這篇論文提出一個固定複雜度的晶格縮減方法運用於多天線廣播系統的TH預編碼器[1]中。晶格縮減方法運用於TH預編碼器中能達到最大多樣性秩序。但晶格縮減演算法的複雜度非固定，不適合用於硬體實現。在固定複雜度的晶格縮減演算法中，我們考慮固定複雜度的LLL-deep演算法[2]用於TH預編碼器中。固定複雜度的LLL-deep演算法在原始基底做SQR排序和長度減縮。然而我們發現在排序TH預編碼器中，在對偶基底做SQR排序會比在原始基底做SQR排序有更好的效能。所以這篇論文提出一個修改版本的固定複雜度LLL-deep演算法，是在對偶基底做SQR排序和長度減縮，不同於原本固定複雜度LLL-deep在原始基底做SQR排序和長度減縮。另外，因為複雜度是固定的，所以適合做硬體的實現。模擬結果顯示在固定複雜度的晶格縮減方法中，對於固定數目的長度減縮次數，本篇論文提出的方法和固定複雜度的LLL[3]演算法以及固定複雜度的LLL-deep演算法比起來會有較低的錯誤率。LLL[4]的平均長度減縮次數為21，在4-正交幅度調變、四發四收通道中，固定長度減縮次數為24，效能能逼近LLL。在16-正交幅度調變、四發四收通道中，固定長度減縮次數為18，效能能逼近LLL。而且有幾乎和LLL演算法一樣的錯誤率和多樣性秩序。

In this thesis, a fixed complexity lattice reduction algorithm with the Tomlinson-Harashima precoding (THP) [1] is proposed for MIMO broadcast channels. Fixed complexity LLL algorithm in deep insertion (LLL-deep) [2] do sorted-QR (SQR) ordering and size reduction on the primal basis. However, we observe that doing SQR ordering on the dual basis is better than doing SQR ordering on the primal basis in ordering THP. So we proposed a modified version of fixed complexity LLL-deep algorithm that do SQR ordering and size reduction on the dual basis, which is different from fixed complexity LLL-deep that do SQR ordering and size reduction on the primal basis. Also, the complexity of proposed algorithm is fixed, which is more sutiable for hardware implementation. Simulation results show that for a fixed number of reduction stages, the performance of proposed lattice reduction algorithm is better than the fixed LLL algorithm [3] and fixed complexity LLL-deep algorithm in THP, and has nearly the same performance and diversity order as LLL [4].

[1] R. Fischer, C. Windpassinger, A. Lampe, and J. Huber, "Mimo precoding for decentralized receivers," in IEEE International Symposium on Information Theory, 2002. Proceedings. 2002, p. 496.
[2] C. Ling and W. H. Mow, "A unified view of sorting in lattice reduction: From v-blast to lll and beyond," in IEEE ITW 2009. Information Theory Workshop, 2009., oct. 2009, pp. 529-533.
[3] H. Vetter, V. Ponnampalam, M. Sandell, and P. Hoeher, "Fixed complexity lll algorithm," IEEE Transactions on Signal Processing, vol. 57, no. 4, pp. 1634-1637, april 2009.
[4] A. K. Lenstra, H. W. L. Jr., and L. Lovasz, "Factoring polynomials with rational coefficients," Mathematische Annalen, vol. 261, pp. 515 - 534, 1982.
[5] E. Telatar, "Capacity of multi-antenna gaussian channels," European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, 1999.
[6] G. J. Foschini, "Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas," Bell Labs Technical Journal, vol. 1, no. 2, pp. 41-59, 1996.
[7] G. Foschini, G. Golden, R. Valenzuela, and P. Wolniansky, "Simplified processing for high spectral efficiency wireless communication employing multi-element arrays," IEEE Journal on Selected Areas in Communications, vol. 17, no. 11, pp. 1841-1852, nov 1999.
[8] H. Harashima and H. Miyakawa, "Matched-transmission technique for channels with intersymbol interference," IEEE Transactions on Communications, vol. 20, no. 4, pp. 774-780, aug 1972.
[9] M. Tomlinson, "New automatic equaliser employing modulo arithmetic," IEEE Electronics Letters, vol. 7, no. 5, pp. 138-139, march 1971.
[10] A. L. Robert F.H. Fischer, Christoph Windpassinger and J. B. Huber, "Space-time transmission using tomlinson-harashima precoding," in In Proc. 4. ITG Conference on Source and Channel Coding, Jan 2002, pp. 139-147.
[11] G. Ginis and J. Cioffi, "A multi-user precoding scheme achieving crosstalk cancellation with application to dsl systems," in IEEE Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, 2000., vol. 2, 2000, pp. 1627-1631 vol.2.
[12] H. Yao and G. Wornell, "Lattice-reduction-aided detectors for mimo communication systems," in Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE, vol. 1, 17-21 2002, pp. 424-428 vol.1.
[13] M. Taherzadeh, A. Mobasher, and A. Khandani, "Communication over mimo broadcast channels using lattice-basis reduction," IEEE Transactions on Information Theory, vol. 53, no. 12, pp. 4567-4582, dec. 2007.
[14] C. Windpassinger and R. Fischer, "Low-complexity near-maximum-likelihood detection and precoding for mimo systems using lattice reduction," in Proceedings. IEEE Information Theory Workshop, 2003., 31 2003, pp. 345-348.
[15] C. Stierstorfer and R. F. Fischer, "Lattice-reduction-aided tomlinson-harashima precoding for point-to-multipoint transmission," AEU - International Journal of Electronics and Communications, vol. 60, no. 4, pp. 328-330, 2006.
[16] G. Golden, C. Foschini, R. Valenzuela, and P. Wolniansky, "Detection algorithm and initial laboratory results using v-blast space-time communication architecture," IEEE Electronics Letters, vol. 35, no. 1, pp. 14-16, 7 1999.
[17] T. Vencel, C. Windpassinger, and R. F. H. Fischer, "Sorting in the vblast algorithm and loading," in In Proceedings of Communication Systems and Networks (CSN 2002), Sept. 2002, pp. 304-309.
[18] C. Windpassinger, T. Vencel, and R. Fischer, "Precoding and loading for blast-like systems," in IEEE International Conference on Communications, 2003. ICC '03., vol. 5, 11-15 2003, pp. 3061-3065 vol.5.
[19] C. Robert F.H. Fischer, Johannes B. Huber, "Precoding for point-to-multipoint transmission," in In Proc. IEEE Eight International Workshop on Signal Processing for Space Communications, September 2003, pp. 137-144.
[20] D.Wubben, R. Bohnke, V. Kuhn, and K.-D. Kammeyer, "Mmse-based lattice-reduction for near-ml detection of mimo systems," in ITG Workshop on Smart Antennas, 2004., 18-19 2004, pp. 106-113.
[21] D.Wubben, R. Bohnke, V. Kuhn, and K.-D. Kammeyer, "Near-maximum-likelihood detection of mimo systems using mmse-based lattice reduction," in IEEE International Conference on Communications, 2004, vol. 2, 20-24 2004, pp. 798-802 Vol.2.
[22] Y. H. Gan and W. H. Mow, "Complex lattice reduction algorithms for low-complexity mimo detection," in IEEE Global Telecommunications Conference, 2005. GLOBECOM '05., vol. 5, 2-2 2005, pp. 5 pp. 2957.
[23] M. Sandell, A. Lillie, D. McNamara, V. Ponnampalam, and D. Milford, "Complexity study of lattice reduction for mimo detection," in IEEE Wireless Communications and Networking Conference, 2007.WCNC 2007., 11-15 2007, pp. 1088-092.
[24] X. Ma and W. Zhang, "Performance analysis for mimo systems with lattice-reduction aided linear equalization," IEEE Transactions on Communications, vol. 56, no. 2, pp. 309-318, feb. 2008.
[25] C. P. Schnorr and M. Euchner, "Lattice basis reduction: Improved practical algorithms and solving subset sum problems," Mathematical Programming, vol. 66, pp. 181-199, 1994, 10.1007/BF01581144.
[26] C. Ling, W. H. Mow, and L. Gan, "Dual-lattice ordering and partial lattice reduction for sic-based mimo detection," IEEE Journal of Selected Topics in Signal Processing, vol. 3, no. 6, pp. 975-985, dec. 2009.