A Novel Detector based on Compressive Sensing for Uplink Massive MIMO Systems
Subject Areas : Communication Systems & DevicesMojtaba Amiri 1 , Amir Akhavan 2
1 - School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
2 - Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran
Keywords: Massive MIMO, MMSE Detector, Error Recovery, Compressive Sensing, Iteratively Reweighted Least Squares (IRLS) Method.,
Abstract :
Massive multiple-input multiple-output is a promising technology in future communication networks where a large number of antennas are used. It provides huge advantages to the future communication systems in data rate, the quality of services, energy efficiency, and spectral efficiency. Linear detection algorithms can achieve a near-optimal performance in large-scale MIMO systems, due to the asymptotic orthogonal channel property. But, the performance of linear MIMO detectors degrades when the number of transmit antennas is close to the number of receive antennas (loaded scenario). Therefore, this paper proposes a series of detectors for large MIMO systems, which is capable of achieving promising performance in loaded scenarios. The main idea is to improve the performance of the detector by finding the hidden sparsity in the residual error of the received signal. At the first step, the conventional MIMO model is converted into the sparse model via the symbol error vector obtained from a linear detector. With the aid of the compressive sensing methods, the incorrectly detected symbols are recovered and performance improvement in the detector output is obtained. Different sparse recovery algorithms have been considered to reconstruct the sparse error signal. This study reveals that error recovery by imposing sparse constraint would decrease the bit error rate of the MIMO detector. Simulation results show that the iteratively reweighted least squares method achieves the best performance among other sparse recovery methods.
[1] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, "Massive MIMO for next generation wireless systems," IEEE communications magazine, vol. 52, no. 2, pp. 186-195, 2014.
[2] E. Björnson, E. G. Larsson, and T. L. Marzetta, "Massive MIMO: Ten myths and one critical question," IEEE Communications Magazine, vol. 54, no. 2, pp. 114-123, 2016.
[3] E. Björnson, J. Hoydis, M. Kountouris, and M. Debbah, "Massive MIMO systems with non-ideal hardware: Energy efficiency, estimation, and capacity limits," IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 7112-7139, 2014.
[4] M. A. Albreem, M. Juntti, and S. Shahabuddin, "Massive MIMO detection techniques: a survey," IEEE Communications Surveys & Tutorials, vol. 21, no. 4, pp. 3109-3132, 2019.
[5] M. A. Albreem et al., "Low complexity linear detectors for massive MIMO: A comparative study," IEEE Access, vol. 9, pp. 45740-45753, 2021.
[6] L. Fang, L. Xu, and D. D. Huang, "Low Complexity Iterative MMSE-PIC Detection for Medium-Size Massive MIMO," IEEE Wireless Communications Letters, vol. 5, no. 1, pp. 108-111, 2016.
[7] M. Wu, C. Dick, J. R. Cavallaro, and C. Studer, "High-Throughput Data Detection for Massive MU-MIMO-OFDM Using Coordinate Descent," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 12, pp. 2357-2367, 2016.
[8] L. Dai, X. Gao, X. Su, S. Han, I. C. L, and Z. Wang, "Low-Complexity Soft-Output Signal Detection Based on Gauss&Side Method for Uplink Multiuser Large-Scale MIMO Systems," IEEE Transactions on Vehicular Technology, vol. 64, no. 10, pp. 4839-4845, 2015.
[9] G. Peng, L. liu, P. Zhang, S. Yin, and S. Wei, "Low-Computing-Load, High-Parallelism Detection Method based on Chebyshev Iteration for Massive MIMO Systems with VLSI Architecture," IEEE Transactions on Signal Processing, vol. PP, no. 99, pp. 1-1, 2017.
[10] A. Elgabli, A. Elghariani, V. Aggarwal, and M. R. Bell, "A low-complexity detection algorithm for uplink massive MIMO systems based on alternating minimization," IEEE Wireless Communications Letters, vol. 8, no. 3, pp. 917-920, 2019.
[11] M. Amiri and M. F. Naeiny, "Low-Complexity Iterative Detection for Uplink Multiuser Large-Scale MIMO," Journal of Information Systems and Telecommunication (JIST), vol. 1, no. 29, p. 25, 2020.
[12] Z. Gao, L. Dai, S. Han, I. Chih-Lin, Z. Wang, and L. Hanzo, "Compressive sensing techniques for next-generation wireless communications," IEEE Wireless Communications, vol. 25, no. 3, pp. 144-153, 2018.
[13] M. Amiri and A. Akhavan, "An iterative detector based on sparse bayesian error recovery for uplink large-scale MIMO systems," AEU-International Journal of Electronics and Communications, vol. 138, p. 153848, 2021.
[14] R. Ran, J. Wang, S. K. Oh, and S. N. Hong, "Sparse-aware minimum mean square error detector for MIMO systems," IEEE Communications Letters, vol. 21, no. 10, pp. 2214-2217, 2017.
[15] S. Kwon, J. Wang, and B. Shim, "Multipath matching pursuit," IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2986-3001, 2014.
[16] D. L. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck, "Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit," IEEE transactions on Information Theory, vol. 58, no. 2, pp. 1094-1121, 2012.
[17] Z. Zhang, Y. Xu, J. Yang, X. Li, and D. Zhang, "A survey of sparse representation: algorithms and applications," IEEE access, vol. 3, pp. 490-530, 2015.
[18] H. Xu, C. Caramanis, and S. Mannor, "Robust regression and lasso," IEEE Transactions on Information Theory, vol. 56, no. 7, pp. 3561-3574, 2010.
[19] M. Tan, I. W. Tsang, and L. Wang, "Matching pursuit LASSO part I: Sparse recovery over big dictionary," IEEE Transactions on Signal Processing, vol. 63, no. 3, pp. 727-741, 2014.
[20] R. Torkamani and R. A. Sadeghzadeh, "Wavelet-based Bayesian Algorithm for Distributed Compressed Sensing," Information Systems & Telecommunication, p. 87, 2019.
[21] W.-C. Chang and Y. T. Su, "Sparse Bayesian Learning Based Tensor Dictionary Learning and Signal Recovery with Application to MIMO Channel Estimation," IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 3, pp. 847-859, 2021.
[22] I. Daubechies, M. Defrise, and C. De Mol, "An iterative thresholding algorithm for linear inverse problems with a sparsity constraint," Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, vol. 57, no. 11, pp. 1413-1457, 2004.
[23] T. Blumensath and M. E. Davies, "Iterative thresholding for sparse approximations," Journal of Fourier analysis and Applications, vol. 14, no. 5-6, pp. 629-654, 2008.
[24] C. J. Miosso, R. von Borries, M. Argaez, L. Velázquez, C. Quintero, and C. Potes, "Compressive sensing reconstruction with prior information by iteratively reweighted least-squares," IEEE Transactions on Signal Processing, vol. 57, no. 6, pp. 2424-2431, 2009.
[25] J. Yang and Y. Zhang, "Alternating direction algorithms for l_1-problems in compressive sensing," SIAM journal on scientific computing, vol. 33, no. 1, pp. 250-278, 2011.
[26] M. Elad, Sparse and redundant representations: from theory to applications in signal and image processing. Springer Science & Business Media, 2010.
[27] S. Boyd, N. Parikh, and E. Chu, Distributed optimization and statistical learning via the alternating direction method of multipliers. Now Publishers Inc, 2011.
[28] A. Beck and M. Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems," SIAM journal on imaging sciences, vol. 2, no. 1, pp. 183-202, 2009.
http://jist.acecr.org ISSN 2322-1437 / EISSN:2345-2773 |
Journal of Information Systems and Telecommunication
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Abstract
Massive multiple-input multiple-output is a promising technology in future communication networks where a large number of antennas are used. It provides huge advantages to the future communication systems in data rate, the quality of services, energy efficiency, and spectral efficiency. Linear detection algorithms can achieve a near-optimal performance in large-scale MIMO systems, due to the asymptotic orthogonal channel property. But, the performance of linear MIMO detectors degrades when the number of transmit antennas is close to the number of receive antennas (loaded scenario). Therefore, this paper proposes a series of detectors for large MIMO systems, which is capable of achieving promising performance in loaded scenarios. The main idea is to improve the performance of the detector by finding the hidden sparsity in the residual error of the received signal. At the first step, the conventional MIMO model is converted into the sparse model via the symbol error vector obtained from a linear detector. With the aid of the compressive sensing methods, the incorrectly detected symbols are recovered and performance improvement in the detector output is obtained. Different sparse recovery algorithms have been considered to reconstruct the sparse error signal. This study reveals that error recovery by imposing sparse constraint would decrease the bit error rate of the MIMO detector. Simulation results show that the iteratively reweighted least squares method achieves the best performance among other sparse recovery methods.
Keywords: Massive MIMO; MMSE Detector; Error Recovery; Compressive Sensing; Iteratively Reweighted Least Squares (IRLS) Method.
1- Introduction
The number of cellular phones and mobile data traffic are extremely growing each year. Telecommunication companies are asked to provide higher data rates, further spectral efficiency, and larger capacity. The fifth-generation (5G) wireless communication systems are being designed to answer excessive data rate demands. Massive MIMO technology is a good candidate for the next-generation of the wireless communication. The base station (BS) in massive MIMO systems equipped with hundreds of antennas are used to serve tens of users simultaneously [1, 2]. But there are some challenges such as pilot contamination, detection performance, channel estimation and detection complexity [3-5].
The purpose of each detection algorithm is to obtain an estimate of the transmit signal, given knowledge of the received signal and the channel state information (CSI). The maximum a posteriori (MAP) and the maximum likelihood (ML) algorithms provide the optimal detectors but they are not practically feasible for the massive MIMO systems since their computational complexity increase exponentially with the number of antennas. Linear MIMO detectors such as zero forcing (ZF) and minimum mean square error (MMSE) receivers can achieve near optimal performance when the number of users is much lower than the number of the antennas in BS [6]. Many methods have been proposed to achieve the performance of MMSE detector with low complexity such as the optimized coordinate descent (OCD) [7], Gauss-Seidel (GS) method [8], parallelizable Chebyshev iteration (PCI) [9] and alternating minimization method (Alt-Min) [10]. The performance of the linear detectors and also the previously mentioned methods degrade when the number of transmitters is close to the number of receiver antennas [11]. Therefore, new and efficient detectors with a low error rate are highly needed to solve this problem. This paper focuses on developing a detector which achieves favorable performance in loaded scenarios.
Recently, compressive sensing (CS) and sparse signal recovery techniques have received much attention in different signal processing applications. Compressive sensing has emerged as a promising approach for use in large MIMO systems [12, 13].
It is noteworthy that the original signals in massive MIMO systems are not intrinsically sparse, but it is expected that the detector output contains an error only for a few number of users. Thus, the error vector resulting from a primary estimator is likely to be sparse, especially in high SNR regime. The motivation of this paper is to improve the performance of the detector by using the sparsity in the residual error of large MIMO systems. In order to exploit the sparsity of the detection errors, the conventional model is converted into a sparse model via the symbol error vector [13, 14]. After that, the error recovery algorithm can be performed to improve the detection performance by recovering the non-zero entries of the error vector.
Sparse signal recovery is basically an optimization problem with -norm and is NP-hard. Therefore, different approaches are proposed to solve this problem. Greedy methods [15-17], -relaxation based optimization [18, 19] and Bayesian methods [20, 21] are the main approaches to estimate the sparse vector. Many different algorithms have been proposed for sparse signal reconstruction. The contribution of this paper is to address the effectiveness of the -relaxation-based sparse recovery methods in massive MIMO detectors for the first time.
The rest of the paper is organized as follows. Section II introduces the system model of the massive MIMO system. Section III presents the conversion of the conventional MIMO system model into the sparse error domain. Sparse recovery algorithms are introduced in section IV. The simulation results and discussions about the performance of the proposed algorithm are presented in section V and finally the paper is concluded in Section VI.
Notation: Boldface capital letters and lowercase letters represent matrices and vectors, respectively. denotes the identity matrix; , , denote the conjugate transposition, the inversion and the transposition, respectively. denotes the complex matrix. The -norm (also called -norm) of vector is .
2- System Model
Consider a multi-user MIMO model with users and receivers in the BS. The received signal can be described as:
(1)
where is the channel matrix between the BS and the users whose entries are modeled as independent and identically distributed (iid) complex Gaussian with zero mean and unit variance., is the complex-valued information vector, and is a white Gaussian noise vector with zero mean and correlation matrix . It is assumed that the channel matrix is known perfectly at the BS but it is unknown at the transmitter side.
2-1- Linear Detection
The maximum likelihood (ML) detector is not suitable for solving large dimensional problems due to the high computational complexity. Therefore, suboptimal detectors such as Minimum Mean Square Error with low complexity are beneficial in operational conditions.
The MMSE detector can be obtained by the solution of the following minimization problem.
(2)
where is the estimation of the user’s data.
3- Massive MIMO Detection in Sparse Domain
This section, presents a class of detectors based on error recovery technique for detection of the transmitted symbols in uplink massive MIMO system. This method iteratively achieves near-optimal performance in terms of bit error rate. In the following the error domain sparse model is introduced.
3-1- Sparse Model
At the first step, the conventional system model (1) should be converted into a sparse model via the symbol error vector obtained from a linear detector.
The error vector , is defined as the difference between the original signal and the recovered one. Therefore, the system model in error domain can be formulated as follows:
(3)
where is the error vector of the primary estimated symbols and its nonzero values correspond to the incorrectly detected symbols.
It is noteworthy that, by recovering the incorrectly detected symbols, the performance of the detector can be improved. Since it is expected that only a few symbols are incorrectly detected, , is a sparse vector. Therefore, the detection operation is equivalent to recover the sparse error vector,, from the difference signal,.
Once the sparse error vector is recovered, the estimation of the transmitted signal,, is obtained by adding the error vector to the initial estimate,. Thus, the final estimation of the user’s data is obtained by
(4)
3-2- Error Recovery
The problem of sparse representation in the MIMO detection is to find the vector . Therefore, we are looking for the sparsest solution which can be done by solving the following optimization problem:
(5)
Where denotes the -norm of and gives the total number of non-zero elements in the vector. Since () is NP-hard, the optimization problem is relaxed with convex -norm. Taking into account the effect of the noise component, the problem () can be converted to the following optimization problem:
(6)
It is assumed that the noise has bounded entries, i.e. for some sufficiently small . Additionally, according to the Lagrange multiplier theorem, there exists an appropriate constant such that the problem () is equivalent to the following unconstrained minimization problem.
(7)
Where the Lagrange multiplier depends on and . Note that the cost function in () is not differentiable with respect to and specific optimization algorithms are required to solve (). The following section addresses three well-known sparse coding algorithms to estimate the error vector . The minimization function in () is composed of two parts. The first term with –norm induces sparsity to the estimated error vector, while the second term, , makes the estimated vector consistent with . In order to investigate the effectiveness of the sparsity promoting term in (), the results of the MIMO detection with the following minimization problem are also considered.
(8)
where the –norm in () is replaced with the –norm. The closed-form solution of the convex minimization problem () can be formulated as follows:
(9)
In the simulation result section, the solution to () is called the regularized least square (RLS) estimation.
4- Sparse Error Reconstruction
The most significant stage in error recovery-based MIMO detection is the sparse error reconstruction. In this study, three algorithms are considered for the sparse coding step. More explicitly, Iterative Re-weighed Least Squares (IRLS), Alternating Direction Method of Multipliers (ADMM), and Iterative Shrinkage-Thresholding Algorithm (ISTA) [22-25] have been applied to reconstruct the error vector. In the following, these three algorithms are introduced briefly.
4-1- IRLS Algorithm
The Iterative Re-weighed Least Squares algorithm is one of the strategies which is able to recover sparse signals. In this algorithm, the -norm in () is replaced by a weighted -norm [26]:
(10)
Where is a diagonal weight matrix and it is updated from the current iterate.
The minimization in (P3) is a quadratic optimization problem, soluble using linear algebra. The pseudo-code for the IRLS error recovery-based MIMO detector has been shown in Algorithm 1.
4-2- ADMM Algorithm
The alternating direction method of multipliers is an alternative algorithm for sparse coding. This algorithm uses the augmented Lagrangian to splits the main optimization problem into two quadratic and separable minimization problems.
In this method, the augmented Lagrangian is defined as [27]
(11)
Algorithm 1: IRLS error recovery-based detector | ||||
Input:, , and Parameters: maximum iteration number ( ), threshold () Output: The estimation of the transmitted symbols:
initialization: 1: , = diag () and 2: ‘Primary Estimation’ 3: 4: The initial weight matrix
Iteration: Increase 5: Regularized Least-Squares: approximately solve the linear system
6: Weight Update: Update the diagonal weight matrix
7: Stopping Rule: if break else go back to step 5 8: Output: . 9: return
where is the Lagrangian multiplier and is a penalty parameter. The pseudo-code of the MIMO detection using the ADMM algorithm has been shown in Algorithm 2. 4-3- ISTA Algorithm Another algorithm which can be used for solving problem (P2) is the iterative shrinkage-thresholding algorithms (ISTA). The solution based on the ISTA algorithm can be written as [23, 28]
(12)
where is the step size and the error vector is updated as
(13)
where is the shrinkage operator and is described by
(14)
where and are the Schur product and the sign function respectively. The parameter represents the threshold value and is the proper scale hyperparameter.
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