${\mathbf{X}}_{k}={\mathsf{diag}\left\{{\mathbf{x}}_{k}\right\}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{c}}}}$ denotes the frequency domain pilot signal sent from the [$k$th]{} UT, ${\mathbf{Z}}_{\ell}$ is the additive white Gaussian noise (AWGN) matrix during the UL pilot segment with elements identically and independently distributed (i.i.d.) as ${\mathcal{CN}\left( 0 , {\sigma_{\mathrm{ztr}}}\right) }$, and ${\sigma_{\mathrm{ztr}}}$ is the noise power.

The proposed APSP over one OFDM symbol for a given UT $k$ is given by $$\begin{aligned}
\label{eq:pls}
  {\mathbf{X}}_{k}\triangleq\sqrt{{\sigma_{\mathrm{xtr}}}}
  \underbrace{{\mathsf{diag}\left\{{\mathbf{f}}_{{N_{\mathrm{c}}},\phi_{k}}\right\}}}_{\triangleq{\mathbf{D}}_{\phi_{k}}}
  {\mathbf{X}},\quad\phi_{k}={0,1,\ldots,{N_{\mathrm{c}}}-1}\end{aligned}$$ where ${\mathbf{X}}={\mathsf{diag}\left\{{\mathbf{x}}\right\}}\in{\mathbb{C}}^{{N_{\mathrm{c}}}\times{N_{\mathrm{c}}}}$ satisfying ${\mathbf{X}}{\mathbf{X}}^{H}={\mathbf{I}}_{{N_{\mathrm{c}}}}$ is the basic pilot matrix shared by all UTs in the same cell, and ${\sigma_{\mathrm{xtr}}}$ is the pilot signal transmit power. The APSP signal given in can be seen as a phase shifted version of $\sqrt{{\sigma_{\mathrm{xtr}}}}{\mathbf{X}}$ with phase shift $\phi_{k}$ in the frequency domain. Note that the proposed APSP has the same PAPR as that of ${\mathbf{X}}$ in the time domain, thus existing low PAPR sequence designs can be easily incorporated into our approach. In addition, as the basic pilot matrix ${\mathbf{X}}$ can be predetermined, only ${\mathbf{X}}$ and the pilot phase shift indices rather than the entire pilot matrices are required to be stored, and the required storage space can be significantly reduced.

From , it can be readily obtained that, for $\forall k,k'\in{\mathcal{K}}$, $$\label{eq:ijcrocor}
  {\mathbf{X}}_{k'}{\mathbf{X}}_{k}^{H}
  ={\sigma_{\mathrm{xtr}}}{\mathbf{D}}_{\phi_{k'}-\phi_{k}}$$ which indicates that cross 