# Precise Channel Estimation Approach for a mmWave MIMO System
## 4 Proposed Extended ADMM (Ex-ADMM)-Based Channel Estimation Scheme
### Procedure for Updating \(I^{(l+1)}\)  
To update \(\mathrm{I}^{(l+1)}\), reformulate \(\mathcal{L}_{\mathrm{A}}\) to \(\mathcal{L}_{2},\text{ and, to get the closed-form solutions for I, consider all terms of I in Equation (9) and differentiate with respect to I:  
\[\begin{split}\mathcal{L}_{2}\triangleq\underset{\mathrm{I}}{ \mathrm{arg\min}}&\frac{1}{2}\|\mathrm{\Psi}\circ\mathrm{I}- \mathrm{H}_{\mathrm{\Psi}}\|_{\mathrm{F}}^{2}+\mathrm{tr}\Big{(}\mathrm{P}_{1} ^{\mathrm{H}}(\mathrm{H}-\mathrm{I})\Big{)}+\frac{\beta}{2}\|\mathrm{H}- \mathrm{I}\|_{\mathrm{F}}^{2}\\ &+\mathrm{tr}\Big{(}\mathrm{P}_{2}^{\mathrm{H}}\Big{(}\mathrm{I}- \mathrm{A}_{\mathrm{R}}\mathrm{D}\mathrm{A}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{ J}\Big{)}\Big{)}+\frac{\beta}{2}\|\mathrm{I}-\mathrm{A}_{\mathrm{R}}\mathrm{D} \mathrm{A}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{J}\|_{\mathrm{F}}^{2}\end{split} \tag{24}\]  
\[\begin{split}\mathcal{L}_{2}=\frac{\partial\mathcal{L}_{\mathrm{A}}}{ \partial\mathrm{I}}&=\frac{\partial}{\partial\mathrm{I}}\big{(} \frac{1}{2}\|\mathrm{\Psi}\circ\mathrm{I}-\mathrm{H}_{\mathrm{\Psi}}\|_{ \mathrm{F}}^{2}+\mathrm{tr}\Big{(}\mathrm{P}_{1}^{\mathrm{H}}(\mathrm{H}- \mathrm{I})\Big{)}+\frac{\beta}{2}\|\mathrm{H}-\mathrm{I}\|_{\mathrm{F}}^{2} \\ &+\mathrm{tr}\Big{(}\mathrm{P}_{2}^{\mathrm{H}}\big{(}\mathrm{I}- \mathrm{A}_{\mathrm{R}}\mathrm{D}\mathrm{A}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{ J}\big{)}\Big{)}+\frac{\beta}{2}\|\mathrm{I}-\mathrm{A}_{\mathrm{R}}\mathrm{D} \mathrm{A}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{J}\|_{\mathrm{F}}^{2})\end{split}\]  
\[\begin{split}\mathcal{L}_{2}\,=\,\mathrm{\Psi}\circ\mathrm{I}- \mathrm{H}_{\mathrm{\Psi}}-\mathrm{P}_{1}-\beta(\mathrm{H}-\mathrm{I})-\mathrm{P}_ {2}-\beta(\mathrm{J}-\mathrm{I}+\mathrm{A}_{\mathrm{R}}\mathrm{D}\mathrm{A}_{ \mathrm{T}}^{\mathrm{H}})\end{split}\]  
Setting \(\mathcal{L}_{2}(\mathrm{I})\) to zero, i.e., \(\mathcal{L}_{2}=0\),  
\[\begin{split}\mathrm{\Psi}\circ\mathrm{I}-\mathrm{H}_{\mathrm{ \Psi}}-\mathrm{P}_{1}-\beta(\mathrm{H}-\mathrm{I})-\mathrm{P}_{2}-\beta( \mathrm{J}-\mathrm{I}+\mathrm{A}_{\mathrm{R}}\mathrm{D}\mathrm{A}_{\mathrm{T}}^{ \mathrm{H}})=0\\ \mathrm{\Psi}\circ\mathrm{I}+2\beta\mathrm{I}=\mathrm{H}_{\mathrm{ \Psi}}+\mathrm{P}_{1}+\beta(\mathrm{H})+\mathrm{P}_{2}+\beta\mathrm{J}+ \mathrm{A}_{\mathrm{R}}\mathrm{D}\mathrm{A}_{\mathrm{T}}^{\mathrm{H}}\end{split}\]\[\mathrm{I}=\left(\mathrm{A}+2\ \beta\mathrm{K}\right)^{-1}(\mathrm{P}_{1}+\beta (\mathrm{H})+\mathrm{P}_{2}+\beta\mathrm{J}+\beta\mathrm{BD}) \tag{25}\]  
where \(\mathrm{K}\) is the identity matrix, \(\mathrm{A}\triangleq\sum_{i=1}^{\mathrm{N}_{\mathrm{R}}}\mathrm{diag}\left( \left[\Psi\right]_{\mathrm{K}}\right)^{\mathrm{T}}\otimes\mathrm{E}_{\mathrm{ kk}}\left[\Psi\right]_{i}\) exhibits the \(\mathrm{k}\)-th row, and \(\mathrm{E}_{\mathrm{kk}}\) is derived by inserting unit values in the \(\mathrm{N}_{\mathrm{R}}\times\mathrm{N}_{\mathrm{R}}\) zero matrix at its \(\mathrm{(k,k)}\)-th position and \(\mathrm{B}\triangleq\mathrm{A}_{\mathrm{T}}^{\star}\otimes\mathrm{A}_{ \mathrm{R}}\).  
Therefore, the closed-form solution of \(\mathrm{i}\) for the \(\left(\mathrm{I}+1\right)\) iteration is  
\[\mathrm{i}^{\left(\mathrm{I}+\mathrm{I}\right)}=\left(\mathrm{A}^{\mathrm{H}} \mathrm{A}+2\beta\mathrm{K}\right)^{-1}\!\left(\mathrm{P}_{1}^{\left(\mathrm{ I}\right)}+\beta\mathrm{h}^{\left(\mathrm{I}+1\right)}+\mathrm{A}^{\mathrm{H}} \mathrm{h}_{\Psi}+\mathrm{P}_{2}^{\left(\mathrm{I}\right)}+\beta\mathrm{j}^{ \left(\mathrm{I}\right)}+\beta\mathrm{B}\mathrm{d}^{\left(\mathrm{I}\right)}\right) \tag{26}\]  
To get the final solution, unvectorized \(\mathrm{i}^{\left(\mathrm{I}+\mathrm{I}\right)}\), i.e.,  
\[\mathrm{I}^{\left(\mathrm{I}+\mathrm{I}\right)}=\mathrm{unvec}(\mathrm{i}^{ \left(\mathrm{I}+\mathrm{I}\right)} \tag{27}\]