# Precise Channel Estimation Approach for a mmWave MIMO System
## 1 Introduction
Side information is very useful to describe the row and column entries of any matrix, and it has been further shown to reduce the complexity of completing a matrix [25; 26]. In this work, side information theory is used to obtain the optimal solution for a joint optimization problem. The main contributions of this work are as follows:  
1. A joint optimization estimation problem for a mmWave massive MIMO system based on the Ex-ADMM algorithm is formulated. With the help of side information in matrix completion theory, a training procedure compatible with the hybrid beamforming (HBF) structure, leveraging the low rank and sparsity in angular domains, is designed. In Ex-ADMM, the joint optimization problem is further divided into several subproblems, which are then solved individually. 2. An Ex-ADMM is proposed for better estimation of the channel matrix. This algorithm is originally derived from [18; 26; 27; 28], and it is more efficient than the other iterative algorithms described in the results section. In the proposed Ex-ADMM algorithm, a nuclear norm is used for the low-rank approximations of the channel matrix, and \(\mathrm{l}_{1}-\mathrm{norm}\) is used to enforce the sparsity, consecutively. In addition, a relaxation factor is introduced to enhance the system performance. Simulation results demonstrate the performance of the proposed Ex-ADMM algorithm, along with orthogonal matching pursuit (OMP) [16], two-stage sparse representation (TSSR) [29], vector approximate message passing (VAMP) [30] and the alternating direction method of multipliers (ADMM) [18] in terms of normalized mean squared error (NMSE), achievable spectral efficiency (ASE) and convergence. This paper is organized as follows. The mmWave channel model is presented in Section 2. In Section 3, a mmWave channel estimation problem based on Ex-ADMM is formulated. Section 4 provides the training procedure details for the proposed Ex-ADMM. A detailed description of the proposed algorithm is provided in Section 5. Section 6 illustrates the computational complexities of proposed algorithm with respect to OMP, TSSR, VAMP and ADMM. Simulation results and discussion, describing the superiority of proposed algorithm over the benchmark algorithms in terms of NMSE, ASE and convergence, are provided in Section 7. Finally, the conclusion is presented in Section 8. **Notation:** We use the following notation throughout the paper. \(\alpha\), \(\mathrm{a}\), and \(\mathrm{A}\) denote a scalar, a vector and a matrix, respectively. \(\mathrm{A}^{\mathrm{T}}\), \(\mathrm{A}^{\mathrm{H}}\) and \(\mathrm{A}^{\mathrm{*}}\) indicate \(\mathrm{A}\)'s transpose, conjugate transpose and conjugate, respectively. \(\|(. )\|_{\rm F},\|(. )\|_{\rm L}\), and \(\|(. )\|_{\rm I}\) indicate the Frobenius norm, nuclear norm (sum of all singular values), and \(\rm I_{1}\) - norm, respectively. The operands \(\circ\) and \(\otimes\) represent matrix Hadamard and Kronecker products. vec(.) and unvec(.) represent the concatenation of columns of the matrix into vectors and the reverse operation, respectively. Expected value is represented by \(\rm E\{.\}\). \(A\in\{0,1\}^{M\times N}\) indicates that A's elements are taken independently with equal probability from the binary set {0,1}.