Dynamic Metasurface Antennas for Uplink Massive MIMO Systems Nir Shlezinger, Or Dicker, Yonina C

Dynamic Metasurface Antennas for Uplink Massive MIMO Systems Nir Shlezinger, Or Dicker, Yonina C. Eldar, Insang Yoo, Mohammadreza F. Imani, and David R. Smith

Abstract—Massive multiple-input multiple-output (MIMO) size and shape [8], [9]. Several signal processing methods have communications are the focus of considerable interest in recent been studied, aimed at tackling these difficulties. The proposed years. While the theoretical gains of massive MIMO have approaches include introducing analog combining to reduce been established, implementing MIMO systems with large-scale antenna arrays in practice is challenging. Among the practical the size and cost of the system [5], [10]; implementing low- challenges associated with massive MIMO systems are increased resolution quantization and/or antenna selection to mitigate cost, power consumption, and physical size. In this work we the power consumption [7], [11]–[15]; and utilizing efficient study the implementation of massive MIMO antenna arrays power amplifiers operating at reduced peak-to-average-power using dynamic metasurface antennas (DMAs), an emerging tech- ratio [16], [17]. Nonetheless, all these approaches assume nology which inherently handles the aforementioned challenges. Specifically, DMAs realize large-scale planar antenna arrays, a fixed optimal antenna array, and attempt to tackle the and can adaptively incorporate signal processing methods such difficulties which arise from this antenna array architecture as compression and analog combining in the physical antenna from a signal processing perspective. structure, thus reducing the cost and power consumption. We In parallel to the ongoing efforts to make massive MIMO first propose a mathematical model for massive MIMO systems feasible using signal processing techniques, a large body of with DMAs and discuss their constraints compared to ideal antenna arrays. Then, we characterize the fundamental limits of research has focused on designing practical antenna arrays for uplink communications with the resulting systems, and propose massive MIMO systems [8], [9], [18], [19]. An emerging tech- two algorithms for designing practical DMAs for approaching nology for realizing large-scale antenna arrays of small phys- these limits. Our numerical results indicate that the proposed ical size uses metamaterial radiators instead of conventional approaches result in practical massive MIMO systems whose ones. Metamaterial antennas consist of array of subwavelength performance is comparable to that achievable with ideal antenna arrays. metamaterial radiators, excited by a waveguide or cavity [20]. Index terms— Massive MIMO, metasurfaces, antenna design. While the resulting antenna arrays typically exhibit mutual coupling and frequency selectivity, they offer comparable beam tailoring capability from a simplified hardware, which I.INTRODUCTION uses much less power and costs less than antenna arrays Future wireless systems are required to support an increas- based on standard antenna arrays [21], [22]. Furthermore, a ing number of end-users with growing throughput demands. large number of tunable metamaterial antenna elements can be Recent years have witnessed a rising interest in massive packed in the same physical area, [9], and metasurfaces can multiple-input multiple-output (MIMO) systems, in which the implement planar antennas, making it an appealing technology base station (BS) is equipped with a large antenna array, for supporting the increased BS deployment of 5G wireless as a method for meeting these demands and increasing the networks [23, Sec. II]. Most previous works on metamaterial spectral efficiency (SE). In particular, it was shown that, antennas for MIMO communications focus on designing the when a sufficiently large number of antennas are utilized, the physical antenna structure and metamaterial substrate to satisfy throughput can be increased in a manner which is scalable desired requirements, such as gain, bandwidth, efficiency, and with the number of BS antennas [1]. level of mutual correlation [9], [18], [19], [24]. Consequently, The theoretical benefits of massive MIMO systems in terms the resulting antenna structure is fixed and independent of the processing which the transmitted and received signals undergo. arXiv:1901.01458v2 [cs.IT] 30 Jun 2019 of SE are well-established [2]–[4]. However, implementing a massive MIMO BS equipped with a standard antenna array, An alternative application for metasurfaces as reflecting ele- capable of achieving these benefits, is still a very challenging ments instead of as transmit or receive antennas, was proposed task. In particular, some of the difficulties which arise when in [25]–[27] as a scheme for improving energy efficiency in realizing large-scale antenna arrays include high cost [5], [6], wireless communication networks. increased power consumption [7], and constrained physical Recently, dynamic metasurface antennas (DMAs) have been proposed as a method for electrically tuning the physical char- Parts of this work were accepted for presentation in the 2019 IEEE Inter- acteristics of metamaterial antennas [20], [28], [29]. DMAs national Conference on Acoustics, Speech, and Signal Processing (ICASSP), inherently implement signal processing techniques such as Brighton, UK. This project has received funding from the Air Force Office of Scientific beamforming, analog combining, compression, and antenna Research under grants No. FA9550-18-1-0187 and FA9550-18-1-0208. selection, without additional hardware. By introducing simple N. Shlezinger and Y. C. Eldar are with the faculty of Mathematics and solid-state switchable components into each metamaterial el- Computer Science, Weizmann Institute of Science, Rehovot, Israel (e-mail: [email protected]; [email protected]). ement and addressing them independently, these capabilities O. Dicker is with the department of Electrical Engineering, Technion, Haifa, can become reconfigurable; i.e. they can adapt to the task Israel (e-mail: [email protected] ). at hand or changes in the environment. The application of I. Yoo, M. F. Imani and D. R. Smith are with the department of Elec- trical and Computer Engineering, Duke University, Durham, NC (e-mail: DMAs was shown to yield simple, fast, planar, and low-power [email protected], [email protected]; [email protected]). systems for microwave imaging [30]–[32], radar systems [33]–

1 [35], and satellite communications [36]. More recently, using ing for the specific characteristics of the metasurfaces. Our cavity-backed DMAs as a novel means to generate desired numerical analysis demonstrates that the achievable perfor- patterns to enhance capacity of MIMO communications in a mance of the resulting massive MIMO systems in which the clustered environment has been proposed and demonstrated in BS implements its large-scale antenna array using DMAs is numerical simulations [37]. Nonetheless, despite the potential comparable to the theoretical fundamental limits of the chan- of DMAs in combining signal processing and antenna design, nel. These limits are achievable using unconstrained antenna their application for realizing massive MIMO systems has not arrays, which are more costly, require more power and are yet been studied. physically larger compared to DMAs with the same number In this work we aim to fill this gap by studying large-scale of radiators. multi-user MIMO networks utilizing DMAs. In particular, The rest of this paper is organized as follows: SectionII we study the achievable performance focusing on the uplink, introduces the mathematical formulation of DMAs, and defines namely, when data is transmitted from the user terminals (UTs) the problem of uplink multi-user MIMO communications with to the BS, and the BS is equipped with a DMA realizing a DMAs. Section III characterizes the fundamental performance large-scale antenna array. The application of DMAs results limits achievable with any antenna array, as well as the per- in a simplified hardware, which inherently implements signal formance limits when utilizing DMA, and derives algorithms processing techniques such as analog combining, subject to for designing DMAs to approach the optimal performance. specific constraints induced by the physics of the metasurfaces. SectionIV provides simulation examples. Finally, SectionV The resulting structure can be thus used for realizing planar, concludes the paper. Proofs of the results are detailed in the compact, low cost, and spectral efficient massive MIMO appendix. BSs. Unlike standard analog combining with conventional Throughout this paper, we use boldface lower-case letters antenna arrays, e.g., [5], [10], DMAs implement adjustable for vectors, e.g., x; the ith element of x is written as (x)i. compression without requiring additional hardware. Matrices are denoted with boldface upper-case letters, e.g., M, We propose a model for MIMO systems with DMAs which (M)i,j denotes its (i, j)th element, rank(M) denotes its rank encapsulates previously proposed mathematical models for the and |M| is its determinant. Sets are denoted with calligraphic unique characteristics and constraints of these metasurfaces, letters, e.g., X . We use In to denote the n×n identity matrix. such as the frequency response of each metamaterial element Stochastic expectation and mutual information are denoted by [20], [38], the propagation inside the waveguide [20], and the E{·} and I (· ; ·), respectively. We use k·k to denote the Eu- mutual coupling induced by the sub-wavelength spacing of clidean norm when applied to vectors and the Frobenius norm the elements [34], [39], [40]. By integrating these established when applied to matrices, ⊗ denotes the Kronecker product, C properties of DMAs into the overall MIMO system model, and N are the sets of complex numbers and natural numbers, we obtain an equivalent communication channel including respectively. For any sequence, possibly multivariate, y[i], and frequency selectivity and constrained linear combining, which integers b < b , yb2 denotes the column vector obtained by 1 2 b1 T can be analyzed using information theoretic tools. Our model h T T i stacking (y[b ]) ,..., (y[b ]) and yb2 ≡ yb2 . also quantifies some of the gains in utilizing DMAs, demon- 1 2 1 strating that they require less RF chains compared to standard II.PRELIMINARIESAND PROBLEM FORMULATION antenna arrays, thus reducing the cost, memory requirement, and power consumption. In the following we model the input-output relationship of Next, we focus on the scenario where the wireless channel DMAs when used on the receive side in a MIMO communi- is frequency flat, and the frequency selectivity, induced by cations scenario. This model is based on previously proposed the physics of the metasurfaces, is identical among all the mathematical models for the electromagnetic properties of radiating elements. We then extend our analysis to the general DMAs, in particular, on the works [20] and [34]. The main scenario of frequency selective channels with an arbitrary contribution of the resulting model lies in its natural integra- frequency selectivity profile among the metamaterial elements. tion into the overall communication system model, discussed For each scenario we characterize the maximal achievable in the following subsection, allowing the properties of DMAs average sum-rate among all UTs in the network, and compare to be incorporated in an equivalent channel which is analyt- it to the fundamental performance limits, which is the maximal ically tractable from an information theoretic perspective, as achievable sum-rate of frequency selective MIMO multiple shown in Section III. To formulate the considered setup, we access channels (MACs) derived in [41], and requires ideal first elaborate on metasurface anteannas and mathematically unconstrained antenna arrays. We show that when channel is express the input-output relationship of DMAs in Subsection frequency flat and the frequency selectivity is identical among II-A. Then, we present the massive MIMO with DMAs system the elements, its effect can be accounted for in the configura- model in Subsection II-B. Finally, in Subsection II-C we tion of the DMAs. Thus, under this setting, when number of discuss the achievable average sum-rate performance metric. DMAs is not smaller than the number of UTs, DMA based antenna arrays can approach the fundamental performance A. Dynamic Metasurface Antennas limits, achievable using ideal unconstrained antenna arrays. Metamaterials are a class of artificial materials whose For each scenario, we derive an alternating optimization physical properties, and particularly their permittivity and algorithm for configuring the DMAs to approach the perfor- permeability, can be engineered to exhibit a broad set of mance achievable with unconstrained antenna arrays, account- desired characteristics [42], [43]. The underlying idea behind

2 L metamaterial elements

K microstrips Metamaterial element Diode Collecting port: zi Microstrip

yi k1 y i   k 2 Fig. 2. Dynamic metasurface microstrip model.

Fig. 1. Metasurface antenna illustration. quency selectivity. Thus, we henceforth model this effect in the discrete-time domain as a causal filter with finite mh metamaterials is to introduce tailored inclusions in a host impulse response {hp,l[τ]}τ=0 whose taps are complex- medium to emulate a desired effective property. This concept valued, i.e., hp,l[τ] ∈ C, and mh represents the memory was later extended to surface configurations (thus ”metasur- of the filter, namely, the number of taps. face”) where the surface effective parameters were tailored to It is worth emphasizing here that we have ignored the realize a desired transformation on the transmitted, received, element-element coupling inside the microstrip for simplicity. or reflected waves [44], [45]. More recently, metasurfaces have This assumption is usually valid when metamaterial elements been implemented as radiative layers on top of a guiding are weakly coupled to the guided mode [20]. For cases with structure, forming a “metasurface antenna”. In a simple con- strong coupling metamaterial element, one can include such figuration, a metasurface antenna consists of microstrips con- coupling using coupled dipole models [39]. This model and its sisting of a multitude of sub-wavelength, frequency-selective implications (if any) on the massive MIMO system is beyond resonant metamaterial radiating elements [28]. To realize a the scope of this paper and are left for future works. larger antenna array, such metasurface antennas can be tiled We can now mathematically formulate the input-output together to form a large array. An illustration of such an array relationship of metasurface antennas. Consider a metasurface is given in Fig.1. antenna with K microstrips, each consisting of L elements, On the receive side, each microstrip feeds a single RF chain, K·L×1 and let y[i] ∈ C be a vector such that (y[i])(p−1)·L+l whose digital output is obtained as a linear combination of denotes the radiation observed at the lth element of the pth the radiation observed by each metamaterial element of the microstrip at time index i. The output of the metasurface microstrip. This linear combination is a result of the following antenna at time index i is the vector z[i] ∈ CK whose entries two physical phenomena: can be written as

• Frequency response of the metamaterial element: This L mh effect can be typically modeled as a bandpass filter whose X X (z[i])p = qp,l hp,l[τ] · (y[i − τ])(p−1)·L+l , (1) quality factor is typically around 30 [46], though higher l=1 τ=0 quality factors of around 100 can also be achieved [28]. with p ∈ {1, 2,...,K}. An illustration of the input-output For example, at carrier frequency of 1.9 GHz, a quality relationship induced by a single manuscript is depicted in factor of 30 would translate into a bandwidth of 63 Fig.2. It is noted that the filters {h [τ]}, representing the MHz. In many relevant communications scenarios, such p,l propagation inside each microstrip, do not depend on the a response can be considered as frequency flat, namely, gains of the metamaterial elements {q }, namely, we assume the gain induced by the metamaterial element is the same p,l that the metamaterial elements do not perturb the feed wave for all the considered frequency range. It is emphasized [20]. Due to the sub-wavelength proximity of the elements in that this does not imply that the communication channel microsrtip, the input vector y[i] is spatially correlated, i.e., its is frequency flat, as the wireless channel gain typically covariance matrix is non-diagonal. We thus do not assume a varies in frequency within this band [47]. specific element spacing and incorporate the resulting coupling • Propagation inside the microstrip: The effect of this into the general covariance of y[i]. phenomenon depends on the location of the elements The relationship between the multivariate processes y[i] and along the waveguide (e.g., the microstrip). In particular, z[i] can thus be written as by letting rp,l denote the location of the lth element on mh the pth microstrip, and βp denote the wavenumber along X the waveguide, which is usually larger than the free space z[i] = Q H[τ]y[i − τ], (2) wavenumber k, the effect of the propagation inside the τ=0 mh waveguide in the frequency domain is proportional to where {H[τ]}τ=0 is a set of N × N diagonal matrices, N , −jβp·r e p,l . Since the wavenumber βp is a linear function K·L, representing the frequency selectivity of the metasurface, of the frequency, this effect induces non-negligible fre- i.e., (H[τ])(p−1)L+l,(p−1)L+l = hp,l[τ], and Q is an K × N

3 matrix representing the configurable weights of the DMAs. which increases the overall size and cost, especially when Using (1), we can write the analog combining should be adjustable in run-time. The ( exact value of the increased cost and size depends on the q p = p (Q) = p1,l 1 2 . (3) specific implementation of the analog combiner. DMAs inher- p1,(p2−1)K+l 0 p1 6= p2 ently implement adjustable analog combining in the physical For mathematical convenience, it is assumed that the coeffi- structure of the metasurfaces without additional hardware. Specifically, multiple metamaterial radiators are fed directly by cients {qp,l} are unitless, i.e., the polarizability of the elements is normalized. While the model detailed above considers a a waveguide structure to simplify the feed structure, avoiding DMA in which the radiating elements are placed along a set of the usage of potentially more expensive and complicated RF one-dimensional microstrip, it can incorporate a broader fam- circuitry. Thus, DMAs have been recognized as a radiative ily of two-dimensional DMAs. In fact, any two-dimensional platform with a simple, energy-efficient, low-cost, and low- DMA in which each element is connected to a single output profile configuration. Furthermore, standard analog combining port can be represented via (2) by modifying the structure implemented using dedicated hardware is typically subject of the weights matrix Q to represent the resulting elements to different constraints than those imposed on Q here. In interconnections. particular, while in DMAs the weights matrix Q must obey the structure in (3) and its entries must be in the feasible DMAs integrate a tuning mechanism into each independent set Q defined above, standard analog combiners must satisfy resonator of a metasurface antenna [29]. The dynamic tuning the architecture-based constraints detailed in [6, Sec. II], such adds the flexibility to adjust the properties of the metamaterial as the commonly used phase shifting network constraint, i.e., elements, namely, to control the values of the coefficients Q = {q ∈ C : |q| = 1}, or the switching network constraint, {q } in (1). The set of possible values of {q }, denoted p,l p,l in which Q = {0, 1}. It is also noted that when L = 1, Q, represents the Lorentzian resonance response [20], and Q = I , and {h [τ]} are Kronecker delta functions, namely, typically consists of a subset of the complex plane C of either K p,l each microstrip realizes a single frequency flat antenna, then of the following forms [20, Sec. III]: z[i] ≡ y[i], and the resulting DMA coincides with the • Amplitude only, namely, Q = [a, b] for some real non- standard antenna array. However, this implementation requires negative a < b. the same amount of RF chains and ADCs as standard arrays, • Binary amplitude, i.e., Q = c ·{0, 1} for some fixed and thus does not result in any gains in terms of cost, c ∈ R+. j+ejφ power consumption, and memory requirement. Finally, it is • Lorentzian-constrained phase, that is Q = {q = 2 : emphasized that while we consider DMAs with frequency flat φ ∈ [0, 2π]}. element response, resulting in the frequency invariant weights In order to quantify the gains of utilizing DMAs, we matrix Q, it is possible to design DMAs to have dynamically next compare these antenna architectures to standard antenna adjustable frequency selective weights. This can be achieved arrays. We use the term standard arrays for systems where the by using elements with different resonance frequency along receiver is capable of directly processing the observed vector the microstrip and turning them on and off to realize a desired y[i], which is the common model in the MIMO communi- frequency selective response. While designing frequency se- cations literature, for both conventional MIMO [48, Ch. 7] lective DMAs is expected to introduce an additional potential as well as massive MIMO [2]–[4]. Clearly, any performance gain over conventional analog combiners, the set of possible achievable with DMA-based antenna arrays is also achievable frequency selectivity profiles which can be realized in DMAs with standard antenna arrays, as z[i] can be obtained from is heavily implementation dependent, and as a result, we leave y[i], but not vice versa. However, unless an additional RF this for future investigation. chain reduction hardware is used, such as analog combiners To summarize, DMAs realize antenna arrays with specific discussed in the sequel, standard antenna arrays require each structure constraints, representing the underlying physics of of the K · L radiating elements to be connected to an RF the metasurface. These constraints include additional filtering chain as well as an analog-to-digital convertor (ADC), while of the received signal due to the propagation inside the DMAs require a single RF chain and ADC per microstrip. microstrip; spatial correlation due to sub-wavelength element Note that RF chain hardware tends to be costly [6], and that spacing; and an inherent adjustable signal compression as the ADCs are typically a dominant source of power consumption signals are combined in each microstrip. The benefits of using [7] and memory usage [49]. Consequently, by utilizing DMAs, DMAs as an antenna array architecture are their low-cost, the resulting cost, memory usage, and power consumption, are power-efficiency, and physical shape and size flexibility. An reduced by a factor of L compared to standard antenna arrays additional benefit in the context of massive MIMO commu- [37]. Additionally, DMAs can realize planar antenna arrays nications, which we exploit in the sequel, is their natural [21], [22], and unlike standard antenna arrays, they can pack ability to implement a form of dynamic analog combining a larger number of elements into a given physical area [9]. as an integral part of the antenna structure without requiring We note that reducing the number of RF chains and ADCs additional dedicated hardware. can also be carried out with standard antenna arrays using dedicated analog combining hardware, see, e.g., [5], [6], [10], B. System Model [49]. However, in the presence of standard antenna arrays, We consider a noncooperative single-cell multi-user MIMO analog combining comes at the cost of additional hardware, system, focusing on the uplink. The BS is equipped with a

4 also consider the case where the BS decodes the transmitted signals based on the output of the wireless channel y[i], instead of z[i]. This scenario is referred to henceforth as optimal MIMO. As discussed in Subsection II-A, the maximal SE of optimal MIMO is not smaller1 than that achievable with DMAs, as the output of the DMA z[i] can be obtained from y[i]. Since different configurations of the same number Fig. 3. System model illustration. of elements results in a different statistical model for the channel output, in order to maintain fair comparison, the DMA, consisting of K microstrips, each with L elements, antenna spacing between the N elements used in the optimal namely, the overall number of radiating elements used by the MIMO setup is identical to that used with DMAs. Under this BS is N , K · L. The number of UTs served by the BS is setting, the resulting wireless channel, namely, the relationship U, assumed to be not larger than N. between x[i] and y[i], is the same as in the DMA setup. mg Our goal is to characterize the performance achievable for Let {G[τ]}τ=0 be a set of N × U matrices representing the multipath channel matrix from the UTs to the BS, where the considered system with DMAs compared to the optimal MIMO case, and to provide guidelines for configuring the mg denotes the length memory of the discrete-time channel DMA weights such that the performance is optimized. In the transfer function, i.e., the number of taps is mg + 1, and following section we properly define the performance metric mg = 0 implies that the channel is memoryless. The channel output at the BS is corrupted by be an i.i.d. zero-mean used henceforth. w[i] ∈ CN proper-complex multivariate Gaussian noise with C. Definitions covariance matrix C . By letting x[i] ∈ CU be the transmitted W In order to rigorously formulate the performance metric signal of the UTs at time index i, the corresponding channel used in the paper, while accounting for the frequency selec- output at the BS is given by tivity induced by the DMAs, we present a set of necessary mg X definitions, which are based on [53, Ch. 4]. We begin with y[i] = G[τ]x[i − τ] + w[i]. (4) the definition of finite-memory multi-user channels: τ=0 Definition 1 (A multi-user MIMO channel with finite-mem- We assume that the UTs utilize Gaussian codebooks, i.e., ory). A discrete-time N × U multi-user MIMO channel with x[i] is a zero-mean Gaussian vector with identity covariance finite memory consists of a set of U scalar input sequences, matrix, and that the BS has full channel state information represented via a multivariate sequence x[i] ∈ RU , i ∈ N , mg (CSI), namely, the matrices {G[τ]}τ=0 are known to the BS. an output sequence y[i] ∈ RN , i ∈ N , an initial state vector N At the BS, the DMA converts the received signal y[i] ∈ C s0 ∈ S0 of finite dimensions, and a sequence of conditional K n n ∞ into the vector z[i] ∈ C , which is used to decode the trans- probabilities p (y |x , s0) . mitted signal. As detailed in Subsection II-A, the relationship n=0 between y[i] and z[i] is given by (2). Note that the frequency Having defined a multi-user MIMO channel with finite- mh memory, we can now introduce the definition of codes for selectivity of the metasurface is modeled via {H[τ]}τ=0. An illustration is given in Fig.3. such channels: U We note that unlike the standard massive MIMO literature, Definition 2 (Multi-user code). A {Rk}k=1, l multi-user U e.g., [2], [3], which models the channel as memoryless, the code with rates {Rk}k=1 and blocklength l ∈ N consists of: lR model in (4) explicitly accounts for the frequency selectivity 1) U message sets Uk , {1, 2,..., 2 k }, with k ∈ of the wireless channel. Frequency selective MIMO models {1, 2,...,U}. as in (4), i.e., without the presence of DMAs, were studied 2) A family of encoders el,k, each maps the message uk ∈ for point-to-point communications [50], MACs [41], broadcast l T Uk into a codeword x(u ) = x(uk) [1] , . . . , x(uk) [l] , i.e., channels [51], and wiretap channels [52]. Our motivation k l for using the model in (4) stems from the fact that the el,k : Uk 7→ R . standard massive MIMO memoryless model is obtained by T The channel input is x[i] = x(u1) [i] , . . . , x(uU ) [i] . assuming orthogonal frequency division multiplexing (OFDM) l modulation with subcarrier spacing smaller than the coherence 3) A decoder dl which maps the channel output y into the bandwidth of the channel, and cyclic prefix of length larger messages uˆ1,... uÛ , i.e., N×l than the number of multipath taps [48, Ch. 3]. However, dl : R 7→ U1 × · · · × UU . when metasurfaces are present, it is no longer reasonable to assume that the moderate frequency variations induced by the The encoders and decoder operate independently of the initial DMA are effectively canceled by the transmission scheme. state s0. Consequently, in order to account for the frequency selectivity 1To be precise, as the considered channel is information stable, the of both the DMA as well as the wireless channel, we explicitly achievable sum-rate can be expressed using the mutual information between incorporate the effect of multipath into the model in (4). its input and output [53, Ch. 4]. Therefore, as z[i] is a deterministic mapping of y[i], it follows from the data processing inequality [53, Ch. 2] that the In order to compare the performance achievable with DMA achievable sum-rate when the output y[i] is not smaller than that achievable to that achievable with ideal unconstrained antenna arrays, we when the output is z[i].

5 2lRk The set xl is referred to as the k-th codebook Theorem 1. The maximal achievable average sum-rate of the (uk) u=1 U channel in (4) and (2) for a fixed weight matrix Q is given by of the {Rk}k=1, l code. Assuming each message uk is uniformly selected from U , the average probability of error, 2π k Z when the initial state is s0 , is given by [51, Sec. III]: 1 H H H 0 Rs = log IK + QΓ(ω)Σ(ω)Σ (ω)Γ (ω)Q 2π · U 2lR1 2lRU 1 X X 0 P l (s0 )= ··· Pr d yl 6= u0 −1 e 0 U l i H H P × QΓ(ω)CW Γ (ω)Q dω. (7) l Rk u1=1 uU=1 2 k=1 0 0 Proof: See AppendixA. ui =ui , s0 =s0 . (5) Using Definition2, we can now properly formulate the def- Theorem1 characterizes the maximal achievable sum-rate inition for achievable sum-rate, which will be used henceforth by incorporating the DMA operation as part of the channel, as our main metric for evaluating multi-user MIMO networks and obtaining the achievable sum-rate of the resulting finite- operating with DMAs. memory MAC as in [41]. Theorem1 can also be used to obtain the fundamental performance limits of the wireless Definition 3 (Achievable average sum-rate). An average sum- channel, achievable with optimal unconstrained antenna ar- rate Rs is called achievable if, for every 1, 2 > 0, there rays, as stated in the following corollary: exists a positive integer l > 0 such that for all integer l > l , 0 0 −1/2 H −1/2 U Corollary 1. Define Σ˜ (ω) C Σ(ω)Σ (ω)C , and there exists a multi-user code, {Rk} , l , which satisfies , W W k=1 U let {λi(ω)} be its eigenvalues arranged in descending l 0 i=1 sup Pe (s0) < 1, (6a) order. The maximal achievable average sum-rate of the optimal s0 ∈S 0 0 MIMO setup is given by and U 2π U 1 X 1 Z X R ≥ R − . (6b) ROM = log 1 + λ (ω)dω. (8) U k s 2 s 2π · U i k=1 0 i=1 Note that the achievable average sum-rate in Definition3 Proof: As noted in Subsection II-A, when L = 1, Q = is a fundamental property of the multi-user MIMO channel. IK , and Γ(ω) ≡ IK , the resulting setup coincides with the In fact, for a given set of multi-user encoders, the decoder optimal MIMO setup. Substituting this into (7) proves (8). which maximizes the achievable average sum-rate typically implements joint decoding [53, Ch. 4]. Such decoders are When DMAs are utilized, we note that due to the integration usually computationally complex, and thus many works on operation and the structure constraints on Q, it is difficult massive MIMO focus on the achievable average sum-rate to determine the DMA weights matrix Q such that (7) is assuming less complex suboptimal separate linear decoding, maximized. Therefore, in order to design Q and obtain the see [2]–[4]. In this work we focus on the implementation of resulting Rs, we first focus on the special case where all the the antenna array using DMAs, with the promise of reducing metasurface elements exhibit the same frequency selectivity cost, size, and power consumption. Consequently, we impose profile, and the wireless channel is frequency flat. For this no constraints on the processing and decoding carried out in case, we derive in Subsection III-A the choice of Q which the digital domain, assume that the BS has perfect knowledge maximizes the achievable sum-rate, ignoring the structure con- of the underlying channel, and characterize the performance straints detailed in Subsection II-A. Then, in Subsection III-B in terms of the maximal achievable average sum-rate. It is we propose an iterative algorithm for configuring practical emphasized that operating under computational complexity constrained DMAs. Finally, in Subsection III-C, we show how constraints and obtaining an accurate channel estimation are these design principles can be extended to arbitrary frequency challenging tasks on their own for massive MIMO BSs with selectivity profiles. DMAs. We consider the design of efficient decoding and channel estimation schemes, as well the analysis of the effect A. Optimal Weights for Flat Channels with Identical Fre- of inaccurate channel knowledge and hardware impairments quency Selectivity on the resulting performance when using DMAs, as potential future research directions, extending the current study. The maximal achievable average sum-rate and the corresponding weights configuration Q which maximizes (7) are in III.ACHIEVABLE AVERAGE SUM-RATES general difficult to compute. Thus, we will first consider the In the following we study the achievable average sum-rate special case where all the metamaterial elements exhibit the and the resulting DMA configuration for the setup presented same frequency selectivity profile, and the wireless channel in Subsection II-B. To formulate the achievable average sum- is frequency flat. Under this model, the multivariate filter rate, let Γ(ω) and Σ(ω), ω ∈ [0, 2π), denote the discrete-time representing the response of the antennas H[τ] can be written as H[τ] = I · h[τ], for some scalar mapping h[τ], and Fourier transforms (DTFTs) of H[τ] and of G[τ], respectively. nt The maximal achievable average sum-rate for a fixed DMA the multipath channel is given by a single tap G = G[0], i.e., weights matrix Q is stated in the following theorem: mg = 0. By letting γ(ω) be the DTFT of h[τ], the multivariate DTFTs of H[τ] and of G[τ] can be written Γ(ω) = Int ·γ(ω)

6 and Σ(ω) = G. Thus, the achievable average sum-rate in (7) of unitary K × K matrix U˜ and diagonal K × K matrix D˜ is given by with positive diagonal entries. OD −1 It follows from (13) that the optimal weights matrix Q 1 H H H Rs = log IK + QGG Q QCW Q . (9) implements the following processing: First, it applies a noise U −1/2 whitening filter, modeled via the matrix CW . Then, it ˜ H In order to find Q which maximizes (9), we define G , utilizes the transformation V˜ to project the output into its −1/2 H −1/2 CW GG CW . We now formulate the dependence of Rs least noisy K ×1 subspace, determined by the largest singular −1/2 on Q in the following lemma: values of the whitened channel transfer matrix CW G, or ˜ Lemma 1. Define Q˜ QC1/2 and let V be its right singular alternatively, by the largest eigenvalues of G. The fact that , W QOD depends on the channel and the statistics of the noise vectors matrix. By letting V˜ be the N × K matrix consisting indicates that the ability to reconfigure the analog combining of the first K columns of the unitary matrix V , the achievable weights, which is inherently supported by DMAs, is vital in sum-rate in (9) can be written as wireless communications. Finally, we note that the remaining 1 H invertible processing, determined by the matrices U˜ , D˜ , has Rs = log IK + V˜ G˜ V˜ . (10) U no effect on the resulting achievable rate, in agreement with ˜ 1/2 the data processing inequality [53, Ch. 2.3]. However, in the Proof: By replacing Q in (9) with Q = QCW it follows from Sylvester’s determinant theorem [54, Ch. 6.2] that following subsection we show that these matrices can be used to facilitate the approximation of QOD via a feasible weights −1 1 ˜ ˜ H ˜ ˜ H ˜ matrix, which satisfies (3) and whose entries belong to Q. Rs = log IN + G Q QQ Q . (11) U −1 B. Practical Design for Flat Channels with Identical Fre- ˜ H ˜ ˜ H ˜ Next, we note that Q QQ Q is a projection matrix, quency Selectivity −1 and can therefore be written as Q˜ H Q˜ Q˜ H Q˜ = V˜ V˜ H The derivation of the optimal sum-rate in Corollary2 [54, Ch. 5.9]. Substituting this into (11) proves (10). ignores the structure constraints on Q, and assumes that the right eigenvectors matrix V˜ can be any set of unitary Lemma1 implies that the achievable average sum-rate Rs depends on the weights matrix Q only through the first K vectors. Nonetheless, as detailed in the problem formulation ˜ 1/2 in Subsection II-B, Q must be written as in (3), and its right eigenvectors of Q = QCW . If we ignore the structure constraints on Q, then the maximal achievable sum-rate and coefficients {qi,l} should belong to the feasible set Q. Since the corresponding choice of V˜ which maximizes (11) are finding the constrained matrix Q which maximizes (9) is a given in the following corollary: difficult task, we propose to set Q to be the closest feasible matrix to the unconstrained QOD in the sense of minimal U ˜ Corollary 2. Let {λi}i=1 be the eigenvalues of G arranged Frobenious norm. Here, as in [10], [55], we exploit the fact OD in descending order. Then, the maximal achievable average that Rs is invariant to the selection of the left singular sum-rate when Q can be any complex matrix is given by matrix U˜ and the diagonal singular values matrix D˜ , and set min(K,U) these matrices such that the Frobenious distance to the feasible OD 1 X approximation is minimized. To formulate the problem, we let R = log(1 + λi). (12) s U K×N i=1 Q be the set of K × N which can be written as in (3) and whose non-zero entries belong to the feasible set Q. Let OD ˜ The rate Rs achieved by setting the columns of V to be the K K U U denote the set of K × K unitary matrices, and D be the eigenvectors corresponding to {λi}i=1. set of K ×K diagonal matrices with positive diagonal entries. Proof: The corollary follows directly from (11). Specifically, we fix some > 0 and restrict the diagonal entires K Note that the number of non-zero eigenvalues of G˜ is of the matrices in D to be not smaller than . We set the equal to its rank, denoted rank(G˜ ), which is at most U. It weights matrix Q to be the solution to: 2 therefore follows from (12) that increasing the number of ˜ ˜ ˜ H −1/2 ˜ min Q − UDV CW . (14) microstrips K to be larger than rank(G) has no effect on the Q∈QK×N ,U˜ ∈U K ,D˜ ∈DK OD optimal sum-rate Rs . In particular, comparing (12) to the K×N K×N ˜ Let PQ : C 7→ Q be the entry-wise projection fundamental limits in (8), we note that when K ≥ rank(G), K×N K×N OD OM into Q . By (3) the entry-wise projection of M ∈ C then Rs achieves the fundamental limits Rs . However, as each microstrip requires a single RF chain and ADC, is given by increasing K implicitly increases the cost, power usage, and (P (M)) = Q p1,(p2−1)K+l memory requirements of the resulting system. Furthermore, by  2 H letting U˜ D˜ V˜ be the compact singular valued decomposition arg min q − (M) p1 = p2 p1,(p2−1)K+l (SVD) of the optimal Q˜ , it follows from Corollary2 that the q∈Q  weights matrix which maximizes (9) can be written as 0 p1 6= p2. −1/2 In order to solve (14), we propose an alternating minimization QOD = U˜ D˜ V˜ H C . (13) W algorithm, based on the properties detailed in the following In particular, the matrix in (13) maximizes (9) for any setting lemma:

7 Lemma 2. For any M ∈ CK×N we have that sum-rate, as the resulting QOD can be better approximated AM 2 using a feasible matrix. Q (M) , arg min kQ − Mk = PQ (M) . (15a) Q∈QK×N K×N C. Practical Design for Arbitrary Frequency Selectivity Additionally, for any M 1, M 2 ∈ C , let U M and V M be the left singular vectors matrix and the right singular vectors In the previous subsections we studied the special case H matrix of M 1M 2 , then where the wireless channel is frequency flat and the metamaterial elements exhibit the same frequency selectivity profile, 2 U˜ AM (M , M ) arg min M − UM˜ i.e., m = 0 and H[τ] = I · h[τ], for some scalar mapping 1 2 , 1 2 g nt U˜ ∈U K h[τ]. Under this setting, we were able to express the integral H = U M V M . (15b) in (7) with a single log-det expression in (9), as the wireless H channel is memoryless and the effect of h[τ] on the transmitted Finally, by letting m1,i and m2,i be the ith columns of M 1 H signal was canceled by its contribution to the effective noise. and M 2 , respectively, i ∈ {1, 2,...,K}, we have that the However, wireless channels are typically frequency selective, diagonal entries of and in practical metasurfaces, each element may exhibit a 2 ˜ AM ˜ different frequency selectivity profile. Here, the frequency D (M 1, M 2) , arg min M 1 − DM 2 , (15c) D˜ ∈DK selectivity cannot be effectively canceled, and the explicit value of H[τ] has to be accounted for. In the following are given by we show how the design principles proposed in the previous ! Re mH m subsections can be extended to this general case. ˜ AM 1,i 2,i D (M 1, M 2) = max 2 , . (15d) To that aim, we fix some positive integer B, and define i,i m2,i 2π·i ωi , B , i ∈ {1, 2,...,B}. We can now approximate (7) as Proof: See AppendixB. B 1 X H H H Rs ≈ log IK +QΓ(ωi)Σ(ωi)Σ (ωi)Γ (ωi)Q Based on Lemma2, we propose to solve the joint opti- B · U mization problem (14) in an alternating fashion, i.e., optimize i=1 −1 ˜ ˜ ˜ ˜ H H over Q for fixed U, D, next optimize over U for fixed Q, D, QΓ(ωi)CW Γ (ωi)Q . (16) then optimize over D˜ for fixed Q, U˜ , and continue until convergence. The resulting alternating minimization algorithm Note that as B increases, (16) approaches the actual sum- is summarized in Algorithm1. As the Frobenious norm ob- rate in (7). We next write (16) in terms of a single log-det B jective in (14) is differentiable, convergence of the alternating expression, as in (9). To that aim, let BlkDiag {Ai}i=1 optimization algorithm is guaranteed [57, Thm. 2]. be a block diagonal matrix with diagonal submatrices B {Ai}i=1, and define the B · N × B · N block diagonal ¯ B ¯ Algorithm 1 DMA weights for identical frequency selectivity matrices G , BlkDiag {Γ(ωi)Σ(ωi)}i=1 and CW , BlkDiag {Γ(ω )C ΓH (ω )}B Q¯ I ⊗ 1: Initialization: Set k = 0 and U˜ k = IK , D˜ k = IK . i W i i=1 . Also, define , B 2: Compute V˜ using Corollary2. Q. Using these notations, it follows from [54, Pg. 122] that AM H −1/2 3: Obtain Q = Q with M = U˜ D˜ V˜ C using −1 k+1 k k W ¯ ¯ ¯ H ¯ H ¯ ¯ ¯ H (15a). IB·K +QGG Q QCW Q AM 4: Set U˜ =U˜ via (15b) with M = Q and M = k+1 1 k+1 2 H H H ˜ ˜ H −1/2 = BlkDiag IK +QΓ(ωi)Σ(ωi)Σ (ωi)Γ (ωi)Q DkV CW . AM H 5: D˜ =D˜ M = U˜ Q −1 Set k+1 via (15c)-(15d) with 1 k+1 k+1 H H B −1/2 ˜ H × QΓ(ωi)CW Γ (ωi)Q i=1 . and M 2 = V CW . 6: If termination criterion is inactive: Set k := k + 1 and go B QB to Step3. Since |BlkDiag {Ai}i=1 | = i=1 |Ai| when Ai are square matrices [54, Pg. 467], it follows that (16) can be written as −1 1 ¯ ¯ ¯ H ¯ H ¯ ¯ ¯ H In Algorithm1 we exploit the fact that the optimal uncon- Rs ≈ log IB·K +QGG Q QCW Q . (17) strained QOD achieves the same sum-rate for any setting of B· U U˜ , D˜ , and use these matrices as optimization variables. Conse- The approximation in (17) implies that the expression for quently, we are able to obtain feasible weight matrices which the achievable sum-rate with arbitrary frequency selectivity are within a small distance from the optimal unconstrained is similar to that with identical frequency selectivity and flat matrix. In our numerical study in SectionIV we demonstrate channels in (9). Consequently, the design principles used for that BSs equipped with DMAs designed via Algorithm1 are configuring the DMA to minimize (9) in Algorithm1 can capable of achieving performance which is within a small gap also be used to minimize (17). The main difference between of the fundamental limits of the channel, achievable using minimizing (9) and (17) is that in (17), the equivalent weights K×N optimal antenna arrays. Furthermore, it is illustrated that, matrix Q¯ has to be written as IB ⊗Q where Q ∈ Q . This unlike the unconstrained case, when Q ∈ QK×N , increasing additional constraint can be accounted for in the alternating the number of microstrips K above U increases the achievable minimization algorithm using the following lemma:

8 Lemma 3. For any M ∈ CB·K×B·N , the weights matrix our design we assume that the DMA weights do not vary which minimizes in frequency, the resulting system in general cannot achieve AM2 2 the bound in Proposition1. Nonetheless, in the numerical Q (M) , arg min k(IB ⊗ Q) − Mk , (18) Q∈QK×N evaluations in SectionIV it is demonstrated that BSs utilizing practical DMAs designed via Algorithm2 are capable of is given by achieving performance which is comparable with the upper QAM2 (M) = bound in (19). In particular, we show that, when using properly p1,(p2−1)K+l configured DMAs, the resulting achievable average sum-rate  B−1 2  P is within a reasonable gap from the upper bound and that arg min q − (M) p1 = p2 i·K+p1,i·N+(p2−1)K+l both curves scale similarly with respect to signal-to-noise ratio q∈Q i=0  (SNR) 0 p1 6= p2. By repeating the arguments in the discussion following Proof: The lemma is obtained by explicitly writing the Corollary2, it holds that when K is not smaller than the Frobenious norm in (18), noting that each element of Q rank of Σ˜ (ω) for each ω ∈ [0, 2π], then the upper bound in OM independently effects the overall norm via the sum of B terms, (19) coincides with the fundamental performance limits Rs as given in the lemma. given in Corollary1. Note that for Q = C, the non-zero entries of QAM2 are given by the sample mean of their corresponding entries in IV. NUMERICAL STUDY M. Using Lemma3, we can now adapt Algorithm1 to In this section we numerically evaluate the achievable account for arbitrary frequency selectivity profiles, resulting performance using the DMA configurations derived in Section in Algorithm2. III. First, in Subsection IV-A we consider frequency flat Algorithm 2 DMA weights for arbitrary frequency selectivity channels with DMAs in which each element exhibits the same frequency selectivity profile, and numerically evaluate 1: k = 0 U˜ = I D˜ = I Initialization: Set and k B·K , k B·K . the average sum-rate achievable using the DMA design in 2: V˜ C G Compute using Corollary2 with W and replaced Algorithm1. Then, in Subsection IV-B, we study frequency C¯ G¯ with W and , respectively. selective channels with DMAs in which each element exhibits AM2 ˜ ˜ ˜ H ¯ −1/2 3: Obtain Qk+1 =Q with M = U kDkV CW using a different frequency selectivity, and compute the achievable (15a). performance of the DMA configuration obtained using Algo- ˜ ˜ AM 4: Set U k+1 = U via (15b) with M 1 = IB ⊗ Qk+1 and rithm2. ˜ ˜ H ¯ −1/2 M 2 = DkV CW . We consider an uplink multi-user MIMO cell in a rich ˜ ˜ AM 5: Set Dk+1 = D via (15c)-(15d) with M 1 = scattering environment. In this setup, a BS equipped with ˜ H ˜ H ¯ −1/2 U k+1 IB ⊗ Qk+1 and M 2 = V CW . a DMA serves U = 10 UTs, uniformly distributed in a 6: If termination criterion is inactive: Set k := k + 1 and go hexagonal cell of radius 400 m, with the exception of a to Step3. circle with radius 20 m around the BS. An illustration of such a system is depicted in Fig.4. We use ρi to denote In order to evaluate the gap of the resulting configuration the distance of the ith UT from the BS. Based on the model from optimality, we wish to characterize the maximal sum-rate for frequency selective wireless MIMO channel proposed in m K×N g achievable when Q can be any matrix in C , not restricted [58], the channel transfer matrices {G[τ]}τ=0 are generated 2 1/2 to satisfy (3). Since Q in (7) does not vary with ω, obtaining as G[τ] = σG[τ]ΣR GR[τ]D[τ], where: 2 mg the optimal performance is a difficult task. Nonetheless, (7) • {σG[τ]}τ=0 is the relative path loss of each tap, given 2 −τ can be used to obtain an upper bound on the optimal average by an exponentially decaying profile, i.e., σG[τ] = e . mg sum-rate, as stated in the following proposition: • {GR[τ]}τ=0 are a set of i.i.d. proper-complex zero- Proposition 1. If Γ(ω) is non-singular for every ω ∈ [0, 2π), mean Gaussian N × U matrices with i.i.d. entires of unit then the maximal achievable average sum-rate is upper- variance. bounded by • ΣR is an N × N representing the correlation induced by the sub-wavelength spacing of the elements in each 2π min(U,K) 1 Z microstrip. Neglecting the coupling between different X L×L Rs ≤ log 1 + λi(ω) dω, (19) microstrips, we set Σ = I ⊗Σ , where Σ ∈ C 2π · U R K M M 0 i=1 models the coupling induced between the elements of the same microstrip. In particular, we use Jakes’ model2 where {λ (ω)}U are defined in Corollary1. i i=1 for the spatial correlation with element spacing of 0.2 Proof: See AppendixC. wavelength for ΣM , i.e., (ΣM )i,l = J0 0.4 · π · |i − l| , It is emphasized that the upper bound in (19) is in general i, l ∈ {1, 2,...,L}, where J0(·) is the zero-order Bessel very difficult to approach in practice, as it is computed by function of the first type [59]. allowing the DMA weights to be frequency selective, thus 2It is noted that Jakes’ model requires the radiating patterns to share the effectively canceling the frequency selectivity of the wireless same azimuth [59], which is a reasonable assumption for metasurface antenna channel and the different elements in the microstrips. As in elements.

9 mg • {D[τ]} are U ×U diagonal matrices representing the τ=0 400 attenuation coefﬁcients, based on the model used in [2]. BS ζi[τ] 300 UTs In particular, we set (D[τ])i,i = 2 , where {ζi[τ]} are ρi the shadow fading coefﬁcients, independently randomized 200

from a log-normal distribution with standard deviation of 100 8 dB. 0 Since the radiating elements in the DMA microstrips are sub-wavelength separated, the additive noise w[i] is inherently -100 spatially correlated. Accounting for the coupling between the -200 2 DMA elements, we set CW = σW · ΣR, where ΣR is -300 the matrix representing the correlation due to sub-wavelength 2 -400 element spacing deﬁned above, and σW > 0 models the -400 -300 -200 -100 0 100 200 300 400 average power of the noise signal. Fig. 4. Multi-user MIMO network illustration. In the following we numerically evaluate the following achievable average sum-rates: UC • R - unconstrained weights, i.e., Q = C. OD s R AO s • RUC R - amplitude only weights, here Q = [0.001, 5]. s s -1 BA 10 RAO • Rs - binary amplitude weights, Q = {0, 0.1}. s RBA LP s • Rs - Lorentzian-constrained phase, namely, Q = RLP j+ejφ s RAC { : φ ∈ [0, 2π]} -2 s 2 . 10 RSN To compare the performance achievable with DMAs to con- s ×10-3 ventional analog combining, as in, e.g., [5], [6], [10], we also 15

10-3 compute the rate when standard analog combining architec- 10

tures with K RF chains are used. In particular, we simulate Achievable average sum-rate [bps/Hz] 5 a fully connected phase shift network (Architecture A.1 in 12 13 14 15 10-4 [6]) and a fully connected switching network (Architecture -5 0 5 10 15 20 25 30 Fig. 5. Rate vs. SNR, flat channel, L = 10. A.3 in [6]), both obtained using MaGiQ algorithm [10, Sec. SNR [dB] V-A]. The resulting achievable average sum-rates are denoted AC SN Rs and Rs , respectively. Since existing works on analog elements, computed via Algorithm1, to the optimal achievable 3 OD combining design assume memoryless channels , we simulate performance Rs computed via Corollary2. In Fig.5 we set these setups only for the frequency flat scenarios in Subsec- L = 10 while in Fig.6 we use L = 15. Recall that since tion IV-A. Our results are averaged over 1000 Monte-Carlo K ≥ U, then, based on the discussion following Corollary OD OM simulations. 2, Rs equals the fundamental performance limit, Rs , LP stated in Corollary1. Observing Figs.5-6, we note that Rs A. Flat Channel with Identical Frequency Selectivity UC approaches Rs for all SNR values when L = 10 and We first consider the case where the channel is frequency- for SNR above 15 dB when L = 15. This indicates that flat, namely, mg = 0, and each element in the DMA ex- the Lorentzian-constrained phase restriction induces negligible hibits the same frequency selectivity profile, as studied in loss when designing the weights using Algorithm1. The Subsections III-A and III-B. In Figs.5-6 we let the SNR, amplitude only restriction, the binary amplitude constraint, the 2 defined as 1/σW , vary in the range [−5, 30] dB. Note that standard phase shifting network, and the standard switching here the term SNR refers only the energy of the noise, and network, all achieve roughly the same performance, which is does not account for the attenuation induced by the channel, within a small gap of that achievable using the Lorentzian- which depends on the specific realization of the location of constrained phase weights. Furthermore, the SNR loss induced each UT. As the generated channels induce severe attenuation, by restricting the weights matrix to satisfy (3), namely, the fact the resulting achievable rate values are significantly smaller that the DMA combines only inputs from the same microstrip, OD than those reported in previous related works, e.g., [6], [7], is approximately 7 dB. In particular, for L = 10, Rs , which in which the SNR encapsulates the channel attenuation. It is is achieved without the structure constraint (3), achieves an UC also noted that in the previous works [6], [7] the rate measure average sum-rate of 0.1 bps/Hz at SNR of 17 dB, while Rs represents the overall achievable rate in point-to-point MIMO achieves the same performance for SNR of 24 dB. For L = 15, OD UC communications, and not the achievable average sum-rate of Rs = 0.1 for SNR of 20 dB, while Rs achieves this a multi-user MIMO network, which can be viewed as the sum-rate at SNR of 27 dB. Furthermore, it is noted that both overall achievable rate divided by the number of UTs. For curves scale similarly with respect to SNR, indicating that each SNR value we compare the average sum-rates achievable any average sum-rate which is achievable using an optimal using DMAs with K = 10 microstrips, each with L radiating unconstrained antenna array, is also achievable using practical DMA setups as the SNR increases. 3While it may be possible to extend analog combining design algorithm such as MaGiQ [10] to frequency selective channels, such an extension is Next, in Fig.7, we fix the SNR to 15 dB, the number of beyond the scope of this work. antennas to N = 90, and compute the achievable average

10 sum-rates for K ∈ [1, 18]. The goal of this study is to ROD numerically evaluate how the number of microstrips effects s RUC s -1 the performance for a given number of antennas. In order 10 RAO s RBA to guarantee that the same channel and noise statistics are s RLP used for each value of K, we fix the coupling matrix ΣR s RAC 10-2 s to ΣR = I15 ⊗ ΣM , where ΣM is a 6 × 6 matrix defined RSN s ×10-3 earlier in this section, representing the element coupling via 8 6 Jakes’ model. Observing Fig.7, we again note that the perfor- 10-3 mance achievable with practical Lornetzian-constrained phase 4 weights approaches that achieved with unconstrained weights Achievable average sum-rate [bps/Hz] 2 LP 10-4 0 1 2 3 4 5 for most considered values of K, where the gap between Rs -5 0 5 10 15 20 25 30 UC −2 SNR [dB] and the unconstrained Rs is at most 1.3 · 10 bps/Hz. As UC Fig. 6. Rate vs. SNR, flat channel, L = 15. expected, for K = 1, Rs , which is subject only to (3), OD coincides with the optimal performance Rs , as (3) imposes no constraint on the weights matrix structure for K = 1. The performance gap of the DMA-based receivers from the OD optimal Rs depends on the number of micropstrips K. For LP −2 ROM example, for K = 6, we observe that Rs = 1.5·10 bps/Hz, s AO BA −2 ROD while R and R are approximately 1 · 10 bps/Hz, s s s RUC s −2 −2 10-2 i.e., gaps of roughly 4 · 10 bps/Hz and 4.5 · 10 bps/Hz, RAO s OD −2 RBA respectively, from the optimal performance Rs = 5.5·10 . s RLP This gap becomes less dominant as K further increases, and s RAC −2 s SN for K = 15 it is reduced to approximately 3.5 · 10 bps/Hz Achievable average sum-rate [bps/Hz] R s for all considered DMA-based receivers. Additionally, we note OD 0 2 4 6 8 10 12 14 16 18 that Rs is monotonically increasing for small values of Number of microstrips K K, and for K > 3 its value remains constant and equals Fig. 7. Rate vs. microstrips, flat channel. OM the fundamental limit of the channel, Rs . This follows OD since, as noted in the discussion following Corollary2, Rs valued amplitude weights, and that under both constraints, ˜ remains constant when K is larger than the rank of G. Our the achievable average sum-rate substantially increases as K numerical study shows that for the considered scenario, most increases where the gap from the optimal MIMO R varies ˜ s realizations of G have at most 3 dominant eigenvalues. This from approximately 6 · 10−2 bps/Hz for K = 1 to 3.5 · 10−2 behavior is due to the fact that the diagonal entries of the bps/Hz for K = 18. The standard analog combining networks attenuation coefficients matrix D, which is randomized using AC SN Rs and Rs , which also depend on K as its value here the statistical model of [2], exhibit notable variations, as determines the number of RF chains, achieve approximately UTs located at different distances from the BS can observe the same performance as DMAs with continuous valued am- substantially different attenuation coefficients. For this reason, plitude weights, where the phase shifters network achieves a OD Rs remains constant for K > 3. Since the constraint induced slightly better performance for K < 9. UC on Rs in (3) becomes less significant as K decreases, and Finally, we observe in Fig.7 that, while ROD remains LP UC s since, as noted in Fig.5, Rs is capable of approaching Rs constant as the number of microstrips K increases above U, at such SNRs, it is shown in Fig.7 that, for a fixed number of the performance achievable with DMAs is monotonically in- UC LP elements N, both Rs and Rs do not necessarily increase creasing. This follows since, as discussed in Subsection III-B, when the number of microstrips K is increased. increasing the number of microstrips K allows designing the While the results in Fig.7 may be in favor of setting matrices U˜ and D˜ in (13), which have no effect on the K = 1 and L = N, in practice increasing the number of OD resulting optimal performance Rs , such that the resulting elements on a single microstrip increases the attenuation which QOD can be better approximated using a feasible weights results from the propagation of the signal inside the microstrip. matrix. This phenomena is not accounted for in the model here, The results presented in this subsection demonstrate that which assumes that the attenuation induced by each element is for frequency flat channels, BSs equipped with DMAs can identical, hence the additional loss by increasing the number achieve a performance which is comparable with costly op- of elements per microstrip is not reflected in Fig.7. This timal unconstrained antenna arrays. Furthermore, by utilizing observation also implies that in a practical implementation, our proposed alternating optimization algorithm, the sum-rate we need to strike a balance between the cost (proportional achievable with DMAs is not smaller and even larger than that to the number of RF ports or K), losses (proportional to the achieved using standard fully connected analog combiners, number of metamaterial elements, L), and the performance obtained using state-of-the-art design algorithms. (related to both quantities). This investigation is left for future works. B. Varying Frequency Selectivity We also note that restricting the weights to binary val- Next, we consider the more general setup where the channel ues achieves roughly the same performance as continuous is frequency selective and each element exhibits a different fre-

11 ROM s RUC 10-1 s RAO s RBA s RLP s 10-2 10-2 ROM s 0.07 Prop. 1 RUC 0.06 s -3 10 RAO 0.05 s RBA 0.04 s LP Achievable average sum-rate [bps/Hz] 0.03 Achievable average sum-rate [bps/Hz] R s 22 24 26 10-4 10-3 -5 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 18 SNR [dB] Number of microstrips K Fig. 8. Rate vs. SNR, frequency selective channel. Fig. 9. Rate vs. microstrips, frequency selective channel.

improves the performance. This follows since increasing the quency selectivity profile. In particular, we consider a channel number of elements in each microsrtip induces additional with two taps, i.e., m = 1, and set Γ(ω) = I ⊗ Γ (ω), g K G attenuation, which impairs the ability of the BS to recover where Γ (ω) ∈ CL×L is a diagonal matrix representing the G the messages from the channel output. It is also observed frequency selectivity profile of single microstrip. Based on in Fig.9 that the performance achievable using DMAs is the model detailed in Subsection II-A, we account for the roughly the same for all settings of Q, as observed in Fig.8. frequency response of the elements and the propagation inside The fact that the binary amplitude setting, which is relatively the waveguide via the setting (Γ (ω)) = e−(α+j·β(ω))·l. In G l,l simple to implement, achieves roughly the same performance β(ω) = 1.592 · ω [ −1] α = 0.0006 particular, we set m and as the other settings, makes it an appealing candidate for future [ −1] m , representing a microstrip with 50 ohm characteristic practical implementations studies. Lastly, we note that there is 1.9 impedance made of Duroid 5880 operating at GHz with a notable gap between the upper bound of Proposition1, which 0.2 element spacing of wavelength (assuming free space is computed by letting the DMA weights be frequency selec- wavelength) [60, Ch. 3.8]. tive, and the actual sum-rate achievable with fixed weights. Fig.8 depicts the average sum-rates achievable using DMAs This indicates that the performance achievable with DMAs configured via Algorithm2 versus SNR, for K = 10 and L = can be substantially improved in frequency selective channels OM 10. The performance is compared to the theoretical limit Rs . by designing the frequency response of each element, which Observing Fig.8 we note that unlike the scenario considered in is modeled in the DMA weights, to vary in frequency as in, the previous subsection, here the performance achievable with e.g., [61]. We leave the analysis and design of such DMAs for DMAs for all considered feasible sets Q is roughly the same. future work. We also note that the achievable performance with DMA is within a notable gap of approximately 10 dB in SNR from V. CONCLUSIONS the upper bound on the maximal achievable performance in In this work we studied massive MIMO systems where Proposition1. The increased gap stems from the fact that, the large-scale antenna array at the BS is implemented using OM a DMA. We characterized the maximal achievable average as shown in the proof of Proposition1, Rs is obtained by mitigating the frequency selectivity of the wireless channel and sum-rate on the uplink, and derived two alternating optimiza- the metamaterial elements. Unlike the scenario considered in tion algorithms for designing practical DMAs to approach Subsection IV-A whose results are depicted in Fig.5, here the the optimal performance: the first algorithm is designed for frequency selectivity induced by the physics of the metasurface frequency flat channels assuming that the frequency selectivity cannot be mitigated by properly setting the coefficients matrix induced by the metasurface is identical among all elements, OM and the second algorithm generalizes the first algorithm to Q, and thus the difference between upper bound Rs and the achievable performance with DMAs increases. Despite this arbitrary multipath channels and frequency selectivity profiles. gap, it is observed in Fig.8 that the performance achievable Our results illustrate the potential gains over standard antenna OM arrays of utilizing DMAs for implementing compact low-cost with DMAs scales similarly to the upper bound Rs with respect to SNR, indicating that the performance achievable and low-power massive MIMO systems. In particular, it is OM shown in our simulations study that by properly adjusting the with DMAs is comparable with Rs . Finally, in Fig.9 we depict the achievable sum-rate versus inherent combining and compression induced by the physics of number of microstrips for fixed number of antennas N = 90 DMAs, a practical massive MIMO system can be constructed and SNR of 15 dB. These rates are compared to the theoretical which is capable of achieving performance comparable to the OM fundamental theoretical limits, which require costly, power limit Rs , as well as to the upper bound on DMA performance computed via Proposition1, which coincides with the consuming, and large-sized optimal antenna arrays. OM theoretical limit Rs for K ≥ U. Observing Fig.9 we note APPENDIX that here, unlike the results depicted Fig.7 which considered a A. Proof of Theorem1 similar scenario but did not account for the signal propagation To prove the theorem, we first write the operation of the inside the microsrtip, increasing the number of microstrips DMA as part of the massive MIMO channel. By combining (4)

12 m Ph ˜ and (2), and defining w˜ [i] , Q H[τ]w[i−τ] and H[τ] , Q(ω)Γ(ω). Since Γ(ω) is non-singular, Q(ω) can be recov- τ=0 ˜ m ered from Q(ω). Under this setting, (7) satisfies Pg Q H[τ − l]G[l], the equivalent input-output relationship 2π Z l=0 1 1 H H can be written as Rs = log IK + Q(ω)Σ(ω)Σ (ω)Q (ω) 2π U mh+mg 0 X ˜ −1 z[i] = H[τ]x[i − τ] + w˜ [i]. (A.1) H × Q(ω)CW Q (ω) dω τ=0 Note that w˜ [i] in (A.1) is a stationary multivariate proper- 2π min(U,K) (a) Z 1 X complex Gaussian process with finite memory mh +mg. Con- ≤ log 1 + λ (ω) dω, (C.1) 2π · U i sequently, (A.1) represents a finite-memory Gaussian MAC. 0 i=1 Thus, by letting Sx(ω), Sw˜ (ω), and Γ˜(ω), ω ∈ [0, 2π), denote the power spectral density (PSD) of x[i], PSD of w˜ [i], and where (a) follows from upper-bounding the integrand for each the DTFT of H˜ [τ], respectively, it follows that the achievable ω ∈ [0, 2π), using Corollary2, thus proving (19). average sum-rate is given by [41]

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