# Precise Channel Estimation Approach for a mmWave MIMO System
## 1 Introduction  
In accordance with recent research trends, millimeter-wave (mmWave) communication has been found to be a potential candidate for next-generation Wireless Local Area Network (WLAN) and 5G cellular systems [1]. mmWave has a large amount of unused bandwidth, which can provide higher frequencies, higher spectral efficiency, and a relevant rise in channel capacity and support to connect millions of devices with high reliability and low latency [2; 3; 4; 5; 6; 7]. However, the poor scattering nature of the mmWave channel causes path loss problems; as a result, fewer dominant paths remain in the mmWave channel [8; 9; 10]. Therefore, the channel matrix can be reconstructed by exploiting information from the remaining paths. In order to reconstruct the mmWave channel, the few dominant paths remaining in the channel matrix have to be estimated at the receiver's end. Conventional channel estimators, such as least-square (LS) estimators and maximum likelihood (ML) estimators, require a higher training overhead, which is not possible for any hybrid mmWave MIMO system. Considering the hardware architecture complexities of mmWave MIMO systems, channel estimation becomes a difficult task. In the literature, two types of approaches are used for mmWave channel estimation. In the first approach, the mmWave channel is estimated by exploiting the sparsity of the channel matrix in the virtual beamspace domain, whereas, in the second approach, the estimation is performed by exploiting the low rank properties of the channel matrix in the antenna domain. In [11; 12; 13; 14], a compressive sensing (CS)-based channel estimation approach was used for a mmWave MIMO system. The basic idea behind this approach is based on the technique in which estimators haveto search for a pair angle in a predefined dictionary matrix, which further depends upon the training information. Such estimators exploit sparsity in a predefined dictionary matrix. For high resolution, another asymmetric approach was proposed in [15]. Under this method, atomic norm is used to formulate the angles of arrival (AoAs)/angles of departure (AoDs), finding problems. A beam codebook design-based approach was proposed in [16], in which, based on the information in static dictionaries, a codebook is designed. All the aforementioned techniques are CS-based approaches. For the case of high-dimensional training information, CS-based estimators suffer from high computational load, which can further reduce the spectral efficiency. A matrix completion (MC)-based approach was proposed in [17; 18], in which the estimator exploits sparsity and low-rank properties of mmWave channels by two independent procedures. In this paper, an extended alternating direction method of multipliers (Ex-ADMM) mmWave massive MIMO channel estimation technique is proposed for more accurate estimation rates. mmWave exhibits some important and peculiar effects over the wireless channel, i.e., the channel experiences an unspecified amount of spread over the angular domain due to its easy scattering nature [19]. As a result, a jointly sparse and low-rank channel matrix in the angular domain can be obtained. This the main motivation behind the formulation of a joint optimization problem for efficient recovery of the channel matrix proposed in this paper. ADMM was recently proposed in [20], has received attention for its easy implementation. Fundamentally, ADMM can exploit both the sparsity and low rank properties of any data matrix. Researchers have found that, in many real-world applications, additional information in a data matrix (known as side information) helps complete the matrix with few entries to obtain more accurate estimations. Therefore, the use of side information for matrix completion and factorization has been introduced in various research areas, like statistical signal processing, image processing, statistical learning, computer vision and so on [21; 22; 23; 24].