Multiuser MIMO with Large Intelligent Surfaces: Communication Model and Transmit Design Robin Jess Williams∗, Pablo Ram´ırez-Espinosa∗, Elisabeth de Carvalho∗ and Thomas L. Marzettay ∗Department of Electronic Systems, Connectivity Section (CNT) Aalborg University, Denmark yTandon School of Engineering, New York University, Brooklyn, NY Email: ∗frjw, pres, [email protected], [email protected] Abstract—This paper proposes a communication model for the received power [12]. However, this effect is not considered multiuser multiple-input multiple-output (MIMO) systems based when designing the linear transmit and receive processing on large intelligent surfaces (LIS), where the LIS is modeled as [13, 14], giving rise to solutions that might not be optimal a collection of tightly packed antenna elements. The LIS system is first represented in a circuital way, obtaining expressions for in realistic conditions. Also, related to mutual coupling is the radiated and received powers, as well as for the coupling the superdirectivity effect, which theoretically allows for the between the distinct elements. Then, this circuital model is design of highly directive (ideally unbounded) arrays of closely- used to characterize the channel in a line-of-sight propagation spaced antennas [9]. However, in practice, achieving such scenario, rendering the basis for the analysis and design of superdirectivity comes at the price of extremely large excitation MIMO systems. Due to the particular properties of LIS, the model accounts for superdirectivity and mutual coupling effects currents, which considerably increases the losses and reduces along with near field propagation, necessary in those situations the efficiency [8], and makes the array sensitive to small random where the array dimension becomes very large. Finally, with the variations in the excitation [15]. proposed model, the matched filter transmitter and the weighted On a related note, as the array dimensions become large and minimum mean square error precoding are derived under both the number of the antennas increases, some of the classical realistic constraints: limited radiated power and maximum ohmic losses. results for MIMO systems are no longer valid. For instance, Index Terms—Beamforming, holographic MIMO, large intelli- in [16], it is proved that the widely accepted scaling law, i.e., gent surfaces, super-directivity. the signal-to-noise ratio scales with the number of antennas, is only valid under the far-field assumption. Therefore, in order I. INTRODUCTION to properly analyze and design LIS-based MIMO systems, it is also necessary to consider near-field effects, specially in Since the seminal paper by Marzetta [1], massive multiple- indoor scenarios or those situations where far-field conditions input multiple-output (MIMO) systems have moved from being cannot be guaranteed due to the large LIS dimensions. an unrealistic idea to becoming a key enabling technology Although some works have considered the effects of superdi- in 5G and future generations of wireless networks [2, 3]. rectivity arrays [5, 17] or mutual coupling [18], to the best The promising gain of these systems have given raise to a of our knowledge, no model accounting for superdirectivity, widespread interest in considering even a larger number of coupling and near-field propagation has been presented in antennas than in conventional massive MIMO. Hence, new the literature. Aiming to fill this gap, we here propose a concepts such as holographic MIMO, large intelligent surfaces communication model for LIS-based MIMO, which considers (LIS) or intelligent reflecting surfaces (IRS) have emerged as a the three aforementioned phenomena. To that end, we merge natural evolution of classical MIMO. electromagnetic theory with classical MIMO system models, The use of LIS (i.e., large arrays) for wireless networks may creating a link that allows to include all these effects in arXiv:2011.00922v1 [cs.IT] 2 Nov 2020 render considerable gains in terms of capacity, interference the channel matrix and paving the way to more detailed reduction and user multiplexing; but it also supposes a new works. As a result, we obtain a characterization based on paradigm from a system design point of view. Introducing a infinitesimal dipoles, which is independent of any physical massive number of antennas in a limited surface leads to a small antenna realization and can be used to model real deployments, inter-element distance (ideally almost-continuous radiating e.g., metasurfaces [19]. Finally, we use the derived model surfaces [4]). Hence, phenomena that have been classically to explore the design and performance of two transmission neglected in the analysis of MIMO systems, such as mutual schemes: matched filtering (MF) and weighted minimum mean coupling [5–7] and superdirectivity effect [5, 8–11], become square error (WMMSE) [13]. now much relevant. Notation: i is the imaginary unit, k · k is the euclidean Mutual coupling is inherent to arrays with closely-spaced 2 norm, j · j is the absolute value and ·T and ·H are the transpose antennas, affecting both the radiation pattern and the impedance and Hermitian transpose respectively. Vectors are denoted by of the antenna element, which implies ultimately a change on bold lowercase symbols, and matrices are denoted by bold This work has been supported by the Danish Council for Independent uppercase symbols. Finally, Re{·}, Tr{·} and E[·] are the real Research under grant DFF-701700271. part, the trace and the expectation operator, respectively. N×1 M×1 rl jt1 jr1 where jt 2 C and jr 2 C are the currents vectors in M×1 the LIS antenna elements and the UEs, vt 2 C and vr 2 vt1 vr1 zl1 CM×1 are the voltage vectors across the LIS and UEs ports, (N+M)×(N+M) rl jt2 jr2 and Z 2 C is the system impedance matrix, which can be split into different submatrices. Specifically, Ztt 2 vt2 vr2 zl2 LIS Z UEs CN×N is the LIS impedance matrix representing the mutual M×M coupling between the different antenna elements, Zrr 2 C r j j M×N l t1 r1 represents the coupling between the UEs, and Zrt 2 C is vtN vrM zlM the LIS to UE impedance matrix, capturing the propagation effects. Eq. (1) is the basis of this paper, allowing us to create a link between the electromagnetic theory and the discrete Fig. 1. Circuit model of the scenario as a multi-port network. The ports on models widely used in communications. the left represent antennas in the LIS where the currents jtn run through the loss resistors rl before entering the network. The ports on the right represent III. SYSTEM ANALYSIS: COUPLING AND RECEIVED POWER the UEs which are terminated in load impedances zlm. A. Transmitted power, received power and efficiency From the circuital model in Fig. 1, the signal power at the II. SYSTEM MODEL receivers is equal to the power dissipated in the attached loads We consider a downlink multi-user MIMO system in which zlm (m = 1;:::;M). By Eq. (1), the voltage across the UE a base station communicates with M user equipments (UEs). ports is given as the sum of the LoS propagation and the All the users are equipped with single-antenna devices, whilst scattered waves originating from the UEs, i.e., an LIS is deployed at the base station. The LIS is modelled vr = Zrtjt + Zrrjr : (2) as a collection of N closely spaced antennas, emulating a |{z} |{z} LoS scattering near-continuous radiating surface, and centered at the origin of a cartesian coordinate system aligned with the yz-plane, Also, applying Ohm’s law at the receiver ports, the received whereas the UEs are arbitrary placed in front of it. voltage is expressed as The antennas composing the LIS are modelled as identical vr = −Zljr; (3) and infinitesimal dipoles carrying a uniform current along a M×M short line segment, where, by definition, the current distribution where Zl 2 C is a diagonal matrix with the m-th diagonal is independent of the surroundings. Note that, with this model, element equal to the load impedance zlm. We consider that we are abstracting from any physical structure for the antennas, the UEs are spaced such that the impedance looking into the and just considering the antennas as uniform current sources multiport network is dominated by the antenna’s self-impedance where the voltage is simply the difference in electrical potential z0 and, therefore, we perform a conjugate matching of the self- ∗ along the length of the dipoles. This keeps the mathematical impedance, i.e., zlm = z0 8 m. Introducing (3) in (2), the complexity of the model under control and allows to capture relation between the transmitted and received currents is given near-field propagation effects without resorting to complicated by electromagnetic simulations. Also, we only consider linear ∗ −1 jr = −(Zrr + IM z0 ) Zrtjt: (4) z-polarized receivers and transmitters, and the effects of the near-field cross-polarization terms is left for future work. As With the relation between jr and jt, the time-averaged power in [4, 16], we assume a pure line-of-sight (LoS) propagation received at the m-th UE is directly expressed as scenario, neglecting fading and shadowing. H 2 Re −jrmvrm jjrmj Refz0g To address the impact of near-field propagation and su- Prm = = ; (5) 2 2 perdirectivity effects, we consider a circuital model for the where j and v for m = 1;:::;M are the elements of j aforementioned MIMO system, similarly as done in [18] to rm rm r and v , respectively. analyze mutual coupling. In the model, represented in Fig. 1, r On the transmitter side, the primary interest is the time- every antenna element in the LIS and every UE is represented averaged power delivered to the network, which is given by1 by individual ports carrying different currents and voltages.