Complex Wishart Matrix Research Articles

Eigenvalue tests and distributions for small sample order determination for complex Wishart matrices

This is a theoretical thesis. The goal is to determine how many signal sources exist in the medium when constrained to using only a few samples. The need to make decisions based on only a few samples is motivated by the slow sound propagation speed and the time urgency to make decisions. This research treats the problem from the point of view of classical hypothesis testing assuming complex multivariate Gaussian random variables. This is the small sample complex principal components analysis problem. The critical issue is the derivation of probability density functions of appropriate test statistics. The goal has been partially achieved. The probability density functions for several important distributions have been derived. In particular, these include the distribution for the set of eigenvalues satisfying the generalized eigenvalue problem of two complex Wishart matrices, the matrix complex Gaussian distribution, a joint distribution needed to derive the density for the sphericity test statistic, the density function for the ratio of averages of disjoint sums of sequential eigenvalues of a complex Wishart matrix, and several tests based on the ratio of an arbitrary eigenvalue to the maximum, minimum, average, or sum of all the eigenvalues for a special case of the complex Wishart matrix. This thesis includes a derivation completely in the context of complex variables of the density function of the complex Wishart distribution and the distribution of its eigenvalues. It also includes a few minor results regarding zonal polynomials of complex matrix argument. A comprehensive development of the tools of statistics of complex variables for engineers and physicists is provided. This includes a study of complex matrix derivatives, changes of complex variables, and properties of the characteristic function of a complex multivariate random variable. A derivation of the complex Hotelling's T$\sp2$ test statistic and distribution useful for tests on means is given. A tutorial on Kiefer and Wolfowitz' application of the Lebesgue-Radon-Nikodym theorem for the estimation approach is provided.

An error bound for a noise canceller

The degradation in performance of the least-mean-square estimation (LMSE) is evaluated after replacing the covariance matrices with estimates. Taking advantage of the statistical properties of the complex Wishart matrix and its inverse, the decrease in performance stemming from this substitution is investigated. The performance criterion is based on the computation of the final output error, which includes covariance deviations. This is carried out for a certain class of linear observation models, involving an underlying problem of noise cancelling using noise alone references (NARs), namely, array processing techniques using a known array manifold. Performance limits of the adaptive noise canceller are investigated for the case of a spectral filter to illustrate the ideas. It is shown that its performance depends on the quadratic coherency between the corrupted signal and the NAR. It is preferable to run no processing when this coherency is not strong enough. A rule is proposed in order to decide when to switch to the no-processing scheme. Contrary to some previous results, the results given are available without any approximations, provided the statistical properties are satisfied. In the case of a special noise canceller, these properties are asymptotically satisfied. In the case of a transversal noise canceller acting in the time domain, these results are valid for white signals and noises.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ratios of Off-Diagonal C-Wishart

In this paper, the density function for the real and imaginary parts of the quotient of some off-diagonal elements of the three-dimensional complex Wishart matrix is derived. Connection to closed-loop transfer function determination is shown.

Some distributions of the latent roots of a complex wishart matrix variate

Some distributions of the latent roots of a complex wishart matrix variate

On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.

On the distribution of the latent roots of a complex wishart matrix (non-central case)

On the distribution of the latent roots of a complex wishart matrix (non-central case)

An approximation to the distribution of the largest root of a complex Wishart matrix

An approximation to the distribution of the largest root of a complex Wishart matrix

A Note on the Bivariate Chi Distribution

A Note on the Bivariate Chi Distribution