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

Revision as of 02:14, 11 August 2015

Revision as of 18:58, 3 September 2015

Line 10:

If we use the variable ''ω''=2''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 [[bilinear time–frequency distribution]]s, can be expressed as

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

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*\left(u-\dfrac{1}{2}\tau\right)s\left(u+\dfrac{1}{2}\tau\right)\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).

For theThe kernel of the [[Wigner distribution function]] (WDF) is one. However, no particular significance should be attached to that, since it is possible to write the general form so that the kernel of any distribution is one, in which case the kernel of the [[Wigner distribution function]] (WDF) would be something else.

==Characteristic function formulation==