Efficient Hybrid Iterative Method for Signal Detection in Massive MIMO Uplink System over AWGN Channel

direction method of multipliers (ADMM) technique and Gauss-Seidel method. The initial vector has a large influence on the performance, complexity, and convergence rate of such iterative algorithms. The proposed detector’s initial solution is determined using the diagonal matrix and MMSE with the ADMM technique. The proposed algorithm’s performance and complexity are compared with existing algorithms based on BER and the real number of multiplications, respectively. The numerical results revealed that the proposed algorithm achieves the desired performance with a small number of iterations and a significant reduction in computational complexity. At 8QAM, SNR (cid:31) 20dB, 80 × 120 massive MIMO antenna configuration, and n (cid:31) 2, the percentage performance improvement of the proposed detector from the GS detector is 99.82%. At 32QAM, SNR (cid:31) 25dB, 120 × 180 antenna configuration, and n (cid:31) 5, performance improvement of the proposed detector is 99.89%. At 64QAM, SNR (cid:31) 28dB, 80 × 120 antenna configuration, and n (cid:31) 3, performance improvement of the proposed detector is 99.93%.


Introduction
Fifth-generation (5G) mobile networks are currently being implemented in order to meet user demands for high performance and high data rates. To achieve high data rates, energy e ciency, and spectral e ciency, 5G used one of the enabling technologies known as massive multiple input multiple output (Massive MIMO) [1]. Massive MIMO is the most enticing technology for 5G and beyond wireless access [2]. Massive MIMO is an advancement of current MIMO systems used in wireless networks that groups together a large number of antennas at the base station and serves many users at the same time as shown in Figure 1.
Massive MIMO technology is being considered by the 5G network as a potential solution to the problem caused by massive data tra c and users [3]. Massive MIMO's extra antennas will help focus energy into a smaller region of space, providing better spectral e ciency and throughput [4]. Radiated beams in a massive MIMO system become narrower and more spatially focused toward the users as the number of antennas increases. ese spatially focused antenna beams improve throughput for the intended user while reducing interference for the neighboring user [5].
Massive MIMO in 5G provides higher spectral e ciency, less radiated power required, higher data rate, low latency, robustness, increased reliability, and enhanced security [6]. Although this massive MIMO scenario bene ts the communication system, it faces di culties in detecting the uplink transmitted signal. Signal detection necessitates advanced signal processing. Several detection methods such as MMSE, ZF, CG, GS, OCD, and MF have been used to mitigate the problem. For massive MIMO systems, minimum means square error and zero-forcing can achieve nearoptimal bit error rate performance [7]. However, due to direct matrix inversion, both MMSE and ZF have high computational complexity which is de ned as O(U 3 ) [8], where U represents the number of users. To avoid a direct matrix inversion, the Gauss-Seidel method decomposed the equalization matrix (A) into three elements: the lower triangular matrix, the upper triangular matrix, and the diagonal matrix. e GS method has a fast convergence rate [9] and low computational complexity. When compared to more complex detectors, the matched lter method performs worse [10]. e use of the optimized coordinate descent method yields an approximate solution with low computational complexity [11]. e conjugate gradient method solves the system equation with low computational complexity through the nth iteration. However, the performance of both the OCD and the CG methods is inferior to that of the MMSE and ZF methods [12]. is manuscript proposes a low complexity and high-performance hybrid detection algorithm based on MMSE with an ADMM method and the GS method. e diagonal matrix is used to compute the initial solution. To avoid a direct matrix inversion, the equalization matrix of the MMSE is decomposed using Cholesky decomposition in the rst iteration, and the ADMM method is applied to the Cholesky decomposed matrix to reduce complexity. e detection is then carried out and iteratively re ned using GS method with the value of the rst iteration serving as an input.
Massive MIMO encountered di culties in detecting uplink signals. Several detection methods were used to solve this problem. e performance and complexity of those methods are the most important factors to consider while evaluating them. e equalization matrix inversion operation is undesirable in massive MIMO detection systems because it greatly increases computational complexity. In [9], a robust and joint low complexity detection algorithm based on the Jacobi and Gauss-Seidel methods are used, and an initial solution is proposed by utilizing the bene ts of a stair to obtain a fast convergence rate and low complexity. For the base, station-to-user antenna ratio (BUAR) 160/30, 160/40, 160/50, and160/60 and the number of iterations (n) 2,3, and 4, the performance of MMSE, NS, GS, JA, and proposed methods is compared using the BER parameter. In [13], a GS-based-soft detection algorithm is proposed to accelerate the convergence rate of the conventional GS method while maintaining an acceptable overhead complexity. e performances of the proposed GS-based algorithm, Cholesky decomposition approach, and Neumann series expansion (NSE)-based algorithm are compared based on the bit error rate for the BUAR � 128/8 and 128/16 using 64-QAM. e complexity of the proposed algorithm reduces from Ο(U 3 ) to Ο(U 2 ). In [14], A soft-output data detection algorithm based on conjugate gradients is used to improve error rate performance for massive MIMO systems with medium BS-to-user antenna ratios. e conjugate gradient is used to reduce the signal detection's high computational complexity. To reduce complexity, a modified version of the conjugate gradient least square (LS) algorithm is used. e performance of the CGLS, Neumann, and Cholesky inversion methods is compared using a block error rate. According to [15], MMSE is a linear detection technique on the receiver side that is critical in terms of implementation complexity and contributes significantly to the improvement of transmission reliability. MMSE and ZF have comparable performance and outperform maximum ratio combination (MRC), particularly in high spectral efficiency. e two received techniques, on the other hand, involve matrix inversion computation, and the complexity grows with the number of users. e adaptive Damped Jacobi (DJ) technique and the conjugate gradient algorithm developed in [7] are combined into a hybrid iterative algorithm for signal detection for uplink.
e CG method is utilized to offer a good search direction for the adaptive DJ algorithm, and the Chebyshev approach is employed to speed up convergence. e initial solution is obtained by the first iteration of the Gauss-Seidel method, and a hybrid detector based on the combined GS and SOR methods is proposed [16]. e signal is then estimated using the iterative SOR approach. In [17], a low complexity softoutput signal detection algorithm based on improved kaczmarz's methods are proposed, which avoids the matrix inversion operation and thus reduces complexity by an order of magnitude. e algorithm is designed for uplink massive MIMO systems to avoid the high dimensional matrix inversion required by the MMSE criterion. An efficient massive MIMO uplinks detection algorithm based on the alternating direction method of multipliers and Huber fitting is proposed in [18]. ADMM makes variable updates much easier in each iteration, and variables are updated during each iteration by solving an unconstrained convex optimization problem. Huber fitting is a robust regression method that reduces the sensitivity of the function to outliers in the data. Matrix inversion is required for minimum mean squared error and zero-forcing detectors, which has significant computational complexity [8]. Proposed a detection algorithm for massive MIMO that computes an approximate inverse using the Cayley-Hamilton theorem and has quadratic complexity in terms of the numbers of users. To reduce the complexity caused by matrix inversion, the Cayley-Hamilton theorem is applied.

System Model.
A massive MIMO system with N total numbers of antennas at the base station (BS) to serve up to U single-antenna users concurrently has been considered, where the number of users is less than the number of base station antennas. e vector x � [x 1 , x 2 , . . . . . . x u ] T represents the signal transmitted by all users and the symbol vector y � [y 1 , y 2 . . . y N ] T represents signal received at the base station as shown in Figure 2. e received data is typically influenced by the channel effect and Gaussian noise (w). e channel matrix (H) entries are assumed to be independent and identically distributed (i.i.d) Gaussian random variables with mean and variance (δ 2 ).
e detection model is determined by y, x, H, and w, where w is additive white Gaussian noise. y is defined as and shown in Figure 3: (1)

Minimum Mean Square
Error. e MMSE detector's main goal is to minimize the mean-square error (MSE) between the transmitted signal x and the estimated signal H H y.
e estimated signal using MMSE method can be expressed as [9] x MMSE � A − 1 y MF , where A � H H H + δ 2 I U and y MF � H H y. δ 2 is the noise variance, I U is the UxU identity and matrix, and H H H is Gram matrix (G), where the exponent H refers to matrix Hermitian, which is the complex conjugate transpose of the matrix. Due to direct computation of A − 1 , MMSE algorithm requires computational complexity of Ο(U 3 ). In an iterative procedure, the alternating direction method of multipliers is used to solve an issue by breaking it down into smaller problems. It is considerably easier to update variables in each iteration, and variables are updated by solving an unconstrained convex optimization problem during each iteration, with the first iteration of MMSE-ADMM, which is used to obtain good initial condition in the proposed detector. MMSE-ADMM is described as [19] x � H H H + βI where β is a scaled version of δ 2 I, and the scaled dual variable λ is associated with the constraint z � x. When z and λ are equal to zero, then equation (3) becomes equation (2).

Gauss-Seidel.
e Gauss-Seidel algorithm, also known as the successive displacement method, is used to solve the linear system depicted in equation (1). e GS method decomposes the equalization matrix A into a diagonal matrix (D), an upper triangular matrix (U), and a lower triangular matrix (L), where A = D + U + L.
e GS method converges quickly if good initialization is considered. e estimated signal using GS algorithm is written as [16] where y MF � H H y.

Conjugate Gradients.
Another method for solving linear equations using n iterations is the conjugate gradients method. e signal calculated using the CG technique is written as [20] x where p (n) is the conjugate direction with respect to A, i.e, where A is the equalization matrix, A H H H + δ 2 I U , and α (n) is a scalar parameter.

Optimized Coordinate Descent.
Optimized coordinate descent is a low complexity iterative approach for inverting a high dimensional linear system. Using a sequence of simple, coordinate-wise updates, it achieves an approximate solution to a wide number of convex optimizations. e estimated solution is as follows [11]: where N o is the noise variance.

Zero
Forcing. e zero-forcing mechanism works by inverting the channel matrix H and so eliminating the channel e ect. e estimated signal is denoted by [10] x ZF Ay, where A (H H H) − 1 H H . e ZF detector clearly ignores the e ect of noise, and it performs well in interference-limited circumstances at the cost of increased computing complexity.

Matched Filter.
By setting A H, the matched lter treats interference from other substreams as pure noise. Using MF, the estimated received signal is given by When the number of users is signi cantly smaller than the number of antennas in the base station, it performs well, but as the number of users grows larger, it performs poorly compared to more complicated detectors [10].

Proposed Method.
e main issues for transmitted signal detection algorithms in massive MIMO systems are performance and complexity. e performance-complexity pro les, as well as the convergence rate, are in uenced by the initialization of detection algorithms. e proposed method takes MMSE equalization matrix and applies Cholesky decomposition to it, then uses the ADMM technique on this decomposed matrix to solve the system for the rst iteration to obtain good initialization, and then the GS algorithm is applied. e ADMM technique is an iterative strategy for solving an issue by breaking it down into smaller problems.
e GS algorithm has low complexity and a high rate of convergence. Based on the diagonal matrix, the starting solution is computed. e proposed detector's block diagram is depicted in Figure 4 and the owchart of the proposed detector's two steps, initialization, and nal detection, is shown in Figure 5. To achieve balanced performance and complexity, the proposed detector employs BUAR ≤ 2.
e initial solution x (0) is calculated as follows: where y MF H H y.
Step 2. Use the MMSE with ADMM method with n 1, where n denotes the number of iterations required to obtain the lowest BER. Compute the rst iteration solution x (1) using equation (3) as follows: where A H H H + βI by applying Cholesky decomposition A LL * to avoid the direct inversion of A. In a signal detection system, the performance of the detector is highly a ected by the initial condition. In this paper, an initial value based on ADMM is used to obtain a good initial value, which helps in achieving less BER. β is a scaled version of δ 2 I, z x (0) and λ is zero vector.
Step 3. Apply the GS algorithm where n ≥ 2 as shown in (4) to estimate the signal. Where c > 0 is an adequate step size for the ADMM technique, proj CO ( x (1) + λ, α) refers to the orthogonal projection of x (1) + λ , and α is the maximum of the real parts of the transmitted symbol.

Results and Discussion
e Simulation parameters used in this paper are shown in Table 1.
Computation complexity of the algorithm is largely depending on the total number of multiplication and division required in the algorithm. To drive the multiplication complexity of the proposed algorithm, consider the formulas that are used in Algorithm 1: , in this formula, U number of division is required to nd (L − 1 ) H and again U number of division is required to nd (L − 1 ), and for the multiplication between (L − 1 ) H and L − 1 , U 2 computation is required, then for ( en for computing, e complexity of projection is negligible. U multiplication is needed to compute λ λ − c( z (1) − x (1) ). U real number of divisions are required to compute the inverse diagonal matrix (D − 1 ) for nding x (0) . For the rst iteration, 2U 2 + 2U + U + U 2U 2 + 4U number of multiplications is required. en the remaining n−1 iterations solution is calculated based on GS method, which is de ned by en for each iteration 2U 2 + U 2 + U 2 4U 2 multiplications are required. Since there are n − 1 iterations, the total computational complexity for GS method is given by en the total computational complexity of the proposed algorithm becomes e proposed algorithm's complexity is reduced to Ο(U 2 ). Table 2 compares the proposed method to other methods in terms of complexity. Figure 6 shows a comparison of the proposed detector's complexity to that of other detection algorithms based on the number of users. e complexity of the proposed algorithm is far lower than that of the MMSE algorithm, as shown in Figure 6. As compared to other methods, the ZF  By considering some numbers of users Figure 6 has been expressed in tabular form. As the number of users increases, the computational complexity also increased as shown in Table 3. Table 3 shows the complexity comparison of the proposed method and other methods for some value of users. e transmission channel is set to additive white Gaussian noise (AWGN) channel, the noise is independent and identically distributed additive Gaussian white noise, and the baseband signal modulation techniques are 8-QAM, 16-QAM, 32-QAM, and 64-QAM, respectively, in order to simulate the performance. e antenna scale is set to 80 × 120, 120 × 180 (BUAR � 1.5), and 128 × 256 (BUAR � 2), where the first number indicates the number of MS user's antennas and the second number represents the number of base station antennas with n representing the number of iterations. e simulation results compare the performance of the proposed algorithm with that of recently introduced massive MIMO (mMIMO) uplink detectors. e performance is shown in terms of bit error rate versus signal-tonoise ratio. e MATLAB software is used to generate simulation results, the method of simulation model is shown in Figure 7. Figure 8 compares the proposed algorithm's BER to that of other currently available methods for an 80 × 120 (BUAR � 1.5) antenna arrangement utilizing 8QAM modulation and n � 2 iterations. At n � 2, the proposed algorithm outperformed MMSE and GS, whereas other methods require more iterations. e proposed algorithm achieved a BER � 10 − 4 at the signal-to-noise ratio (SNR) � 20 dB, whereas the GS algorithm achieved a BER � 5.45 × 10 − 2 at the same SNR � 20 dB and BER � 5 × 10 − 2 even when SNR was increased to 30 dB. e detectors based on MF, and conjugate-gradient methods perform the worst. e detectors based on OCD and GS methods perform moderately well with low computational complexity. e detectors based on MMSE and ZF algorithms have good performance ALGORITHM 1: proposed detection method based on hybrid MMSE with ADMM and GS methods. 6 Journal of Engineering but the computational complexity is very high. e proposed algorithm outperformed as compared to other methods with low computational complexity. Table 4 shows that the performance comparison of proposed detector and other methods for the number of users 80, number of base station antennas 120, and number of iterations 2 over 8QAM at SNR 11 dB to 20 dB. As it has been observed from the table, the performance of the proposed detector is better than the other methods. Figure 9 depicts the BER performance of the proposed method and other algorithms for a 120 × 180 antenna con guration, n 3 iterations, and the same modulation technique as in the previous gure. e proposed algorithm reached a BER 10 − 4 at SNR 20 dB, while the GS method acquired a BER 3.08 × 10 − 2 at the same SNR 20 dB and BER 2.5x10 − 2 at SNR 30 dB. e proposed algorithm requires only two iterations to achieve the desired performance, whereas other algorithms require more iterations, resulting in an increase in computational complexity. Table 5 shows that the performance comparison of proposed detector and other methods for the number of users 120, number of base station antennas 180, and number of iterations 3 over 8QAM at SNR 16 dB to 20 dB. As we observed from the table the BER performance of proposed detector is better than that from other methods. Figure 10 compares the performance of the proposed algorithm to other algorithms for an 80 × 120 massive MIMO system with n 2 and 16QAM modulation. e proposed detector obtained a BER 10 − 4 at SNR 21 dB, whereas the GS algorithm reached a BER 8.37 × 10 − 2 at the same SNR 21 dB. As shown in the gure, as the SNR

Algorithm
Computational complexity CG [14] nU 2 + 6nU GS [16] 4nU 2 OCD [11] 2nNU + nU ZF [8] 1/2 (U 3 + NU 2 + 5U 2 + 3NU) MMSE [19,20] 4U 3 Proposed  MMSE  ZF  GS  OCD  CG  76  34,960  6169984  594168  46208  36632  12464  77  35,882  6298292  612689  47432  37114  12782  78  36,816  6428448  631566  48672  37596  13104  79  37,762  6560476  650802  49928  38078  13430  80  38,720  6694400  670400  51200  38560  13760 increased, the proposed detector outperformed the MMSE and ZF detectors. Table 6 indicates the performance comparison of proposed detector and other detectors for SNR 17 dB to 21 dB, at the number of users 80, number of base station antennas 120 and n 2 over 16QAM. As we observed from the table the proposed detector achieved the best performance. CG and MF methods are the poor performance method as indicated in the table. Figure 11 compares the performance of the proposed algorithm with others recently used detectors for a massive MIMO system with a 120 × 180 antenna con guration and n 3 over 16QAM modulation. e proposed detector had a BER 10 − 4 at SNR 21 dB, while the GS detector had a BER 5.43 × 10 − 2 at the same SNR 21 dB and a BER 4.8 × 10 − 2 at SNR 30 dB. Table 7 represents the BER comparison of proposed detector and other detectors at SNR 17 dB to 21 dB, the number of users 120, number of base station antennas 180, and n 3 over 16QAM. As shown in the table, the proposed detector outperformed over the MMSE and the ZF detectors. Figure 12 shows a performance comparison of the proposed detector with other currently available massive MIMO detectors for an 80 × 120 antenna con guration, n 2, and 32QAM. e proposed algorithm achieved a BER 10 − 4 at SNR 24 dB, but the GS method achieved a BER 2.29 × 10 − 1 at SNR 24 dB, indicating that the GS method required additional iterations to achieve the target performance. Table 8 depicts the BER performance comparison of the proposed detector and other detectors at SNR 20 dB to  24 dB, the number of users 80, the number of base station antennas 120, n 2 over 32QAM. As indicated in the table, the proposed algorithm outperformed from the other methods. Figure 13 demonstrates a BER performance comparison of a proposed detector with other detectors for a massive MIMO system with a 120 × 180 antenna con guration, n 5, and 32QAM. e proposed detector obtained a BER 10 − 4 at SNR 25 dB, while the GS method achieved a BER 9.27 × 10 − 2 at the same SNR 25 dB and a BER 7.57 × 10 − 2 at SNR 30 dB. Table 9 shows the performance comparison of the proposed algorithm and other algorithms at SNR 21 dB to 25 dB, the number of users 120, the number of base station antennas 180, and n 5 over 32QAM. As shown from the table, even though the number of iterations is increased from 2 to 5, the performance of OCD and CG is poor, which means they require an additional number of iterations. Figure 14 depicts a BER performance comparison of a proposed detector with other detectors for a massive MIMO system with an 80 × 120 antenna con guration, n 2, and 64QAM. e proposed detector achieved a BER 10 − 4 at SNR 27 dB, whereas the GS detector achieved a BER 1.805 × 10 − 1 at SNR 27 dB. Table 10 demonstrates the BER performance comparison of the proposed algorithm and other currently available methods at SNR 23 dB to 27 dB, the number of users 80, the number of base station antennas 120, and n 2 over  e proposed algorithm achieved a target performance with a small number of iterations 2. Figure 15 shows a BER performance comparison of a proposed detector with other detectors for massive MIMO system with an 80 × 120 antenna con guration, n 3, and 64QAM. e proposed detector achieved a BER 10 − 4 at SNR 28 dB, while the GS detector achieved a BER 1.348 × 10 − 1 at SNR 28 dB, even though the iteration number increased from 2 to 3 the GS method shows only a small improvement in a BER performance. Table 11 compares the performance of the proposed algorithm with other existing algorithms at SNR 24 dB to 28 dB, the number of users 80, the number of base stations antenna 120, and n 3 over 64QAM. e proposed algorithm outperformed from the other methods. e performance of CG and MF detectors is very poor. Figure 16 compares the BER performance of a proposed detector to that of other detectors for a massive MIMO system with a 120 × 180 antenna con guration, n 5, and 64QAM. e proposed detector achieved a BER 10 − 4 at SNR 27 dB, whereas the GS detector reached a BER 8.24 × 10 − 2 at the same SNR 27 dB. Table 12 shows the performance comparison of the proposed detector to that of other detectors at the number of   Figure 17 depicts a BER performance comparison of a proposed detector with other detectors for a massive MIMO system with an antenna con guration of 128 × 256 (BUAR 2), n 5, and 32QAM. e proposed detector achieved a BER 10 − 4 at SNR 21 dB, while the GS detector achieved a BER 6.7 × 10 − 3 at SNR 21 dB. Table 13 compares the performance of the proposed detector and the other existing detectors at the number of users 128, the number of base station antennas 256, n 5 and 32QAM for SNR 17 dB to 21 dB. e proposed algorithm achieved the target performance. Figure 18 compares the BER performance of proposed detectors to that of other detectors for a massive MIMO system with a 128 × 256 (BUAR (2) antenna con guration, n 5, and 64QAM. e proposed algorithm achieved a BER 10 − 4 at SNR 24 dB, whereas the GS algorithm obtained a BER 1.14 × 10 − 2 at the same SNR 24 dB. Table 14 shows the performance comparison of the proposed detector with the other detectors at SNR 20 dB to 24 dB, the number of users 128, the number of base station  As shown in the above all-performance simulation results, the proposed detector has achieved the best BER, MSE, and SER performance as compared to, GS, OCD, and CG and achieved comparable performance with MMSE and Zero forcing detectors. e proposed detector achieved the optimal BER 10 − 4 at the SNR 20 dB, whereas GS detector achievedBER 5.45 × 10 − 2 for 8-QAM, 80 × 120 antenna con guration, and n 2. At this condition, the percentage performance improvement of the proposed detector from the GS detector is given (5.45 × 10 − 2 − 10 − 4 )/ 5.45 × 10 −2 × 100 99.82%. e proposed detector achieved the optimal BER 10 − 4 at the SNR 25 dB, whereas GS detector achieved BER 9.27 × 10 − 2 for 32-QAM, 120 × 180 antenna con guration, and n 5. At this  condition, the percentage performance improvement of the proposed detector from the GS detector is 99.89%. e proposed detector achieved the optimal BER 10 − 4 at the SNR 28 dB, whereas GS detector ach-ievedBER 1.348 × 10 − 1 for 64-QAM, 80 × 120 antenna con guration, and n 3. At this condition, the percentage performance improvement of the proposed detector from the GS detector is 99.93%.

Number of users Number of multiplications in algorithms
Unlike the GS method, the proposed algorithm achieved the target performance with a small number of iterations, as shown in all of the gures. For example, at n 2, the proposed detector achieved a BER 10 − 4 . According to the gures, the MMSE and ZF algorithms also perform well, but they have high computational complexity. With a low computational complexity, the proposed algorithm outperformed the MMSE and ZF methods. For instance, the proposed algorithm required 38,720 multiplications at n 2, U 80, and N 120, whereas the MMSE algorithm required 6,694,400 multiplications, and the ZF algorithm required 670,400 multiplications.

Conclusion
A low complexity, e cient hybrid MMSE with ADMM technique and GS-based signal detection algorithm for massive MIMO uplink systems was proposed in this paper. Initialization using the MMSE with ADMM technique and estimation using the GS algorithm were the two stages of the proposed hybrid detector. In addition, a diagonal matrix was also used to initialize the proposed detector. e proposed detector had improved performance with a small number of iterations and low computational complexity. e proposed algorithm complexity was reduced from Ο(U 3 ) to Ο(U 2 ). e proposed algorithm achieved the target BER performance with only two iterations, as demonstrated by the  numerical results. In this paper, the proposed detector performance was only evaluated at the simulation level. In the future scope of this paper, the practical performance of the proposed detector can also be investigated by designing a very large-scale integration (VLSI) architecture and implementing it on a Xilinx Virtex-7 eld-programmable gate array (FPGA). e channel used in this paper was the AWGN channel. In future work, fading channels such as  Nakagami fading, Rayleigh fading, and Rician fading will be used.
Data Availability e data used to support this study are included within the article.

Conflicts of Interest
e authors declare that they have no con icts of interest.