, while the same MSE-CE performance can be maintained. In the following, we focus on estimation of the ADCRM ${\mathbf{H}}_{k,\ell}$ under the MMSE criterion.

Recalling , the received pilot signal at the BS in can be rewritten as $$\begin{aligned}
\label{eq:recsigad}
    {\mathbf{Y}}_{\ell}
    =\sum_{k'=0}^{K-1}{\mathbf{V}}_{M}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}
    {\mathbf{X}}_{k'}+{\mathbf{Z}}_{\ell}.\end{aligned}$$ After decorrelation and power normalization of ${\mathbf{Y}}_{\ell}$, the BS can obtain an observation of the UL channel ${\mathbf{H}}_{k,\ell}$, given by shown at the top of the next page,

$$\begin{aligned}
\label{eq:ykltr}
    {\mathbf{Y}}_{k,\ell}
    &={\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Y}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&={\frac{1}{{\sigma_{\mathrm{xtr}}}}}\sum_{k'=0}^{K-1}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{X}}_{k'}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}
    +{\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Z}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}{\mathbf{H}}_{k,\ell}+\underbrace{\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}}_{\mathrm{pilot\ interference}\triangleq\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}}
    +\underbrace{{\frac{1}{{\sigma_{\mathrm{xtr}}}}}{\mathbf{V}}_{M}^{H}{\mathbf{Z}}_{\ell}{\mathbf{X}}_{k}^{H}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}}_{\mathrm{pilot\ noise}}\end{aligned}$$

where (a) follows from . Using the unitary transformation property, it can be readily shown that the pilot noise term in exhibits a Gaussian distribution with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , {\sigma_{\mathrm{ztr}}}/{\sigma_{\mathrm{xtr}}}\right) }$, and can be simplified as $$\begin{aligned}
\label{eq:simykltr}
    {\mathbf{Y}}_{k,\ell}
    ={\mathbf{H}}_{k,\ell}+\sum_{k'\neq k}{\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}
    +{\frac{1}{\sqrt{{\rho_{\mathrm{tr}}}}}}{{\mathbf{Z}}_{\mathrm{iid}}}\end{aligned}$$ where ${\rho_{\mathrm{tr}}}\triangleq{\sigma_{\mathrm{xtr}}}/{\sigma_{\mathrm{ztr}}}$ is the signal-to-noise ratio (SNR) during the pilot segment, and ${{\mathbf{Z}}_{\mathrm{iid}}}\in{\mathbb{C}}^{M\times{N_{\mathrm{g}}}}$ is the normalized AWGN matrix with i.i.d. elements distributed as ${\mathcal{CN}\left( 0 , 1\right) }$.

Note that the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ defined in satisfies $$\begin{aligned}
\label{eq:hulphi}
  {\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}
  &={\mathbf{H}}_{k',\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{*}{\notag\\}&={\mathbf{H}}_{k',\ell}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{*}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}{\notag\\}&{\mathop{=}^{(\mathrm{a})}}{\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{T}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}{\mathbf{F}}_{{N_{\mathrm{c}}}}^{*}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}{\notag\\}&{\mathop{=}^{(\mathrm{b})}}{\bar{{\mathbf{H}}}}_{k',\ell,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}-\phi_{k}}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\end{aligned}$$ where (a) follows from , and (b) follows from the permutation matrix definition given in [Section \[sec:prwb\_not\]]{}. Thus, the pilot interference term ${\mathbf{H}}_{k',\ell}^{\phi_{k'}-\phi_{k}}$ in is a column truncated version of the 