# Precise Channel Estimation Approach for a mmWave MIMO System
## 4 Proposed Extended ADMM (Ex-ADMM)-Based Channel Estimation Scheme
In contrast, variables I, D and J are known as essential variables, and Lagrange multipliers \(P_{1}\) and \(P_{2}\) are recognized as dual variables. Another ADMM algorithm mentioned in [18] is also able to solve the optimization problem in (9) but the proposed Ex-ADMM algorithm converges at a lower NMSE as compared to the ADMM and provides better ASE performance as well. Basically, the Ex-ADMM algorithm is a combination of cyclical ADMM and a relaxation factor [28, 38, 39, 40]. In general, ADMM updates its order in an arbitrary way, i.e., \(H\to I\to D\to J\to P_{1}\to P_{2}\). In contrast, the Ex-ADMM updates its order in a cyclical way, i.e., \(H\to P_{1}\to P_{2}\to I\to D\to J\). In the proposed Ex-ADMM, reordering of the ADMM is only done to ensure the channel matrix H satisfies the first-order optimality conditions [39]; therefore, right after the updating of \(H^{(l+1)}\), dual variables \(P_{1}\) and \(P_{2}\) are updated and then, at last, the essential variables I, D and J are updated. A relaxation technique, along with a relaxation parameter, is used to relax the essential variables [41, 42]. Consequently, for the relaxation of the essential variables, we can assume that the cyclical order of dual and essential variables in Equations (11)-(15) is a block variable, i.e., \(\overline{z}^{(l)}=\Big{(}\overline{P}_{1}^{(l)},\overline{P}_{2}^{(l)}, \overline{I}^{(l)},\overline{D}^{(l)},\overline{J}^{(l)},\Big{)}\). Thus, the final relaxed variable, \(z^{(l+1)}=\Big{(}P_{1}^{(l+1)},P_{2}^{(l+1)},I^{(l+1)},D^{(l+1)},J^{(l+1)} \Big{)}\), can be generated as  
\[z^{(l+1)}=z^{l}-\gamma(z^{l}-\overline{z}^{(l)})\]  
Here, roughly speaking, the tilde (\(\sim\)) variables (i.e., \(\overline{P}_{1}^{(l+1)},\overline{P}_{2}^{(l+1)},\overline{I}^{(l+1)}, \overline{D}^{(l+1)}\) and \(\overline{J}^{(l+1)}\)) are auxiliary variables, and they can be updated using the main variables (i.e., \(P_{1}^{(l+1)},P_{2}^{(l+1)},I^{(l+1)},D^{(l+1)}\)and \(J^{(l+1)}\)), and these variables can be obtained using the Equations (11)-(15) as follows:  
\[\overline{P}_{1}^{(l+1)}=P_{1}^{(l)}+\beta\Big{(}H^{(l+1)}-I^{(l)}\Big{)} \tag{16}\]  
\[\overline{P}_{2}^{(l+1)}=P_{2}^{(l)}+\beta\Big{(}I^{(l)}-A_{R}D^{l}A_{T}^{H}- J^{l}\Big{)} \tag{17}\]  
\[\overline{I}^{(l+1)}=\underset{I}{argmin}\ \mathcal{L}_{A}\Big{(}H^{(l+1)},P_{1}^{(l+1)},P_{2}^{(l+1)},I,D^{(l)},J^{(l)}\Big{)} \tag{18}\]  
\[P_{1}^{(l+1)}=P_{1}^{(l)}+\beta\Big{(}H^{(l+1)}-I^{(l)}\Big{)} \tag{11}\]0  
\[P_{1}^{(l+1)}=P_{1}^{(l)}+\beta\Big{(}H^{(l+1)}-I^{(l)}\Big{)} \tag{11}\]1Therefore, the final relaxed variable can be expressed as  
\[\left( \begin{array}{c}\mathrm{P}_{1}^{(1+1)}\\ \mathrm{P}_{1}^{(1+1)}\\ \mathrm{P}_{2}^{(1+1)}\\ \mathrm{I}^{(1+1)}\\ \mathrm{D}^{(1+1)}\\ \mathrm{J}^{(1+1)}\end{array} \right)=\left( \begin{array}{c}\mathrm{P}_{1}^ {(1)}\\ \mathrm{P}_{2}^{(1)}\\ \mathrm{I}^{(1)}\\ \mathrm{D}^{(1)}\\ \mathrm{J}^{(1)}\end{array} \right)-\gamma\left[\left( \begin{array}{c} \mathrm{P}_{1}^{(1)}\\ \mathrm{P}_{2}^{(1)}\\ \mathrm{I}^{(1)}\\ \mathrm{D}^{(1)}\\ \mathrm{J}^{(1)}\end{array} \right)-\left( \begin{array}{c}\mathrm{\widetilde{ P}}_{1}^{(1)}\\ \mathrm{\widetilde{P}}_{2}^{(1)}\\ \mathrm{\widetilde{I}}^{(1)}\\ \mathrm{\widetilde{D}}^{(1)}\\ \mathrm{\widetilde{J}}^{(1)}\end{array} \right)\right] \tag{21}\]  
From the above discussion, it is clear that the Equations (10) and (21) have several advantages: i.e., the optimization problem described in Equation (9) turns into six individual subproblems, and these can be solved further without any strict conditions. Hence, the global optimal solution can be derived effortlessly. This helps reduce the computational complexity and storage requirements. For the next iteration, the values of the subproblems described in Equations (10) and (21) have to be updated. In order to do so, first of all, the closed-form solutions of Equations (10)-(15) need to be obtained. The advantages of Ex-ADMM over ADMM are described further in Section 5.