Multiuser Detection and Statistical Mechanics

  • Dongning Guo
  • Sergio Verdú
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 712)


A framework for analyzing multiuser detectors in the context of statistical mechanics is presented. A multiuser detector is shown to be equivalent to a conditional mean estimator which finds the mean value of the stochastic output of a so-called Bayes retrochannel. The Bayes retrochannel is equivalent to a spin glass in the sense that the distribution of its stochastic output conditioned on the received signal is exactly the distribution of the spin glass at thermal equilibrium. In the large-system limit, the performance of the multiuser detector finds its counterpart as a certain macroscopic property of the spin glass, which can be solved using powerful tools developed in statistical mechanics. In particular, the large-system uncoded bit-error-rate of the matched filter, the MMSE detector, the decorrelator and the optimal detectors is solved, as well as the spectral efficiency of the Gaussian CDMA channel. A universal interpretation of multiuser detection relates the multiuser efficiency to the mean-square error of the conditional mean estimator output in the many-user limit.


Statistical Mechanic Spin Glass Spectral Efficiency Network Security Matched Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    S. Verdú, Multiuser Detection. Cambridge University Press, 1998.zbMATHGoogle Scholar
  2. [2]
    D. Guo, L. K. Rasmussen, and T. J. Lim, “Linear parallel interference cancellation in long-code CDMA multiuser detection,” IEEE J. Selected Areas Commun., vol. 17, pp. 2074–2081, Dec. 1999.CrossRefGoogle Scholar
  3. [3]
    A. J. Grant and P. D. Alexander, “Random sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 44, pp. 2832–2836, Nov. 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: effective interference, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, pp. 622–640, March 1999.MathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Verdú and S. Shamai, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inform. Theory, vol. 45, pp. 622–640, March 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    D. N. C. Tse and S. Verdu, “Optimum asymptotic multiuser efficiency for randomly spread CDMA,” IEEE Trans. Inform. Theory, vol. 46, pp. 2718–2722, Nov. 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    R. R. Müller, “Multiuser receivers for randomly spread signals: fundamental limits with and without decision-feedback,” IEEE Trans. Inform. Theory, vol. 47, pp. 268–283, Jan. 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    P. Viswanath, D. N. C. Tse, and V. Anantharam, “Asymptotically optimal water-filling in vector multiple-access channels,” IEEE Trans. Inform. Theory, vol. 47, pp. 241–267, Jan. 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. Zhang, E. K. P. Chong, and D. N. C. Tse, “Output MAI distribution of linear MMSE multiuser receivers in DS-CDMA systems,” IEEE Trans. Inform. Theory, vol. 47, pp. 1128–1144, March 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    S. Shamai and S. Verdú, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inform. Theory, vol. 47, pp. 1302–1327, May 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    D. Guo, S. Verdú, and L. K. Rasmussen, “Asymptotic normality of linear CDMA multiuser detection outputs,” in Proceedings 2001 IEEE International Symposium on Information Theory, p. 307, Washington, D.C., June 2001.Google Scholar
  12. [12]
    L. Li, A. Tulino, and S. Verdú, “Asymptotic eigenvalue moments for linear multiuser detection,” Communications in Information and Systems, vol. 1, pp. 273–304, Sept. 2001.MathSciNetzbMATHGoogle Scholar
  13. [13]
    D. Guo, S. Verdú, and L. K. Rasmussen, “Asymptotic normality of linear multiuser receiver outputs,” accepted for IEEE Trans. on Inform. Theory, 2001.Google Scholar
  14. [14]
    U. Grenander and J. W. Silverstein, “Spectral analysis of networks with random topologies,” SIAM J. Appl. Math., vol. 32, pp. 499–519, March 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    D. Jonsson, “Some limit theorems for the eigenvalues of a sample covariance matrix,” Journal of Multivariate Analysis, pp. 1–38, Dec. 1982.Google Scholar
  16. [16]
    J. W. Silverstein and Z. D. Bai, “On the empirical distribution of eigenvalues of a class of large dimensional random matrices,” Journal of Multivariate Analysis, vol. 54, pp. 175–192, Aug. 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.zbMATHCrossRefGoogle Scholar
  18. [18]
    N. Sourlas, “Spin-glass models as error-correcting codes,” Nature, vol. 339, pp. 693–695, June 1989.CrossRefGoogle Scholar
  19. [19]
    N. Sourlas, “Spin-glasses, error-correcting codes and finitetemperature decoding,” Europhysics Letters, vol. 25, Jan. 1994.Google Scholar
  20. [20]
    A. Montanari and N. Sourlas, “The statistical mechanics of turbo codes,” Eur. Phys. J. B, vol. 18, no. 1, pp. 107–119, 2000.CrossRefGoogle Scholar
  21. [21]
    Y. Kabashima and D. Saad, “Statistical mechanics of errorcorrecting codes,” Europhysics Letters, vol. 45, no. 1, pp. 97–103, 1999.CrossRefGoogle Scholar
  22. [22]
    H. Nishimori and K. Y. M. Wong, “Statistical mechanics of image restoration and error-correcting codes,” Pysical Review E, vol. 60, pp. 132–144, July 1999.Google Scholar
  23. [23]
    K. Nakamura, Y. Kabashima, and D. Saad, “Statistical mechanics of low-density parity check error-correcting codes over Galois fields,” Europhysics Letters, vol. 56, pp. 610–616, Nov. 2001.CrossRefGoogle Scholar
  24. [24]
    H. Nishimori, Statistical Physics of Spin Glasses and Information Processing: An Introduction. Number 111 in International Series of Monographs on Physics, Oxford University Press, 2001.zbMATHCrossRefGoogle Scholar
  25. [25]
    G. I. Kechriotis and E. S. Manolakos, “Hopfield neural network implementation of the optimal CDMA multiuser detector,” IEEE Trans. Neural Networks, Jan. 1996.Google Scholar
  26. [26]
    V. Dotsenko, The Theory of Spin Glasses and Neural Networks. World Scientific, 1994.zbMATHGoogle Scholar
  27. [27]
    T. Tanaka, “Analysis of bit error probability of direct-sequence CDMA multiuser demodulators,” in Advances in Neural Information Processing Systems (T. K. Leen et al., ed.), vol. 13, pp. 315–321, The MIT Press, 2001.Google Scholar
  28. [28]
    T. Tanaka, “Average-case analysis of multiuser detectors,” in Proceedings 2001 IEEE International Symposium on Information Theory, p. 287, Washington, D.C., June 2001.Google Scholar
  29. [29]
    T. Tanaka, “Large-system analysis of randomly spread CDMA multiuser detectors,” preprint, 2001.Google Scholar
  30. [30]
    J.-P. Bouchaud and M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management. Cambridge University Press, 2000.Google Scholar
  31. [31]
    Y. C. Eldar and A. M. Chan, “On wishart matrix eigenvalues and eigenvectors and the asymptotic performance of the decorrelator,” IEEE Trans. Inform. Theory, 2001. submitted.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Dongning Guo
    • 1
  • Sergio Verdú
    • 1
  1. 1.Dept. of Electrical EngineeringPrinceton UniversityPrincetonUSA

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