## 5 Conclusions  
This paper studied the numerical calculation of mutual information for finite-alphabet-based transmissions over doubly correlated MIMO fading channels. The average mutual information was dominated by statistical CSI, and the obstacle of computation was complexity. We examined the appropriateness of the saddle point method first. Then mutual information over any known channel model was calculated by saddle point approximation. Furthermore, we induced the expectation of mutual information over doubly correlated MIMO Rayleigh-fading channels. Numerical results for various MIMO scenarios showed the efficacy of the proposed method. Compared to existing conclusions, the proposed approximation is of considerable accuracy in estimating the average mutual information with radically reduced complexity. It is promising that its accuracy and convenience will facilitate the practical application of mutual information.  
Conceptualization, J.Z., D.Z., and Y.L.; methodology, J.Z. and Y.L.; validation, J.Z. and Y.L.; investigation, J.Z. and D.Z.; writing--original draft preparation, Y.L.; writing--review and editing, J.Z. and D.Z. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Operations** & **Monte Carlo Method** & **Formula (15)** \\ \hline Exponential operation & N\({}_{\text{W}}\)\({}^{*}\) (N\({}^{2}\) + 1) & 0 \\ Logarithm operation & N\({}_{\text{W}}\)\({}^{*}\) N & 2N + 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of computational complexity between Monte Carlo method and saddle point approximation method according to the number of operations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & **Input Type** & **Monte Carlo Method** & **Formula (15)** \\ \hline N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 2 & BPSK & 3.043959 & 0.001049 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 3 & BPSK & 4.444435 & 0.019326 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 4 & BPSK & 8.427808 & 0.029586 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 2 & QPSK & 7.457080 & 0.027120 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 3 & QPSK & 69.558626 & 0.081062 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 4 & QPSK & / & 0.558082 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 2 & 8 PSK & 58.627341 & 0.070549 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 3 & 8 PSK & / & 1.573137 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 4 & 8 PSK & / & / \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 2 & 16 QAM & 1281.853 & 0.612448 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 3 & 16 QAM & / & 255.736633 \\ N\({}_{\text{T}}\) = N\({}_{\text{R}}\) = 4 & 16 QAM & / & / \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of computational complexity between Monte Carlo method and saddle point approximation method according to CPU time (seconds) under different input types and correction parameters (\(\rho_{\text{T}}=\rho_{R}=0.4\)). The symbol/indicates the CPU time is more than half an hour.  
**Funding:** This research was funded by the Basic Research Business Expenses of Beijing Jiaotong University--Special Project of Frontier Science Center of Smart High Speed Railway System (number 2020 JBZD004, 2020 JBZD010); and Enterprise project (number AIPEC/BJJT/20 BW1219).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors wish to thank the reviewers for their valuable comments and suggestions concerning this manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.