Time-frequency analysis: theory and applications
Time-frequency analysis: theory and applications
Generalization of the Gram-Charlier/Edgeworth Series andApplication to Time-Frequency Analysis
Multidimensional Systems and Signal Processing - Special issue on recent developments in time-frequency analysis
IEEE Transactions on Signal Processing
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We present a simple unified approach for the transformation of probability densities. The approach relates any two probability densities by an operator transformation using the standard differentiation operator. We obtain a generalization of the Gram-Charlier and Edgeworth series. We further generalize the approach by showing that the differentiation operator can be replaced by an arbitrary Hermitian operator. As a further special case we obtain the operator transformation that scales the mean and standard deviation of a probability density. Furthermore, we show how the unusual formalism of quantum mechanics for constructing densities can be related to standard probability theory. We do so by generalising the theorem of Khinchin regarding characteristic functions.