## 3 Saddle Point Approximation for Mutual Information
### Saddle Point Approximation  
We first considered the expectation over the AWGN vector \(\boldsymbol{w}\). The Taylor series of (4) is expanded to  
\[\mathcal{I}(\boldsymbol{x};\boldsymbol{y}|\boldsymbol{H})=\log_{2}N-\frac{1}{ N\ln 2}\sum\limits_{m=1}^{N}\sum\limits_{\sigma=1}^{+\infty}\frac{1}{ \sigma}\sum\limits_{q=0}^{\sigma}\frac{\sigma!(-1)^{q}}{q!(\sigma-q)!}\frac{1}{ \pi^{N_{\text{R}}}\sigma^{2N_{\text{R}}}}\int_{\boldsymbol{w}}\left(\sum \limits_{k=1}^{N}\sigma_{m,k}(\boldsymbol{w})\right)^{-q}\text{d}\boldsymbol{w} \tag{5}\]where  
\[p_{m,k}(\mathbf{w})=\exp\left(\frac{||\mathbf{w}-q\epsilon_{m,k}||^{2}}{q\sigma^{2}}- \frac{(q+1)||\epsilon_{m,k}||^{2}}{\sigma^{2}}\right)\text{ and }\mathbf{H}=\mathbf{\Sigma}_{\text{R}} ^{1/2}\mathbf{H}_{\text{WG}}\mathbf{\Sigma}_{\text{T}}^{1/2} \tag{6}\]  
Before proceeding to the saddle point approximation, we needed to establish the following lemma, which guarantees the existence of the saddle point. **Lemma 1**. : _For non-zero natural number \(q\), the maximum of \(\left(\sum_{k=1}^{N}\sigma_{m,k}(\mathbf{w})\right)^{-q}\) exists and is achieved at \(\mathbf{w}=\mathbf{w}_{0}\), where \(\mathbf{w}_{0}\) is the weighted average vector of \(\mathbf{c}_{m,1}\), \(\mathbf{c}_{m,2}\),..., and \(\mathbf{c}_{m,N}\)._  
\[\mathbf{w}_{0}=\sum_{k=1}^{N}q\rho_{m,k}\mathbf{c}_{m,k}\triangleq q\overset{-}{\mathbf{ c}}_{m} \tag{7}\]  
_where \(\rho_{m,k}=\sigma_{m,k}(\mathbf{w}_{0})/\sum_{k=1}^{N}\sigma_{m,k}(\mathbf{w}_{0})= \sigma_{m,k}(\mathbf{w}_{0})/\sigma_{m}(\mathbf{w}_{0})\) is a positive real number over an open interval (0, 1) and satisfies \(\sum_{k=1}^{N}\rho_{m,k}=1\)._  
The proof of Theorem 1 is shown in Appendix A. We are now at \(\mathbf{w}=\mathbf{w}_{0}\) to perform saddle point approximation. **Proposition 1**. : _For non-zero natural number \(q\), integral over complex AWGN vector \(\mathbf{w}\) is approximated by_  
\[\int_{\mathbf{w}}\frac{1}{\pi^{N_{\text{R}}}\sigma^{2N_{\text{R}}}}\Bigg{(}\sum_{k =1}^{N}\sigma_{m,k}(\mathbf{w})\Bigg{)}^{-q}\text{d}\mathbf{w}\approx\Bigg{[}\sum_{k =1}^{N}\exp\left(-\frac{\alpha_{m,k}||\mathbf{c}_{m,k}||^{2}}{\sigma^{2}}\right) \Bigg{]}^{-q} \tag{8}\]  
\[\alpha_{m,k}=\Bigg{[}1-q\frac{||\mathbf{c}_{m}||^{2}}{||\mathbf{c}_{m,k}||^{2}}+q\frac {\mathbf{c}_{m,k}\overset{-}{\mathbf{c}}_{m}\overset{-}{\mathbf{c}}_{m}\overset{-}{\mathbf{ c}}_{m,k}}{||\mathbf{c}_{m,k}||^{2}}-\left(\frac{||\mathbf{c}_{m,k}||^{2}}{\sigma^{2}} \right)^{-1}\ln\left(\sum_{k=1}^{N}\frac{\rho_{m,k}||\mathbf{c}_{m,k}||^{2}}{ \sigma^{2}}-\frac{||\mathbf{c}_{m}||^{2}}{\sigma^{2}}\right)\Bigg{]} \tag{9}\]  
The proof of Proposition 1 is shown in Appendix B. Mutual information is approximated by (5) and (8) as  
\[\mathcal{I}(\mathbf{x};\mathbf{y}|\mathbf{H})\approx\mathcal{I}(\alpha_{m,k},\mathbf{H})=\log_ {2}\mathbf{N}-\frac{1}{N}\sum_{m=1}^{N}\log_{2}\Bigg{[}\sum_{k=1}^{N}\exp\left(- \frac{\alpha_{m,k}||\mathbf{c}_{m,k}||^{2}}{\sigma^{2}}\right)\Bigg{]} \tag{10}\]  
Generally speaking, the close-form solution is hardly obtained. That is to say, we cannot write down the exact expression of \(\alpha_{m,k}\). Optionally, we adopt a numerical method to obtain approximated \(\alpha_{m,k}\),  
\[\underset{\alpha_{m,k}:k=1,2}{\text{argmin}}\left\{|\mathcal{I}(\mathbf{x};\mathbf{y }|1)-\mathcal{I}(\alpha_{m,k},1)\right\}\approx\left\{3-\exp[-||\mathbf{c}_{m,k}|| ^{2}/(4\sigma^{2})]\right\}^{-1} \tag{11}\]  
where \(\mathcal{I}(\mathbf{x};\mathbf{y}|1)\) is computed by Monte Carlo method by taking BPSK over single-input and single-output (SISO) over AWGN channel (that is \(\mathbf{H}=1\)) as an example, and \(\alpha_{m,k}\) is fixed at each signal to noise ratio (SNR). Thus, (10) can be written as  
\[