## Appendix A Proof of Lemma 1  
To prove Lemma 1, we first consider the asymptotic property for dominant eigenvector and eigenvalue of channel covariance matrix \(\mathbf{C}_{l}\). According to (5) and the equality \((\mathbf{A}\otimes\mathbf{B})(\mathbf{C}\otimes\mathbf{D})=(\mathbf{AC}) \otimes(\mathbf{BD})\), we have  
\[\mathbf{C}_{l}^{\text{LoS}}=\frac{\beta_{l}K_{l}}{K_{l}+1}\left(\mathbf{b} \left(\rho_{x,l}\right)\otimes\mathbf{a}\left(\rho_{y,l}\right)\right)\left( \mathbf{b}\left(\rho_{x,l}\right)\otimes\mathbf{a}\left(\rho_{y,l}\right) \right)^{H}. \tag{1}\]From (A1), it is easy to verify that  
\[ \begin{split}\mathbf{u}_{\max}\left(\mathbf{C}_{l}^{\text{LoS}} \right)&=\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{b}\left(\rho_{x,l} \right)\otimes\mathbf{a}\left(\rho_{y,l}\right),\\ \lambda_{\max}\left(\mathbf{C}_{l}^{\text{LoS}}\right)& =\frac{\beta_{l}K_{l}}{K_{l}+1}N_{x}N_{y}.\end{split} \] (A2)  
Moreover, using the equality \(\left(\mathbf{A}\otimes\mathbf{B}\right)\left(\mathbf{C}\otimes\mathbf{D} \right)=\left(\mathbf{AC}\right)\otimes\left(\mathbf{BD}\right)\) on (5) repeatedly, we can obtain  
\[ \begin{split}&\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{C}_{l}^{\text{NLoS }}\mathbf{b}\left(\rho_{x}^{\prime}\right)\otimes\mathbf{a}\left(\rho_{y}^{ \prime}\right)\\ &=\frac{1}{\sqrt{N_{x}N_{y}}}\frac{\beta_{1}}{K_{l}+1}\int\limits \limits_{\rho_{x,l}^{\prime}}^{\rho_{x,l}^{\prime}}\int\limits_{\rho_{y}^{ \prime}}^{\rho_{x,m}^{\prime}}S_{l}\left(\rho_{x,p_{y}}\right)\mathbf{b}\left( \rho_{x}\right)\otimes\mathbf{a}\left(\rho_{y}\right)\mathbf{b}^{H}\left(\rho _{x}\right)\mathbf{b}\left(\rho_{y}^{\prime}\right)\mathbf{a}^{H}\left(\rho_ {y}\right)\mathbf{a}\left(\rho_{y}^{\prime}\right)\mathbf{d}\rho_{x}\mathrm{d} \rho_{y}\\ &=\frac{1}{\sqrt{N_{x}N_{y}}}\frac{\beta_{1}}{K_{l}+1}\int\limits \limits_{\rho_{x,l}^{\prime}}^{\rho_{x,l}^{\prime}}\int\limits_{\rho_{y}^{ \prime}}^{\rho_{x,m}^{\prime}}S_{l}\left(\rho_{x,p_{y}}\right)\mathbf{b}\left( \rho_{x}\right)\otimes\mathbf{a}\left(\rho_{y}\right)\sum\limits_{i=0}^{N_{x}- 1}\exp\left(2\pi i\left(\rho_{x}^{\prime}-\rho_{x}\right)\right)\sum\limits_{i =0}^{N_{x}-1}\exp\left(2\pi i\left(\rho_{y}^{\prime}-\rho_{y}\right)\right) \mathrm{d}\rho_{y}\mathrm{d}\rho_{y}\\ &=\frac{1}{\sqrt{N_{x}N_{y}}}\frac{\beta_{1}}{K_{l}+1}\int\limits _{\rho_{x,l}^{\prime}}^{\rho_{x,l}^{\prime}}\int\limits_{\rho_{y}^{\prime}}^{ \rho_{x,m}^{\prime}}S_{l}\left(\rho_{x,p_{y}}\right)\mathbf{b}\left(\rho_{x} \right)\otimes\mathbf{a}\left(\rho_{y}\right)\frac{\exp\left(\eta\pi\left(N_{x }-1\right)\left(\rho_{x}^{\prime}-\rho_{x}\right)\right)}{\exp\left(\eta\pi \left(1-N_{y}\right)\left(\rho_{y}^{\prime}-\rho_{y}\right)\right)}\frac{\sin \left(N_{x}\pi\left(\rho_{x}^{\prime}-\rho_{x}\right)\right)}{\sin\left(\pi \left(\rho_{x}^{\prime}-\rho_{x}\right)\right)}\\ &\times\frac{\sin\left(N_{y}\pi\left(\rho_{y}^{\prime}-\rho_{y} \right)\right)}{\sin\left(\pi\left(\rho_{y}^{\prime}-\rho_{y}\right)\right)} \mathrm{d}\rho_{x}\mathrm{d}\rho_{y}.\end{split} \] (A3)  
According to [43], the function \(\frac{1}{N}\frac{\sin(N\pi x)}{\sin(\pi x)}\) (this function is called aliased sinc function in [43]) converges to standard sinc function \(\mathrm{sinc}(Nx)\) as \(N\rightarrow\infty\). Therefore, using the limit \(\frac{1}{a}\mathrm{sinc}\left(\frac{x}{a}\right)\overset{a\to 0}{\rightarrow} \delta\left(x\right)\), we can obtain \(\frac{1}{N}\frac{\sin(N\pi x)}{\sin(\pi x)}\overset{N\rightarrow\infty}{ \rightarrow}\delta\left(x\right)\). With the above result, in the large \(\left\{N_{x},N_{y}\right\}\) limit we can rewrite (A3) as  
\[ \begin{split}&\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{C}_{l}^{\text{NLoS }}\mathbf{b}\left(\rho_{x}^{\prime}\right)\otimes\mathbf{a}\left(\rho_{y}^{ \prime}\right)\\ &=\frac{1}{\sqrt{N_{x}N_{y}}}\frac{\beta_{l}}{K_{l}+1}S_{l}\left( \rho_{x,^{\prime}}^{\prime}\rho_{y}^{\prime}\right)\mathbf{b}\left(\rho_{x}^{ \prime}\right)\otimes\mathbf{a}\left(\rho_{y}^{\prime}\right).\end{split}