MMSE estimation [@Kailath00Linear].

Before we proceed, we define the sum MSE-CE of all the UTs as $$\begin{aligned}
\label{eq:epsicesum}
  {{\epsilon}^{\mathrm{CE}}}\triangleq&\sum_{k=0}^{K-1}{\epsilon_{k}^{\mathrm{CE}}}.\end{aligned}$$ Due to the incurred pilot interference, performance of the APSP-based channel estimation might deteriorate. However, we will show in the following proposition that such effects can be eliminated with proper phase shift scheduling for different pilots.

\[prop:mmsetrseqcon\] The sum MSE-CE ${\epsilon^{\mathrm{CE}}}$ is lower bounded by $$\label{eq:varepsilonl}
  {\epsilon^{\mathrm{CE}}}\geq{\varepsilon^{\mathrm{CE}}}=\sum_{k=0}^{K-1}\sum_{i=0}^{M-1}\sum_{j=0}^{{N_{\mathrm{g}}}-1}
  \left\{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}-\frac{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}^{2}}{{\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}+{\frac{1}{{\rho_{\mathrm{tr}}}}}}\right\}$$ and the lower bound can be achieved under the condition that, for $\forall k,k'\in{\mathcal{K}}$ and $k\neq k'$, $$\label{eq:condokou}
  \left({\bar{{{\boldsymbol\Omega}}}}_{k,{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k}}\right)
  \odot\left({\bar{{{\boldsymbol\Omega}}}}_{k',{\left({{N_{\mathrm{c}}}}\right)}}{{\boldsymbol\Pi}}_{{N_{\mathrm{c}}}}^{\phi_{k'}}\right)
  ={\mathbf{0}}.$$

See [Appendix \[app:prop\_mmsetrseqcon\]]{}.

[Proposition \[prop:mmsetrseqcon\]]{} shows that with the proposed APSPs, the sum MSE-CE can be minimized when phase shifts for different pilots are properly scheduled according to the condition given in . The interpretation is very intuitive. With frequency domain phase shifted pilots, equivalent channels will exhibit corresponding cyclic shifts in the delay domain, as seen from . If the equivalent channel power distributions in the angle-delay domain for different UTs can be made non-overlapping after pilot phase shift scheduling, the pilot interference effect can be eliminated, and the sum MSE-CE can be minimized.

Wireless channels are approximately sparse in the angle-delay domain in many practical propagation scenarios, and typically only 