ADCPMs of all the UTs are known by the BS.

Before we conclude this section, we define the extended ADCRM as follows $$\begin{aligned}
\label{eq:extadcrm}
  {\bar{{\mathbf{H}}}}_{k,\ell,{\left({{N_{\mathrm{c}}}}\right)}}
  &\triangleq{\mathbf{H}}_{k,\ell}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\notag\\}&=\left[{\mathbf{H}}_{k,\ell}
  \quad{\mathbf{0}}_{M\times\left({N_{\mathrm{c}}}-{N_{\mathrm{g}}}\right)}\right]
  \in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}.\end{aligned}$$ Similarly, the extended ADCPM, which corresponds to the power distribution of the extended ADCRM ${\bar{{\mathbf{H}}}}_{k,\ell,{\left({{N_{\mathrm{c}}}}\right)}}$, is defined as $$\begin{aligned}
\label{eq:extadcpm}
  {\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}
  &\triangleq{{\boldsymbol\Omega}}_{k}{\mathbf{I}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}{\notag\\}&=\left[{{\boldsymbol\Omega}}_{k}
  \quad{\mathbf{0}}_{M\times\left({N_{\mathrm{c}}}-{N_{\mathrm{g}}}\right)}\right]
  \in{\mathbb{R}}^{M\times{N_{\mathrm{c}}}}.\end{aligned}$$ Such definitions will be employed to simplify the analyses in the following sections.

Channel Acquisition with APSPs over One Symbol {#sec:one_tr}
==============================================

Based on the sparse massive MIMO-OFDM channel model presented in the previous section, we propose APSP-CA for massive MIMO-OFDM, including channel estimation and prediction. In this section, we first investigate the case where the APSPs are sent over one OFDM symbol, while the multiple symbol case will be investigated in the next section.

APSPs over One Symbol
---------------------

We assume that all the UTs are synchronized. During the UL pilot segment, namely, the [$\ell$th]{} OFDM symbol of each frame, all the UTs transmit the scheduled pilots simultaneously, and the space-frequency domain signal received at the BS can be represented as $$\begin{aligned}
\label{eq:recsigult}
    {\mathbf{Y}}_{\ell}
    =\sum_{k'=0}^{K-1}{\mathbf{G}}_{k',\ell}{\mathbf{X}}_{k'}+{\mathbf{Z}}_{\ell}\in{\mathbb{C}}^{M\times{N_{\mathrm{c}}}}\end{aligned}$$ where ${\left[{\mathbf{Y}}_{\ell}\right]_{i,j}}$ denotes the received pilot signal at the [$i$th]{} antenna over the [$j$th]{} subcarrier, ${\mathbf{G}}_{k,\ell}$ is the SFCRM defined in , 