Showing papers on "Complex normal distribution published in 1993"
TL;DR: The method used is based on inverting the characteristic function of the quadratic form by solving a number of convolution integrals, which results in two forms for the probability density function, an expression for the cumulative distribution function, and expressions for the distribution moments and cumulants.
Abstract: Decision variables in numerous practical systems can frequently be characterized using a Hermitian quadratic form in complex Gaussian variates. Performance analysis involving these variates requires a complete description of the statistical distribution of the quadratic form. Such a complete description is presented. The method used is based on inverting the characteristic function of the quadratic form by solving a number of convolution integrals. The results presented include two forms for the probability density function (PDF), an expression for the cumulative distribution function (CDF), and expressions for the distribution moments and cumulants. These results are shown to reduce to previously known results for some special cases. Relations of the quadratic form and its CDF to the noncentral chi /sup 2/ (chi-square) and the complex noncentral Wishart distributions are exposed. Evaluation of the CDF at the origin is shown to reduce to the doubly noncentral F-distribution due to R. Price (1962, 1964). A generalization of the Marcum Q-function is also identified and suggested. >
59 citations
10 citations
27 Apr 1993
TL;DR: A maximum likelihood estimator of spectral moments of a zero-mean complex Gaussian vector process, immersed in independent additive Gaussian white noise is proposed, which provides estimates consistent with probability one, and that are asymptotically efficient, which are the asymPTotic properties of the ML estimator.
Abstract: A maximum likelihood (ML) estimator of spectral moments of a zero-mean complex Gaussian vector process, immersed in independent additive Gaussian white noise is proposed. The covariance function is assumed known in advance, apart from a vector of parameters which are related with the spectral moments. Since the maximization of the log-likelihood function yields a highly cumbersome algorithm, a more manageable objective function is considered. This objective function provides estimates consistent with probability one, and that are asymptotically efficient, which are the asymptotic properties of the ML estimator. For finite sample sizes, and signal to noise ratio (SNR) tending to zero, the results are similar to the ML results. Statistical characterization and simulation examples are presented. >
5 citations
12 Jul 1993
TL;DR: In this article, the left and right eigenvectors of generalized characteristic value problems with non-Hermitian complex matrices are reviewed, and diagonal expansions of various matrix combinations and inverses are derived.
Abstract: : The properties of the left and right eigenvectors of generalized characteristic value problems with non-Hermitian complex matrices are reviewed, and diagonal expansions of various matrix combinations and inverses are derived. The interrelationships of the solutions of two related non-Hermitian characteristic value problems are determined and found to be simply related to the earlier eigenvectors and their properties. These results are then applied to the evaluation of the characteristic function of a general complex second-order form in correlated Gaussian random variables. The end result is a compact closed form result involving a finite product and a finite sum, along with one square root and exponential. This expression allows for rapid evaluation of the characteristic function for numerous values of its complex argument.... Characteristic function, Complex form, Second-order form, Correlated variables, Gaussian variables, Closed form.
1 citations
13 Sep 1993
TL;DR: The complex multivariate Gaussian and Wishart densities are often encountered in the analysis of statistical signal processing algorithms in frequency domain array processing as discussed by the authors, and the derivation of the probability density function of forms involving complex Gaussian random vectors and complex Wishart matrices may be simplified by the use of standard theorems found in multivariate statistics for real Gaussian matrix and real Wishart matrix.
Abstract: : The complex multivariate Gaussian and Wishart densities are often encountered in the analysis of statistical signal processing algorithms in frequency domain array processing. The derivation of the probability density function of forms involving complex Gaussian random vectors and complex Wishart matrices may be simplified by the use of standard theorems found in multivariate statistics for real Gaussian random vectors and real Wishart matrices. This memorandum contains the extensions to the complex case of a selection of these theorems.
1 citations
06 Sep 1993
TL;DR: Tight upper and lower bounds are obtained, which together provide very accurate estimates of the actual error probability, and some applications are presented to show their general applicability and to illustrate their tightness.
Abstract: The problem of efficiently evaluating the probability that the magnitude squared of one complex Gaussian random variable is less than the magnitude squared of another, possibly correlated, complex Gaussian random variable is addressed. This is known as an error probability of the Rician-type. In this paper, a bounding approach is taken. Tight upper and lower bounds are obtained, which together provide very accurate estimates of the actual error probability. Some applications of the bounds are presented to show their general applicability and to illustrate their tightness.