correlations of the proposed APSPs for different UTs depend only on the associated phase shift difference. It is worth noting that, for conventional PSOPs, the phase shift differences for different pilots are set to satisfy the orthogonality condition ${\left|\phi_{k'}-\phi_{k}\right|}\geq{N_{\mathrm{g}}}\ \forall k'\neq k$. However, for our APSPs, the phase shifts for different pilots are adjustable, and pilots for different UTs can even share the same phase shift, which leads to more available pilots, and thus pilot overhead can be significantly reduced.

Channel Estimation with APSPs
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In this section we investigate channel estimation during the pilot segment under the minimum MSE (MMSE) criterion using the proposed APSPs. Direct MMSE estimation of the SFCRM ${\mathbf{G}}_{k,\ell}$ requires information about the large dimensional SFCCM ${\mathbf{R}}_{k}$ and a large dimensional matrix inversion, which is difficult to implement in practice. However, with the sparse massive MIMO-OFDM channel model presented above, when we shift our focus from the space-frequency domain to the angle-delay domain, channel estimation can be greatly simplified. The BS can first estimate the ADCRM to obtain ${\hat{{\mathbf{H}}}}_{k,\ell}$, then the SFCRM estimates can be readily obtained as ${\hat{{\mathbf{G}}}}_{k,\ell}={\mathbf{V}}_{M}{\hat{{\mathbf{H}}}}_{k,\ell}{\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}^{T}$ via exploiting the unitary equivalence between the angle-delay domain channels and the space-frequency domain channels given in 