On error rate performance of multi-cell massive MIMO systems with linear receivers

Abstract

For an uplink of multi-cell multiuser (MU) Multiple-Input, Multiple-Output (MIMO) system, where each cell has a Base Station (BS) with $M$ antennas and $K$ users with single antenna, the Zero-Forcing (ZF) decoder and the Minimal Mean-Square Error (MMSE) decoder are considered. Upper bounds, lower bounds on Pair-wise Error Probability (PEP) of these decoders are derived. Moreover, analytic expressions of approximations on PEP are given. These show that, if the BS knows Channel State Information (CSI) in its own cell only and does not have CSI in other cells, error floors will occur even when Signal-to-Noise Ratio (SNR) goes to infinity for both the ZF decoder and the MMSE decoder, while these error floors disappear when $M$ goes to large. All theoretical results above are confirmed by simulations. Especially, the approximations of PEP match up with simulation results very well.

Introduction

Wireless communication systems with a high transmission rate, such as, a Multiple-Input, Multiple-Output (MIMO) system, is urgently required to satisfy high-speed transmission requirements of various applications. For such a system, a decoder with low decoding complexity becomes more and more important. Thus, linear decoders, such as Zero-Forcing (ZF) decoder or Minimal Mean-Square Error (MMSE) decoder, have attracted much attention. Although these decoders are classical, and a lot of investigations have been conducted, it is still interesting to find out various properties of these decoders in new environments of their applications. In fact, recently, new work on performance analysis of ZF or MMSE decoder still can be found in literature.

For binary inputs and non-fading channel, an exact formula for computing the bit error rate (BER) can be found in  [1], while high computing complexity is required to calculate this formula. This work was generalized to more general cases, such as, multiuser detection (see  [2]) or interference channels (see  [3]). Instead, for a fading channel, researches on the asymptotic properties of multiuser decoders have received a lot of attention recently. An asymptotic first moment of the Signal-to-Interference-plus Noise Ratio (SINR) for uncorrelated channels is derived by D. Tse and S. Hanly  [4] and S. Verdu and S. Shamai  [5]. Also D. Tse and O. Zeitouni in  [6] have proved the asymptotic normality of SINR for the equal power case. After that, for a multiple-access interference channel, the asymptotic normality has been also proved in  [7]. This asymptotic normality has been generalized to a variety of linear multiuser decoders in  [8].

In paper  [9], authors considered performance of MMSE decoder in a MIMO system with uncorrelated and correlated Rayleigh flat-fading channels. At first, SINR is decomposed into two independent variables: $\mathrm{SINR}={\mathrm{SINR}}^{\mathrm{ZF}}+F$, where ${\mathrm{SINR}}^{\mathrm{ZF}}$ is the SINR of ZF decoder, and it has been shown that this part has an exact Gamma distribution. After that, various asymptotic moments and approximations of distributions of $F$ are provided. Also in paper  [10], authors tried to derive an exact distribution of SINR. Unfortunately, only limited cases are successful. In recent paper  [11], an in-depth analysis of performance was introduced. Conditioning on that SNR goes to infinite, authors have obtained an exact distribution of the random variable $F$ for the uncorrelated channel. Furthermore, tight approximations of the uncoded error probability for a MIMO system can be obtained by a numerical integration, rather than by Monte Carlo simulations.

Recently, there are a great deal of research in massive MIMO systems, where Base Station (BS) is equipped with very large antenna arrays, see  [12], [13] and references therein. It is reported in  [12] that the very large antenna arrays can substantially reduce intra-cell interference with linear decoders. This result comes from an qualitative observation that the channel vectors are nearly orthogonal based on the Large Number Theorem when the number of antennas is large enough. Also in  [12], the asymptotic SINR was derived for the maximum-ratio combining (MRC) in the uplink system. In paper  [14], a deterministic approximations of the SINR for uplink with MRC and MMSE were derived by using random matrix theory under an assumption that the number of antennas at the BS and the number of users become large at a fixed rate. For the ZF decoder, in paper  [15], an analytic formula of calculating exact average symbol error rate (SER) based on finite Phase-Shift Keying (PSK) constellations was given in a form of integral of hypergeometric functions. Also it was reported that the inter-cell interference can cause an error floor. In paper  [16], the energy and spectral efficiency of a massive MIMO system based on linear decoders were discussed, and in paper  [17], impact on the sum rate of the ZF decoder was analyzed.

In this paper, we make an analysis on Pair-wise Error Probability (PEP) performance of linear decoders. Specifically, for ZF and MMSE decoders in a multi-cell uplink of MU-MIMO system, lower bounds, upper bounds and estimations of PEP are considered, and their analytic formulas are derived based on finite SNR for any numbers $M$ and $K$ with $M\ge K$. These formulas show that, if channel state information (CSI) from the users in its own cell to the BS is available at the BS, and the BS does not have CSI from the users in other cells to the BS, an error floor occurs inevitably. But when $M$ goes to infinity, this error floor disappears, even when the received SNR is scaled as $E_u/M^{\alpha }$ ($0<\alpha <1$), where $E_u$ is a fixed constant. Again, these judgements tell us that increasing the number $M$, the number of antennas at the BS, is an effective way to improve the performance of the system. These generalize results given in  [15], where ZF decoder based on finite PSK only were considered.

The paper is organized as follows. In Section  2, we introduce models of systems with single cell and multi-cell. In Section  3, for a multi-cell system, lower bounds, upper bounds and estimations of PEP based on ZF and MMSE decoders are presented. Simulation results are shown in Section  4, and the last section concludes the paper.

Notations. The transpose of matrix $A$ is denoted as $A^t$, and its Hermitian transpose is denoted as $A^{†}$. $\Vert A\Vert$ denotes the Frobenius norm of the matrix $A$. Also ${\left[A\right]}_k$ denotes the $k$th row of the matrix $A$, and ${\left[A\right]}_{j,i}$, or simply $A_{j,i}$, is the $(j,i)$-th component of $A$. For a random variable $X$, $E\left[X\right]$ or $E\left(X\right)$ denotes the mean of this random variable.