# Computationally Efficient Soft Detection Schemes for Coded Massive MIMO Systems +
## 1 Introduction  
In recent years, the massive multiple-input multiple-output (MIMO) technology which utilizes hundreds of antennas at the base station (BS) has attracted great interests due to its high data rates for a given bandwidth [1]. In coded MIMO systems, the joint iterative detection and decoding (JIDD) method can produce high performance gain [2]. The performance gain from the JIDD method could be achieved by iteratively exchanging soft information between the maximum likelihood (ML) detector and decoder, at the cost of computational complexity. Minimum mean square error (MMSE)-based detection schemes were often considered for JIDD due to their reasonable complexity and performance tradeoff. These were called the minimum mean square error parallel interference cancellation (MMSE-PIC) methods [3].  
A complexity reduced JIDD scheme with the MMSE-PIC was proposed in [4], and later JIDD with three loops were proposed in [5]. Even though the MMSE-PIC method for JIDD provides a trade-off between the performance and computational complexity, its computational complexity is still too high for massive MIMO systems. The main computational burden during the MMSE-PIC process is incurred from the estimation of the Gram matrix and matrix inversion [6]. Several attempts were made to apply the MMSE-PIC based detection for massive MIMO systems. Large efforts were made toapproximate matrix inversion process by using iteration based methods [7; 8; 9; 10]. In addition, Neumann series expansion (NSE) method [11] and Newton iteration method [12] were proposed.  
Considering that the channel matrix is asymptotically orthogonal in massive MIMO systems [13], a diagonal-like matrix was proposed to approximate the MMSE filtering matrix to reduce the complexity [14]. However, these methods still suffer from too much complexity when the number of antennas at the BS is massive. Not only the the complexity caused by calculating the Gram matrix and the matrix inversion, but also the post-equalization signal-to-interference-plus-noise ratio (PE-SINR) estimation at every layer causes a lot of computations [10]. For this reason, our previous study presented a complexity reduced method to approximate PE-SINR, but the investigation was limited to non-iterative detection and decoding schemes [15].  
In this paper, we propose an efficient MMSE-PIC-based JIDD scheme with huge reduction in computational complexity for coded massive MIMO systems. The proposed scheme first derives a universal constant diagonal matrix which eventually results in a simplified filtering matrix as well as compact PE-SINR estimation. In addition, an efficient soft-symbol estimation method is proposed to reduce the number of computations. Especially, we present compact equations for \(M\)-ary quadrature amplitude modulation (QAM) schemes which require less than 1% of the computational complexity compared to the conventional scheme. Finally, soft bit information (SBI) to the decoder is extracted by using a simple symbol mapping method to further reduce the complexity.  
The remainder of this paper is organized as follows. In Section 2, we first review an existing massive MIMO system with JIDD combined with MMSE-PIC, and introduce a simple soft demapping method which can extract soft bit information with a linear-order complexity. Section 3 presents the proposed MMSE-PIC-based JIDD scheme with a number of techniques to reduce the computational complexity. In addition, the complexity reduced equations to estimate soft symbols for MMSE-PIC process are derived. Simulation results are shown in Section 4, and Section 5 concludes the paper.  
_Notation_: Lowercase and uppercase boldface letters denote vectors and matrices, respectively, lower and upper case letters denote scalars. Transpose, conjugate transpose, matrix inversion, and norm operations are denoted by \((\cdot)^{T}\), \((\cdot)^{H}\), \((\cdot)^{-1}\) and \(|\cdot|\), respectively. In addition, \(M\times M\) identity matrix is represented by \(\mathbf{I}_{M}\).