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FIGURE 13. SER performance of the proposed method and conventional method versus a number of iterations is evaluated under 16 x 64 Rayleigh MIMO channels with 256-QAM modulation. FIGURE 12. SER performance of the proposed method and conventional method versus a number of iterations is evaluated under 16 x 64 Rayleigh MIMO channels with 64-QAM modulation.

Figure 13 SER performance of the proposed method and conventional method versus a number of iterations is evaluated under 16 x 64 Rayleigh MIMO channels with 256-QAM modulation. FIGURE 12. SER performance of the proposed method and conventional method versus a number of iterations is evaluated under 16 x 64 Rayleigh MIMO channels with 64-QAM modulation.

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where H € CN**% represents the channel between the base station and users, mn € C%%*! is the additive white Gaussian noise (AWGN) that follows a CN(0, ¢?Iy,) distribution, and x € AK*! is the transmitted symbol vector from all users, where A denotes the constellation alphabet. Further, we can transform equation (1) into the real domain as follows:
where H € CN**% represents the channel between the base station and users, mn € C%%*! is the additive white Gaussian noise (AWGN) that follows a CN(0, ¢?Iy,) distribution, and x € AK*! is the transmitted symbol vector from all users, where A denotes the constellation alphabet. Further, we can transform equation (1) into the real domain as follows:
two algorithms to improve the convergence rate of AOR and
two algorithms to improve the convergence rate of AOR and
SOR. In Algorithm 1, we present the AC-AOR algorithm. The inputs of the algorithm are y, H, D, L, U, 8, a, 4, and p (here we abbreviated p(G) as p). The parameters a and 6 can be determined using (16) and (17), respectively. The parameter y can be determined using (36) or can be approximated from the range [0.6, 0.9]. The spectral radius p of the iteration matrix can be estimated using numerical simulation. For VN >> K with a given a and £, p can be approximated as a constant for all channel matrices. Hence, to avoid the complexity burden for exact p computation, in this work, we have used an approximated value of p. The ilnitialization is presented between lines 4-7. On line 4 we calculate the matched filter output y“"” = Hy. The initial value presented in line 5 is based on the chan- nel hardening property of massive MIMO systems. Since N > K, the channel matrix H is asymptotically orthogonal, and the matrix W is diagonally dominant, which leads us to —y * (more details can be found in [26]). On lines 6 and 7 we have the initial value of the Accelerated Chebychev method, and the initial value of under-relaxation. Afterward, we present the iterative part of AC-AOR, which includes two main parts: the AOR calculation at line 10, which is adopted from (7), and the Accelerated Chebyshev method in line 11 based on equation (35). Finally, the AC- SOR algorithm is presented in Algorithm 2, which is very similar to AC-AOR except that we have the parameter w as input and the SOR equation at line 10, which is adopted from (5);
SOR. In Algorithm 1, we present the AC-AOR algorithm. The inputs of the algorithm are y, H, D, L, U, 8, a, 4, and p (here we abbreviated p(G) as p). The parameters a and 6 can be determined using (16) and (17), respectively. The parameter y can be determined using (36) or can be approximated from the range [0.6, 0.9]. The spectral radius p of the iteration matrix can be estimated using numerical simulation. For VN >> K with a given a and £, p can be approximated as a constant for all channel matrices. Hence, to avoid the complexity burden for exact p computation, in this work, we have used an approximated value of p. The ilnitialization is presented between lines 4-7. On line 4 we calculate the matched filter output y“"” = Hy. The initial value presented in line 5 is based on the chan- nel hardening property of massive MIMO systems. Since N > K, the channel matrix H is asymptotically orthogonal, and the matrix W is diagonally dominant, which leads us to —y * (more details can be found in [26]). On lines 6 and 7 we have the initial value of the Accelerated Chebychev method, and the initial value of under-relaxation. Afterward, we present the iterative part of AC-AOR, which includes two main parts: the AOR calculation at line 10, which is adopted from (7), and the Accelerated Chebyshev method in line 11 based on equation (35). Finally, the AC- SOR algorithm is presented in Algorithm 2, which is very similar to AC-AOR except that we have the parameter w as input and the SOR equation at line 10, which is adopted from (5);
FIGURE 1. AC-SORNet architecture. Each of the / cascading layers has the same structure. where MM is the batch size. In backpropagation, the adap- tive moment estimation (ADAM) optimizer [43] has been employed to optimize the learnable parameters. Once the training process is over, the optimized parameters are stored and used in respective detection algorithms. Consequently, the AC-SOR, AC-AOR, SOR, and AOR can retrieve the rele- ‘vant parameters automatically based on the MIMO system configuration. The proposed methods used offline training to optimize the iterative methods’ training phase, the model is trained parameters. During the using multiple batches of random channel matrices at a range of SNR values. After training convergence, the optimized should be noted that we do not need and AC-AORNet methods for each c the channel coherence). In addition, tion plan is illustrated in Figure 4. parameters are stored. It to retrain AC-SORNet hannel matrix (say, after a feasible implementa-
FIGURE 1. AC-SORNet architecture. Each of the / cascading layers has the same structure. where MM is the batch size. In backpropagation, the adap- tive moment estimation (ADAM) optimizer [43] has been employed to optimize the learnable parameters. Once the training process is over, the optimized parameters are stored and used in respective detection algorithms. Consequently, the AC-SOR, AC-AOR, SOR, and AOR can retrieve the rele- ‘vant parameters automatically based on the MIMO system configuration. The proposed methods used offline training to optimize the iterative methods’ training phase, the model is trained parameters. During the using multiple batches of random channel matrices at a range of SNR values. After training convergence, the optimized should be noted that we do not need and AC-AORNet methods for each c the channel coherence). In addition, tion plan is illustrated in Figure 4. parameters are stored. It to retrain AC-SORNet hannel matrix (say, after a feasible implementa-
FIGURE 3. Block diagram of training scheme of AC-SORNet, AC-AORNet, SORNet, and AORNet. FIGURE 2. AC-AORNet architecture. Each of the 1 cascading layers has the same structure.
FIGURE 3. Block diagram of training scheme of AC-SORNet, AC-AORNet, SORNet, and AORNet. FIGURE 2. AC-AORNet architecture. Each of the 1 cascading layers has the same structure.
FIGURE 5. Complexity comparison against the number of users. and y are multiplied with 2/ dimensional vectors. Thus, an additional 4K real multiplications are required, bringing the total number of real multiplications for the AC-SOR to 4L (4K? + 8K) and for the AC-AOR to 2L(3K? + 15K). The complexity of the training and testing phases of a deep learning network is different. Generally, the complexity of the training phase is higher than that of the testing phase, as the training phase requires MSE computation and back- propagation to optimize the learnable parameters. From the training scheme presented in figure 3, the proposed AC- SORNet and AC-AORNet need the ZF Output. Therefore, the training phase complexity of the proposed AC-SORNet (or AC-AORNet) will be approximately equal to the sum of ZF Output complexity and AC-SOR (or AC-AOR) complexity. It should be noted that the computations required for back- propagation are not considered here, as the number of layers and the number of trainable parameters are small enough. On the other hand, the testing phase of DUN does not require any ZF operation, as shown in Figure 4. Therefore, the evaluation phase of AC-SORNet and AC-AORNet is identical to AC- SOR and AC-AOR, respectively. In the evaluation phase, the only difference between AC-SORNet (AC-AORNet) and AC-SOR (AC-AOR) is the iteration parameters optimized by the DUN method. As a result, the computational complexity of AC-SORNet and AC-AORNet is identical to AC-AOR and AC-SOR in the evaluation phase.
FIGURE 5. Complexity comparison against the number of users. and y are multiplied with 2/ dimensional vectors. Thus, an additional 4K real multiplications are required, bringing the total number of real multiplications for the AC-SOR to 4L (4K? + 8K) and for the AC-AOR to 2L(3K? + 15K). The complexity of the training and testing phases of a deep learning network is different. Generally, the complexity of the training phase is higher than that of the testing phase, as the training phase requires MSE computation and back- propagation to optimize the learnable parameters. From the training scheme presented in figure 3, the proposed AC- SORNet and AC-AORNet need the ZF Output. Therefore, the training phase complexity of the proposed AC-SORNet (or AC-AORNet) will be approximately equal to the sum of ZF Output complexity and AC-SOR (or AC-AOR) complexity. It should be noted that the computations required for back- propagation are not considered here, as the number of layers and the number of trainable parameters are small enough. On the other hand, the testing phase of DUN does not require any ZF operation, as shown in Figure 4. Therefore, the evaluation phase of AC-SORNet and AC-AORNet is identical to AC- SOR and AC-AOR, respectively. In the evaluation phase, the only difference between AC-SORNet (AC-AORNet) and AC-SOR (AC-AOR) is the iteration parameters optimized by the DUN method. As a result, the computational complexity of AC-SORNet and AC-AORNet is identical to AC-AOR and AC-SOR in the evaluation phase.
FIGURE 4. Block diagram of the testing phase of AC-SORNet, AC-AORNet, SORNet, and AORNet.
FIGURE 4. Block diagram of the testing phase of AC-SORNet, AC-AORNet, SORNet, and AORNet.
TABLE 1. Computational complexity comparison
TABLE 1. Computational complexity comparison
TABLE 2. Simulation setup
TABLE 2. Simulation setup
TABLE 3. List of trainable parameters for different algorithms Properly initializing the trainable parameters is crucia in achieving good performance with a small training data set. Our proposed method initializes the trainable parameters with randomly generated Gaussian variables. The mean and variance of the Gaussian distribution are predefined and can significantly affect the convergence and accuracy of the training process. Furthermore, appropriate initialization of the parameters can also prevent the model from getting stuck in local minima and help it converge faster to the globa minimum. Therefore, we carefully choose the initial values of the parameters to ensure the best performance of the proposed method. According to [40], the under-extrapolation coefficient +; is suitable for sufficient convergence in the (0.6, 0.9]. Therefore, in our training process, we set the mean initial value of 7; to 0.7. Next, we set assign the mean value of the under-relaxation parameter pj4; to 1. Based on the numerical analysis presented in [45], SOR achieves near- optimal detection performance when w; ~ 1.05. The AOR
TABLE 3. List of trainable parameters for different algorithms Properly initializing the trainable parameters is crucia in achieving good performance with a small training data set. Our proposed method initializes the trainable parameters with randomly generated Gaussian variables. The mean and variance of the Gaussian distribution are predefined and can significantly affect the convergence and accuracy of the training process. Furthermore, appropriate initialization of the parameters can also prevent the model from getting stuck in local minima and help it converge faster to the globa minimum. Therefore, we carefully choose the initial values of the parameters to ensure the best performance of the proposed method. According to [40], the under-extrapolation coefficient +; is suitable for sufficient convergence in the (0.6, 0.9]. Therefore, in our training process, we set the mean initial value of 7; to 0.7. Next, we set assign the mean value of the under-relaxation parameter pj4; to 1. Based on the numerical analysis presented in [45], SOR achieves near- optimal detection performance when w; ~ 1.05. The AOR
FIGURE 6. Average loss vs training iterations for different deep unfolded networks with 64-QAM.
FIGURE 6. Average loss vs training iterations for different deep unfolded networks with 64-QAM.
FIGURE 7. Average loss vs. training iterations for different deep unfolded networks with 256-QAM.
FIGURE 7. Average loss vs. training iterations for different deep unfolded networks with 256-QAM.
FIGURE 8. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 Rayleigh MIMO channel with 64-QAM modulation.
FIGURE 8. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 Rayleigh MIMO channel with 64-QAM modulation.
FIGURE 9. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 Rayleigh MIMO channel with 256-QAM modulation.
FIGURE 9. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 Rayleigh MIMO channel with 256-QAM modulation.
FIGURE 10. SER performance of the proposed method and conventional method versus the number of antenna at the base station is evaluated under K = 32 and SNR=16 dB with Rayleigh MIMO channels and 64-QAM modulation.
FIGURE 10. SER performance of the proposed method and conventional method versus the number of antenna at the base station is evaluated under K = 32 and SNR=16 dB with Rayleigh MIMO channels and 64-QAM modulation.
FIGURE 11. SER performance of the proposed method and conventional method versus a number of users is evaluated under N = 128 and SNR=16 dB with Rayleigh MIMO channels and 64-QAM modulation.
FIGURE 11. SER performance of the proposed method and conventional method versus a number of users is evaluated under N = 128 and SNR=16 dB with Rayleigh MIMO channels and 64-QAM modulation.
TABLE 5. Number of multiplications and SNR required to achieve an SER of 1073 for 16 x 64 and 256-QAM
TABLE 5. Number of multiplications and SNR required to achieve an SER of 1073 for 16 x 64 and 256-QAM
TABLE 4. Number of multiplications and SNR required to achieve an SER of 107 for 16 x 64 and 64-QAM
TABLE 4. Number of multiplications and SNR required to achieve an SER of 107 for 16 x 64 and 64-QAM
FIGURE 14. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 correlated MIMO channels with 64-QAM modulation. a correlated channel as L = 4 and L = 5 for 64-QAM and 256-QAM constellations, respectively. It is observed that the detection performance is affected significantly by spatial correlation variation, as shown in Figures 14 and 15. The proposed AC-SORNet and AC-AORNet methods converge to the ZF performance at the low correlation (¢ = 0.3). Although, for higher correlations (¢ = 0.7) the performance of the proposed methods is worse than the ZF method, it still performs better than the state-of-the-art methods.
FIGURE 14. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 correlated MIMO channels with 64-QAM modulation. a correlated channel as L = 4 and L = 5 for 64-QAM and 256-QAM constellations, respectively. It is observed that the detection performance is affected significantly by spatial correlation variation, as shown in Figures 14 and 15. The proposed AC-SORNet and AC-AORNet methods converge to the ZF performance at the low correlation (¢ = 0.3). Although, for higher correlations (¢ = 0.7) the performance of the proposed methods is worse than the ZF method, it still performs better than the state-of-the-art methods.
FIGURE 15. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 correlated MIMO channels with 256-QAM modulation.
FIGURE 15. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 correlated MIMO channels with 256-QAM modulation.
FIGURE 16. Impact of the correlated magnitude on the SER performance for 16 x 64 correlated MIMO channels with 64-QAM modulation. Figures 16 and 17 present the SER versus the ¢ variation plot for 64-QAM and 256-QAM constellations, respectively. Increasing the correlation leads to significant performance degradation for all methods. However, the proposed meth- ods perform better, outperforming the other algorithms. For instance, Figure 16 demonstrates that the proposed methods achieve near-ZF performance when the correlation magni- tude is between ¢ = 0.1 and 0.4. Similarly, Figure 17 shows that for 256-QAM modulation, the proposed methods pro- vide near-ZF performance for correlation values of ¢ = 0.1 to 0.3.
FIGURE 16. Impact of the correlated magnitude on the SER performance for 16 x 64 correlated MIMO channels with 64-QAM modulation. Figures 16 and 17 present the SER versus the ¢ variation plot for 64-QAM and 256-QAM constellations, respectively. Increasing the correlation leads to significant performance degradation for all methods. However, the proposed meth- ods perform better, outperforming the other algorithms. For instance, Figure 16 demonstrates that the proposed methods achieve near-ZF performance when the correlation magni- tude is between ¢ = 0.1 and 0.4. Similarly, Figure 17 shows that for 256-QAM modulation, the proposed methods pro- vide near-ZF performance for correlation values of ¢ = 0.1 to 0.3.
FIGURE 17. Impact of the correlated magnitude on the SER performance for 16 x 64 correlated MIMO channels with 256-QAM modulation.
FIGURE 17. Impact of the correlated magnitude on the SER performance for 16 x 64 correlated MIMO channels with 256-QAM modulation.
FIGURE 18. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 realistic channel with 64-QAM modulation.
FIGURE 18. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 realistic channel with 64-QAM modulation.
FIGURE 19. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 realistic channel with 256-QAM modulation.
FIGURE 19. SER performance of the proposed method and conventional method versus SNRs is evaluated under 16 x 64 realistic channel with 256-QAM modulation.
Related topics:
Computer ScienceComputational Complexity TheorySuccessive over-relaxationChebyshev filter
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