# LDPC Coded Massive MIMO Systems
## 1 Introduction  
The massive multiple-input multiple-output (MIMO) system, whose transmitter and receiver are equipped with tens to hundreds of antennas, has recently attracted many researchers and engineers because it can vastly improve the transmission data rate and spectral efficiency [1; 2; 3; 4; 5; 6; 7]. Massive MIMO technology has recently found successful applications in cellular networks, known as the fifth generation (5G) systems [5; 6; 7; 8; 9; 10; 11], as well as in energy-efficient wireless sensor networks [12; 13; 14; 15]. Recovering multiplexed data from signals received by many antennas in an optimal manner requires tremendously high amount of computations, so the reduction of detection complexity has been a great concern for utilizing the massive MIMO technique in practical communication systems [3; 4]. As an approach to reduce the detection complexity, suboptimal linear detection algorithms have been intensively studied [7; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26], where matched filter (MF) detection, zero forcing (ZF) detection and minimum mean squared error (MMSE) detection are well known examples. Nevertheless, these linear detection schemes cannot lower the computational complexity of the massive MIMO receiver to an acceptable level because the inversion of high dimensional matrices is still required. Then, low complexity detection algorithms based on approximate matrix inversion [18; 19], low complexity factor graph (FG) based belief propagation (BP) algorithms [20; 21; 22] and pairwise Markov random fields (MRF) based MIMO detection algorithms [20; 21] have been proposed. Tree-searching soft-input soft-output (SISO) MIMO detection algorithms have also been proposed in various forms [23; 24; 25; 26]. The FG based BP detection with Gaussian approximation of interference (GAI), called FG-GAI BP detector, was proposed as one of the promising solutions to reduce the computational complexity of the massive MIMO receiver to the practically allowable level [21; 22]. Low-density parity-check (LDPC) codes have been widely used in various communication systems thanks to the powerful error correction capability [27; 28; 29]. It is well known that LDPC codes can be efficiently designed by using the density evolution algorithm [29; 30] or the extrinsic information transfer (EXIT) chart [31]. There have been many research activities regarding the analysis and design of LDPC coded MIMO systems with various forms of detection and decoding mechanisms [32; 33; 34; 35; 36]. It is a natural approach to apply LDPC codes to the massive MIMO system to improve the transmission reliability, where a joint detection and decoding (JDD) algorithm of low complexity, of course, needs to be considered. In [37], non-binary LDPC codes are designed for coded massive MIMO systems considering modified MMSE and MF soft-output detectors. In [22; 38], binary LDPC codes and non-binary LDPC codes, respectively, are designed by considering the FG-GAI BP detection algorithm through the degree distribution optimization based on the EXIT chart analysis. To make the LDPC coded massive MIMO technology more applicable in practical communication systems, the convergence of JDD has to be sped up for a given amount of computational complexity. Note that the faster JDD convergence results in the lower BER if the computational complexity is limited to finite amount. In [39], a modified FG-GAI BP detection algorithm is proposed to improve the convergence rate of JDD in LDPC coded massive MIMO systems at the cost of increased computational complexity. It is notable that one JDD iteration can be composed of multiple detection iterations followed by multiple decoding iterations. Thus, the JDD strategy, specified by the ratio of the number of detection iterations and the number of decoding iterations composing one JDD iteration, can be used as a design parameter to obtain a good trade-off between the error correction performance and the computational complexity.