Millimeter-wave massive MIMO channel estimation based on majorization–minimization approach

Abstract

Millimeter-wave massive MIMO systems acquire a high dimensional distorted channel matrix to perform channel estimation. The captured channel matrix is sparse and rank deficient due to low scattering at millimeter-wave bands. The reduced number of RF chains enable the indirect access of signal at each antenna. This makes it difficult to estimate the channel accurately. To address this issue, researchers have focussed on compressive sensing and convex regularization methods. The first one performs well in noiseless scenario and while the latter suffers from over shrinking of all eigenvalues equally and is not suitable for large dimensional matrices. In this paper, a majorization minimization based fast non-convex (MM-FNC) algorithm is proposed to solve non-convex regularization based channel estimation. The proposed algorithm estimates the low-rank channel with fast convergence properties for large dimensional arrays. Simulation results show that the proposed approach exhibits superior performance over the CS-based orthogonal matching pursuit (CS-OMP) method, nuclear norm minimization methods (NNM) and matrix factorization (MF) methods in terms of normalized mean square error and achievable spectral efficiency with respect to different signal-to-noise ratio level.

Introduction

Millimeter-wave massive MIMO system is the key technology to next generation 5G cellular systems to compensate the demand for high data rate which achieves enormous transmission capacity at millimeter-wave band [1], [2]. Due to poor scattering, the wireless channel at these bands suffer from higher path loss attenuation, which is and compensated using large antenna arrays packed with a small form factor [3], [4]. A fully digital architecture consumes enormous power and needs an equal number of RF chains, increasing implementation costs. The processing overhead in the digital baseband is intensive and the alternate solution is to discharge part of the computations to an analog domain via a network of phase shifters or switches at the base station and mobile station respectively [5], [6]. Different types of hybrid beamforming architectures, including phase shifters, lens arrays [7] switches that use fewer RF chains, have been studied very recently. The benefits of switched-based systems includes low power consumption and low complexity in implementation. For millimeter-wave higher frequency systems, the limitation in RF chain is studied with the hybrid beamforming architecture for precoding problem, which shows significant improvement in the throughput of the system [8], [9], [10], [11], [12]. Reliable directional beamforming is achieved using accurate channel estimation. Channel state information is acquired using channel estimation. The channel estimation is limited for the millimeter-wave massive MIMO system as it has limited RF chains and channel-use overhead in the system.

The conventional MIMO channel estimation techniques are not applicable for the low signal-to-noise ratio and poor scattering channels. As the digital baseband is not directly accessible with analog beamformers, with partial channel state information, the channel estimation methods merge into the subspace sampling technique for beam alignment [8]. The closed-loop adaptive methods such as hierarchical multi-resolution codebook [11], Arnoldi iteration [12], adaptive compressive sensing [13], multilevel beam sequence design [14] and two-way channel estimation [15] apply subspace sampling to determine the best beamformer combiner pair during beam-alignment between the base station and mobile station. The closed-loop methods witnesses improved accuracy over the non-adaptive techniques such as training-based channel estimation [16], time division multiplexed switching based channel sounder [17], off-grid channel estimation [18], block sparsity technique [19], training sequence design [20], closed-loop beam alignment [21] and downlink–uplink training strategies [22] with an increase in channel use overhead. The short coherent interval in the millimeter-wave channel and repeated sampling for channel estimation limit the reduction in channel use overhead. However, these techniques does not scale up well with large-dimensional channels due to their inaccuracy.

The millimeter-wave channels exhibit rank sparsity relative to the dimension of antenna elements [19]. The millimeter-wave channel’s empirical studies report that the channel is sparse and low rank [23], [24]. The channel measurement strategies conducted at 28 GHz and 73 GHz in New York City demonstrated that the millimeter-wave energy in the channel exists in the form of a minimal number of narrow beam clusters [25]. The rank sparsity level is often 3 or 4 and is termed as low-rank for the millimeter-wave channel [26]. The observed low-rank property confirms fewer channel parameters that characterize the channel measurement matrix to estimate the millimeter-wave channel [27]. The low-rank spatial structure has been used to solve classical channel estimation problems in wireless systems [28], [29], [30]. The low-rank property can be portrayed by a low dimensional sub-space that is independent of array response and calibration errors. Compressive sensing methods have been widely studied to exploit and estimate the sparse nature of the channel [31]. These methods performs well in a noiseless scenario and causes less training overhead. However, their performance depends on the basis function and the dictionary codebook. With higher resolution in an angle of arrival and departure angle, the basis function will not match with the array vector responses and will increase the overhead computation [32].

The CS-OMP methods are susceptible to phase error in antenna arrays. In contrast, the singular value projection algorithm related to the low-rank estimation method reveals lower complexity in its implementation [33]. The singular value thresholding algorithm (SVT) [34] and fixed-point continuation algorithm (FPC) [35] have been widely studied to solve low-rank matrix in the area of image processing, computer vision and machine learning. Moreover, considerable interest toward low-rank matrix reconstruction methods such as NNM [36], quadratic sampling via convex programming [37], affine rank minimization [38], matrix recovery using restricted isometric property [39], MF [40], power factorization method [41] and alternating minimization [42] perform well with rank sparsity matrices.

The rank minimization problem, which is NP-hard [43] can be relaxed to a convex nuclear norm minimization problem [36]. It does not promise a low-rank solution for large dimensional matrices. The drawback is the over-shrinking of eigenvalues and high computational complexity. A matrix factorization method [41] is proposed, which outperforms the NNM method in terms of computational complexity and provides low-rank estimation in high signal-to-noise ratio. However, it shows unsatisfactory performance in a low signal-to-noise ratio [44]. The general challenge of the techniques in [36], [37], [41] reports the over-matching problem for distorted noisy observations scenario.

Thus, there is a need for reformulating the minimization problem considering requirements such as reduced computational complexity, improved low signal-to-noise ratio performance and scaling up to large dimensional arrays. For such scenario, a majorization–minimization based fast non-convex algorithm is proposed in this paper. The main contribution in this paper includes

• The low rank millimeter wave channel estimation minimization problem is reformulated as a non-convex based weighted nuclear norm minimization problem which is solved using the majorization minimization framework based fast non-convex algorithm. The channel subspace sampling is adopted to capture the distorted noisy channel matrix. A numerical minimizer function is proposed using a derivative which possesses only a local optimum due to its non-convex objective function.

• In the proposed algorithm, the matrix shrinkage operator is defined and the singular value decomposition property is applied to obtain the optimal solution for minimization. The adaptive soft thresholding and weight adaptation significantly overcomes the shrinkage of eigen values heavily and thus improves the estimation accuracy. The incomplete SVD computation is considered to reduce the computational complexity of the proposed algorithm.

• A fixed point acceleration scheme is adopted to improve the speed of convergence and this overcomes the drawback of majorization minimization framework because of a constrained upper bound condition.

• The performance of the algorithm is assessed in terms of normalized mean square error and spectral efficiency. The signal-to-noise ratio and computation overhead results are better than the existing algorithms.

The paper is organized as follows. Section 2 discussed the system overview and system model. Section 3 deliberates the background of low rank matrix completion and formulates the channel estimation problem. Section 4 elaborates the proposed majorization minimization framework to obtain the optimal minimizer function and fast non-convex algorithm. Section 5 illustrates the Monte Carlo simulation results and discussion. Finally, conclusions and future work are given in Section 6

Notations: A bold lower case letter a is a vector, a bold capital letter A is a matrix. ${\left[\mathbf{A}\right]}_{i,j}$ is the $i$th row and $j$th column entry of A, and ${\left[\mathbf{A}\right]}_i$ is the $i$th column. ${\mathbf{A}}^{\mathit{T}}$, ${\mathbf{A}}^{-\mathbf{1}}$, ${\Vert \mathbf{A}\Vert }_F$, ${\Vert \mathbf{a}\Vert }_2$, ${\Vert \mathbf{A}\Vert }_{\mathrm{\ast }}$, ${\Vert \mathbf{A}\Vert }_{\mathrm{w},\text{*}}$, respectively, are the complex transpose, inverse, Frobenius norm of A, $l_2$-norm of a, nuclear norm of A (which is the sum of its singular values) and weighted nuclear norm of A. ${\mathbb{R}}^{M\times M}$ is the linear space of all $M\times M$ real matrices and ${\mathit{I}}_M$ is the identity matrix. $\mathbf{A}⨂\mathbf{B}$ is the Kronecker product of A and B. Let $\mathrm{vec}\left(\mathbf{A}\right)$ be the operator that stacks columns of A into a column vector, $\text{diag}\left(\mathbf{A}\right)$ be the operator that collects the diagonal elements of a square matrix A. $\mathbb{E}\left[\cdot \right]$ is used to denote the expectation.