在無線通訊系統下，分集結合技術對於降低多路徑衰落的效應而言是不可或缺的，並且分集技術被期望為可以增加系統的效能。然而在相關性通道分支的情況下，分集技術的作用力會被降低。在空間分集技術的應用上，若天線沒有被間隔至少同調頻寬以上，通道之間產生相關性的現象是時常發生的。對於無線通訊系統效能的估測而言，二階統計特性是一個重要的指標。我們所提出的方法是利用Karhunen-Loeve 展開式，將原本具有相關性的隨機變數轉換為獨立的隨機變數，則這些獨立的隨機變數可以接著以分集結合技術處理，並期望觀察通道之統計特性。

For wireless communication systems, diversity combining techniques are indispensable for reducing the multipath fading effects, and it is expected to enhance the system performance. However, diversity techniques are diminished over correlated channel branches, which are commonly occurred in space diversity if the antennas are not separated at least coherence bandwidth. Second-order statistics are crucial performance criterion to evaluate the diversity effects of fading channels, whereas the performance in a correlated diversity system must be worse than that on independent multipath fading channels. A method which is to transform the correlated diversity branches into uncorrelated ones by the Karhunen-Loeve expansion, and then the transformed independent random variables are fed into the diversity combiner, and is expected to study the statistics of the fading channels.

[1] S. Haykin, Communication Systems, 4th edition, John Wiley & Sons, 2001.
[2] A. Goldsmith, Wireless Communications, Stanford University Press, 2003.
[3] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, 2nd edition, John Wiley & Sons, 2005.
[4] J. G. Proakis, Digital Communications, 4th edition, McGraw-Hill, 2001.
[5] D. G. Brennan, “Linear diversity combining techniques,” Proc. IRE, vol.47, pp. 1075-1101, Jun. 1959.
[6] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, Jul. 1999.
[7] W. C. Jakes, Jr., Ed., Microwave Mobile Communications. New York: Wiley, 1974.
[8] M. D. Yacoub, J. E. Vargas B. and L. G. de R. Guedes, “On higher order statistics of the Nakagami- distribution,” IEEE Trans. Veh. Technol., vol. 48, pp. 790-794, May 1999.
[9] M. Nakagami, “The -distribution – a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, Ed. Elmsford, NY: Pergamon, 1960.
[10] M. D. Yacoub, C. R. C. M. da Silva and J. E. Vargas B., “Second-order statistics for equal gain and maximal ratio diversity-combining reception,” Electron. Lett., vol. 36, no. 4, pp. 382-384, Feb. 2000.
[11] H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 3rd edition, Prentice Hall, 2002.
[12] A. L.-Garcia, Probability and Random Processes for Electrical Engineering, 2nd edition, Addison Wesley, 1994.
[13] M. D. Yacoub, C. R. C. M. da Silva and J. E. Vargas B., “Second-order statistics for diversity-combining techniques in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, Nov. 2001.
[14] C.-D. Iskander and P. T. Mathiopoulos, “Analytical level crossing rates and average fad durations for diversity techniques in Nakagami fading channels,” IEEE Trans. Commun., vol. 50, no. 8, Aug. 2002.
[15] M. D. Yacoub, C. R. C. M. da Silva and J. E. Vargas B., “Level crossing rate and average fade duration for pure selection and threshold selection diversity-combining systems,” Intl. J. on Commmun. Systems, vol. 14, pp. 897-907, Dec. 2001.
[16] X. Dong and N. C. Beaulieu, “Average level crossing rate and average fade duration of selection diversity,” IEEE Commun. Lett., vol. 5, pp. 396-398, Oct. 2001.
[17] M. Z. Win and J. H. Winters, “Analysis of hybrid selection/maximal-ratio combining in Rayleigh fading,” IEEE Trans. Commun., vol. 47, pp. 1773-1776, Dec. 1999.
[18] L. Yang and M.-S. Alouini, “An exact analysis of the impact of fading correlation on the average level crossing rate and average outage duration of selection combining,” Proc. Veh. Technol. Conf. (VTC2003), vol. 1, pp. 241-245, Apr. 2003.
[19] K. Zhang, Z. Song and Y. L. Guan, “Cholesky decomposition model for correlated mrc diversity systems in Nakagami fading channels,” Proc. Veh. Technol. Conf. (VTC2002), vol. 3, pp. 1515-1519, Sep. 2002.
[20] J. C. S. S. Filho, G. Fraidenraich and M. D. Yacoub, “Exact crossing rates of dual diversity over unbalanced correlated Rayleigh channels,” IEEE Commun. Lett., vol. 10, no. 1, Jan. 2006.
[21] D. Li and V. K. Prabhu, “Average level crossing rates and average fade durations for maximal-ratio combining in correlated Nakagami channels,” Proc. Wireless Commun. Network Conf. (WCNC2004), vol. 1, pp. 339-344, Mar. 2004.
[22] Q. T. Zhang, “Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1141-1150, Jul. 1999.
[23] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun., vol. 43, no. 8, Aug. 1995.
[24] G. K. Karagiannidis, D. A. Zogas and S. A. Kotsopoulos, “On the multivariate Nakagami-m distribution with exponential correlation,” IEEE Trans. Commun., vol. 51, no. 8, Aug. 2003.
[25] R. K. Mallik, “The uniform correlation matrix and its application to diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 5, May 2007.
[26] Q. T. Zhang, “Efficient generation of correlated Nakagami fading channels with arbitrary fading parameter,” Proc. Int. Conf. Commun. (ICC2002), no. 1, pp. 1358-1362, Apr. 2002.
[27] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd Ed., Springer, 1994.
[28] Q. T. Zhang, “A decomposition technique for efficient generation of correlated Nakagami fading channels,” IEEE Journ. Sel. Area. Commun., vol. 18, no. 11, Nov. 2000.