Approximate iteration detection with iterative refinement in massive MIMO systems
2017; Institution of Engineering and Technology; Volume: 11; Issue: 7 Linguagem: Inglês
10.1049/iet-com.2016.0826
ISSN1751-8636
AutoresChuan Tang, Cang Liu, Luechao Yuan, Zuocheng Xing,
Tópico(s)Advanced Wireless Communication Techniques
ResumoIET CommunicationsVolume 11, Issue 7 p. 1152-1157 Research ArticleFree Access Approximate iteration detection with iterative refinement in massive MIMO systems Chuan Tang, Corresponding Author Chuan Tang tc8831@nudt.edu.cn National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorCang Liu, Cang Liu National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorLuechao Yuan, Luechao Yuan National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorZuocheng Xing, Zuocheng Xing National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this author Chuan Tang, Corresponding Author Chuan Tang tc8831@nudt.edu.cn National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorCang Liu, Cang Liu National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorLuechao Yuan, Luechao Yuan National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this authorZuocheng Xing, Zuocheng Xing National Laboratory for Parallel and Distributed Processing, National University of Defense Technology, Changsha, 410073 People's Republic of ChinaSearch for more papers by this author First published: 06 April 2017 https://doi.org/10.1049/iet-com.2016.0826Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract To improve energy efficiency and spectral efficiency, massive multiple-input–multiple-output (MIMO) is proposed and becomes a promising technology in the next generation mobile communication. However, massive MIMO systems equip with scores of or hundreds of antennas which induce large-scale matrix computations with tremendous complexity, especially for matrix inversion in data detection. Thus, many detection methods have been proposed using approximate matrix inversion algorithms, which satisfy the demand of precision with low complexity. In this study, the authors focus on the approximate detection method based on Newton iteration (NI), and propose upgraded methods named NI method with iterative refinement (NIIR) and diagonal band NIIR (DBNIIR) which combine NI method and DBNI method with iterative refinement (IR). The results show that their proposals provide about 2 dB improvement on bit error rate (BER) for 16-quadrature amplitude modulation (QAM), and could even break the error floor existing in NI and DBNI methods for 64-QAM modulation. Furthermore, the BER of their proposals could provide almost the same performance as the exact method. Moreover, in contrast with NI and DBNI methods, NIIR and DBNIIR methods require quite few extra complexity cost and no extra hardware resource which is quite suitable for data detection in massive MIMO. 1 Introduction As the explosive increase of the mobile devices, high-speed and efficient data transmission is required in the next generation mobile communication [fifth generation (5G)]. Massive multiple-input–multiple-output (MIMO) [1] is one of the promising techniques for 5G [2], which has the feature of high data rate and high energy efficiency. Massive MIMO was first proposed by Marzetta in 2010 [3]. Compared to traditional MIMO (4 × 4, 8 × 8), the massive MIMO system has hundreds of antennas in base station (BS) and serves scores of users. More antennas can provide multiplexing gain and diversity gain, which results in higher performance of data rate and reliability, respectively. Moreover, massive MIMO has improved energy efficiency and is not sensitive to noise, so just using linear methods of signal processing (e.g. precoding, detection) can achieve approximate optimal performance with perfect channel state information [4]. However, the number of antennas corresponds to the dimensions of matrix handled in signal processing, hence more antennas require more powerful hardware chips in the physical layer, especially the ability of dealing with large-scale matrix operations, which is difficult and even an important research point for traditional small-scale MIMO [5, 6]. Data detection is a critical and complex task for massive MIMO signal processing [7, 8], which involves matrix inversion or solving linear system of equations. Owing to the enormous complexity, some research has been carried on about low-complexity detection using approximate iteration methods [9–14], which decrease the complexity from to where M is the dimensions of the matrix to be inversed. Matrix inversion based on polynomial expansion is used for massive MIMO data detection in [9], it can meet the demand of bit error rate (BER) using 1 or 2 order Neumann series to approximate matrix inversion. Instead of calculating matrix inversion in [10], data detection is directly implemented by solving the linear system of equations with conjugate gradient method, and especially conjugate gradient method can obtain exact results for M iterations theoretically. Richardson method [11, 12] is used to obtain the detection symbol vector directly avoiding the complicated matrix inversion, and the convergence of Richardson method is determined by a weight coefficient which is around a certain value for a given antennas configuration. Similarly in [13], it directly solves linear system of equations using Gauss–Seidel method, which has the best precision for first few iterations at the cost of parallelism. Lately, matrix inversion based on Newton iteration (NI) is adopted in [14], and it has the best iteration efficiency, i.e. the fastest convergence speed. As a balance of complexity and precision, a further approximation for NI method is proposed in [14] named diagonal band NI (DBNI) method; however, it reaches a high error floor for more than two iterations. In this paper, focusing on latest approximate data detection based on NI, we propose NIIR and DBNIIR methods combining NI and DBNI methods with iterative refinement (IR) [15–17] without any extra hardware resource. The precision of our methods is very close to that of the exact method for both 16-quadrature amplitude modulation (QAM) and 64-QAM modulation schemes, and the extra complexity of IR is only complex-value multiplications. Furthermore, compared to NIIR method, we find DBNIIR method has the lower complexity and nearly the same precision, so DBNIIR method is the best choice for data detection of massive MIMO systems in contrast with NI, DBNI and NIIR methods. The remainder of this paper is organised as follows. Section 2 introduces the system model and linear detection based on minimum mean square error (MMSE). In Section 3, we introduce the NI and DBNI methods, and explain how NIIR and DBNIIR methods work, and then analyse the complexity of different methods. The simulation results are shown in Section 4 and we conclude in Section 5. Notations: Column vectors and matrices are represented by lower case and upper case boldface, respectively, x and X. denote transpose, conjugate transpose and inverse. denotes the Euclidean norm. denotes the expectation. denotes the boundary of complexity is for some C > 0. means octal number. 2 Background In this section, we would introduce the system model and MMSE linear detection method. 2.1 System model for uplink In this paper, we consider a multi-user massive MIMO system. Concerning about the uplink data detection, we assume there are receive antennas in BS and users with single antenna. Vectors and represent the complex-value transmitted symbol vector and received symbol vector, respectively, the Rayleigh fading channel matrix can be represented as , where the , and means the complex channel gain from transmit antenna n to receive antenna m. The system can be modelled as (1) where the is the complex additive white Gaussian noise vector whose elements are mutually independent and zero mean with variance . We assume that the transmitted signals are independent and identically distributed Gaussian distribution and satisfy . 2.2 Linear detection for massive MIMO In massive MIMO systems, non-linear detection methods induce incredible complexity, so linear detection methods such as MMSE [18] are adopted mostly. The MMSE estimation of the transmitted symbol vector x can be represented as (2) where is the identity matrix. Since the different are independent, Gram matrix is a diagonally dominant matrix [3] when and is constant, and obviously, A is also diagonally dominant. Given the MMSE data detection, the inversion of A is the main complex computation because the algorithm requires number of operations for traditional methods, where means the dimensions of the matrix A. 3 Approximate data detection based on NI In this section, we first introduce the NI and DBNI methods. Then, we explain how the NIIR and DBNIIR methods work. Finally, we discuss about the complexity of these methods. 3.1 NI method NI method [19, 20] is an approximate method for calculating matrix inversion. It is derived from the first-order Taylor series, and can improve the precision by the iteration. If is the rough and original estimation of , then the (K + 1)th iteration estimation is (3) For convergence, the initial input must satisfy the condition (4) Then (3) converges quadratically to . For an arbitrary matrix, it usually requires a complex calculation to get an appropriate initial input . However, due to the diagonally dominant feature of matrix A in the data detection of massive MIMO system, the initial input is easy to obtain. We represent , where D and E own the main diagonal elements and the off diagonal elements of A, respectively, then the initial input can be set as which can meet the convergence condition easily [14], and only two diagonal-matrix-by-matrix multiplications [in the complexity of ] are required for calculating according to (3). NI method converges very fast; however, two matrix-by-matrix multiplications are calculated for each iteration when the number of iterations is bigger than one, which induces complexity for each iteration. Thus, in massive MIMO system we usually adopt as the estimation of [14] for the modulation scheme of 16-QAM. 3.2 DBNI method Although induces complexity, it has higher precision for K > 1. Regarding the feature of diagonal dominance for , we could neglect the elements far away from the main diagonal by setting them to be zero to get the approximate input for the next iteration. In this way, the complexity is for each iteration, where E is the bandwidth meaning that we only consider E elements adjacent with the diagonal element in a row for both right-hand and left-hand sides. This is the DBNI method, and E can be set as 2 in the majority situations [14]. The estimation inversion of DBNI method can be represented as (5) where (6) However, the precision of reaches an error floor which is a little worst than that of when K > 2. The little worst effect for K > 2 is because of the error propagation induced by omitting the elements far away from the diagonal elements. Meanwhile, due to the diagonally dominant feature and small parameter E, the error is mainly concentrated on the elements far away from the diagonal elements, which would be omitted in the next iteration, so the average error of the results converges at a constant error level. Thus, usually only two iterations are considered for DBNI method. Fortunately, the good thing about the error floor of DBNI method is that the divergence of DBNI method is not as sensitive as NI method which is shown in Section 4. 3.3 NI and DBNI methods with IR IR is a method which could refine the error of the results of linear system of equations, it was first proposed by Wilkinson [15] in 1963. Considering the linear system of equations derived from (2), if is an approximate solution of , then the remnant r is (7) where e is the error of the estimation symbol vector . Ideally, assuming no round-off error in (7), the exact solution of e would be obtained by (7) and then exact is obtained. Actually, the round-off error induces an approximate estimation of the error e through calculating the solution of , and can be regarded as an improved estimation of symbol vector . For better precision, a better could be obtained by executing (7) recursively. Although calculating also induces some round-off error, IR can effectively improve the estimation of , especially when the method for calculating has low complexity and low precision [17] such as the NI method with few iterations and the DBNI method. Given that the complexity of calculating the solution of is the same (high) as that of in massive MIMO system, we can also adopt the identical approximate method to solve the process of IR. In this way, we could share the resource of the hardware in both data detection and IR without extra hardware cost. Furthermore, when the approximate estimation of is known, the IR can be just implemented by , which requires a matrix-by-vector multiplication with low complexity of complex-value multiplications. Since it improves the precision at the cost of very low complexity, we propose to use the methods combining NI and DBNI with IR (NIIR and DBNIIR for short, respectively) in the data detection of massive MIMO systems, and concerning the complexity limit we only conduct iterations for no more than twice. In addition, IR is also feasible for other iteration methods mentioned in Section 1. Concerning (2) and (3), the symbol vector of (K + 1)th iteration detection obtained by NI method can be represented as Compared to the Kth iteration detection of NIIR method, the detection symbol vector can be represented as (8) where means the correction of error using NI method with K iterations. Seeing from (8), the effect of IR for NI method is equivalent to one extra iteration. Compared to complexity of IR, an extra iteration for NI method requires two matrix-by-matrix multiplications whose complexity is , so NIIR method has lower complexity than NI method at the same precision. Given DBNI method for more than two iterations, no improvement of precision is obtained due to the error floor of DBNI method. However, through estimating the error of DBNI method, DBNIIR could provide higher precision. What is more, due to lower precision, DBNI method would obtain more obvious improvement in precision than NI method when combining with IR. Especially, we find that the precision of DBNIIR method is nearly the same as the precision of NIIR method, and also very close to the precision of the exact method. In addition, because the convergence of DBNI method is not as sensitive to as NI method, so DBNIIR also works well on the case of small ratio of . Thus, synthetically considering the complexity and precision, DBNIIR method is the best choice in data detection of massive MIMO systems compared with NI, DBNI and NIIR methods. 3.4 Complexity analysis In traditional MIMO system, data detection usually adopts exact inversion based on Cholesky factorisation, whose complexity is . However, in massive MIMO systems, the complexity of is unacceptable as the number of antennas increases markedly. So, approximate methods are adopted to limit the complexity on . In this paper, we evaluate the complexity of the approximate methods in terms of the number of complex-value multiplications, and we assume that matrixes A and are known. Concerning the first NI iteration, requires two diagonal-matrix-by-matrix multiplications with the complexity of . For more iterations of NI method, it requires two matrix-by-matrix multiplications with the complexity of for each iteration. Concerning the DBNI method, has the same complexity with ; however, for more iterations of DBNI method, it only requires two band-matrix-by-matrix multiplications for each iteration [The first iteration indeed requires two band-matrix-by-diagonal-matrix multiplications which complexity is [14], because D is a diagonal matrix.]. The complexity of band-matrix-by-matrix multiplication is which is determined by the number of non-zero matrix elements . Then to obtain detection symbol vector or , an extra matrix-by-vector multiplication ( or with complexity) is required. Given the NIIR and DBNIIR methods, assuming the or is known, it requires to calculate a matrix-by-vector multiplication for obtaining the remnant r and solve linear system of equations for obtaining the estimation error . As mentioned above, with known or , we could just obtain or with a matrix-by-vector multiplication. So, for NIIR and DBNIIR methods, we just require extra complexity than NI and DBNI methods with the same number of iterations. The complexity of different methods are shown in Table 1, where E is the bandwidth of DBNI method and set as 2 for most cases, and the approximate complexity of different methods when M≫E and concerning only the highest order of M are shown in Table 2. Table 1. Complexity of different algorithms Algorithms The number of complex-value multiplications NI method DBNI method DBNI method NIIR method DBNIIR method Table 2. Approximate complexity of different algorithms when M≫E Algorithms The number of complex-value multiplications NI method DBNI method DBNI method NIIR method DBNIIR method 4 Simulation results In this section, we evaluate the performance for different approximate methods. If s represents the approximation estimation of the MMSE detection symbol vector , then the mean-squared error (MSE) of the approximate method is , and we use the normalised MSE (NMSE) which is equal to MSE/M to represent the precision of algorithms, where is the length of vector s. We also conduct experiments on detection with 16-QAM and 64-QAM demappers (Since the performance of soft decision demapper also depends on , so we use hard decision demapper to survey the effect of approximate detection independently.) combining with the rate-1/2 Viterbi decoder of [ ] generator polynomial, and we show the BER of different algorithms for different signal-to-noise ratios (SNRs). All BER simulations are averaged over 810,000 channel realisations. First, we show the NMSE of the detection symbol vector for different detection methods in Fig. 1. The configuration of antennas is . Impressively, the NI method converges very fast, and the DBNI method has an obvious error floor; however, the precision of DBNI method can provide an acceptable BER performance for 16-QAM modulation scheme [14]. When combining with the IR, both NI and DBNI methods have a marked improvement on NMSE. What is more, the relationship of (8) is clearly shown in Fig. 1. Especially, when the number of iterations is more than ten, the error of NI method approximates to the round-off error which induces an error floor, but NIIR method still provides a little improvement on NMSE which proves that IR could improve the precision effectively. For sake of low complexity, we mainly concern about the first two iterations, and the partial enlarged drawing is shown in the southwest of Fig. 1. When the number of iterations is two, though the precision of NIIR method is the best, the NMSE of DBNIIR method with lower complexity is very close to that of NIIR method. Figure 1Open in figure viewerPowerPoint NMSE of different detection methods Fig. 2 shows the NMSE of DBNI method and DBNIIR method on different ratios, the number of iterations is two for all cases in Fig. 2. Similar to NI method, the ratio of is bigger, the convergence is better, and usually the ratio smaller than 6 will induce bad performance due to divergence [14] for DBNI method. As shown in Fig. 2 for 4× (), the NMSE is notable, and the error decreases markedly for DBNIIR method with same ratio 4×. Moreover, when the user number is <40, the NMSE of 4×-dbniir is even better than 8×-dbni which could provide enough good BER performance [14]. In other words, DBNIIR method could be feasible for a more widely range of the ratio of . Figure 2Open in figure viewerPowerPoint NMSE of DBNI and DBNIIR methods for different ratios of Fig. 3 shows the NMSE of DBNI method and DBNIIR method for different numbers of iterations. As shown in Fig. 3, more iterations for small ratio of induce not only no good effect, but also a little bad effect for DBNI method as mentioned before, and the only good thing is that it would not diverge like NI method on the ratio of 4×. Furthermore, the NMSE of DBNIIR method on the ratio of 4× is lower than that of 8× DBNI method for the number of iterations bigger than 1 in Fig. 3, which proves that DBNIIR method could work well even for small ratio of again as shown in Fig. 2. Figure 3Open in figure viewerPowerPoint NMSE of DBNI, DBNIIR and NI methods for different numbers of iterations () Fig. 4 shows the BER performance for different antennas configurations using the modulation scheme of 16-QAM, and we only consider two iterations as mentioned above. As the ratio of increases, the diversity gain increases, so the BER performance of detection using exact matrix inversion is improved. Meanwhile, bigger ratio of provides better convergence for NI and DBNI methods, so the BER performance of NI and DBNI methods are more close to that of the exact method. Concerning our proposals, the BER performance is improved markedly for both NIIR and DBNIIR methods, and the effect is more notable for DBNIIR method which has about 2 dB improvement than NI method (not to mention DBNI method) as shown in Fig. 4 for . Moreover, as the number of users increases in the same ratio of (comparing with ), the difference of BER performance between NI and DBNI methods is bigger due to fixed bandwidth E = 2; however, the BER performance of DBNIIR is not only almost the same as that of NIIR method, but also much close to that of the exact method. Figure 4Open in figure viewerPowerPoint BER for data detection with different configurations and methods using the modulation scheme of 16-QAM. The legend NI(x, y) means the number of iterations for matrix inversion is x and the number of iterations for IR is y. If y = 0, it is NI or DBNI method, else it is NIIR or DBNIIR method. The unit of SNR is decibels For further demonstration of the improvement induced by IR, we conduct simulations using the modulation scheme of 64-QAM for the antennas configuration . The results are shown in Fig. 5. Higher-order modulation scheme induces worst precision of detection at the same SNR. Although NIIR method has slightly better BER performance than DBNIIR method, both NIIR and DBNIIR methods have the BER performance which is close to that of the exact method. Concerning about NI and DBNI methods, they both reach an error floor, and the error floor of DBNI method is expressly high which could not meet the requirement of the systems. Thus, DBNIIR method, which has lower complexity than NI and NIIR methods and has nearly the same BER performance with exact matrix inversion, is more suitable in massive MIMO systems than the other three approximate methods for arbitrary antennas configuration. Figure 5Open in figure viewerPowerPoint BER for data detection with different methods using the modulation scheme of 64-QAM. The unit of SNR is decibels 5 Conclusion To balance the complexity with precision, many approximate MMSE data detection methods are proposed in massive MIMO systems. In this paper, NIIR and DBNIIR methods are proposed to improve the precisions of NI and DBNI approximate methods using IR. We find that NIIR and DBNIIR methods require only two extra matrix-by-vector multiplications with complexity and no extra hardware resources, and can provide 2 dB improvement on BER performance for 16-QAM modulation. Moreover, for 64-QAM modulation, NI and DBNI methods are not feasible due to BER error floor, but NIIR and DBNIIR methods could break the error floor providing BER performance close to that of the exact method. Especially, the DBNIIR method with two iterations provides approximate exact BER performance at the complexity of instead of [for NIIR (2, 2) and exact methods], and even works well on the case of small , so it is more suitable for data detection in massive MIMO systems. 6 Acknowledgment This work was supported in part by the Specialized Research Fund for the Doctoral Programme of Higher Education (grant no. 20114307110001). References 1Larsson E.G., Edfors O., Tufvesson F. et al.: ‘Massive MIMO for next generation wireless systems’, IEEE Commun. Mag., 2014, 52, (2), pp. 186– 195 2Wang C.X., Haider F., Gao X. et al.: ‘Cellular architecture and key technologies for 5G wireless communication networks’, IEEE Commun. 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