On the distribution of positive-definite Gaussian quadratic forms

Quadratic signal processing is used in detection and estimation of random signals. To describe the performance of quadratic signal processing, the probability distribution of the output of the processor is needed. Only positive-definite Gaussian quadratic forms are considered. The quadratic form is diagonalized in terms of independent Gaussian variables and its mean, moment-generating function, and cumulants are computed; conditions are given for the quadratic form to be
\chi^{2}
distributed and distributed like a sum of independent random variables having a Gamma distribution. A new method is proposed to approximate its probability distribution using an expansion in Laguerre polynomials for the central case and in generalized
\chi^{2}
distributions in the noncentral case. The series coefficients and bounds on truncation error are evaluated. Some applications in average power and power spectrum estimation and in detection illustrate our method.