, Volume 4, Issue 1–4, pp 303–312 | Cite as

Computational complexity of optimum multiuser detection

  • Sergio Verdú


Optimum centralized demodulation of the independent data streams transmitted simultaneously by several users through a Code Division Multiple-Access channel is considered. Each user sends an arbitrary assigned signal waveform, which is linearly modulated by symbols drawn from a finite alphabet. If the users are asynchronous, the optimum multiuser detector can be implemented by a Viterbi algorithm whose time-complexity is linear in the number of symbols transmitted by each user and exponential in the number of users. It is shown that the combinatorial problem of selecting the most likely transmitted data stream given the sufficient statistics (sequence of matched filter outputs), and the signal energies and cross-correlations is nondeterministic polynomial-time hard (NP-hard) in the number of users. And it remains so even if the users are restricted to be symbol-synchronous.

The performance analysis of optimum multiuser detection in terms of the set of multiuser asymptotic efficiencies is equivalent to the computation of the minimum Euclidean distance between any pair of distinct multiuser signals. This problem is also shown to be NP-hard and a conjecture on a longstanding open problem in single user data communication theory is presented.

Key words

NP-complete Hypothesis testing Code Division Multiple Access Gaussian communication channels Maximum-likelihood sequence detection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Verdú, Minimum Probability of Error for Asynchronous Gaussian Multiple-Access Channels,IEEE Trans. Inform. Theory,32, pp. 85–96 (January 1986).MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. D. Forney, The Viterbi Algorithm,Proc. IEEE,61, pp. 268–278 (March 1973).CrossRefMathSciNetGoogle Scholar
  3. [3]
    K. S. Schneider, Optimum Detection of Code Division Multiplexed Signals,IEEE Trans. Aerospace Electron. Systems,15, pp. 181–185 (January 1979).CrossRefGoogle Scholar
  4. [4]
    M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco (1979).MATHGoogle Scholar
  5. [5]
    S. Verdú, Optimum Multi-user Signal Detection, Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. Report T-151, Coordinated Science Laboratory, Urbana, IL (August 1984).Google Scholar
  6. [6]
    S. Verdú, Optimum Multiuser Asymptotic Efficiency,IEEE Trans. Comm.,34, pp. 890–897 (September 1986).MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    C. H. Papadimitrou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ (1982).Google Scholar
  8. [8]
    S. Verdú, Multiple-Access Channels with Point-Process Observations: Optimum Demodulation,IEEE Trans. Inform. Theory,32, pp. 642–651 (September 1986).MATHCrossRefGoogle Scholar
  9. [9]
    M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt,Spread Spectrum Communications, Vol. 3, Computer Science Press, Rockville, MD (1985).MATHGoogle Scholar
  10. [10]
    R. Lupas and S. Verdú, Linear Multiuser Detectors for Synchronous Code-Division Multiple-Access Channels,IEEE Trans. Inform. Theory,35 (January 1989).Google Scholar
  11. [11]
    A. J. Viterbi and J. K. Omura,Principles of Digital Communication and Coding, McGraw-Hill, New York (1979).MATHGoogle Scholar
  12. [12]
    G. J. Foschini, A Reduced State Variant of Maximum Likelihood Sequence Detection Attaining Optimum Performance for High Signal-to-Noise Ratios,IEEE Trans. Inform. Theory,23, pp. 605–609 (September 1977).CrossRefMathSciNetGoogle Scholar
  13. [13]
    A. P. Clark,Advanced Data-Transmission Systems, Halsted Press, New York (1977).MATHGoogle Scholar
  14. [14]
    D. Kazakos, Computational Savings and Implementation of Maximum Likelihood Detectors,IEEE Trans. Inform. Theory,24, pp. 124–126 (January 1978).MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • Sergio Verdú
    • 1
  1. 1.Department of Electrical EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations