The Multivariate Gram-Charlier Series Applied to Random Signal Detection.
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This report comprises two parts. The first part focuses on extending the well known Gram Charlier series expansion technique for univariate probability density functions to multivariate functions. The approach employed here is particularly attractive because it avoids explicit use of tensor analysis and multivariate Hermite polynomials. The desired expansion is accomplished via straightforward application of Kronecker products and matrix calculus formulae. The second part of this report addresses the problem of signal detection in the presence of background noise. A canonical representation of the likelihood ratio is derived that applies to a broad class of multivariate random signals embedded in Gaussian noise. The representation takes the form of an infinite series whose terms depend on the received measurement vector and the signal and noise statistics. It prescribes a solution to the binary detection problem in a variety of sonar applications.
- Acoustic Detection and Detectors