J an 2 01 5 On the Convergence of Massive MIMO Systems

In this paper we examine convergence properties of massive MIMO systems with the aim of determining the number of antennas required for massive MIMO gains. We consider thr ee characteristics of a channel matrix and study their asymptotic behaviour. Furthermore, we derive ZF SNR and MF SINR for a scenario of unequal receive powers. In our results we includ e the effects of spatial correlation. We show that the rate of convergence of channel metrics is much slower than that of th e ZF/MF precoder properties. I. I NTRODUCTION Multiple-Input Multiple-Output (MIMO) has been incorporated into emerging wireless broadband standards such as LT E. In general, in time-division duplexing (TDD) mode, as we add more and more antennas to the base station, the performance in terms of data rate, enhanced reliability, improved energ y efficiency and interference increases [1]. Consequently, m assive MIMO is an emerging technology, where the number of antennas is scaled up by 1-2 orders of magnitude [2]–[8] with the aim of even greater performance benefits [1]. Motivating the surge in research activities into massive MIMO are the additional gains resulting from a large channel matrix, due to the asymptotics of random matrix theory. For example, the effect of small-scale fading can be averaged ou t [3], beacuse random elements tend to become deterministic and matrix operations can be computed much easier due to matrices becoming well conditioned [9]. Consequently as we let the number of antennas tend to infinity, system analysis becomes simplified. However, as a very large number of antennas may be impractical due to physical constraints, we would like to determine the point where MIMO systems begin to exhibit these additonal benefits. This is the focus of the paper. The following are the contributions: • We study the convergence properties of the channel matrix by considering three metrics defined in Sec. III. • We investigate precoder properties by deriving new analytical expressions for SNR and SINR of ZF and MF precoders for unequal power gains. • We demonstrate the difference in convergence rate of channel properties and that of precoder characteristics 1. II. SYSTEM MODEL A. System Description We consider a massive MIMO system with M co-located antennas at the base station serving K single-antenna users. 1Note: Imperfect CSI is considered in future work On the down-link (where TDD is assumed) the K terminals collectively receive aK × 1 vector xf = √ ρfG T sf +wf , (1) whereρf is the transmit SNR, [·]T represents matrix transpose, sf is aM×1 precoded vector of data symbols, qf , andwf is a K×1 noise vector with independent and identically distributed (iid) CN (0, 1) entries. The transmit power is normalized such thatE{‖sf‖} = 1. TheM ×K channel matrix,G, is G = HD 1 2 β , (2) whereH is a M ×K matrix which accounts for small-scale Rayleigh fading and transmit spatial correlation, and Dβ is a diagonal matrix modelling large-scale effects. The diago nal elements ofDβ are represented by a K × 1 vector,β, with entriesβj representing the link gains. B. Power Model We consider cases with equal and unequal link gains. The qual power case models a single-user MIMO system where one user has K co-located antennas. This is used as a reference case. While not considered by other authors, the unequal power case models K distributed UEs where each user has a different link gain due to path-loss, shadowing, etc. Since we are considering convergence issues as the system dimension grows large, it is not convenient to generate the link gains using classic log-normal shadowing, path loss models. This is because the variation in the link gains will confound the limiting effects. Also, with random link gains, as the syste m size increases we may obtain an artificially large number of igh gains or small gains. We propose a power model to counter these two problems. We select theβj values from the limiting functionβ(x) = Aη, whereη is arbitrarily set such that 0 < η < 1, A = βmax, and0 < x < x0, such thatx0 = log(βmin/βmax) log(η) . This allows us to control the range of the link gains in the interva l [βmin, βmax] = [Aη x0 , A] and also to control the rate of decay of the link gains byη. Given the parameters βmin, βmax, η and the number of users, K, the βj values are given byβj = β( x0 2K (2j − 1)) which gives theK values ofβ1, β2, . , βK as the values of β(x) usingK values ofx evenly spread over [0, x0]. C. Convergence Model Many features of massive MIMO systems are driven by the convergence of various functions as the system size increas es. A key issue is the convergence of 1 MG G to I. This is examined in Sec. III-A and III-B. Also of interest is the converge nce of ZF and MF precoder properties to their asymptotic limits. Thus, in Section III-C, we examine the convergence of the SNR and SINR of ZF and MF precoders, respectively. In doing so we will consider the following two types of convergence scenarios: 1) K is fixed,M → ∞, 2) K M = α ,K → ∞, α is fixed. D. Correlation Model In massive MIMO, if we deploy more and more antennas in a fixed volume, correlation amongst antenna elements will increase. Hence, as we allow the array sizes to grow, we also allow correlations to increase. We consider the Kronec ker correlation model, where H = R 1/2 t Hiid, (3) whereHiid is theM×K iid channel matrix,H is theM×K correlated channel matrix and Rt is theM×M transmit spatial correlation matrix. The elements of Rt, rij , are based on the exponential correlation matrix model [10] rij = ρ dij , |ρ| ≤ 1, (4) wheredij is the distance between antennas i and j, ρ is the correlation decay constant and rij is the correlation coefficient between antennas i and j. We will assume theM colocated antennas are in a Uniform Linear Array (ULA) at the base station. III. M ETHODOLOGY In this section we focus on the second convergence scenario, where bothK andM grow large withα = M/K remaining fixed.2 A. Convergence of GG/M First, consider the entries ofHH. Let H = [h1 h2 . hK ], wherehi is an M × 1 column vector with entries,hij , then it is well known that