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
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In this paper, we present a general formulation that unifies the probabilistic characterization of Hermitian random matrices with a specific structure. Based on a general expression for the joint pdf of the ordered eigenvalues, we obtain i) the joint cdf; ii) the marginal cdfs; and iii) the marginal pdfs of the ordered eigenvalues, where ii) and iii) follow as simple particularizations of i). Our formulation is shown to include the distribution of some common MIMO channel models such as the uncorrelated, semicorrelated, and double-correlated Rayleigh MIMO fading channel and the uncorrelated Rician MIMO fading channel, although it is not restricted only to these. Hence, the proposed formulation and derived results provide a solid framework for the simultaneous analytical performance analysis of MIMO systems under different channel models. As an illustrative application, we obtain the exact outage probability of a spatial multiplexing MIMO system transmitting through the strongest channel eigenmodes. This paper characterizes the eigenvalue distributions of full-rank Hermitian matrices generated from a set of independent (non)zero-mean proper complex Gaussian random vectors with a scaled-identity covariance matrix. More specifically, the joint and marginal cumulative distribution function (CDF) of any subset of unordered eigenvalues of the so-called complex (non)central Wishart matrices, as well as new simple and tractable expressions for their joint probability density function (PDF), are derived in terms of a finite sum of determinants. As corollaries to these new results, explicit expressions for the statistics of the smallest and largest eigenvalues, of (non)central Wishart matrices, can be easily obtained. Moreover, capitalizing on the foregoing distributions, it becomes possible to evaluate exactly the mean, variance, and other higher order statistics such as the skewness and kurtosis of the random channel capacity, in the case of uncorrelated multiple-input multiple-output (MIMO) Ricean and Rayleigh fading channels. Doing so bridges the gap between Telatar's initial approach for evaluating the average MIMO channel capacity (Telatar, 1999), and the subsequently widely adopted moment generating function (MGF) approach, thereby setting the basis for a PDF-based framework for characterizing the capacity statistics of MIMO Ricean and Rayleigh fading channels. This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type @math , where the rank of @math stays bounded as the dimension goes to infinity and where the matrix @math is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if @math has some eigenvalues out of the maximal circle of the single ring, then @math has some eigenvalues (called outliers) in the neighborhood of those of @math , which is not the case for the eigenvalues of @math in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of @math around the eigenvalues of @math and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of @math ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new. Random matrices play a crucial role in the design and analysis of multiple-input multiple-output (MIMO) systems. In particular, performance of MIMO systems depends on the statistical properties of a subclass of random matrices known as Wishart when the propagation environment is characterized by Rayleigh or Rician fading. This paper focuses on the stochastic analysis of this class of matrices and proposes a general methodology to evaluate some multiple nested integrals of interest. With this methodology we obtain a closed-form expression for the joint probability density function of k consecutive ordered eigenvalues and, as a special case, the PDF of the lscrth ordered eigenvalue of Wishart matrices. The distribution of the largest eigenvalue can be used to analyze the performance of MIMO maximal ratio combining systems. The PDF of the smallest eigenvalue can be used for MIMO antenna selection techniques. Finally, the PDF the kth largest eigenvalue finds applications in the performance analysis of MIMO singular value decomposition systems.