# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 1 Introduction  
Millimeter wave (mmWave) communication has been widely considered as a key technology for B5G due to its potential to provide gigabits-per-second data rates by exploiting the large unused bandwidth [1]. Unfortunately, it is challenging for long range wireless communication because of the huge paths loss in mmWave band. To compensate the paths loss and extend the coverage, massive MIMO antennas are supposed to be deployed in mmWave communication systems. However, in fell-digital MIMO structure, the radio frequency (RF) chains will increase as the antennas become large scale, which results in unaffordable hardware cost and power consumption [2]. Thus, massive MIMO with phase shifter network based hybrid precoding and electromagnetic lens based beamspace has been considered [1; 2; 3]. Generally, it is difficult for such an efficient structure to obtain the full channel state information (CSI), since the limited number of RF chains only combines the measurement signal with smaller dimension compared to the entire channel matrix. Meanwhile, many pilot symbols need to be assigned during the channel estimation phase, leading to overwhelming pilot overhead. Fortunately, some inherent properties including low rank and structural sparsity of mmWave massive MIMO channels have been demonstrated in practical measurement environments [4; 5]. By exploiting low rank and structural sparsity, some channel estimation algorithms have been proposed to efficiently recover the sparse channel matrix [6; 7; 8; 9; 10; 11; 12; 13; 14], including blind, semi-blind and pilot-aided estimation techniques. For blind and semi-blind channel estimation techniques, the basic idea is to exploit some inherent features in a modulated signal or use a decision feedback method rather than many pilots to acquirethe CSI [6; 7; 8]. Specifically, the blind 1-norm regularized channel estimation algorithm was proposed to solve the narrowband channel estimation problem in [6]. In [7], a novel blind signal detection scheme that simultaneously estimates the channel and data was proposed. In [8], based on certain available prior data obtained from the output of a soft-input soft-output channel decoder, a data-aided channel estimation scheme was proposed. In such a channel estimation scheme, the CSI can be coarsely estimated via a few pilot symbols, and then its precision can be further improved by data symbols. However, for such blind or semi-blind channel estimation schemes, only after a long time receiving the data symbols can the CSI be estimated accurately. In addition, the great computational complexity will lead to a long time delay, thus such algorithms cannot be applied to estimate the channel with relatively short coherence time or meet the demand of low time delay. To efficiently acquire the CSI, pilot-aided channel estimation algorithms are widely studied [9; 10; 11; 12; 13; 14], which can be divided into three categories, i.e., convex relaxation algorithms, greedy iterative algorithms and Bayesian inference algorithms. Specifically, based on the frequency-flat geometric channel model, the CSI of complex gains, angles of arrival (AoAs) and angles of departure (AoDs) were estimated through gradient descent algorithm in [9]. In [10], by exploiting sparse and low-rank structures of the mmWave massive MIMO channel with angle spread, the two-stage compressed sensing algorithm was proposed to reduce the sample complexity. In [11], the angle spread at receiver was estimated by greedy iterative algorithms and the normalized mean squared error (NMSE) was compared among different greedy iterative algorithms. In [12], the frequency selective channel was recovered with orthogonal matching pursuit (OMP) algorithm in the single carrier (SC) system and orthogonal frequency division multiplexing (OFDM) system, respectively. In [13], the off-grid sparse Bayesian learning algorithm was proposed to estimate CSI in an uplink multiuser mmWave massive MIMO system.