A Novel Detector based on Compressive Sensing for Uplink Massive MIMO Systems
Research Areas : Communication Systems & Devices
Mojtaba Amiri
^{
1
}
Amir Akhavan
^{
2
}
Keywords: Massive MIMO, MMSE Detector, Error Recovery, Compressive Sensing, Iteratively Reweighted Least Squares (IRLS) Method.,
Abstract :
Massive multipleinput multipleoutput 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 nearoptimal performance in largescale 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.
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http://jist.acecr.org ISSN 23221437 / EISSN:23452773 
Journal of Information Systems and Telecommunication

A Novel Detector based on Compressive Sensing for Uplink Massive MIMO Systems 
Mojtaba Amiri1, Amir Akhavan2*

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 8415683111, Iran 
Received: 19 Feb 2022/ Revised: 07 Apr 2022/ Accepted: 13 Apr 2022 

Abstract
Massive multipleinput multipleoutput 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 nearoptimal performance in largescale 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 fifthgeneration (5G) wireless communication systems are being designed to answer excessive data rate demands. Massive MIMO technology is a good candidate for the nextgeneration 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 [35].
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], GaussSeidel (GS) method [8], parallelizable Chebyshev iteration (PCI) [9] and alternating minimization method (AltMin) [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 nonzero entries of the error vector.
Sparse signal recovery is basically an optimization problem with norm and is NPhard. Therefore, different approaches are proposed to solve this problem. Greedy methods [1517], 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 relaxationbased 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 multiuser 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 complexvalued 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.
21 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 nearoptimal performance in terms of bit error rate. In the following the error domain sparse model is introduced.
31 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)
32 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 nonzero elements in the vector. Since () is NPhard, 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 wellknown 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 closedform 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 recoverybased MIMO detection is the sparse error reconstruction. In this study, three algorithms are considered for the sparse coding step. More explicitly, Iterative Reweighed Least Squares (IRLS), Alternating Direction Method of Multipliers (ADMM), and Iterative ShrinkageThresholding Algorithm (ISTA) [2225] have been applied to reconstruct the error vector. In the following, these three algorithms are introduced briefly.
41 IRLS Algorithm
The Iterative Reweighed 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 pseudocode for the IRLS error recoverybased MIMO detector has been shown in Algorithm 1.
42 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 recoverybased 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 LeastSquares: 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 pseudocode of the MIMO detection using the ADMM algorithm has been shown in Algorithm 2. 43 ISTA Algorithm Another algorithm which can be used for solving problem (P2) is the iterative shrinkagethresholding 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.
