## Appendix C  
Obviously, when SNR varies from \(-\infty\) to \(+\infty\), \(\alpha_{m,k}\) satisfies \(1/3<\alpha_{m,k}<1/2\) by (11), so the average mutual information over doubly correlated MIMO Rayleigh-fading channels is approximated by  
\[ \begin{array}{rl}\mathcal{I}(\mathbf{x};\mathbf{y})\approx&-\frac{1}{2N} \sum\limits_{m=1}^{N}\mathbb{E}_{H_{\text{WG}}}\left\{\log_{2}\left[\frac{1}{N }\sum\limits_{k=1}^{N}\exp(-\frac{||\mathbf{c}_{m,k}||^{2}}{2\sigma^{2}})\right] \right\}\\ &-\frac{1}{2N}\sum\limits_{m=1}^{N}\mathbb{E}_{H_{\text{WG}}}\left\{\log_{2} \left[\frac{1}{N}\sum\limits_{k=1}^{N}\exp(-\frac{||\mathbf{c}_{m,k}||^{2}}{3 \sigma^{2}})\right]\right\}\end{array}  \tag{17}\]  
Because \(\log_{2}(\mathbf{x})\) is a concave function, by Jensen's inequality, we have  
\[\mathbb{E}_{H_{\text{WG}}}\left\{\log_{2}\left[\sum\limits_{k=1}^{N}\exp(- \frac{||\mathbf{c}_{m,k}||^{2}}{\alpha_{m,k}\sigma^{2}})\right]\right\}\leq\log_{ 2}\left[\sum\limits_{k=1}^{N}\mathbb{E}_{H_{\text{WG}}}\left\{\exp(-\frac{|| \mathbf{c}_{m,k}||^{2}}{\alpha_{m,k}\sigma^{2}})\right\}\right] \tag{18}\]  
where \(\mathbf{c}_{m,k}=\mathbf{\Sigma}_{\text{R}}{}^{1/2}\mathbf{H}_{\text{WG}}\mathbf{ \Sigma}_{\text{T}}{}^{1/2}\mathbf{U}_{\text{T}}{}^{\text{H}}\mathbf{d}_{m,k}\), so expectation in (18) is rewritten as  
\[\mathbb{E}_{H_{\text{WG}}}\left\{\exp(-\frac{||\mathbf{c}_{m,k}||^{2}}{\alpha_{m,k }\sigma^{2}})\right\}=\mathbb{E}_{H_{\text{WG}}}\left\{\exp\left(-\sum\limits _{l=1}^{N_{\text{R}}}\frac{[H_{\text{WG}}]_{l}([\mathbf{\Sigma}_{\text{R}}]_{ l,l}\mathbf{q}_{m,k})[H_{\text{WG}}]_{l}^{\text{H}}}{\alpha_{m,k}\sigma^{2}} \right)\right\} \tag{19}\]  
where  
\[\mathbf{q}_{m,k}=\mathbf{\Sigma}_{\text{T}}{}^{1/2}\sigma\mathbf{d}_{m,k}\mathbf{d}_{m,k}\mathbf{\sigma}^{\text{H}}\mathbf{\Sigma}_{\text{T}}{}^{1/2}\text{ and }\mathbf{\sigma}=\mathbf{U}_{\text{T}}{}^{\text{H}} \tag{20}\]  
\([\mathbf{H}_{\text{WG}}]_{l}\) stands for the \(l^{\text{th}}\) row of \(H_{\text{WG}}\). Since \(\mathbf{H}_{\text{WG}}\) is an independent and identically distributed complex AWGN matrix, \([\mathbf{H}_{\text{WG}}]_{l}\) is an independent and identically distributed complex AWGN vector. So  
\[ \begin{array}{rl}\mathbb{E}_{H_{\text{WG}}}\left\{\exp\left(-\frac{||\mathbf{c}_ {m,k}||^{2}}{\alpha_{m,k}\sigma^{2}}\right)\right\}&=\mathbb{E}_{H_{\text{WG}}} \left\{\exp\left(-\frac{1}{\alpha_{m,k}\sigma^{2}}\sum\limits_{l=1}^{N_{\text{ R}}}[H_{\text{WG}}]_{l}([\mathbf{\Sigma}_{\text{R}}]_{l,l}\mathbf{Q}_{m,k})[H_{\text{WG}}]_{l} ^{\text{H}}\right)\right\}\\ &=\prod\limits_{l=1}^{N_{\text{R}}}\mathbb{E}_{[H_{\text{WG}}]_{l}}\left\{\exp \left(-\frac{[H_{\text{WG}}]_{l}([\mathbf{\Sigma}_{\text{R}}]_{l,l}\mathbf{q}_{ m,k})[H_{\text{WG}}]_{l}^{\text{H}}}{\alpha_{m,k}\sigma^{2}}\right)\right\}\\ &=\prod\limits_{l=1}^{N_{\text{R}}}\int_{[H_{\text{WG}}]_{l}}p([H_{\text{WG}}]_{l })\exp\left(\begin{array}{c}-\frac{1}{\alpha_{m,k}\sigma^{2}}\\ \left[H_{\text{WG}}\right][l,\left(\left[\mathbf{\Sigma}_{\text{R}}\right]_{l,l} \mathbf{q}_{m,k}\right)\\ \left[H_{\text{WG}}\right][l^{\text{H}}\end{array}\right)\\ &=\prod\limits_{l=1}^{N_{\text{R}}}\int_{[H_{\text{WG}}]_{l}}\frac{1}{\pi^{N_{ \text{T}}}}\exp\left(\begin{array}{c}-[\mathbf{\Sigma}_{\text{R}}]_{l,l}\\ \left(I_{\text{WG}}\right][l^{\text{H}}\end{array}\right)\\ \left[H_{\text{WG}}]_{l}^{\text{H}}\end{array} \right)\mathrm{d}[\mathbf{H}_{\text{WG}} ]_{l} \tag{21}\]According to [24],  
\[\int_{h\in\mathbb{C}^{N\times 1}}\exp\Bigl{(}-h^{\rm h}(A+{\rm j}B)h\Bigr{)}{ \rm d}h=\frac{\pi^{N}}{\det(A+{\rm j}B)} \tag{100}\]  
where \([h_{1}]\) is iid \(\mathcal{CN}(0,\sigma)\). (100) can be written as  
\[