptor elements of future wireless transmitters and receivers, replacing conventional metallic antennas. This can be attributed to their unique properties, such as the ability to be reconfigured in real-time and to reduce the radio frequency chains, resulting in low implementation cost. However, the Lorentzian constraint associated with the DMA elements poses a challenge to real-time configuration and limits the application of the DMA. In this study, we propose a DMA-based wireless network, wherein a DMA-equipped base station (BS) communicates with single and multiple users. For the single-user scenario, we develop an optimal algorithm to maximize the signal-to-noise ratio of the user, which provides the weight of each DMA element in closed form. Furthermore, for multiple users, we formulate the weighted sum rate (WSR) problem and employ techniques from the single-user case to develop an efficient alternating optimization algorithm, which optimizes both the transmit precoders and DMA weights, to enhance the WSR of the system under the transmit power constraint of the BS. The numerical results demonstrate the effectiveness of the proposed algorithms in achieving better performance than that of the benchmark schemes.


I. INTRODUCTION
T HE world has been witnessing fast-growing throughput demands. A significant amount of the data traffic is channeled through wireless networks owing to their flexibility and low cost compared with wired networks. To meet these demands, wireless networks have constantly been evolving, with each change improving upon the preceding technology. For example, massive multiple-input multiple-output (MIMO), which employs many antennas, is presently the best technology for enabling the delivery of high data traffic and increased spectral efficiency (SE) through the wireless channel [1], [2], [3], [4]. Furthermore, the combination of massive MIMO and high frequencies such as millimeter wave (mmWave) and terahertz (THz) is considered a breakthrough technology for traffic-intensive wireless networks. However, each constituent piece of this combination has its limitations. The use of large antenna arrays in massive MIMO can inevitably increase the complexity, power consumption, and hardware cost. Additionally, because of large antenna apertures, massive MIMO can potentially extend the near-field region [5], [6], [7], [8], [9], necessitating entirely different signal processing techniques. This is the case because of the fundamental change in the electromagnetic wave properties from those of the approximated planar wavefronts in the far-field to those of the actual spherical wavefronts in the nearfield. For instance, the near-field spherical waves are able to focus the signal energy in terms of both angle and distance via the envisioned approach called beam focusing, whereas the far-field techniques can only direct a signal toward a given angle via a technique commonly known as beam steering. Moreover, channel estimation [10] and beam split problem in wideband systems [7] still remain pressing challenges in near-field wireless communication. Similarly, there are several issues associated with the use of high frequencies that hinder large-bandwidth communication. Among them, the high path loss is important because it significantly reduces the communication range and presents difficulties for heuristic signal processing techniques [11], [12], [13].
Several efforts have been made to address some of these challenges. The power consumption issue in massive MIMO, for example, has been intensively addressed using hybrid architectures, which reduce the number of radio frequency (RF) chains by adopting relatively less expensive and low-power phase shifters [14], [15], [16]. Thanks to ultra-dense networks [17] and cell-free massive MIMO [18], [19], [20], [21] technologies, which share the same philosophy of creating a user-centric architecture that significantly reduces signal propagation distance, high-frequency communication is still a possible vision. In this study, we extend the prior efforts of realizing large antenna arrays with low power consumption based on an emerging technology, known as dynamic metasurface antennas (DMAs). The DMA comprises a multitude of artificial metasurface elements capable of realtime reconfiguration. The reconfiguration, which enables the programmable control of the transmitted and received signal, is achieved by incorporating solid-state switchable components into each element. Notably, the adjustable properties of metasurface materials are being exploited not only by DMA but by other technologies as well. In the literature, it has also been suggested that other technologies, such as intelligent reflecting surfaces (IRS) [22], [23], [24], [25] and intelligent transmitting surfaces (ITS) [26], also make use of customizable metamaterials in somewhat different ways. For instance, in wireless communication, IRS is commonly placed near the receiver or transmitter for controlled reflection of signals to the desired destination. By contrast, DMA, which is fundamentally an array of antennas, must be attached to either the transmitter or receiver to facilitate the necessary signal interactions for transmission and reception. In addition, as the IRS only reflects signals rather than retransmitting them, it is made up of semi-passive elements, which have extremely low power consumption. The same objective is achieved in DMA via natural reduction of radio frequency (RF) chains, which reduces the hardware complexity and helps realize large-scale antenna arrays as in massive MIMO. Furthermore, for the same array aperture, DMA can pack a larger number of elements than the conventional massive MIMO, owing to the small size of the elements. Moreover, in DMA, signal processing techniques such as analog combining, beamforming, and antenna selection are implemented automatically without additional hardware as in hybrid architectures.
Since its inception, the DMA technology has mostly been explored for use in satellite communications, radar systems, and microwave imaging [27]. The performance of DMAs in such applications has been reported to be good in terms of simplicity, power consumption, and flexibility. However, its use in cellular communication, particularly with massive MIMO, remains limited. A few studies have examined the use of DMAs in cellular communication and have served as a reference [6], [27], [28], [29], [30]. For example, Shlezinger et al. [27] studied the multi-user MIMO system, in which a base station (BS) uses DMA to receive signal streams sent from different user terminals. They proposed two alternating optimization (AO) algorithms for frequency-flat and frequency-selective channels of DMA. Their results revealed the potential advantages of using DMAs over standard antennas for realizing large-scale and low-power massive MIMO. This work was further extended in [28], wherein a similar DMA-equipped BS was bit-constrained and used to operate multi-user MIMO orthogonal frequency division multiplexing (OFDM) communications. The authors formulated a model of coarsely quantized DMA outputs and developed an algorithm for OFDM symbol detection. These two studies focused on uplink communication. However, the researchers in [6] and [30] employed the opposite approach, that is, downlink communication. As stated earlier, large-scale antenna arrays have the potential to extend the near-field regions; these studies specifically investigated the use of DMA for beam focusing in near-field communication regions. They formulated a mathematical model for the near-field wireless channel, which was later used in the considered antenna architectures, which are as follows: fully digital antennas, hybrid antennas, and DMA. They finally adopted the AO algorithm assisted by other techniques to solve the resulting optimization problems for different architectures.
Most of the previous studies relaxed the Lorentzian constraint on the DMA elements to make the optimization problems tractable. Consequently, the analysis of the problem was simplified, which led to the development of feasible algo-rithms; however, this simplification resulted in performance degradation. We learned that such problems can possibly be solved with manageable complexity without employing these performance-degrading relaxations. Thus, the focus of this study is to investigate the use of DMA in downlink multipleinput single-output (MISO) systems for both single-user and multiple-user environments. Unlike prior studies that focused on the near-field communication region, our study focuses on the general far-field communication scenario.
The main contributions of this study can be summarized as follows: • We propose a novel technique of splitting the problematic Lorentzian constraint into two parts, which significantly simplifies the analysis of various system setups. In addition, this technique also avoids the adoption of relaxation procedures when formulating the corresponding optimization algorithms. • The effectiveness of the splitting technique is evaluated by analyzing the single-user MISO systems. Using this method, we develop a simple algorithm that provides the optimal solution for DMA configurable weights in a closed form. • We then extend the analysis to the multi-user system, wherein we introduce the weighted sum rate (WSR) maximization problem for the DMA. We use the aforementioned splitting technique to simplify the development of the proposed tractable AO algorithm, which alternately optimizes the precoders and DMA configurable weights. Specifically, the precoders are optimized via the minimum mean square error (MMSE) scheme when the DMA configurations are fixed, and when the precoders are set, DMA configurations are solved via manifold optimization (MO). • Finally, after analyzing the average computational complexities of the proposed algorithms and benchmark schemes, extensive numerical results are provided to validate the accuracy of the analysis and demonstrate the effectiveness of the proposed algorithms.
The remainder of this paper is organized as follows. Section II provides a brief introduction to DMA architecture and its behavior when interacting with electromagnetic waves. Furthermore, the signal model and problem formulation for both single-user and multi-user systems are presented. Section III presents the optimal solution for singleuser systems, whereas Section IV provides a sub-optimal yet highly competitive solution for multi-user systems. Section V presents the simulation results. Finally, Section VI concludes the paper.
Hereinafter, R and C denote the real and complex domain, respectively. Scalars are denoted by lower-case italic letters. The bold-face lower-case (a) and upper-case (A) letters denote a vector and a matrix, respectively. For a matrix G, G i,j denotes the element in the i-th row and j-th column. The transpose, conjugate transpose, and complex conjugate are denoted by (·) T , (·) H and (·) * , respectively. We use I n and 0 n,m to denote an n×n identity matrix and an n×m zero matrix, respectively. For any vector x, [x] i is the i-th element of x; |x| and x denote its absolute value and Euclidean norm, respectively. The remainder of the division of a by b is denoted by a modulus operator, mod (a, b). E (·) denotes the expectation operator. arg(·) represents the phase extraction operator, which returns the phase of its argument, whereas · represents a rounding-up operator, which rounds its argument to the nearest larger integer. A circularly symmetric complex Gaussian random variable with mean υ and variance σ 2 is denoted by CN υ, σ 2 .

II. SYSTEM MODEL AND PROBLEM FORMULATION
In this section, we present the mathematical model of the proposed DMA-based communication setups. First, we provide a summary of the DMA architecture and signal behavior in Section II-A. Next, in Section II-B, we present the signal model and problem formulation for a downlink single-user MISO system. We then extend this model in Section II-C to a multi-user MISO model and formulate its corresponding sum rate maximization problem.

A. DMA Architecture and Signal Model
DMAs belong to a class of artificial metamaterials that allow engineering of their physical properties, such as permittivity and permeability, to attain a certain desired behavior toward electromagnetic waves. Recently, these materials have received considerable research attention in the wireless communication field owing to their simple and flexible structure, as well as the introduction of new signal processing abilities, which facilitate several new use cases.
Architecturally, DMA is made up of microstrips/ waveguides, containing multiple metamaterial elements arranged vertically. One end of the microstrip is connected to the RF chain that is responsible for the baseband processing of the signal. The elements inside the microstrip, which are generally sub-wavelength spaced, are arranged horizontally, hence, making a planar surface of radiating metamaterial elements. As shown in Fig. 1, multiple elements in a single microstrip are connected to the same RF chain through a waveguide. This arrangement resembles the partially connected hybrid architecture in massive MIMO [31], [32], [33]. However, the former is more efficient as it naturally possesses this capability, whereas the latter incurs additional hardware costs of using unit-norm constrained phase shifters. This gives the DMA a major advantage in terms of hardware size and signal processing flexibility.
Owing to its distinctive architecture, the DMA exhibits unique signal interactions. For example, during transmission, each RF chain feeds its corresponding microstrip with the same baseband-processed signal. As the signal traverses through the waveguide, it is radiated to the wireless channel by each element. Depending on the frequency response (state) of the element and the characteristics of the signal reaching the element, each radiation takes on a distinct form. The signal characteristics inside the microstrip are normally dictated by the nature and size of the waveguide material [34], whereas the state of the element is configurable. These two properties primarily govern the signal behavior in the DMA. Thus, we explain them further in the following sections.

1) Frequency Response of a Radiating Element:
Each radiating element is considered as a resonant electrical circuit, whose frequency response q, for frequency f , is described by the Lorentzian form [35] and [36] where Ω ∈ R is the oscillator strength, f 0 is the resonance frequency, and Γ is the damping factor. These parameters can be configured for each element to attain the desired outcome. Nonetheless, for narrowband systems, previous studies have reported that it is safe to assume that the elements exhibit flat frequency responses [28]. With this assumption, the state configuration of each DMA element takes the following form where θ is the configurable phase shift on each element.
2) Signal Propagation Inside the Microstrip: Similar to a wireless channel, when the signal propagates through the microstrip, it undergoes two main effects: attenuation and phase shift. The attenuation level is mainly influenced by the material characteristics and size, whereas the phase shift typically depends on the wavenumber and signal location. Therefore, if α i and β i denote the attenuation coefficient and wavenumber of microstrip i, respectively, the signal observed by the l-th element, located at ρ i,l , from the input port of the microstrip is given by: Consequently, the relationship between the signal x ∈ C N d ×1 , which is input to N d DMA microstrips, each containing N e elements, and the corresponding transmitted signal t ∈ C N ×1 from all N = N d × N e elements is given by: is a diagonal matrix of dimension N × N , whose diagonal elements represent the signal propagation effects h i,l , ∀ i,l , and is the block-diagonal matrix of dimension N ×N d that collects the configurable weights of each element, with q i,l denoting the weight of l-th element in the i-th microstrip.

B. Single-User Model
We consider a system with a BS comprising N radiating metasurface elements as its transmit antennas, serving one single-antenna user. When the BS transmits a precoded unit-power symbol d by a precoder f ∈ C N d ×1 with a power P , the received signal y by the user can be expressed as where g ∈ C N ×1 is the channel between the BS and the user; z ∼ CN (0, σ 2 ) is the additive white Gaussian noise at the receiver of the user. The elements of H ∈ C N ×N and Q ∈ C N ×N d are defined in (4) and (5), respectively. For single-user systems, we aim to maximize the signal-tonoise ratio (SNR), which is given by This problem is formulated as follows: where f 1 (f , Q) = |g T HQf| 2 is the objective function obtained by dropping the constant terms, P and σ 2 , from the SNR. For a single user MISO system, the optimal precoder is given by maximum ratio transmission (MRT), which in this Substituting this into f 1 reduces this two-variables expression into a single-variable one, f 2 (Q) = g T HQ 2 , which eventually simplifies problem (P1) to

C. Multi-User Model
For the multi-user scenario, we consider the same setup as in the single-user system, except that the number of single-antenna users is increased to K. Similar to the singleuser scenario, the BS uses power P to transmit a precoded signal is the precoder for a unit-power information symbol d i , intended for user i. The signal received by user k from all radiating elements of the BS is given by where g k ∈ C N ×1 is the channel between the BS and the user k. To obtain a more manageable expression that simplifies the subsequent analysis, we exploit the unique structures of matrices H and Q to exchange the shapes of Q and g k . For this, the DMA weight matrix Q is vectorized to a column vector q ∈ C N ×1 , whose i-th element is given by N]; similarly, the channel vector g k is recast to a block-diagonal matrix G k ∈ C N d ×N , whose element in the m-th row and n-th column is given by This exchange transforms (11) into One of the major challenges in multi-user communication is the interference from other users, which can complicate the system analysis. A popular scheme for dealing with multi-user MIMO systems is based on weighted MMSE (WMMSE) sumutility maximization, proposed in [37]. In this scheme, the WSR for our system is formulated as where and 0 ≤ ω k ≤ 1 denote the signal-to-interference-plus-noise ratio (SINR) and the priority of user k, respectively, and J k ∈ C N ×N d = H H G H k is used to simplify the notation. Let F = [f 1 , f 2 , · · · , f K ] ∈ C N d ×K be a collection of all precoders for all users in the network. In this study, we aim to find the precoder F and DMA configurable weights Q that maximize the sum rate R 1 of the system, without exceeding the total transmit power of the BS while ensuring that all reflecting elements operate within the Lorentzian region. This problem is represented by

III. OPTIMAL SOLUTION FOR SINGLE-USER MISO SYSTEMS
In DMA-based systems, the physical constraint on the radiating element, which allows operation only within the Lorentzian region, is a major challenge. The Lorentzian constraint restricts the phase and amplitude of its resonator, in this case q, to the range 0 ≤ arg (q) ≤ π and 0 ≤ |q| ≤ 1 , respectively. In general, these kinds of limitations complicate signal processing. To resolve these complications, we can consider a three-step process, that is, i) relax the constraints into more manageable ones, ii) solve the problem with relaxed conditions, iii) recast the solution to a close approximation of a real solution. Most of the early works in DMA relied heavily on this technique. For instance, in a study [6], the Lorentzian constraint was relaxed to a phase-only constraint with a constant amplitude. Essentially, this approach focused only on the phase, whereas the amplitude had a constant value of 1. To make the problem more tractable, the optimizable range of the phase was expanded to 2π [6]. The solution was recast to the original Lorentzian restriction in the final step to make it practical.
As stated previously, this approach has an undesired effect on the system performance, which raised the need for novel techniques to achieve better performance with reasonable complexity. In this section, we present one such technique for the single-user downlink MISO system. The key idea is to split the Lorentzian constraint in (2) into two terms, that is, the complex constant term j 2 and the exponential term e jθ 2 , containing the optimizable phase. The unique structures of matrices H and Q allow (P2) to be decomposed into N d independent subproblems, which can be solved individually, as each optimizable weight appears in only one term and one subproblem. Essentially, each subproblem optimizes the weights of a single microstrip. To get more insight, the objective function of (P2) is rewritten as Taking a closer look at the first subproblem, we have the following: where θ n,m is the configurable phase shift of the n-th element in the m-th microstrip. Let c = j 1 . By dropping the constant term 1 4 , the right-hand side term of (20) can be further simplified as: By dropping the constant term |c| 2 , (23) can be written as with {·} denoting the real part of a complex number.
In the first term of (24), the real part is maximum when the imaginary part is zero, which is achieved by setting arg (b i ) = arg (c) , ∀ i . By aligning all of the phases, this assignment also complies with the second term. As the second sum contains conjugated values, the phases cancel each other and maximize the amplitude. Following this observation, the phase shift of Q i,1 is given by This process is repeated for all other subproblems in (19). We summarize the procedure in Algorithm 1. As this technique provides a closed-form solution for each subproblem without relaxing the Lorentzian constraint, we regard it as an optimal solution for the DMA-based single-user downlink MISO system.

A. Algorithm Description
This section provides the joint optimization algorithm of the precoders and DMA weights for the multi-user scenario. We specifically develop a solution for (P3), which is more challenging than (P2). The difficulty arises from the fact that apart from solving for DMA weights, we also have to solve for precoders of all transmitted symbols, which unlike the single-user scenario, the solution cannot be obtained in a closed-form. Previous studies [6], [30] have also arrived at formulations similar to (P3). Despite proposing well-tractable Algorithm 1 Optimal Algorithm for Single-User MISO Systems Input : g, H, N, N Output: Q algorithms for solving their problems, the aforementioned relaxation approach severely affects the performance. Thus, we propose an AO algorithm, which optimizes the precoder F and DMA weights q in two steps alternatively. In the first step, we optimize the precoders while fixing the DMA weights at their last updated values, and in the second step, we do the opposite. This process is repeated until convergence. In the following, we describe these steps in more detail. When q is fixed, the resulting subproblem to solve for F is reduced to a WSR maximization problem for conventional MISO systems [37]. A popular method for solving such problems is the WMMSE algorithm [38], which iteratively updates the following values until they converge: where μ ≥ 0 is the Lagrangian variable for the transmit power constraint, obtained by line search techniques such as the bisection method. Once F is obtained, we fix it and focus on optimizing q. However, this optimization is not straightforward because of the performance degrading Lorentzian constraint on the optimizable variable. Therefore, in this step, we adopt the same technique used in the single-user scenario of splitting the Lorentzian constraint into two parts. For ease of representation, we employ the following notations: s ∈ C N ×1 = e jθ1,1 , · · · , e jθN e,Nd T , that help to compactly transform the SINR expression of the k-th user into The new SINR expression in (32) presents a different optimizable variable θ n,m with a manageable constraint. Consequently, the inputs of the objective function (when F is fixed) in (14) must be modified to which also transforms the problem statement. The new problem takes the following form: The problem (P4) has a less restrictive constraint on θ n,m . In fact, the new constraint forms a complex circle manifold with a continuous and differentiable objective function R 2 (s), which allows the easy adoption of the MO algorithm [39], [40], [41]. MO can conceptually be broken in three main steps 1) Computation of Riemannian Gradient: The Riemannian gradient of a function R 2 at point s k , denoted by grad s k R 2 , is the orthogonal projection of the Euclidean gradient ∇ s k R 2 onto the tangent space. For our case, this is given by where the Euclidean gradient is The Euclidean gradient expression is derived using the natural logarithm version of (33) because of its computation simplicity and the independence of the solution from the base of the logarithm [37]. 2) Finding the Search Direction: The search direction of the conjugate gradient at point s k can be computed as where T (·) is the vector transport function given by k is the conjugate gradient update parameter at point s k , chosen as the Polak-Ribiere parameter [39], and d k is the search direction at point s k . 3) Retraction: This is the process of finding the next point s k+1 in the manifold by mapping the current point s k on the tangent space. Mathematically, it is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where τ is the step size, normally given by Armijo rule [42]. The steps above, which are summarized in Algorithm 2, are repeated until convergence, that is achieved when the Riemannian gradient in (36) approaches zero [41], [43]. Input : g, H, N, N d , N e  1 Find the initial s 0 and let k = 0. 2 Compute grad s k R 2 using (36).

end
Optimization of q when F is fixed. 8 Update q by using Algorithm 2. 9 until Objective function R 1 (f k , q) converges Output: F and q.
The complete proposed algorithm for the multi-user DMAbased downlink MISO system is summarized in Algorithm 3. As this algorithm employs AO and MO to optimize the precoders and the DMA weights, we abbreviate it as AO-MO. In the first step of this algorithm, we start by randomly initializing the DMA weights q. The initial values of q must be within the Lorentzian region to ensure the convergence of both the MMSE and MO schemes. Steps 3 -7 focus on optimizing the precoders of all users by using MMSE when the DMA weights are fixed, whereas, in step 8, we use the MO algorithm to update the values of q, keeping the precoders fixed. This process of optimizing F and q is repeated until the objective function converges.

B. Complexity Analysis
To conclude the section, we provide complexity analysis of our proposed scheme in terms of floating-point operations (FLOPS) [44], [45]. Additionally, for performance comparison, we also analyze the complexities of the algorithms proposed in [6] and [30], which are used in Section V as the benchmark schemes. Zhang et al. [30] proposed an algorithm that uses AO and MO for its optimization process but with the relaxed Lorentzian constraint on Q. Therefore, we call it "Relaxed AO-MO". Similarly, as in [6] the AO algorithm is used twice for optimizations, we call it "double AO". It begins by optimizing the precoders in the outer AO loop, then relaxes the constraints on Q, before deploying another alternate procedure to individually optimize the weight of each DMA element by using line search techniques.
Considering the proposed algorithm, complexity in the first step, which computes the precoders for the user equipment, can be attributed to matrix inversion in (28), which is given by where I W and I μ respectively denote the number of iterations for the WMMSE iterative process and for searching μ. In the second step, where MO is used to update the DMA weights, the dominant complexity arises from the Euclidean gradient computation, and is given by where I MO is number of iterations accumulated in Algorithm 2. If I o denotes the number of outer iterations for AO-based algorithms, the total complexity of AO-MO can be given by O (I o (C W + C MO )). As the relaxed AO-MO also uses AO and MO for its optimization, its complexity is calculated similarly. Despite sharing the same complexity expression, these two algorithms are expected to have different numbers of iterations for their convergence, resulting in different complexities.
Next, we analyze the complexity of the double AO scheme. During the optimization of precoders, it uses the same WMMSE technique as the AO-MO; hence its complexity is also given by C W . For the DMA weights optimization, the complexity is largely dominated by the matrix Kronecker product and the iterative line search, which updates the weight of each DMA element. The combined complexity for these two procedures is given by C KL = O(K 2 2N 2 + N N d −N +2I ls N 2 (K + 1)), where I ls is the average number of iterations needed by the line search process. Therefore, O (I o (C W + C KL )) is the total complexity of the double AO scheme.

V. SIMULATION RESULTS
In this section, we present the numerical results to demonstrate the effectiveness of our proposed schemes. The algorithms developed in Sections III and IV are used to configure a BS using DMA to communicate with user terminals. We use relaxed AO-MO and double AO algorithms as the benchmark schemes to compare their performance with our proposed scheme. However, these algorithms have been designed specifically for multi-user scenarios. For the singleuser scenario, we use the single-user counterpart of double AO, which iteratively optimizes the relaxed DMA weights, as the benchmark scheme. We abbreviate it as "Relaxed DMA-SU." Additionally, the performance of a "Random weight" scheme, which randomly chooses the weight of each DMA element, is compared with other single-user schemes.

A. Simulation Setup
In this study, we consider a BS with a planar array of metasurface elements that communicates with K users using a carrier frequency of 28 GHz. The spacing between the microstrips and the elements therein is set to λ/2, where λ is the wavelength of the carrier wave. We set the DMA attenuation coefficient and wavenumber to α = 0.6 m −1 and β = 827.67 m −1 , respectively [30]. The transmitter power is set to 23 dBm, and the receiver noise is set to −80 dBm.
Moreover, we adopt the practical narrowband Saleh-Valenzuela mmWave channel model [16], [46] throughout our simulations. The channel is further multiplied by the square root of the distance-dependant pathloss, which is given as where ∂ is the distance in meters between the BS and the user, and η = 3.5 is a pathloss exponent.

B. Single User Scenario, K = 1
In this scenario, a BS centered at the origin of the xy plane is used to serve a single-antenna user located at (D, 0).

1) SNR Versus Number of Microstrips:
We start by positioning a user at a distance D = 100 m from the BS and set the number of elements in each microstrip to N e = 10. Fig. 2 demonstrates the expected trend that the performance of all algorithms increases with the number of microstrips. However, it is observed from the performance of the random weight scheme that if the weights of the DMA elements are not optimized, the system performance is poor. Furthermore, it is clearly seen from this figure that our proposed optimal algorithm attains the highest SNR for all considered sizes of DMA. The suggested scheme provides approximately a 2-dB SNR gain over the relaxed scheme.
2) SNR Versus BS-User Distance (D): Next, we evaluated the impact of the separation between the BS and user. Fig. 3 illustrates the performance of various algorithms as the user is moved in a straight line from D = 50 m to D = 300 m. It can be seen from Fig. 3 that as the distance between the BS and the  user increases, the performance of all the compared schemes decreases. This can be attributed to the increased pathloss that inevitably reduces the signal strength. Nevertheless, our proposed algorithm still demonstrates the best performance compared to the other benchmark schemes for the entire range of the distance. It provides an SNR gain of 2 dB compared to the relaxed scheme.
3) SNR Versus BS' Transmit Power: Fig. 4 presents the variation of the SNR as the BS transmit power is altered over the range P = [5 − 40] dBm. The number of microstrips and their radiating elements are set to N d = 20 and N e = 10, respectively. As expected, the performance of each scheme improves as the transmit power increases. Furthermore, following the prior trend, the proposed algorithm attains the best performance compared to the benchmark schemes. This further solidifies the superiority of our proposed algorithm.

C. Multi-User Scenario
For the multi-user system, we adopted a different BS and users setup, where the BS is positioned at the center of a 300 m-radius cell, radiating signals omnidirectionally to serve K = 5 single-antenna users, that are randomly distributed Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. within the cell, with an exception of a circular area of radius 35 m from the center. Each user is then assigned a priority proportional to its pathloss Λ i from the BS, which is, ω k = Λ k È K i Λi . Moreover, we adopt a fully digital (FD) antenna architecture to act as the performance upper bound. For this architecture, we assume the BS is ideally equipped with the same number of antenna elements as in the DMA case, and each antenna is connected to a dedicated RF chain. As conventional antennas do not have optimizable weights, only the precoder is optimized, using the same WMMSE algorithm [38].
1) Convergence Behavior: Fig. 5 presents the convergence behavior of our proposed algorithm compared to the benchmark schemes. For this, we assume N d = 15, N e = 10, K = 5, and P = 23 dBm. From Fig. 5, it can be observed that the proposed scheme and the relaxed AO-MO converge faster than the double AO, and approximately at the same number of iterations. According to our analysis, double AO has the slowest convergence because of its large number of iterative processes as it optimizes the weight of each DMA element by using line search techniques. Additionally, we also observe that the FD converges at a higher value than any other schemes-this is expected because of its larger number of RF chains. Furthermore, our proposed algorithm converges at the highest objective value compared to the other benchmark schemes using the same number of RF chains.
2) WSR Versus the Number of Microstrips and Elements: We continue to assess the performance of our proposed algorithm for various sizes of the DMA array. We begin by varying the number of microstrips from 5 to 20 and setting other parameters to their default values. In Fig. 6(a), where we plot the performance of various schemes for these settings, it is seen that our proposed algorithm attains higher performance than both the relaxed AO-MO and double AO. Quantitatively, for a small number of microstrips, e.g., N d = 5, the proposed algorithm provides a 12.6% gain of WSR over the benchmark algorithms. This gain is expected because unlike the benchmark schemes the proposed algorithm optimizes the DMA weights without relaxing its constraints. Despite the decrease in performance gain as the number of microstrips increase, the proposed algorithm still provides a significant gain of approximately 10.8% over the benchmark DMA-based schemes for N d = 20. Moreover, when we vary the size of the DMA by changing the number of elements in the same range, that is, N e = {5, 10, 15, 20}, while keeping the number of microstrips at N d = 10, the performance of our proposed algorithm as shown in Fig. 6(b), is significantly higher than the benchmark DMA schemes. However, the FD achieves the highest performance among all other schemes because of the aforementioned reason.
3) WSR Versus the Number of Users: Next, we vary the number of users in the network and observe its impact on the performance of our proposed algorithm. Fig. 7, which plots the WSR of various schemes for different numbers of users in the network, is obtained by setting N d = 16 and N e = 10, and the rest of parameters take their default values. Fig. 7 shows that the performance of each algorithm decreases as the number of users increases. This can be credited to increased user interference as well as the natural phenomenon of decrease in quality of service when demand increases but the supply, such as the BS power and number of antenna elements, remains constant. However, the proposed AO-MO algorithm still provides remarkable performance gains over  the benchmark DMA-based schemes, which increases with the number of users. For example, the performance gain increases from 10.7%, for K = 5, to 27.1% for K = 16. 4) Comparison of Complexity: Finally, we compare the complexities of our proposed algorithm and all the benchmark schemes, except FD. The comparison presented in Fig. 8 was attained by configuring the network with the same parameters as those used for Fig. 6(a). As shown in Fig. 8 the complexity of each algorithm increases with the size of the DMA. This behavior is predictable, as the number of elements increases, more computational resources are needed to optimize their weights. Despite using the computationally expensive Kronecker product and a separate line search-based iterative procedure for optimization of each DMA element's weight, the double AO scheme is less complex than relaxed AO-MO. However, this comes at the cost of poorer performance than all other schemes. Nevertheless, our proposed algorithm is the least complex among the compared schemes.

VI. CONCLUSION
This study examined a DMA-based downlink MISO system, in which a DMA-equipped BS serves single-antenna user(s). We split the analysis into single-user and multiple-user sce-narios and formulated the corresponding problem for each. First, we analyzed the single-user case, wherein we aimed to maximize the SNR of the user, considering the physical constraints of the DMA elements. We developed an optimal algorithm for this problem, which provides a closed-form solution. Next, we analyzed the multi-user scenario, wherein the rates of the users are first weighted proportionally to create a WSR problem. We then developed an AO algorithm to solve this WSR problem; this algorithm alternately optimizes the precoders of the system based on the MMSE technique and the DMA weights based on the MO algorithm. Although the proposed algorithms solve problems in different scenarios, they both share the same novel technique of splitting the Lorentzian constraint into two parts, which greatly facilitates the algorithm development.
The proposed optimal algorithm in the single-user case offered a significant performance gain over those of the benchmark schemes. For example, when the size of the DMA was varied by changing the number of microstrips, our proposed optimal algorithm maintained an SNR gain of approximately 2 dB throughout the range. Apart from the performance superiority of our proposed optimal algorithm in single-user systems, we observe similar behaviour in the multiuser scenario. Specifically, a gain of more than 27% in the system's WSR is attained by our proposed AO-MO algorithm over the benchmark schemes when the BS' precoders and DMA weights are configured to serve sixteen users in the network.
Most future wireless communication standards, such as 6G, are expected to employ wide bandwidths in order to deliver high data rates. In such cases, the operational bandwidth may significantly exceed the coherence bandwidth of the channel. In this study, we assumed a narrowband system; extending the proposed formulations to the wideband system constitutes an interesting research direction for future work.