在此論文中我們基於雙向放大轉送多輸出多輸入中繼系統下設計低複雜度的中繼預編碼器。最近被提出的特徵模選擇法（eigenmode-selection）用較低的計算複雜度即可達到接近最佳的效能，在這個方法中，作者基於對應的中繼系統下有效多輸出多輸入通道（effective multiple input multiple output channel）的特徵空間（eigenspace）先建構出一個中繼預編碼器的集合，其中集合中可能的中繼預編碼器數是有限的；再根據不同需求的目標函數來找到最好的那一個當作所要使用的中繼預編碼器，也就是選到使得目標函數最好的特徵模。此外，為了更進一步降低目標函數的計算量，分別為最大化通道容量（channel capacity）或是最小化均方誤差（mean-squared error），作者發展基於條件數（condition number）底下的特徵模選擇法，其定義為一矩正之最大與最小特徵值的比例。
然而，隨著系統的自由度（degrees of freedom）越高，中繼預編碼器集合的數目變得很多，也就是計算次數相當高；為了改善這個問題，首先我們利用條件數的特性並請找到條件數跟奇異值（singular value）的關聯，接下來，我們提出針對兩種目標函數分別提出兩種降低集合數的方法，一是最大化條件數和，二是最小化條件數和。從模擬的結果顯示我們提出的方法可以達到跟原本的特徵模選擇法相近的效能，但我們減少搜尋複雜度K(K-1)之一倍。

Recently, an eigenmode-selection approach was suggested for relay precoding in a two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relay system, where a finite set of relay precoders are constructed based on the eigenspaces of the effective MIMO channels and one of them with appropriate eigenmodes is selected to meet a specific design criterion. To design a relay precoder subject to the minimization/maximization of the sum of the condition numbers of the effective MIMO channels, this method needs to search for the solution in a relay procoding set of size 2K!, where the condition number is defined as the ratio of the largest to the smallest singular value of a MIMO channel and K denotes the degrees of freedom of the system. Although the eigenmode-selection precoding scheme can achieve close-to-optimal performance, the required search complexity would become prohibitively high when K is large. In this thesis, we propose some low-complexity relay precoder designs based on eigenmode selection for a two-way AF MIMO relay system. We first exploit properties of the condition numbers of the effective MIMO channels, and then present two set-reduction methods based on these properties to reduce the search complexity in selecting appropriate eigenmodes for the minimization/maximization of the sum of the condition numbers. As compared to the original eigenmode-selection precoding scheme, the proposed approaches reach similar performance but reduce the search complexity by a factor of K(K-1).

REFERENCE
[1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, "Cooperative diversity in wireless networks: Efficient protocols and outage behavior," IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.
[2] B. Rankov and A. Wittneben, "Spectral efficient protocols for half-duplex fading relay channels," IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp. 379–389, Feb. 2007.
[3] S. Rankov and A. Wittneben, “Embracing wireless interference: Analog network coding,” in Proc. SIGCOMM, Kyoto, Japan, pp. 397–408, Aug. 2007.
[4] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, "Joint Tx-Rx beamforming de sign for multicarrier MIMO channels: A unified framework for convex optimization," IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2381–2401, Sep. 2003.
[5] L. Sanguinetti, A. A. D’Amico, and Y. Rong, “A tutorial on the optimization of amplify-and-forward MIMO relay networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 8, pp. 1331–1346, Sep. 2012.
[6] Y. Rong, X. Tang and Y. Hua, "A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems," IEEE Trans. Signal Process., vol. 57, no. 12, pp. 4837–4851, Dec. 2009.
[7] R. Wang and M. Tao, "Joint source and relay precoding designs for MIMO two-way relaying based on MSE criterion," IEEE Trans. Signal Process., vol. 60, no. 3, pp. 1352–1365, Mar. 2012.
[8] S. Xu and Y. Hua, "Optimal design of spatial source-and-relay matrices for a non-regenerative two-way MIMO relay system," IEEE Trans. Wireless Commun., vol. 10, no. 5, pp. 1645–1655, May 2011.
[9] H. Park, H. J. Yang, J. Chun, and R. Adve, "A closed-form power allocation and signal alignment for a diagonalized MIMO two-way relay channel with linear receivers," IEEE Trans. Signal Process., vol. 60, no. 11, pp. 5948–5962, Nov. 2012.
[10] C.-L. Wang, J.-Y. Chen, and J.-J. Jheng, “A precoder design for two-way amplify-and-forward MIMO relay systems with linear receivers,” in Proceedings of the 2014 IEEE Veh. Technol. Conf.–Fall (VTC 2014–Spring), Vancouver, Canada, Sep. 2014.
[11] C.-L. Wang, J.-Y. Chen, and Y.-H. Peng, “Precoder designs for two-way amplify-and-forward MIMO relay systems: An eigenmode-selection approach,” to be submitted.
[12] K.-J. Lee, H. Sung, E. Park, and I. Lee, "Joint optimization for one and two-way MIMO AF multiple-relay systems," IEEE Trans. Wireless Commun., vol. 9, no. 12, pp. 3671–3681, Dec. 2010.
[13] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.
[14] A. Hjorungnes, and D. Gesbert, "Complex-valued matrix differentiation: Techniques and key results," IEEE Trans. Signal Process., vol.55, no.6, pp.2740–2746, Jun. 2007.
[15] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cilffs, NJ: Prentice Hall, 1993.
[16] Y. Rong, "Joint source and relay optimization for two-way linear non-regenerative MIMO relay communications," IEEE Trans. Signal Processing, vol. 60, no. 12, pp. 6533–6546, Dec. 2012.
[17] S. Barbarossa, Multiantenna Wireless Communications Systems, Boston, Mass.: Artech House, 2005.
[18] W. Guan and H. Luo, “Joint MMSE transceiver design in non-regenerative MIMO relay systems,” IEEE Trans. Commun. Lett., vol. 12, no. 7, pp. 517–519, Jul. 2008.
[19] A. Edelman, “Eigenvalues and condition numbers of random matrices,” SIAM J. Matrix Analysis and Applications, vol. 9, no. 4, pp. 543–560, Oct. 1988.