and the power angle-delay spectrum for massive MIMO-OFDM channels.

\[prop:Decomp\_cov\] Define ${\mathbf{V}}_{M}\in{\mathbb{C}}^{M\times M}$ as ${\left[{\mathbf{V}}_{M}\right]_{i,j}}\triangleq\frac{1}{\sqrt{M}}\cdot{\exp\left(-{\bar{\jmath}}2\pi\frac{i\left(j-M/2\right)}{M}\right)}$, and ${{\boldsymbol\Omega}}_{k}\in{\mathbb{R}}^{M\times{N_{\mathrm{g}}}}$ as $$\begin{aligned}
\label{eq:cov_eigen_ele}
  {\left[{{\boldsymbol\Omega}}_{k}\right]_{i,j}}\triangleq M{N_{\mathrm{c}}}\left(\theta_{i+1}-\theta_{i}\right)\cdot {{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta_{i},\tau_{j}\right)\end{aligned}$$ where $\theta_{m}\triangleq\arcsin\left(2m/M-1\right)$, and $\tau_{n}\triangleq n{T_{\mathrm{s}}}$. Then when the number of antennas $M\to\infty$, the SFCCM ${\mathbf{R}}_{k}$ tends to $\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}$ in the sense that, for fixed non-negative integers $i$ and $j$, $$\begin{aligned}
\label{eq:Decomp_cov}
  \lim_{\substack{{M \to\infty}}}&\Big[{\mathbf{R}}_{k}
  -\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)
  {\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}{\notag\\}&\quad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}\Big]_{i,j}=0.\end{aligned}$$

See [Appendix \[app:prop\_Decomp\_cov\]]{}.

The relationship between the space-frequency domain channel joint correlation property and the channel power distribution in the angle-delay domain for massive MIMO-OFDM is established in [Proposition \[prop:Decomp\_cov\]]{}. Specifically, for massive MIMO-OFDM channels in the asymptotically large array regime, the eigenvectors of the SFCCMs for different UTs tend to be the same, which shows that massive MIMO-OFDM channels can be asymptotically decorrelated by the fixed space-frequency domain statistical eigendirections, while the eigenvalues depend on the corresponding channel power angle-delay spectra.

[Proposition \[prop:Decomp\_cov\]]{} indicates that, for massive MIMO-OFDM channels, when the number of BS antennas $M$ is sufficiently large, the SFCCM can be well approximated by $$\begin{aligned}
\label{eq:cov_appr}
  {\mathbf{R}}_{k}&\approx\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right){\mathsf{diag}\left\{{\mathsf{vec}\left\{{{\boldsymbol\Omega}}_{k}\right\}}\right\}}{\notag\\}&\qquad\cdot\left({\mathbf{F}}_{{N_{\mathrm{c}}}\times{N_{\mathrm{g}}}}\otimes{\mathbf{V}}_{M}\right)^{H}.\end{aligned}$$ It is worth noting that the approximation in is consistent with existing results in the literature. For frequency-selective single-input single-output channels, agrees with the results in [@Li98Robust; @van95channel]. For frequency-flat massive MIMO 