# Precise Channel Estimation Approach for a mmWave MIMO System
## 2 mmWave Channel Model  
Let us consider a \(\rm N_{R}\times N_{T}\) mmWave massive MIMO system based on the recent models described in [5; 6; 7]. This system is equipped with \(\rm N_{T}\) transmitters at the base station (BS) and \(\rm N_{R}\) receivers at the mobile station (MS), \(\rm N_{S}\) parallel data streams, and radio frequency (RF) chains, such that \(\rm N_{RF}\leq min(N_{T},N_{R})\) at the transmitter and receiver. At the transmitter, \(\rm N_{RF}\) chains are present, such that \(\rm N_{S}<N_{RF}<N_{T}\). The mmWave MIMO system shown in Figure 1 is a combination of two consecutive joint segments: a digital MIMO baseband \(\rm F_{BB}\in\mathbb{C}^{N_{RF}\times N_{S}}\) and analog RF precoder \(\rm F_{RF}\in\mathbb{C}^{N_{RF}\times N_{RF}}\). Similarly, at the receiver, the signal is processed by the consecutive joint segments of the RF combiner \(W_{\rm RF}\in\mathbb{C}^{N_{RF}\times N_{RF}}\) and baseband combiner \(\rm W_{BB}\in\mathbb{C}^{N_{RF}\times N_{S}}\), respectively. In order to get the closest estimation, the transmitter employs \(\rm N_{T}^{Beam}\leq N_{T}\) pilot beam patterns, denoted as \(\rm\{f_{a}\in\mathbb{C}^{N_{T}\times 1}:\|f_{a}\|_{2}^{2}=1\}\). Meanwhile, at the receiver end, the receiver employs \(\rm N_{R}^{Beam}\leq N_{R}\) pilot beam patterns, denoted as \(\rm\{w_{b}\in\mathbb{C}^{N_{R}\times 1}:\|w_{b}\|_{2}^{2}=1\}\)[16], where a and b are the transmitter's training precoding vector and receiver's training combining vector, respectively. For enabling communication, an analog HBF [11] is used, which opens a path for the transmitter to apply a baseband precoder by steering the beam to the receiver. The b-th received vector for the a-th transmitted beam is determined by  
\[y_{b,a}=w_{b}^{H}Hf_{a}x_{a}+w_{b}^{H}\overline{v} \tag{1}\]  
Here, \(\rm x_{a}\) is the pilot vector symbol emitted from the transmitter, \(\overline{v}\) is the observation noise, with zero mean and \(\rm\sigma_{v}^{2}K_{N_{R}}\) variance, i.e., \(CN\big{(}0,\rm\sigma_{v}^{2}K_{N_{R}}\big{)}\), and \(\rm H\in\mathbb{C}^{N_{R}\times N_{T}}\) is the channel matrix of the mmWave MIMO system. The received vector \(\rm y\) in Equation (1), which is expressed as \(\rm y\triangleq\Big{[}y_{1,b}\ldots y_{N_{T}^{Beam},b}\Big{]}^{T}\in\mathbb{C }^{N_{R}^{Beam}\times 1}\), can be written in generalized form as follows:  
\[\rm y=w^{H}Hfx+q \tag{2}\]  
Here, the combiner noise vector \(\rm q=w_{b}^{H}\overline{v}\in\mathbb{C}^{N_{R}^{Beam}\times 1}\). Therefore, the received signal matrix \(\rm Y\) can be expressed as  
\[\rm Y=W^{H}HFX+Q \tag{3}\]  
Here, the received signal matrix \(\rm Y\triangleq\Big{[}y_{1},\ldots,\ y_{N_{T}^{Beam}}\Big{]}\in\mathbb{C}^{N _{R}^{Beam}\times N_{T}^{Beam}}\), complex combining matrix \(\rm W\triangleq\Big{[}w_{1},\ldots,\ w_{N_{R}^{Beam}}\Big{]}\in\mathbb{C}^{N _{R}\times N_{R}^{Beam}}\), and precoding matrix \(\rm F\triangleq\Big{[}f_{1},\ldots,\ f_{N_{T}^{Beam}}\Big{]}\in\mathbb{C}^{N _{T}\times N_{T}^{Beam}}\). Figure 1: Block structure of a typical mmWave Multiple Input Multiple Output (MIMO) system. \(\mathbb{C}^{\rm{N}^{Beam}_{T}\times\rm{N}^{Beam}_{T}}\) and \(\rm{Q}\in\mathbb{C}^{\rm{N}^{Beam}_{R}\times\rm{N}^{Beam}_{T}}\) are independent and identically distributed (I.I.D) complex additive white gaussian noise (AWGN), with zero mean and \(\rm{\sigma}^{2}_{q}\) variance \(\mathcal{CN}\Big{(}0,\rm{\sigma}^{2}_{q}\Big{)}\). In this paper, we assume pilot symbols are identical, so \(\rm{X}=\sqrt{P_{t}}K_{N^{Beam}_{T}}\), where \(P_{t}\) is the average transmitted pilot power [12, 16]. According to the hybrid mmWave MIMO structure, the transmitted and received matrices are decomposed at the transmitter and receiver ends, such that \(\rm{F}=F_{BB}F_{RF}\), and \(\rm{W}=W_{RF}W_{BB}\), respectively. Hence, Equation (3) of the received signal matrices \(\rm{Y}\) can be re-written as  
\[ \begin{array}{rl}\rm{Y}\triangleq&\sqrt{P_{t}}\,W_{BB}^{H}W_{RF}^{H}HF_{BB} F_{RF}+Q\\ &\rm{Y}\triangleq&\sqrt{P_{t}}\,W^{H}HF+Q\end{array}