An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

  • Z. Zhang
  • S. W. Cheung
  • T. I. Yuk


A close-form expression for the exact Pair-wise Error Probability (PEP) of Space-Time (S-T) codes in Rayleigh fading channel is derived using the general and close-form solution for the probability-density function (PDF) of a sum of independent exponential distributed random variables. The expression requires evaluating the coefficients for partial fraction expansion, so an easy analytical way is proposed for doing this. The exact PEP is subsequently used to develop a simple PEP using the upper bound. Both PEPs are used in the Union bound for error rate evaluation. Numerical calculations and Monte Carlo computer simulation are used to study the accuracies of these Union bounds for error rate evaluation of a rotation-based diagonal S-T code (D code) in Rayleigh fading channels. Four other PEPs based on different bounds, i.e., the Chernoff bound, the asymptotic bound, the tight asymptotic bound, and the Eigen-Geometric-Mean (EGM) bound, are also studied for comparison. Results show that our derived close-form PEP is an exact PEP and our proposed PEP is a very tight bound to the exact PEP.


Pair-wise error probability Rayleigh fading channel Space-time codes 


  1. 1.
    V. Tarokh, N. Seshadri and A. R. Calderbank, Space–time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Transactions on Information Theory, Vol. 44, No. 3, pp. 744–765, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Benedetto and E. Biglieri, Principles of Digital Transmission with wireless applications, Kluwer, New York, 1999.zbMATHGoogle Scholar
  3. 3.
    G. Taricco and E. Biglieri, Exact pairwise error probability of space-time codes, IEEE Transactions on Information Theory, Vol. 48, No. 2, pp. 510–513, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Uysal and C. N. Georghiades, On the error performance analysis of space-time trellis codes, IEEE Transactions on Wireless Communications, Vol. 3, No. 4, pp. 1118–1123, 2004.CrossRefGoogle Scholar
  5. 5.
    H. F. Lu, Y. K. Wang, P. V. Kumar and K. M. Chugg, Remarks on space-time codes including a new lower bound and an improved code, IEEE Transactions on Information Theory, Vol. 49, No. 10, pp. 2752–2757, 2003.CrossRefMathSciNetGoogle Scholar
  6. 6.
    M. K. Simon and M. S. Alouini, Digital Communication Over Fading Channels, vol. 2nd, Wiley, New York, 2005.Google Scholar
  7. 7.
    M. P. Fitz, J. Grimm, and S. Siwamogsatham, A new view of performance analysis techniques in correlated Rayleigh fading, 1999 IEEE Wireless Communications and Networking Conference, Vol. 1, pp. 139–144, 1999.Google Scholar
  8. 8.
    M.-K. Byun and B. G. Lee, New bounds of pairwise error probability for space-time codes in Rayleigh fading channels, IEEE Transactions on Communications, Vol. 55, pp. 1484–1493, 2007. doi: 10.1109/TCOMM.2007.902532.CrossRefGoogle Scholar
  9. 9.
    B. Vucetic and J. H. Yuan, Space-time coding, Wiley, UK, 2003.CrossRefGoogle Scholar
  10. 10.
    J. G. Proakis, Digital Communications, vol. 4th, McGraw-Hill, Boston, USA, 2001.Google Scholar
  11. 11.
    H. V. Khuong and H. Y. Kong, General expression for pdf of a sum of independent exponential random variables, IEEE Communications Letters, Vol. 10, pp. 159–161, 2006. doi: 10.1109/LCOMM.2006.1603370.CrossRefGoogle Scholar
  12. 12.
    A. V. Oppenheim and A. S. Willsky, Signals and Systems, Prentice-Hall, Englewood Cliffs, 1983.zbMATHGoogle Scholar
  13. 13.
    T. Eng and L. B. Milstein, Coherent DS-CDMA Performance in Nakagami Multipath fading, IEEE Transactions on Communications, Vol. 43, No. 2, pp. 1134–1143, 1995.zbMATHCrossRefGoogle Scholar
  14. 14.
    J. W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations, in IEEE MILCOM 1991, Vol. 2, pp. 571–575, 1991.Google Scholar
  15. 15.
    S. G. Wilson, Digital Modulation and Coding, Prentice Hall, NJ, 1996.zbMATHGoogle Scholar
  16. 16.
    R. K. Mallik and Q. T. Zhang, A tight upper bound on the PEP of a Space-Time Coded system, IEEE Transactions on Wireless Communications, Vol. 6, pp. 3238–3247, 2007.CrossRefGoogle Scholar
  17. 17.
    H. Yao and G. Wornell, Achieving the full MIMO diversity-multiplexing frontier with rotation-based space-time codes, Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, 2003.Google Scholar
  18. 18.
    P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel, 1998 URSI International Symposium on Signals Systems and Electronics, ISSSE 98, pp. 295–300, 1998.Google Scholar
  19. 19.
    Z. Zhang, S. W. Cheung, T. I. Yuk, and H. Kuo, Union bounds for BER evaluation and code optimization for space-time codes in 2-by-2 MIMO systems, 2006 IEEE Spring Vehicular Technology Conference, Vol. 3, pp. 1506–1510, 2006.Google Scholar
  20. 20.
    Z. Zhang, X. G. Dai, S. W. Cheung, and T. I. Yuk, A common function of different union bounds for optimization and BER evaluation of space-time codes, Wireless Personal Communications, Vol. 46, pp. 493–506, 2008.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Electronic & Electrical EngineeringThe University of Hong KongHong KongHong Kong

Personalised recommendations