# Precise Channel Estimation Approach for a mmWave MIMO System
## 4 Proposed Extended ADMM (Ex-ADMM)-Based Channel Estimation Scheme  
In the following section, the proposed Ex-ADMM is described in detail. From Section 3, Equation (7) is clearly a convex function with two objectives. Therefore, there are many possible ways to get the global optimal solution. However, the best ways to solve the convex problem are first-order methods, which only require the first-order information of the optimization problem. Since these methods are computationally expensive, alternating optimization techniques (AOTs) [26] are used to obtain the optimal solution to a convex optimization problem, due to their less complex structure and easy handling. ADMM [20] is one very popular and efficient AOT. To solve the convex optimization problem described in Equation (7), an extension of ADMM [28] known as extended ADMM (Ex-ADMM) is proposed for the channel estimation of a mmWave MIMO system. In order to solve the optimization problem described in Equation (7) by Ex-ADMM, we reformulate the optimization problem, and two auxiliary matrices, \(\text{I}\in\mathbb{C}^{\text{N}_{\text{R}}\times\text{N}_{\text{T}}}\) and \(\text{J}\triangleq\text{I}-\text{A}_{\text{R}}\text{DA}_{\text{T}}^{\text{H}}\), are introduced. As a result, the reformulated optimization problem can be re-written as,  
\[ \begin{split}\underset{\text{H,J,D,J}}{\text{minimize}}\ \Gamma_{\text{H}}\|\text{H}\|_{\star}+\Gamma_{\text{D}}\|\text{D}\|_{1}+\frac{1 }{2}\|\|\|_{\text{F}}^{2}+\frac{1}{2}\|\Psi\circ\text{I}-\text{H}_{\Psi}\|_{ \text{F}}^{2}\\ \text{Subject to}\ \text{H}=\text{I}\ \text{and}\ \text{J}=\ \text{I}- \text{A}_{\text{R}}\text{DA}_{\text{T}}^{\text{H}}\end{split}  \tag{8}\]  
The first term in the optimization problem described in Equation (8) is the side information of the matrix, i.e., \(\text{A}_{\text{R}}\) and \(\text{A}_{\text{T}}\), along with the virtual channel gain in matrix Z. The second term is the information on subsampled virtual channel gain. The third term in the optimization problem is described by Equation (8), and considers the discretization error, and the fourth term is the AWGN noise. Afterwards, Equation (8) can be re-written with consideration of the augmented Lagrangian function (ALF) as follows:  
\[ \begin{split}\mathcal{L}_{\text{A}}(\text{H,I,D,J,P}_{1},\text{ P}_{2})\\ \triangleq\Gamma_{\text{H}}\|\text{H}\|_{\star}+\Gamma_{\text{D}} \|\text{D}\|_{1}+\frac{1}{2}\|\|\|_{\text{F}}^{2}+\frac{1}{2}\|\Psi\circ\text{ I}-\text{H}_{\Psi}\|_{\text{F}}^{2}+\text{tr}\Big{(}\text{P}_{1}^{\text{H}}( \text{H}-\text{I})\Big{)}\\ +\frac{\beta}{2}\|\text{H}-\text{I}\|_{\text{F}}^{2}+\text{tr} \Big{(}\text{P}_{2}^{\text{H}}\big{(}\text{I}-\text{A}_{\text{R}}\text{DA}_{ \text{T}}^{\text{H}}-\text{J}\big{)}\Big{)}+\frac{\beta}{2}\|\text{I}-\text{A}_ {\text{R}}\text{DA}_{\text{T}}^{\text{H}}-\text{J}\|_{\text{F}}^{2}\end{split}  \tag{9}\]Here, \(P_{1}\) and \(P_{2}\in\mathbb{C}^{N_{R}\times N_{T}}\) are Lagrange multipliers. \(\beta\) is the step size of Ex-ADMM and \(\beta>0\). For the next iteration, the Ex-ADMM can split Equation (9) into six sub-equations and solve it alternatively, i.e.,  
\[H^{(l+1)}=\underset{H}{argmin}\ \mathcal{L}_{A}\Big{(}H,P_{1}^{(l)},P_{2}^{(l)},I^{(l)},D^{(l)},J^{(l)}\Big{)} \tag{10}\]  
\[P_{1}^{(l+1)}=P_{1}^{(l)}+\beta\Big{(}H^{(l+1)}-I^{(l)}\Big{)} \tag{11}\]  
\[P_{2}^{(l+1)}=P_{2}^{(l)}+\beta\Big{(}I^{(l)}-A_{R}D^{(l)}A_{T}^{H}-J^{(l)} \Big{)} \tag{12}\]  
\[I^{(l+1)}=\underset{I}{argmin}\ \mathcal{L}_{A}\Big{(}H^{(l+1)},P_{1}^{(l+1)},P_{2 }^{(l+1)},I,D^{(l)},J^{(l)}\Big{)} \tag{13}\]  
\[D^{(l+1)}=\underset{D}{argmin}\ \mathcal{L}_{A}\Big{(}H^{(l+1)},P_{1}^{(l+1)},P_{2 }^{(l+1)},I^{(l+1)},D,J^{(l)}\Big{)} \tag{14}\]  
\[J^{(l+1)}=\underset{J}{argmin}\ \mathcal{L}_{A}\Big{(}H^{(l+1)},P_{1}^{(l+1)},P_{2 }^{(l+1)},I^{(l+1)},D^{l+1},J\Big{)} \tag{15}\]  
In Equations (10)-(15), \(H^{(l+1)}\) is involved in every step; henceforth, \(H^{(l+1)}\) is known as an intermediate variable.