# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems
## 3 Formulation of the Channel Estimation Problem via Compressed Sensing  
In this section, we formulate a technique via CS theory to estimate the complex gain \(a_{l}\), the angle of arrival \(\theta_{l}^{R}\) and the angle of departure \(\theta_{l}^{T}\) with a unified framework, leveraging the sparsity of mmWave channel.  
To formulate a sparse signal recovery problem of channel estimation, we substitute Equation (9) into Equation (2) and use the result of Kronecker product (i.e., \(vec(\mathbf{ABC})=(\mathbf{C}^{T}\otimes\mathbf{A})vec(\mathbf{B})\)) to transform it as  
\[vec\left(\mathbf{y}\right)=\left(\left(\mathbf{F}\mathbf{s}+\mathbf{e}_{I} \right)^{T}\otimes\mathbf{W}^{H}\right)\left(\mathbf{\tilde{A}}_{T}^{*} \otimes\mathbf{\tilde{A}}_{R}\right)vec(\mathbf{\Delta}_{d}^{p})+\left( \mathbf{F}\mathbf{s}+\mathbf{e}_{I}\right)^{T}\otimes\left(\mathbf{W}\right)^ {H}vec(\mathbf{E}_{Q})+vec\left(\mathbf{W}^{H}\mathbf{n}\right) \tag{10}\]  
According to CS theory, sufficient measurement vectors are needed to guarantee the sparse vector recovered with high probability. Assume that \(M\) training frames are occupied and both sides use pseudorandomly precoders and combiners during the channel estimation phase. After collecting the combined signals at the receiver, Equation (10) can be extended to a standard expression based on CS theory.  
\[\underbrace{\begin{bmatrix}vec\left(\mathbf{y}^{(1)}\right)\\ \vdots\\ vec\left(\mathbf{y}^{(M)}\right)\end{bmatrix}}_{\mathbf{Y}}=\underbrace{ \begin{bmatrix}\left(\left(\mathbf{F}^{(1)}\mathbf{s}^{(1)}+\mathbf{e}_{I} \right)^{T}\otimes\mathbf{W}^{(1)H}\right)\\ \vdots\\ \left(\left(\mathbf{F}^{(M)}\mathbf{s}^{(M)}+\mathbf{e}_{I}\right)^{T}\otimes \mathbf{W}^{(M)H}\right)\end{bmatrix}}_{\mathbf{\Phi}}\underbrace{\begin{bmatrix} \left(\mathbf{\tilde{A}}_{T}^{*}\right)^{(1)}\otimes\left(\mathbf{\tilde{A}}_ {R}\right)^{(1)}\\ \vdots\\ \mathbf{\tilde{A}}_{T}^{(M)}\otimes\left(\mathbf{\tilde{A}}_{R}\right)^{(M)} \end{bmatrix}}_{\mathbf{Y}}\underbrace{\begin{bmatrix}vec\left(\mathbf{\Delta }_{d}^{e}\right)^{(1)}\\ \vdots\\ vec\left(\mathbf{\Delta}_{d}^{e}\right)^{(M)}\end{bmatrix}}_{\mathbf{h}}+ \underbrace{\begin{bmatrix}vec\left(\mathbf{e}_{N}\right)^{(1)}\\ \vdots\\ vec\left(\mathbf{e}_{N}\right)\end{bmatrix}}_{\mathbf{E}_{N}} \tag{11}\]  
where \(\mathbf{\Phi}^{(m)}\overset{\Delta}{=}(\mathbf{F}^{(m)}\mathbf{s}^{(m)}+( \mathbf{e}_{I})^{(m)})^{T}\otimes\left((\mathbf{W})^{(m)}\right)^{H},m\in[1,M]\) is defined as the measurement matrix, which is assumed to be known at the receiver for standard CS-based model. However, the transmitter hardware impairments will prevent the receiver from fully knowing the real measurement matrix. \(\mathbf{\Psi}^{(m)}\overset{\Delta}{=}\left(\mathbf{\tilde{A}}_{T}^{*} \right)^{(m)}\otimes\left(\mathbf{\tilde{A}}_{R}\right)^{(m)}\) is defined as the orthogonal dictionary matrix. \(\mathbf{h}^{(m)}\overset{\Delta}{=}vec\left(\mathbf{\Delta}_{d}^{(m)}\right) \in\mathbb{C}^{G\times 1}\) is a sparse vector that contains all complex channel gains. \(\mathbf{e}_{N}^{(m)}\overset{\Delta}{=}(\mathbf{F}^{(m)}\mathbf{s}^{(m)}+ \mathbf{e}_{I}^{(m)})^{T}\otimes(\mathbf{W}^{(m)})^{H}vec(\mathbf{E}_{Q}^{(m) })+vec\left(\mathbf{W}^{(m)}{}^{H}\mathbf{n}^{(m)}\right)\) is defined as the additive noise vector that contains the quantization error, hardware impairments and additive white Gaussian noise (AWGN) at the receiver.  
For such a sparse signal recovery problem, it has been widely accepted [9; 10; 11; 12; 13; 14] that the parameters, including the complex gain \(a_{I}\), the angle of arrival \(\theta_{I}^{R}\) and the angle of departure \(\theta_{I}^{T}\), can be estimated by the following optimization problem  
\[\begin{array}{l}\underset{\mathbf{h}}{min}\left\|\mathbf{h}\right\|_{0}\\ s.t.\ \|\mathbf{y}-\mathbf{\Phi}\mathbf{\Psi}\mathbf{h}\|_{2}^{2}<\varepsilon \end{array} \tag{12}\]  
where \(\varepsilon\) is a parameter that denotes the tolerance of estimation error and the variance of the noise is a suitable choice for it.