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

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  • Smith, Peter J.
  • Gao, Hongsheng

Abstract

Consider the quadratic form Z=YH(XLXH)-1Â 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.

Suggested Citation

  • Smith, Peter J. & Gao, Hongsheng, 2000. "A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 41-54, April.
  • Handle: RePEc:eee:jmvana:v:73:y:2000:i:1:p:41-54
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    References listed on IDEAS

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    1. Muirhead, Robb J., 1975. "Expressions for some hypergeometric functions of matrix argument with applications," Journal of Multivariate Analysis, Elsevier, vol. 5(3), pages 283-293, September.
    2. Mara L. McLaren, 1976. "Coefficients of the Zonal Polynomials," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 25(1), pages 82-87, March.
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    Cited by:

    1. Smith, Peter J. & Garth, Lee M., 2007. "Distribution and characteristic functions for correlated complex Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 661-677, April.

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