Eigenvalue Dynamics of a Central Wishart Matrix with Application to MIMO Systems

F. Javier Lopez-Martinez, Eduardo Martos-Naya, Jose F. Paris, Andrea Goldsmith

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart matrix W(t) as it varies along a certain dimension t. We characterize the second-order joint cumulative distribution function (cdf) of the largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue of this matrix. We show that both cdfs can be expressed in exact closed-form in terms of a finite number of well-known special functions in the context of communication theory. As a direct application, we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the complex random matrix that defines the MIMO channel, we characterize the second-order joint cdf of the signal-to-noise ratio (SNR) for the best and worst channels. We use these results to study the rate of change of MIMO parallel channels, using different performance metrics. For a given value of the MIMO channel correlation coefficient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel. This different dynamic behavior is much more appreciable when the number of transmit ( NT) and receive (NR) antennas is similar. However, as NT is increased while keeping NR fixed, we see how the best and worst channels tend to have a similar rate of change.

Original languageEnglish (US)
Article number7055347
Pages (from-to)2693-2707
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number5
StatePublished - May 1 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Complex Wishart matrix
  • Cumulative Distribution Function
  • MIMO systems
  • Mutual Information
  • Outage probability
  • Random Matrices
  • Statistics


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