# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 1 Introduction
In [14], the Bayesian compressed sensing (BCS) algorithm was adopted to recover the channel matrix based on a geometric channel model, where the transmitter's hardware impairments were taken into consideration. Note that the prior works have investigated these channel estimation algorithms based on different assumptions and drawn some conclusions. However, these conclusions are only for specific parameter ranges under different scenarios. To the best of our knowledge, there is no comprehensive comparison among CS-based channel estimation algorithms with a unified framework for mmWave massive MIMO systems. Inspired by the above analysis, in this paper, we formulate the sparse channel estimation as a normal CS problem and provide a general overview of the current mmWave massive MIMO channel estimation approaches. Different from the previous works, the purpose of our work is a comprehensive comparison among existing algorithms with a unified framework, including their basic assumptions, application conditions and key results. The main contributions and key results of this work are listed as follows. * In this paper, with a unified massive MIMO framework, we compare the NMSE performance among three categories of algorithms, which is affected by the received SNR, the number of resolvable paths and pilot symbols, angular quantization error, hardware impairments and computational complexity. Through comprehensive comparison, the characteristics and application conditions of each algorithm are revealed and the factors that affect the estimated error and computational complexity are also presented. * Through theoretical analysis and simulation results, we show that convex relation algorithms achieve the best estimation accuracy at the high SNR range and it is mainly affected by the received SNR and transmitter's hardware impairments. At the low SNR range, greedy iteration algorithms outperform others and the estimated accuracy is then limited by the angle quantization error. Furthermore, a tradeoff between the estimated error and complexity is achieved in Bayesian inference algorithms, although its estimated error is sensitive to the number of available pilot symbols. * We also analyze the overall computational complexities of three categories of algorithms and visually represent them by the running time. Through illustrating the runtime of different algorithms versus the sparseness, we show that the computational complexity in the convex relaxation algorithm is the highest, and it even squarely increases with the sparseness in the gradient descent-based convex algorithm, while that in greedy iteration algorithms is minimum and grows linearly with the sparseness. In contrast to them, the computational complexities of Bayesian inference algorithm decreases as the sparseness increases. The rest of the this paper is organized as follows. After describing the system model and channel model in Section 2, we formulate the channel estimation as a compressed sensing (CS)-based sparse signal recovery problem in Section 3. A comparison of sparse signal recovery algorithms, namely convex relaxation algorithms, greedy iteration algorithms and Bayesian inference algorithms, are presented in Section 4. Simulation results are provided in Section 5. Finally, we draw conclusions in Section 6. Throughout our discussions, bold uppercase \(\mathbf{A}\) denotes a matrix and lowercase \(\mathbf{a}\) denotes a vector. Superscripts \(\mathbf{A}^{*}\), \(\mathbf{A}^{T}\), \(\mathbf{A}^{H}\), and \(\mathbf{A}^{-1}\) denote the conjugate, the transpose, the conjugate transpose, and the inverse of a matrix \(\mathbf{A}\), respectively. _vec_ (\(\mathbf{A}\)) denotes a vector obtained through the vectorization of a matrix \(\mathbf{A}\). For \(M\times N\) matrices \(\mathbf{A}\) and \(\mathbf{B}\), \(\mathbf{A}\otimes\mathbf{B}\) denotes the \(M^{2}\times N^{2}\) matrix of Kronecker product between \(\mathbf{A}\) and \(\mathbf{B}\).