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A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors

By Peter J. Smith and Hongsheng Gao


Consider the quadratic form Z=YH(XLXH)-1 Y where Y is a p-m complex Gaussian matrix, X is an independent p-n complex Gaussian matrix, L is a Hermitian positive definite matrix, and m[less-than-or-equals, slant]p[less-than-or-equals, slant]n. The distribution of Z has been studied for over 30 years due to its importance in certain multivariate statistics but no satisfactory numerical methods for computing this distribution appear to be available. Hence this paper deals with a representation for the density function of Z in terms of a ratio of determinants which is shown to be more amenable to numerical work than previous representations, at least for small values of p. Also for m=1 this work has applications in digital mobile radio for a specific channel where p antennas are used to receive a signal with n interferers. Some of these applications in radio communication systems are discussed.quadratic form, complex normal vector, hypergeometric functions, distributions

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