Statistical distributions of Hermitian quadratic forms in complex Gaussian variables

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. The purpose is to present such a complete description since only some special cases have been treated in the past. The method employed 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 obtained for some special cases. Relations of the quadratic form and its cdf to the noncentral χ² (chi-square) and the complex noncentral Wishart distributions are exposed. Evaluating of the cdf at the origin is shown to reduce to the doubly noncentral F-distribution due to Price. A generalization of the Marcum Q-function is also identified and suggested.