ROD-based hybrid TH precoding and combining for mmWave large-scale MIMO systems☆
Introduction
Recent proliferation of smart mobile devices has resulted in an unprecedented growth of data traffic in wireless communications. For example, Cisco Visual Networking Index shows that global mobile data traffic will increase sevenfold between 2017 and 2022 [1]. Since the spectrum resources below 6 GHz have been almost completely occupied, it is very challenging to fulfill the increasing communication demands with the conventional commercial frequency bands. To meet the incredible increase of mobile data traffic, one of the most efficient resolutions is to transmit data with millimeter wave (mmWave) large-scale multiple-input multiple-output (MIMO) systems, due to its high data rate, enormous idle spectrum resources and compact hardware structure [2].
Fully digital precoding scheme for MIMO systems requires one individual radio frequency (RF) chain per antenna element [3], [4], [5], which is prohibitively complex and costly at mmWave frequencies. To address this issue, a hybrid precoding architecture is widely considered for mmWave MIMO systems in recent years. The hybrid precoding architecture divides the precoder into a low-dimensional digital signal processor and a high-dimensional analog signal processor, so that it only requires a significantly lower number of RF chains in comparison to the fully digital counterpart [6], [7], [8], [9].
In the past few years, several hybrid precoding algorithms for multi-user mmWave MIMO systems have been proposed. The first prevalent category of hybrid precoding is codebook-based method [10], [11], [12], in which the columns of the RF precoding matrix are selected from a predefined codebook. An equally spaced grouping scheme based on the discrete Fourier transform (DFT) codebook is proposed in [10], the presented algorithm firstly divides the DFT codebook into groups, and then the group which maximizes the sum rates is selected to construct the RF precoding matrix. A spatial rotation algorithm is proposed in [11], which can refine the angles of the DFT beams and improve the performance of the DFT codebook-based hybrid precoding method effectively. Besides, the impacts of the instantaneous channel state information (CSI) and hybrid CSI are studied for codebook-based hybrid precoding method in [12], and it is shown that the hybrid CSI is sufficient to achieve the first-order gain provided by massive MIMO systems for most of cases. The second category of hybrid precoding is non-codebook based method, which usually solves a relaxation problem of the hybrid precoding matrix firstly, and then regulates the solution according to the hardware constraints [13], [14], [15], [16]. For example, a singular value decomposition (SVD)-based hybrid precoding algorithm is given in [13], which derives the analog precoding matrix and the digital precoding matrix via the SVD and the zero forcing (ZF) precoding, respectively. A hybrid block diagonalization (BD) precoding algorithm is developed in [14], the proposed algorithm firstly maximizes the effective channel gain via the RF precoding matrix, and then the BD precoding is implemented in the digital domain to suppress the inter-user interference. A joint channel estimation and hybrid precoding algorithm is given in [15], which exploits the strongest angle-of-arrival to design the analog precoding matrix and optimizes the digital precoding matrix through the ZF precoding. Additionally, an iterative hybrid precoding algorithm for low resolution RF phase shifters is developed in [16], which refines the RF precoding matrix and digital precoding matrix via block coordinated ascent and minimum mean square error (MMSE) precoding, respectively.
However, all the aforementioned algorithms are linear precoding, which may incur a performance loss, especially for an ill-conditioned channel matrix [17]. Fortunately, it has been shown in [18] that such a phenomenon could be effectively avoided by adding a perturbation vector to data streams in advance. Based on this idea, a nonlinear hybrid MMSE-vector perturbation (MMSE-VP) scheme is proposed in [19]. However, the RF precoding matrix in [19] is implemented by phase shifters and power amplifiers simultaneously, which is energy hungry. To solve this problem, a DFT codebook-based nonlinear hybrid MMSE-VP precoding algorithm is further given in [20], whose RF precoding matrix is only based on energy-efficient phase shifters. Though the hybrid MMSE-VP precoding algorithms offer significant improvement compared with linear methods, a Lenstra-Lenstra-Lovász basis reduction and a Branching-Reduction-and-Bounding method are used to solve the perturbation vector and RF precoding matrix in [19], [20], which involve high complexities. In order to reduce the computational costs, a low-complexity nonlinear hybrid block diagonal geometric mean decomposition (BD-GMD) Tomlinson-Harashima (TH) precoding algorithm is proposed [21], in which an orthogonal matching pursuit (OMP) algorithm is used to decompose the fully digital TH precoding matrix into the product of the RF precoding matrix and the digital precoding matrix. However, the OMP algorithm restricts the column vectors of the RF precoding matrix to belong to a predefined codebook, if the codewords within the codebook are far from the optimal solution of the RF precoding matrix, the system performance will inevitably decays.
With this backdrop, a novel hybrid TH precoding and combining algorithm is proposed in this paper, the main contributions can be summarized as follows:
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A tractable optimization problem of the precoding and combining matrices is firstly constructed through the lower bound of the sum rates, and then the problem is further transformed into an equivalent three-stage optimization problem which allows the digital precoding matrix, the RF precoding matrix and the RF combining matrix to be optimized sequentially and independently.
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To solve the aforementioned three-stage optimization problem effectively, a novel row orthogonal decomposition (ROD) which represents the orthonormal bases of the row space of a matrix is defined. Based on the newly defined ROD, it is interesting that the necessary and sufficient condition for the optimal digital precoding matrix can be derived and a near-optimal RF precoding matrix can be given. Then, by utilizing the asymptotic orthogonality of different user channels, the optimization of the RF combining matrix is reformulated as a unimodular quadratic programming and solved by a generalized power method.
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The sum rates and bit error rate (BER) of the presented algorithm are evaluated by theoretic analyses and simulations. Results indicate that the performance loss of the proposed algorithm is slight. Compared with the previous hybrid precoding methods, it is observed that the proposed algorithm can improve the sum rates and reduce the BER significantly with comparable computational costs.
The rest of this paper is organized as follows. In Section 2, the system and channel models are described. Section 3 explains the proposed hybrid TH precoding algorithm. In Section 4, the asymptotic performance of the proposed algorithm is analyzed. Section 5 evaluates the performance of the proposed algorithm through several simulations, and Section 6 concludes the whole paper.
Throughout this paper, A is a matrix, a is a vector, a is a scalar. , ∠a and are the magnitude, argument and conjugate of the complex number a, respectively. The field of complex numbers is represented by . denotes its determinant, is its Frobenius norm, stands for the rank of A, represents the trace of A, , and are its inverse, Moore-Penrose pseudo-inverse, transpose and conjugate transpose, respectively. and are the column space and nullspace of A. stands for the th element of the matrix A. is a matrix whose th element is equal to . I and 0 stand for the identity matrix and zero matrix, respectively. is a diagonal matrix with the entries in on its diagonal, represents a diagonal matrix with diagonal elements given by , is a block-diagonal matrix with the elements in as the diagonal blocks. denotes the Euclidean norm of the vector a, is the ith entry of the vector a. is used to denote expectation, is the complex Gaussian distribution with the mean a and the covariance b.
Section snippets
System model
Consider a multi-user mmWave MIMO system with a hybrid TH precoder as shown in Fig. 1 [21], in which a base station (BS) equipped with antennas and RF chains simultaneously serves K users with antennas and one RF chain. The BS employs a hybrid TH precoder to send an signal vector to users, where each entry of s is chosen from an M-ary quadrature amplitude modulation (M-QAM) constellation set . It is assumed that all
Hybrid TH precoding algorithm
In this section, the nonlinear unit, the hybrid precoder at BS and the hybrid combiner at users will be considered sequentially. For convenience, the diagram of the hybrid TH precoder is redescribed in Fig. 2, where the digital precoding matrix is decomposed into a product of the matrices and , i.e., .
Asymptotic sum rates analysis
The asymptotic sum rates of the proposed ROD-based hybrid TH precoding and combining algorithm is analyzed in this section. For simplicity, we consider the LOS component and NLOS component, respectively. The sum rates achieved by cooperative users serves as a benchmark.
For LOS component, the asymptotic sum rates loss of the proposed ROD-based hybrid TH precoding algorithm is discussed by the following theorem. Theorem 4 For a multi-user mmWave MIMO system as shown in Fig. 1 under an LOS channel
Simulation results
In this section, we present simulation results to evaluate the performance of the proposed ROD-based hybrid TH precoding and combining algorithm. The channel matrix between the BS and the kth user is modeled as a Rician fading channel, the Rician factor is uniformly distributed between 1 and 10. The AOA and AOD of the LOS component are uniformly distributed in . The NLOS component consists of clusters, each cluster is composed of rays. The average AOA
Conclusions
A novel nonlinear hybrid ROD-based TH precoding and combining algorithm is presented in this paper. In the presented algorithm, the necessary and sufficient condition of the optimal digital precoding matrix and a near-optimal RF precoding matrix are derived via a newly defined ROD, and the RF combining matrix is given by a generalized power method. Theoretical analyses and simulations show that compared with the optimal sum rates of cooperative users, the proposed algorithm only yields a slight
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Xiaoyu Bai received the B.S. degree from Nanjing University of Science and Technology, Nanjing, China, in 2010 and the M.S. degree from Shanghai Academy of Spaceflight Technology, Shanghai, China, in 2013. Currently, he is pursuing the PhD at the School of Computer Science and Engineering, Northeastern University, Shenyang, China. His research interests include array signal processing and millimeter wave MIMO systems. He is a recipient of the Best Student Paper Award at 2017 Progress in
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Xiaoyu Bai received the B.S. degree from Nanjing University of Science and Technology, Nanjing, China, in 2010 and the M.S. degree from Shanghai Academy of Spaceflight Technology, Shanghai, China, in 2013. Currently, he is pursuing the PhD at the School of Computer Science and Engineering, Northeastern University, Shenyang, China. His research interests include array signal processing and millimeter wave MIMO systems. He is a recipient of the Best Student Paper Award at 2017 Progress in Electromagnetics Research Symposium (PIERS).
Fulai Liu received the M.S. degree and Ph.D. degree from Northeastern University, Shenyang, China, in 2002 and in 2005, respectively. Since 2010, he is a professor in Northeastern University, Qinhuangdao, China. His research interests include array signal processing and its applications, cognitive radio, millimeter wave MIMO systems, etc.
Ruiyan Du received her B.S. degree from Hebei Normal University, Shijiazhuang, China, in 1999, the M.S. degree from Yanshan University, Qinhuangdao, China, in 2006, and the Ph.D. degree from Northeastern University, Shenyang, China, in 2012. Since 2012, she is an assistant professor in Northeastern University, Qinhuangdao, China. Her research interests include wireless communications, signal processing for communications.
Xiaodong Kan received the B.S. degree from Anhui Normal University, Wuhu, China, in 2017. She is currently pursuing the M.S. degree at Northeastern University. Her research interests include millimeter wave communications and array signal processing.
Yixin Xu received the M.S. degree from Yanshan University, Qinhuangdao, China, in 2011. He is currently pursuing the Ph.D. degree in signal and information processing with Northeastern University. His research interests include signal processing for communications and massive MIMO systems.
Yanshuo Zhang received the B.S. degree from Northeast Petroleum University, Daqing, China, in 2017. She is currently pursuing the M.S. degree at Northeastern University. Her research interests include millimeter wave communications and massive MIMO systems.
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This work was supported by the Natural Science Foundation of Hebei Province (Grant No. F2016501139) and the Fundamental Research Funds for the Central Universities (Grant No. N172302002 and No. N162304002).