# Precise Channel Estimation Approach for a mmWave MIMO System
## 4 Proposed Extended ADMM (Ex-ADMM)-Based Channel Estimation Scheme
### Procedure for Updating \(\mathrm{D}^{\left(l+1\right)}\)  
To get the solution for \(\mathrm{D}\), reformulate \(\mathcal{L}_{\mathrm{A}}\) to \(\mathcal{L}_{3}\) and consider all terms corresponding to \(\mathrm{D}\) in Equation (9). Keeping only the terms that are related to it,  
\[\mathcal{L}_{3}\triangleq\underset{\mathrm{D}}{\mathrm{argmin}} \ \mathrm{\Gamma}_{\mathrm{D}}\|\mathrm{D}\|_{1}+\mathrm{tr}\!\left( \mathrm{P}_{2}^{\mathrm{H}}\!\left(\mathrm{I}-\mathrm{A}_{\mathrm{R}}\mathrm{ DA}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{J}\right)\right)+\frac{\beta}{2}\|\mathrm{I}- \mathrm{A}_{\mathrm{R}}\mathrm{DA}_{\mathrm{T}}^{\mathrm{H}}-\mathrm{J}\|_{ \mathrm{F}}^{2} \tag{28}\] \[=\mathrm{\Gamma}_{\mathrm{D}}\|\mathrm{D}\|_{1}+\frac{\beta}{2}\| \mathrm{A}_{\mathrm{R}}^{\mathrm{H}}\!\left(\frac{1}{\beta}\mathrm{P}_{2}^{ \left(\mathrm{I}+1\right)}-\mathrm{J}^{\left(\mathrm{I}\right)}+\mathrm{I}^{ \left(\mathrm{I}+1\right)}\right)\mathrm{A}_{\mathrm{T}}\|_{\mathrm{F}}^{2}\]  
Here, \(\mathrm{A}_{\mathrm{R}}\) and \(\mathrm{A}_{\mathrm{T}}\) are the unitary matrices. To get the closed-form solution of \(\mathcal{L}_{3}\), Equation (28) is transformed into the standard least absolute shrinkage and selection operator (LASSO) problem [44]. In order to do this, vectorization is performed on Equation (28):  
\[\underset{\mathrm{D}}{\mathrm{argmin}}\ \mathrm{\Gamma}_{\mathrm{D}}\|\mathrm{D}\|_{1 }+\frac{\beta}{2}\|\mathrm{A}_{\mathrm{R}}^{\mathrm{H}}\!\left(\frac{1}{\beta }\mathrm{P}_{2}^{\left(\mathrm{I}+1\right)}-\mathrm{J}^{\left(\mathrm{I} \right)}+\mathrm{I}^{\left(\mathrm{I}+1\right)}\right)\mathrm{A}_{\mathrm{T}}\|_{ \mathrm{F}}^{2} \tag{29}\]  
Let us assume \(\mathrm{V}^{\left(\mathrm{I}+1\right)}=\mathrm{A}_{\mathrm{R}}^{\mathrm{H}} \mathrm{A}_{\mathrm{T}}\!\left(\frac{1}{\beta}(\mathrm{P}_{2}^{\left(\mathrm{I }+1\right)}-\mathrm{J}^{\left(\mathrm{I}+1\right)}+\mathrm{I}^{\left(\mathrm{I }+1\right)}\right)\) and \(\mathrm{v}^{\left(\mathrm{I}+1\right)}=\mathrm{vec}\!\left(\mathrm{V}^{\left( \mathrm{I}+1\right)}\right)\). Therefore, Equation (29) can be written as  
\[\underset{\mathrm{d}}{\mathrm{argmin}}\ \mathrm{\Gamma}_{\mathrm{d}}\|\mathrm{d}\|_{1}+ \frac{\beta}{2}\|\mathrm{d}-\mathrm{v}^{\left(\mathrm{I}+1\right)}\|_{\mathrm{F }}^{2} \tag{30}\]  
Thus, to obtain the estimate of \(\mathrm{d}\) in Equation (30), a soft thresholding operator is imposed for \(\left(\mathrm{I}+1\right)\) iterations:  
\[\begin{array}{c}\mathrm{d}^{\left(\mathrm{I}+\mathrm{I}\right)}=\mathrm{ sign}\!\left(\mathrm{Re}\!\left(\mathrm{v}^{\left(\mathrm{I}+1\right)}\right) \right)\circ\mathrm{max}\!\left(\left|\mathrm{Re}\!\left(\mathrm{v}^{\left( \mathrm{I}+1\right)}\right|-\Gamma_{\mathrm{d}}^{\prime},0\right)\right.\\ \left.+\mathrm{sign}\!\left(\mathrm{Im}\!\left(\mathrm{v}^{\left(\mathrm{I}+1 \right)}\right)\right)\circ\mathrm{max}\!\left(\left|\mathrm{Im}\!\left( \mathrm{v}^{\left(\mathrm{I}+1\right)}\right|-\Gamma_{\mathrm{d}}^{\prime},0 \right)\right.\end{array} \tag{31}\]  
\(\Gamma_{\mathrm{d}}^{\prime}\) is the scaled version of \(\mathrm{\Gamma}_{\mathrm{d}}\) and \(\Gamma_{\mathrm{d}}^{\prime}\triangleq\frac{\mathrm{\Gamma}_{\mathrm{d}}}{ \beta}\). Hence, the value of \(\mathrm{D}^{\left(\mathrm{I}+1\right)}\) is obtained by  
\[\mathrm{D}^{\left(\mathrm{I}+1\right)}\!=\mathrm{unvec}\ \left(\mathrm{d}^{\left(\mathrm{I}+\mathrm{I}\right)}\right) \tag{32}\]