於本論文，我們探討放大前送多重輸入多重輸出(Multiple-Input and Multiple-Ou-tputpp簡稱MIMO)多天線系統的互資訊(Mutual Information)，研究重點著重在互資訊的統計特性。我們所考慮的中繼系統為兩段式跳躍的傳輸模式:一個傳送端、一個接收端及一個中繼站。我們假設天線間的傳輸通道具相關性，其中通道係數是根據瑞雷衰落(Rayleigh Fading)或是萊斯衰落(Rician Fading)的統計特性。具我們所知，於此設定下互資訊的統計特性仍是一個未解的問題。於分析時，我們加入適當的假設，我們的作法如下:我們假設各節點的天線數在一定的比例下漸漸趨近無窮大，接著在分析過程中，我們利用鞍點近似(Saddlepoint Approximation)和超級分析(Superanalysis)兩種數學分析工，求得互資訊的期望值與變異數，並證實當天線數趨近無窮大時，其高次統計特性趨近於零，根據此特性，我們可以利用高斯機率分布來近似互資訊的統計特性。

整體來說，本論文的主要貢獻有兩項: 1)我們指出文獻上，於瑞雷衰落通道上結果的錯誤，並提供修正的結果，2)我們也將之前文獻上考慮的環境做更進一步的推廣，得到在萊斯衰落通道上互資訊的統計特性，這是在現今研究中尚未出現的
近似結果。雖然我們的推導是基於天線數趨近無窮大的假設，但我們利用推出來的期望值及變異數作模擬分析，發現即使天線數目很小，也可得到精確的模擬結果。

In this thesis, we address the statistics of mutual information of amplify-and-forward (AF) multiple-input multiple-output (MIMO) two-hop relay channels, where the
source terminal (ST), relay terminal (RT) and destination terminal (DT) are equipped with a number of correlated antennas and there is a line-of-sight (LOS)
component (Rician fading) of each link. To the best of our knowledge, deriving analytical expressions for the statistics of mutual information of the relay channel
is difficult and still unsolvable. To circumvent the mathematical difficulties, we consider this problem under the large-system regimen in which the numbers of
antennas at the transmitter and receiver go to infinity with a fixed ratio. In the large-system regimen, this thesis has made the following contributions: 1) We
get the mean and the variance of the mutual information of the concerned relay channel. 2) We show that the mutual information distribution converges to the
Gaussian distribution. The analytical results are derived by mean of two powerful tools developed in the context of theoretical physics: emph{saddle-point
approximation} and emph{superanalysis}. The derived analytical results are very general and can degenerate to several previously results as special cases. From a
degenerated case, we realize that the previous result by Wagner {em et al.} cite{Wag-08} is wrong and thus
we provide the corrected result. Finally, Simulation results demonstrate that even for a moderate number of antennas at each end, the proposed analytical results
provide undistinguishable results as those obtained by Monte-Carlo results.

摘要 i
Abstract ii
目錄 iv
圖次 vi
表次 viii
1 緒論 1
2 常用數學工具 4
3 多重輸入多重輸出萊斯中繼站系統 10
3.1 系統模型 10
3.2系統互資訊近似解分析 13
4 多重輸入多重輸出瑞雷中繼站系統 81
4.1系統模型 81
4.2系統互資訊近似解分析 82
5 模擬結果 87
5.1平均互資訊模擬分析 87
5.2隨機互資訊機率分布 89
5.3模擬比較 93
6 結論 97
參考文獻 98

[1] C. K. Wen, J. C. Chen and P. Ting, Robust transmitter design for amplify-andforward MIMO relay systems exploiting only channel statistics," IEEE Trans.
Wireless Commun., vol. 11, no 2, pp. 668-682, Feb. 2012.
[2] E. Riegler, and G. Taricco, Asymptotic statistics of the mutual information
for spatially correlated Rician fading MIMO channels with interference," IEEE
Trans. Inf. Theory., vol. 56, no. 4, pp. 1542-1559, Apr. 2010.
[3] G. Taricco, Asymptotic mutual information statistics of separately correlated
Rician fading MIMO channels," IEEE Trans. Inf. Theory., vol. 54, no. 8, pp.
3490-3504, Aug. 2008.
[4] A. L. Moustakas, S. H. Simon, and A. M. Sengupta, MIMO capacity through
correlated channels in the presence of correlated interferers and noise: A (not
so) large n analysis," IEEE Trans. Inf. Theory., vol. 49, no. 10, pp. 2545-2561,
Oct. 2003.
[5] J. Wagner, B. Rankov, and A. Wittneben, Correspondence-Large n analysis
of amplify-and-forward MIMO relay channels with correlated Rayleigh fading,"
IEEE Trans. Inf. Theory., vol. 54, no. 12, pp. 5735-5746, Dec. 2008.
[6] F. A. Berezin, and A. A. Kirillov, Introduction to Superanalysis. Dordrecht,
Netherlands: Reidel, 1987.
[7] S. F. Edwards, and P. W. Anderson, Theory of spin glasses," J. Phys. F: Metal
Phys., vol. 5, no. 52, pp. 965-974, May 1975.
[8] T. Cover, and J. A. Thomas, Elements of Information Theory, New York: Wiley,
1991.