# Accurate Performance Analysis of Coded Large-Scale Multiuser MIMO Systems with MMSE Receivers
## 1 Introduction  
Large-scale multiuser multiple-input multiple-output (MU-MIMO) systems are characterized by consisting of a hundred, or even more antennas at the base station (BS) and multiple users being served simultaneously, which dramatically improves the link reliablility and spectral efficiency of wireless communication systems [1]. More attractively, combining with channel coding techniques and linear signal processing technologys, e.g., Zero-Forcing (ZF) and Minimum-Mean-Square-Error (MMSE) can further enhance the performance of massive MIMO. With the appropriate user scheduling scheme [2], all communicating entities can be obtained an equal opportunity without sacrificing sum-rate performance. For MU-MIMO uplink orthogonal frequency division multiple access (OFDMA) video transmission systems, the authors of [3] propose an average peak-signal-to-noise-ratio (PSNR) optimized cross-layer resource allocation and user grouping scheme, in which zero-forcing MU-MIMO detector is considered. When the number of scatterers is small compared to the base station antenna and single antenna users in the cell, the authors of [4] proposes a fast iterative weighted singular value thresholding (WSVT) algorithm to obtain a more accurate channel estimate.It is worth emphasizing that, when the elements of the BS antenna array become large, the column-vectors of the propagation matrix from BS to users tend to be orthogonal [5; 6]. Considering a ZF detector, an approximate distribution for the received SINR and an approximated symbol error probability (SEP) were explored [7]. When an MMSE receiver with beamforming is considered in massive MIMO links, the asymptotic evaluation of the target SINR was investigated in [8]. Work that relates directly to the throughput bound for convolutionally encoded large-scale MIMO-OFDM links was investigated in [9], by using the approximate noise distribution for the ZF detector. Moreover, the energy and spectral efficiency of the uplink large multiuser MIMO wireless networks are analyzed in [10]. We note that the authors only discuss the rate by a lower bound result, based on the approximated gamma distribution of SINR for the MMSE receiver considered in [11]. According to the exact SINR distribution provided in [12], the authors of [13] studied the performance of convolutionally encoded MMSE-MIMO Rayleighfading channels. However, for large-scale multi-user MIMO systems, this distribution cannot be analyzed analytically. By considering additive quantization noise model (AQNM), the authors of [14] studied the MMSE detection performance of a hybrid analog-to-digital converters (ADC) massive MIMO system, which can reduce energy consumption without reducing spectral efficiency compared to a full-resolution ADC system. Impressively, a system named cell-free massive MIMO with many distributed access points was investigated in [15]; by applying MMSE detection, the author comprehensively analyzed the spectral efficiencies of the system, and the analysis results showed that the cell-free massive MIMO is a more promising technology than conventional Cellular massive MIMO in beyond-5G networks. Furthermore, a closed-form distribution for the output SINR of the MMSE receiver was derived in [16], for uncoded MIMO systems in ideal fast Rayleigh fading, and accurate symbol error performance was evaluated. In this paper, we focus on the investigation of the uplink performance for coded cellular large-scale multiuser MIMO systems with an MMSE receiver. Our research work so far makes the following specific contributions:  
1. We derive a novel statistical distribution of the SINR for any user when the channels are subject to path loss, shadowing and Rayleigh fading. 2. Using the derived SINR density, we further derive the PEP, which is then used to obtain analytical expressions relating to upper-bounds on BER of the convolutionally coded massive multiuser MMSE-MIMO systems. Compared with [16], the channel considered in this paper is more in line with the actual multi-user MIMO scenario.