On the Distribution of Quadratic Expressions in Various Types of Random Vectors

Several approximations to the distribution of indefinite quadratic expressions in possibly singular Gaussian random vectors and ratios thereof are obtained in this dissertation. It is established that such quadratic expressions can be represented in their most general form as the difference of two positive definite quadratic forms plus a linear combination of Gaussian random variables. New advances on the distribution of quadratic expressions in elliptically contoured vectors, which are expressed as scalar mixtures of Gaussian vectors, are proposed as well. Certain distributional aspects of Hermitian quadratic expressions in complex Gaussian vectors are also investigated. Additionally, approximations to the distributions of quadratic forms in uniform, beta, exponential and gamma random variables as well as order statistics thereof are determined from their exact moments, for which explicit representations are derived. Closed form representations of the approximations to the density functions of the various types of quadratic expressions being considered herein are obtained by adjusting the base density functions associated with the quadratic forms appearing in the decompositions of the expressions by means of polynomials whose coefficients are determined from the moments of the target distributions. Quadratic forms being ubiquitous in Statistics, the proposed distributional results should prove eminently useful

On the distribution of indefinite quadratic forms in Gaussian random variables

In this work, we propose a transparent approach to evaluating the CDF of indefinite quadratic forms in Gaussian random variables and ratios of such forms. This quantity appears in the analysis of different receivers in communication systems and in various applications in signal processing. Instead of attempting to find the pdf of this quantity as is the case in many papers in literature, we focus on finding the CDF. The basic trick that we implement is to replace inequalities that appear in the CDF calculations with the unit step function and replace the latter with its Fourier transform. This produces a multi-dimensional integral that can be evaluated using complex integration. We show how our approach extends to nonzero mean Gaussian real/complex vectors and to the joint distribution of indefinite quadratic forms

What is the probability that a random integral quadratic form in nn variables has an integral zero?

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$.Comment: 17 pages. This article supercedes arXiv:1311.554

Outage Probability and Outage-Based Robust Beamforming for MIMO Interference Channels with Imperfect Channel State Information

In this paper, the outage probability and outage-based beam design for multiple-input multiple-output (MIMO) interference channels are considered. First, closed-form expressions for the outage probability in MIMO interference channels are derived under the assumption of Gaussian-distributed channel state information (CSI) error, and the asymptotic behavior of the outage probability as a function of several system parameters is examined by using the Chernoff bound. It is shown that the outage probability decreases exponentially with respect to the quality of CSI measured by the inverse of the mean square error of CSI. Second, based on the derived outage probability expressions, an iterative beam design algorithm for maximizing the sum outage rate is proposed. Numerical results show that the proposed beam design algorithm yields better sum outage rate performance than conventional algorithms such as interference alignment developed under the assumption of perfect CSI.Comment: 41 pages, 14 figures. accepted to IEEE Transactions on Wireless Communication

Detection and Estimation of Abrupt Changes contaminated by Multiplicative Gaussian Noise

The problem of abrupt change detection has received much attention in the literature. The Neyman Pearson detector can be derived and yields the well-known CUSUM algorithm, when the abrupt change is contaminated by an additive noise. However, a multiplicative noise has been observed in many signal processing applications. These applications include radar, sonar, communication and image processing. This paper addresses the problem of abrupt change detection in presence of multiplicative noise. The optimal Neyman Pearson detector is studied when the abrupt change and noise parameters are known. The parameters are unknown in most practical applications and have to be estimated. The maximum likelihood estimator is then derived for these parameters. The maximum likelihood estimator performance is determined, by comparing the estimate mean square errors with the Cramer Rao Bounds. The Neyman Pearson detector combined with the maximum likelihood estimator yields the generalized likelihood ratio detector