We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart matrix @math as it varies along a certain dimension @math . 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 ( @math ) and receive ( @math ) antennas is similar. However, as @math is increased while keeping @math fixed, we see how the best and worst channels tend to have a similar rate of change.
If we consider two samples of a stationary CW random process @math , namely @math and @math , the dynamics of @math are captured by the joint distribution of @math and @math . More precisely, the dynamics of the MIMO parallel channels (or ) can be studied separately by studying the joint distribution of the eigenvalues of @math and @math . Along these lines, the statistical analysis of CW matrices was tackled in @cite_12 @cite_37 , deriving the @math -dimensional joint pdf of the @math eigenvalues of a CW matrix and a perturbed version of it. Rather than the , we consider the of a particular eigenvalue as our metric to capture the dynamic behavior of a CW random process, since this distribution allows for the separate statistical characterization of all @math eigenvalues. Therefore, we will focus our attention on this set of second-order (or bivariate) distributions.
@cite_37 @cite_12
2098213935 2099298993
In this letter, the joint probability density function (PDF) for the eigenvalues of a complex Wishart matrix and a perturbed version of it are derived. The latter version can be used to model channel estimation errors and variations over time or frequency. As an example, the joint PDF is used to calculate the transition probabilities between modulation states in an adaptive MIMO system. This leads to a Markov model for the system. We then use the model to investigate the modulation state entering rates (MSER), the average stay duration (ASD), and the effects of feedback delay on the accuracy of modulation state selection in mobile radio systems. Other applications of this PDF are also discussed. Let A(t) be a complex Wishart process defined in terms of the MxN complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is @S. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t"1, t"2, where t"1