多天線(Multiple-input-multiple-output)通訊系統比傳統的單天線(Single-input-single-output)通訊系統擁有更多的分集增益(diversity gain)和多工增益(multiplexing gain)，因此多天線通訊系統擁有較好的通訊品質。為了享有多天線系統的好處，在接收端必須要有一些容易計算的檢測方法可以檢測出傳送的訊號。除了改善接收端的檢測器(detector)，我們也可以藉由在傳送端加入一個預編碼器(precoder)來改善多天線系統的通訊品質。如果一個最佳化的問題可以被寫成凸集最佳化問題(convex optimization problem)的形式，我們就可以利用內部點方法(interior point method)快速的解出最佳解。因此，最近有許多關於多天線系統的檢測問題或者是預編碼問題都被寫成凸集最佳化問題的形式。在這篇論文裡，我們結合半定放寬(semidefinite relaxation)檢測器提出了一個預編碼器來改善錯誤率。此預編碼器的限制條件是全部傳送的能量，我們利用凸集最佳化問題來解出此預編碼器。此編碼器除了可以用於半定放寬檢測器，它也可以用於其它的檢測器。從模擬的結果來看，此預編碼器可以有效的降低錯誤率。

1] C.-C. Weng and P. Vaidyanathan, “Mimo transceiver optimization with linear con¬straints on transmitted signal covariance components,” IEEE Trans. on Signal Process¬ing, vol. 58, no. 1, pp. 458 –462, jan. 2010.
[2] A. Klein, G. Kaleh, and P. Baier, “Zero forcing and minimum mean-square-error equal¬ization for multiuser detection in code-division multiple-access channels,” IEEE Trans. on Veh. Technol., vol. 45, no. 2, pp. 276 –287, may 1996.
[3] B. S. Thian and A. Goldsmith, “A reduced-complexity mimo receiver via channel or¬dering,” in IEEE GLOBECOM, nov. 2009, pp. 1 –6.
[4] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in lattices,” IEEE Trans. on Inform. Theory, vol. 48, no. 8, pp. 2201 – 2214, aug 2002.
[5] A. Wiesel, Y. C. Eldar, and S. Shitz, “Semidefinite relaxation for detection of 16-qam signaling in mimo channels,” IEEE Signal Processing Lett., vol. 12, no. 9, pp. 653 – 656, sept. 2005.
[6] N. Sidiropoulos and Z.-Q. Luo, “A semidefinite relaxation approach to mimo detection for high-order qam constellations,” IEEE Signal Processing Lett., vol. 13, no. 9, pp. 525 –528, sept. 2006.
[7] Y. Yang, C. Zhao, P. Zhou, and W. Xu, “Mimo detection of 16-qam signaling based on semidefinite relaxation,” IEEE Signal Processing Lett., vol. 14, no. 11, pp. 797 –800, nov. 2007.
[8] J. Jalden and B. Ottersten, “An exponential lower bound on the expected complexity of sphere decoding,” in Proc. IEEE ICASSP, vol. 4, 2004, pp. 393 – 396.
[9] S. Boyd and L. Vandenberghe, Convex Optimization, ch. 5, Ed. New York: Combridge, 2004.
[10] R. V. C. Helmberg, F. Rendl and H. Wolkowicz, “An interior-point method for semidef¬inite programming,” in SIAM Journal on Optimization 6: 342-361, 1996.
[11] J. Jalden and B. Ottersten, “The diversity order of the semidefinite relaxation detector,” IEEE Trans. on Inform. Theory, vol. 54, no. 4, pp. 1406 –1422, april 2008.
[12] ——, “High diversity detection using semidefinite relaxation,” ACSSC, pp. 2082 – 2086, oct. 2006.
[13] M. Kisialiou and Z.-Q. Luo, “Performance analysis of quasi-maximum-likelihood detec¬tor based on semi-definite programming,” in Proc. IEEE ICASSP, vol. 3, march 2005, pp. 433 – 436.
[14] A.-C. So, “On the performance of semidefinite relaxation mimo detectors for qam con-stellations,” in IEEE ICASSP, april 2009, pp. 2449 –2452.
[15] W.-K. Ma, C.-C. Su, J. Jalden, and C.-Y. Chi, “Some results on 16-qam mimo detection using semidefinite relaxation,” in IEEE ICASSP, march 2008, pp. 2673 –2676.
[16] D. Palomar, J. Cioffi, and M. Lagunas, “Joint tx-rx beamforming design for multicarrier mimo channels: a unified framework for convex optimization,” IEEE Trans. on Signal Processing, vol. 51, no. 9, pp. 2381 – 2401, sept. 2003.
[17] A. W. Marshall and I. Olkin, “Inequalityies: Theory of majorization and its applica¬tions.” in New York: Academic, 1979.
[18] D. Palomar, “Unified framework for linear mimo transceivers with shaping constraints,” IEEE Commun. Lett., vol. 8, no. 12, pp. 697 – 699, dec. 2004.
[19] C.-C. Weng and P. Vaidyanathan, “Per-antenna power constrained mimo transceivers optimized for ber,” in ACSSC, oct. 2008, pp. 1300 –1304.
[20] Z.-Q. Luo, T. Davidson, G. Giannakis, and K. M. Wong, “Transceiver optimization for block-based multiple access through isi channels,” IEEE Trans. on Signal Processing, vol. 52, no. 4, pp. 1037 – 1052, april 2004.
[21] D. Palomar, M. Lagunas, and J. Cioffi, “Optimum linear joint transmit-receive process¬ing for mimo channels with qos constraints,” IEEE Trans. on Signal Processing, vol. 52, no. 5, pp. 1179 – 1197, may 2004.
[22] D. Palomar, M. Bengtsson, and B. Ottersten, “Minimum ber linear transceivers for mimo channels via primal decomposition,” IEEE Trans. on Signal Processing, vol. 53, no. 8, pp. 2866 – 2882, aug. 2005.
[23] Y. Huang and D. Palomar, “Rank-constrained separable semidefinite programming for optimal beamforming design,” in IEEE ISIT, june 2009, pp. 2432 –2436.
[24] M. Bengtsson and B. Ottersten, “Optimal transmit beamforming using convex opti¬mization,” 1999.
[25] ——, “Optimal downlink beamforming using semidefinite optimization,” in in Proc. 37th Annual Allerton Conference on Communication, Control and Computing, pp. 987¬996, 1999.
[26] A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed mimo receivers,” IEEE Trans. on Signal Processing, vol. 54, no. 1, pp. 161 – 176, jan. 2006.
[27] C.-C. Weng, C.-Y. Chen, and P. Vaidyanathan, “Gtd-based transceivers for decision feedback and bit loading,” in IEEE ICASSP, april 2009, pp. 2661 –2664.
[28] Y. Jiang, J. Li, and W. Hager, “Uniform channel decomposition for mimo commu¬nications,” IEEE Trans. on Signal Processing, vol. 53, no. 11, pp. 4283 – 4294, nov. 2005.
[29] ——, “Transceiver design using generalized triangular decomposition for mimo commu-nications with qos constraints,” in ACSSC, vol. 1, nov. 2004, pp. 1154 – 1157.
[30] T. Liu, J.-K. Zhang, and K. M. Wong, “Efficient design of optimal transmitter for mimo systems using decision feedback receivers,” in 24th Biennial Symposium on Communi¬cations, june 2008, pp. 138 –141.
[31] ——, “Optimal precoder design for mimo systems using decision feedback receivers,” in IEEE SAM, july 2008, pp. 113 –117.
[32] W. W. H. Y. Jiang and J. Li, “Generalized triangular decomposition,” in Mathematics of computation., Oct. 2007.
[33] M. Grant and S. Boyd, “Cvx user’s guide for cvx version 1.2,” 2009.
[34] J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. Meth. Software., vol. 11-12, pp. 625–653, 1999.