Low-complexity soft-output signal detector for massive MIMO with higher order QAM constellations

Abstract

As the number of base station (BS) antennas in massive multiple-input multiple-output (MIMO) systems is dramatically increased, linear detection methods are able to achieve the near optimal performance at the expense of performing the complicated matrix inversion of a high dimensional matrix, which makes hardware implementation infeasible. Meanwhile, the straightforward application of the hard symbol-to-bit decision algorithm yields a very poor performance for the higher order quadrature amplitude modulation (QAM) that is widely adopted as an efficient tool to accommodate high demand of data throughput in modern wireless systems. In this work, we propose a low-complexity soft-output signal detector for multiuser massive MIMO communication systems employing Gray-coded higher order QAM with square constellations. In the proposed detector, the two-dimensional double successive projection (2D-DSP) algorithm is utilized to iteratively realize zero-forcing (ZF) algorithm for multiuser signal recovery, which circumvents the matrix inverse operation. Moreover, a simplified implementation of Max-Log-MAP algorithm is developed to calculate log likelihood ratios (LLRs) for soft-input channel decoding by fully exploiting the bit-flipping property of Gray-coded modulation scheme along with the utilization of the channel hardening property of the massive MIMO systems. Numerical results show that the proposed soft-output detector provides a relatively good tradeoff between the complexity and performance compared with several existing detectors, and approaches the performance of the ZF algorithm with only 3 iterations.

Introduction

The booming developments of mobile Internet and Internet of Things directly drive an intensive research on the fifth generation (5G) mobile communication systems [1]. Massive multiple-input multiple-output (MIMO), as one of the key transmission technologies of the physical layer in the 5G wireless communication systems, employs a large number of antennas (e.g., in the order of hundreds) at the base station (BS) to simultaneously serve a set of users (e.g., in the order of tens) [2]. The high diversity gain and spatial resolution provided by the large-scale antenna array greatly improve the spectral efficiency and energy efficiency of the system while achieving transmission reliability [3]. In 5G technology, the mobile communication systems need to support the digital baseband modulation techniques with large constellation size in order to provide higher data transmission rates. Among the baseband modulation techniques, higher order Gray-coded quadrature amplitude modulation (QAM) with the square constellations, attracts more attentions because its modulation and demodulation are relatively simple owing to the symmetry of the in-phase and quadrature components. For example, 256-QAM signaling is finalized as a standard in 5G mobile communication systems [4]. However, realizing the attractive benefits of the massive MIMO and higher order QAM modulation in practice faces some challenges in wireless transmission, and one of which is the uplink multiuser signal detection due to the increased multiuser interference and the shrunk spacing among the modulation constellation points.

A review of various detection techniques for massive MIMO systems was provided in [5]. The optimal uplink detector is the maximum likelihood (ML) algorithm, which performs an exhaustive search in the whole solution space [6]. However, its computational complexity scales exponentially with the product of the number of users and the modulation order, which is feasible for small-scale MIMO systems with low order modulation but imposes a formidable complexity to practical implementations of the massive MIMO systems, especially with a higher order modulation. Several non-linear signal detectors were proposed to relieve the computational burden, among which the likelihood ascent search (LAS) detector [7] and reactive tabu search (RTS) detector [8] were two typical representatives. Given the initial solution, these two detectors avoided the exhaustive search over all the possible transmit symbol vectors and obtained the optimal estimate of the transmitted signal vector by multiple iterations with the computational complexity $\mathcal{O}(K^2)$ and $\mathcal{O}(MK+K^3)$ with K and M denoting the number of the single-antenna users and the modulation order, respectively. The near-optimal performance for these two detectors was achieved only for low order modulation, such as BPSK and 4-QAM. An iterative detector based on belief propagation (BP) with the parallel interference cancellation (PIC) was reported in [9]. In [10], a low-complexity algorithm based on Monte Carlo sampling for signal detection was proposed with complexity $\mathcal{O}(K^3)$, which employs a novel mixed sampling technique to effectively alleviate the stalling problem encountered at high signal-to-noise ratios (SNRs) in the conventional Gibbs-sampling-based detection scheme. The authors in [11] studied the block-wise sampling method using the Markov chain Monto Carlo algorithm with Gibbs sampler for the detection of large-scale (underdetermined) MIMO systems. A hybrid approach of particle swarm optimization (PSO) and ant colony optimization (ACO) was presented in [12] to implement the signal detection of the massive MIMO systems. The authors in [13] proposed an approximate message passing algorithm for the detection of massive multiuser MIMO-OFDM systems, which offered a desirable tradeoff between performance and complexity. However, it should be pointed out that a low order modulation modes were usually adopted for these researches, and that the near-optimal performance was achieved only when the number of BS antennas N was assumed to be equal to the number of users K, i.e., the system loading factor $\alpha =\frac{K}{N}=1$. Based on the assumption that $N\ge K$, a soft-output-sparse-aware (SOSA) detector was proposed in [14], where the ZF/MMSE detector was used to obtain the initially estimated symbol vector, and with this initial result, an improved soft-output was generated by exploiting the sparsity of a residual error vector (i.e., the difference between the transmit vector and the detected one). In practical multiuser massive MIMO systems, those assumptions rarely hold true, where the system is usually of the low system loading factors [1].

For the multiuser massive MIMO systems with low system loading factors, i.e., $\alpha \ll 1$, the random channel vectors between the users and the BS become asymptotically pairwisely orthogonal, i.e., the channel hardening behavior [15]. Thanks to that, the traditional linear detectors, such as the matched filtering (MF), the zero-forcing (ZF) and minimum mean square error (MMSE) detectors, can achieve the near-optimal performance [1]. Unfortunately, the systems with realistic antenna configurations (e.g., with a few hundred BS antennas) are far from the large-antenna limit. As a consequence, a large performance gap between the MF and ZF/MMSE detectors incurs because the MF detector minimizes the noise but doesn't deal with the multiuser interferences very well in contrast to ZF/MMSE detectors [16]. However, ZF/MMSE detectors require direct matrix inversion operations [17], which bears a complexity of $\mathcal{O}(K^3)$. To avoid the expensive matrix inversion, the truncated Neumann series approximation was proposed to convert the matrix inversion into a series of matrix-vector multiplications [18], which still had complexity of $O(K^3)$ when the order of the Neumann series was equal to or greater than 3. Based on the eigenvalue decomposition of the interference and noise covariance matrix, an minimum mean square error-interference rejection combining (MMSE-IRC) signal detection algorithm was presented in [19], which had a complexity of $O(K^2)$ with identical performance to the conventional MMSE algorithm. In [20], the fast matrix inversion updates in the context of massive MIMO were investigated, in which the complexity of the signal detection is $O(K^2)$ by using the Sherman-Morrison formula. The authors in [21] adopted Landweber method to design a low-complexity uplink detection algorithm without the involvement of matrix inversion but requiring optimizing the relax factor. A diagonal band Newton iteration (DBNI) method was proposed for signal detection in massive MIMO systems [22]. Although the DBNI achieved a higher precision, the complexity of the signal detection were very sensitive to the bandwidth of DBNI, and it still had a complexity of $O(K^3)$ when the bandwidth of DBNI was larger than 2. Using the conjugate gradient (CG) method, an iterative detector was proposed for signal recovery in massive MIMO systems [23], which reduced the computational complexity by about one order of magnitude but achieved the near-optimal performance of the classical MMSE algorithm by using only a small number of iterations. In [24], an near-optimal approach was proposed based on joint steepest descent and Jacobi (J-SDJ) algorithm, in which the steepest descent provided an efficient search direction for Jacobi iteration. It obtained the near-optimal performance of linear detectors with a small number of iterations with the same order of magnitude as [23]. An iterative sequential detection algorithm was proposed for near-optimal detection in uplink massive MIMO systems with $O(K^2)$ complexity in [25]. Utilizing the symmetric positive definitive property of the MMSE filtering matrix, two iterative detectors were presented based on Richardson and successive overrelaxation (SOR) methods in [26] and [27], respectively. Although these two detectors reduced the computational complexity from $O(K^3)$ to $O(K^2)$, the relaxation parameters in them should be carefully selected. With an equivalent objective function by QR-decomposition and further relaxation on the constrains, the authors developed an improved semidefinite further relaxation detector for high-order QAM (e.g., 64-QAM) in massive MIMO systems [28]. In [29], an algorithm based on the alternating minimization technique was proposed to solve the uplink massive MIMO detection problem, which had performance advantage over the CG-based method [23] and the J-SDJ algorithm [24] but required more computational complexity. An approach based on error recovery for detection in uplink massive MIMO systems was presented in [30]. In all these works, the modulation order considered was generally not high enough. The authors in [31] used real-domain Schnorr Euchner enumeration with K-best algorithm to reduce computational complexity of hard decision detector for the high order QAM modulation in massive MIMO and small-scale MIMO systems. However, it is well-known, with the increase of the modulation order, the hard-output detection deteriorates sharply due to the decrease in the spacing between signal points of the modulation constellation. Therefore, the practical systems often use the soft-output information to exploit gains from forward-error-correcting codes to achieve an acceptable performance.

In [32], a Gauss-Seidel (GS)-based soft-output signal detector was proposed, in which the GS method was exploited to avoid the unfavorable matrix inversion required in the MMSE algorithm. The performance/complexity trade-offs associated with CG-based soft-output detection and precoding were investigated in [33]. According to MMSE criterion, an iterative signal detector that exploited the coordinate descent method (CDM)-based algorithmic framework for uplink multiuser massive MIMO systems was proposed in [34]. According to ZF criterion, a soft-output detector based on Lanczos algorithm was presented for uplink massive multiuser MIMO system with near-optimal performance [35], where the recursive relationship of the intermediate variables between consequent two iterations was exploited in order to decrease the storage load. However, in these works, the Log-domain maximum a posterior probability (Max-Log-MAP) algorithm was adopted to calculate the Max-Log approximated LLRs of the coded bits, which also involves a very high computational complexity for higher order QAM. This brings great challenges to the efficient implementations of these soft-output detectors in multiuser Massive MIMO with higher order QAM constellations.

In this paper, according to ZF criterion, we propose a low-complexity soft-output signal detection method based on two-dimensional double successive projection (2D-DSP) algorithm [36] for massive MIMO communication systems employing Gray-coded higher order QAM constellations. In the developments of the proposed method, the unfavorable matrix inversion, required in traditional ZF algorithm, is avoided, and the LLRs calculation is simplified by taking the bit-flipping property of the Gray-coded QAM along with the utilization of the hardening property of the massive MIMO channel. We verify through simulations that the proposed method approaches the performance of the traditional ZF algorithm with soft-output at most 3 iterations. To the best of our knowledge, this work is the first to utilize the 2D-DSP algorithm for signal detection in uplink massive MIMO systems. The key results and contributions of this study are summarized as follows.

• A low-complexity soft-output signal detector based on 2D-DSP algorithm is presented for multiuser massive MIMO communication systems employing Gray-coded higher order QAM with square constellations. Compared with several soft-output detectors, including the conventional ZF detector with Cholesky decomposition, and the detectors in [18], [23], and [24], our proposed soft-output signal detector has obvious superiority in terms of computational complexity while achieving the near optimal performance of the conventional ZF detector with Cholesky decomposition with a small number of iterations.

• The 2D-DSP algorithm, which was originally presented in [36], is utilized to iteratively realize ZF algorithm for multiuser signal recovery, which circumvents the matrix inverse operation required in the conventional ZF detector.

• A simplified implementation of Max-Log-MAP algorithm for higher order QAM with square constellation is developed to reduce the complexity of calculating LLRs by taking the bit-flipping property of the Gray-coded QAM along with the utilization of the channel hardening property of the massive MIMO systems.

The remainder of this paper is organized as follows. System model is presented in Section 2. In Section 3, the soft-output signal detector based on ZF algorithm is introduced and the existing problem is formulated. In Section 4, the proposed low-complexity soft-output signal detector based on 2D-DSP algorithm is described. We present numerical simulation results in Section 5 and provide the conclusion in Section 6.