extended ADCRM ${\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}$ with a cyclic column shift, where the shift factor depends on the corresponding pilot phase shift difference $\phi_{k'}-\phi_{k}$. Thus elements of ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ can be readily obtained as $$\begin{aligned}
  {\left[{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}
  =\left\{ {\begin{array}{l}
{\left[{\mathbf{H}}_{k',\ell}\right]_{i,{\left\langlej-\left(\phi_{k'}-\phi_{k}\right)\right\rangle_{{N_{\mathrm{c}}}}}}}, \\
 \quad {\left\langlej-\left(\phi_{k'}-\phi_{k}\right)\right\rangle_{{N_{\mathrm{c}}}}}\leq{N_{\mathrm{g}}}-1 \\
0, \quad\textrm{else}.
\end{array}} \right.\end{aligned}$$

Recalling [Proposition \[prop:adcmsts\]]{}, elements of the ADCRM ${\mathbf{H}}_{k',\ell}$ are statistically uncorrelated. Consequently, elements of the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$, a column truncated copy of ${\mathbf{H}}_{k',\ell}$ with cyclic column shift, are also statistically uncorrelated. Thus, using the same methodology as in the previous section, the corresponding power matrix of the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ can be defined as $$\begin{aligned}
\label{eq:omeuphuk}
  {{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}
  &\triangleq{{{\mathsf{E}}}\left\{{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\odot\left({\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}\right)^{*}\right\}}{\notag\\}&={\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\end{aligned}$$ which is a column truncated version of the extended ADCPM ${\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}$ defined in with cyclic column shift $\phi_{k'}-\phi_{k}$.

With the channel observation ${\mathbf{Y}}_{k,\ell}$ in , and the fact that the angle-delay domain channel elements are uncorrelated as derived in [Proposition \[prop:adcmsts\]]{}, the MMSE estimate ${\hat{{\mathbf{H}}}}_{k,\ell}$ can be obtained in an element-wise manner as follows [@Kailath00Linear] $$\begin{aligned}
\label{eq:hathkl}
    {\left[{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}=\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}}
  {\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}
  {\left[{\mathbf{Y}}_{k,\ell}\right]_{i,j}}.\end{aligned}$$ Let ${\tilde{{\mathbf{H}}}}_{k,\ell}={\mathbf{H}}_{k,\ell}-{\hat{{\mathbf{H}}}}_{k,\ell}$ be the angle-delay domain channel estimation error of the [$k$th]{} UT, then the corresponding MSE-CE can be obtained as $$\begin{aligned}
\label{eq:epsilonkl}
  {\epsilon_{k}^{\mathrm{CE}}}
  &\triangleq\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
  {{{\mathsf{E}}}\left\{{\left|{\left[{\tilde{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
  {{{\mathsf{E}}}\left\{{\left|{\left[{\mathbf{H}}_{k,\ell}\right]_{i,j}}\right|^{2}}-{\left|{\left[{\hat{{\mathbf{H}}}}_{k,\ell}\right]_{i,j}}\right|^{2}}\right\}}{\notag\\}&=\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}\left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}
  -\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{\sum_{k'=0}^{K-1}{\left[{{\boldsymbol\Omega}}_{k'}^{\phi_{k'}-\phi_{k}}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}\end{aligned}$$ where (a) follows from the orthogonality principle of 