Transformation between distributions in time–frequency analysis: Difference between revisions

Revision as of 13:19, 16 August 2010

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==General class==

If we use the variable ''ω''=2πf2''πf'', then, borrowing the notations used in the field of quantum mechanics, we can show that time-frequency representation, such as [[Wigner distribution function]] (WDF) and other [[quadraticbilinear TFDstime-frequency distribution]]s, can be expressed as

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*(u-\dfrac{1}{2}\tau)s(u+\dfrac{1}{2}\tau)\phi(\theta,\tau)e^{-j\theta t-j\tau\omega+j\theta u}\, du\,d\tau\,d\theta ,</math> (1)

where <math>\phi(\theta,\tau)</math> is a two dimensional function called the kernel, which determines the distribution and its properties (for a signal processing terminology and treatment of this question, the reader is referred to the references already cited in the introduction).