Parameters: maximum iteration number () ), threshold (),
All simulations are carried out in Matlab 2015b on a processor Intel(R) Core (TM) i5-6200U CPU at 2.30 GHz and 8GB RAM and all results are averaged over 10000 iterations.
Prior to apply the minimization problems () or () for the MIMO detection, the coefficient should be adjusted. Fig. 1 shows the BER of the MIMO detectors for ISTA, ADMM, and IRLS algorithms with respect to different values of . In this simulation, the SNR has been fixed at 15 dB and the parameter varies from 0 to 50. The simulations are conducted with for 4-QAM modulation. According to this figure, the values of the parameters in the following simulations are set to , And .
Fig .2 (a) shows the error of the primary detector in an uplink massive MIMO system with 16-QAM modulation with . This simulation shows that the error of the estimated user symbols is sparse. Fig. 2 (b), (c) and (d) illustrate the recovered error vector using the ADMM, ISTA, and IRLS respectively. It can be seen that only the error corresponding to the 40th user is not completely recovered. To further investigate the performance of different error recovery methods, various detection scenarios are simulated.
Fig. 1 BER performance versus in the uplink massive MIMO for 4-QAM modulation with SNR = 15 dB.
Fig.3-Fig .6 shows the bit error rate (BER) of the MIMO detection for and -QAM modulations. In Fig. 3 and Fig. 4, 4-QAM constellation with and are considered respectively. In comparison to the MMSE detector, performance improvement of the error recovery methods are markedly evident. It can be seen that the IRLS method has the best performance among other error recovery methods. In addition, all sparsity-based error recovery methods lead to lower BER in comparison to the RLS.
Fig. 5 and Fig. 6 show the detection performance for 16-QAM modulation with and respectively. Although the detection improvement is decreased but still all error recovery detectors achieved better performance than the MMSE detector.
Fig.7 compares the run time of the previously mentioned error recovery algorithms for different number of transmitters and
The times are averaged over 10000 iterations. It can be seen that the run time of the all methods increase with the system dimensions. Generally, the run time of the IRLS method is less than that of the ADMM and ISTA algorithms. Since the RLS method has a close form solution, it leads to the least run time.
The total computational complexity of the methods can be analyzed with respect to the number of multiplications in the Big-O notation. Since in the simulations is close to , it can be easily shown that the computational complexity of all methods is of order which is similar to that of the MMSE MIMO detector. In order to summarize the results, it was demonstrated that the IRLS method leads to the best MIMO detection performance. Note that, since the IRLS method is an iterative algorithm and also it requires the matrix inversion operation in each iteration, the run time of the proposed algorithm is more than that of the MMSE detector. Applying the approximation methods in matrix inversion computation such as Gauss-Seidel, Chebyshev, and conjugate gradient methods would decrease the run time of the IRLS sparse recovery method.
The performance of the large-scale MIMO systems depends on the accuracy of the channel state information (CSI). In future works, an algorithm for joint channel estimation and signal detection in sparse error domain would be considered.
This paper focused on the problem of detection in massive MIMO systems. The main idea of this algorithm is to improve the performance of the detector by finding the hidden sparsity in the residual error of the received signal. In this paper, three sparse recovery algorithms, i.e. Iterative Re-weighed Least Squares (IRLS), Alternating Direction Method of Multipliers (ADMM), and Iterative Shrinkage-Thresholding Algorithm (ISTA) have been applied to reconstruct the error of the primary detector. It is noteworthy that the iteratively reweighted least-squares (IRLS) method achieved the best performance among other sparse recovery methods. The proposed methods outperform the MMSE detector but it is obvious that the complexity of the sparse error recovery-based MIMO detectors is more than that of the MMSE detector. Consequently, more efforts are needed to decrease the computational burden of the sparse error recovery algorithms.
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* Amir Akhavan
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