a few elements of the ADCPM ${{\boldsymbol\Omega}}_{k}$ are dominant in massive MIMO-OFDM. When such channel sparsity is properly taken into account, the equivalent angle-delay domain channels for different UTs are almost non-overlapping with high probability, assuming proper pilot phase shifts. This suggests the feasibility of the proposed APSPs for massive MIMO-OFDM.

Note that performance of the proposed APSP approach is related to the channel sparsity level. For the case where channels of different UTs have a sparse common support with $s\left(\leq{N_{\mathrm{g}}}\right)$ representing the number of the columns containing non-zero elements in the ADCPM [@Barbotin12Estimation; @Rao14Distributed], the maximum number of UTs that can be served without pilot interference is $\left\lfloor{N_{\mathrm{c}}}/s\right\rfloor$. However, for practical wireless channels, most of the channel elements in the angle-delay domain are close to zero, and the condition in usually cannot be satisfied exactly, which will lead to degradation of the channel acquisition performance. In such cases, it is clear that the more sparse the channels are, the better performance can be achieved by the proposed APSP approach.

Before we conclude this subsection, we remark here that several existing pilot approaches satisfy the optimal condition given in [Proposition \[prop:mmsetrseqcon\]]{}. For the case where channel sparsity property is not 