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.
This problem was addressed in @cite_10 when studying the mutual information distribution in orthogonal frequency division multiplexing (OFDM) systems operating under frequency-selective MIMO channels; specifically, a closed-form expression for the joint second-order pdf was given for arbitrarily-selected eigenvalues of the equivalent frequency-domain Wishart matrix. However, in order to obtain the joint bivariate cdf or the correlation coefficient for a particular eigenvalue, a two-fold numerical integration with infinite limits was required. In @cite_22 , an expression for this bivariate cdf was derived for the extreme eigenvalues (i.e. the largest and the smallest) in terms of the determinant of a matrix whose entries are expressed as infinite series of products of incomplete gamma functions; hence, its evaluation is highly impractical as the number of antennas is increased.
@cite_10 @cite_22
2132291816 2144100131 2132158614 2153637161
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 paper extends Khatri (1964, 1969) distribution of the largest eigenvalue of central complex Wishart matrices to the noncentral case. It then applies the resulting new statistical results to obtain closed-form expressions for the outage probability of multiple-input-multiple-output (MIMO) systems employing maximal ratio combining (known also as "beamforming" systems) and operating over Rician-fading channels. When applicable these expressions are compared with special cases previously reported in the literature dealing with the performance of (1) MIMO systems over Rayleigh-fading channels and (2) single-input-multiple-output (SIMO) systems over Rician-fading channels. As a double check these analytical results are validated by Monte Carlo simulations and as an illustration of the mathematical formalism some numerical examples for particular cases of interest are plotted and discussed. These results show that, given a fixed number of total antenna elements and under the same scattering condition (1) SIMO systems are equivalent to multiple-input-single-output systems and (2) it is preferable to distribute the number of antenna elements evenly between the transmitter and the receiver for a minimum outage probability performance. This communication considers the distribution of the mutual information of frequency-selective spatially uncorrelated Rayleigh fading multiple-input-multiple-output (MIMO) channels. Results are presented for orthogonal frequency-division multiplexing (OFDM)-based spatial multiplexing. New exact closed-form expressions are derived for the variance of the mutual information. In contrast to previous results, our new expressions apply for systems with both arbitrary numbers of antennas and arbitrary-length channels. Simplified expressions are also presented for high and low signal-to-noise ratio (SNR) regimes. The analytical variance results are used to provide accurate analytical approximations for the distribution of the mutual information, and the outage capacity.