the temporal correlations and joint space-frequency domain correlations of the channels can be separated [@Li98Robust; @Auer12MIMO], i.e., $$\begin{aligned}
\label{eq:uncorcha}
  &{{{\mathsf{E}}}\left\{g_{k}\left(\theta,\tau,\nu\right)g_{k}^{*}\left(\theta',\tau',\nu'\right)\right\}}{\notag\\}&={{\mathtt{S}}_{k}^{\mathrm{ADD}}}\left(\theta,\tau,\nu\right)\cdot{\delta\left(\theta-\theta'\right)}{\delta\left(\tau-\tau'\right)}{\delta\left(\nu-\nu'\right)}{\notag\\}&={{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau\right)\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right)\cdot{\delta\left(\theta-\theta'\right)}{\delta\left(\tau-\tau'\right)}{\delta\left(\nu-\nu'\right)}\end{aligned}$$ where ${{\mathtt{S}}_{k}^{\mathrm{ADD}}}\left(\theta,\tau,\nu\right)$, ${{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,\tau\right)$, and ${{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right)$ represent the power angle-delay-Doppler spectrum, power angle-delay spectrum, and power Doppler spectrum of UT $k$, respectively [@Patzold12Mobile; @Clerckx13MIMO].

From and , we can obtain the following channel statistical property (see [Appendix \[app:der\_sta\]]{} for the derivations) $$\begin{aligned}
\label{eq:expbgkldel}
  {{{\mathsf{E}}}\left\{{\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell+\Delta_{\ell}}\right\}}{\mathsf{vec}^{H}\left\{{\mathbf{G}}_{k,\ell}\right\}}\right\}}=\varrho_{k}\left(\Delta_{\ell}\right)\cdot{\mathbf{R}}_{k}\end{aligned}$$ where $\varrho_{k}\left(\Delta_{\ell}\right)$ is the channel temporal correlation function (TCF) given by $$\varrho_{k}\left(\Delta_{\ell}\right)\triangleq\int\limits_{-\infty}^{\infty}\!{\exp\left({\bar{\jmath}}2\pi\nu\Delta_{\ell}{T_{\mathrm{sym}}}\right)}\cdot{{\mathtt{S}}_{k}^{\mathrm{Dop}}}\left(\nu\right){{\mathrm{d}}\nu}$$ and ${\mathbf{R}}_{k}$ is the space-frequency domain channel covariance matrix (SFCCM) given by $$\begin{aligned}
\label{eq:spa_fre_cov}
  {\mathbf{R}}_{k}&\triangleq\sum_{q=0}^{{N_{\mathrm{g}}}-1}\int\limits_{-\pi/2}^{\pi/2}\!
  \left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]
  \left[{\mathbf{f}}_{{N_{\mathrm{c}}},q}\otimes{{\mathbf{v}}_{M,\theta}}\right]^{H}{\notag\\}&\qquad\cdot{{\mathtt{S}}_{k}^{\mathrm{AD}}}\left(\theta,q{T_{\mathrm{s}}}\right)
  {{\mathrm{d}}\theta}\in{\mathbb{C}}^{M{N_{\mathrm{c}}}\times M{N_{\mathrm{c}}}}.\end{aligned}$$

In this work, we consider the widely accepted Clarke-Jakes channel power Doppler spectrum,[^7] with the corresponding channel TCF given by [@Jakes94Microwave; @Patzold12Mobile] $$\label{eq:timcor}
  \varrho_{k}\left(\Delta_{\ell}\right)=\mathrm{J}_{0}\left(2\pi\nu_{k}{T_{\mathrm{sym}}}\Delta_{\ell}\right)$$ where $\mathrm{J}_{0}\left(\cdot\right)$ is the zeroth-order Bessel function of the first kind, and $\nu_{k}$ is the Doppler frequency of UT $k$. Note that the Clarke-Jakes power Doppler spectrum is an even function, i.e., $\varrho_{k}(\Delta_{\ell})=\varrho_{k}(-\Delta_{\ell})$, and satisfies $\varrho_{k}\left(0\right)=1$. Also, we assume that according to the law of large numbers, the channel elements exhibit a joint Gaussian distribution, i.e., ${\mathsf{vec}\left\{{\mathbf{G}}_{k,\ell}\right\}}\sim{\mathcal{CN}\left( {\mathbf{0}}, {\mathbf{R}}_{k}\right) }$.

Before proceeding, we investigate in the following proposition a property of the large dimensional SFCCM, and present a relationship between the SFCCM 