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

Revision as of 08:50, 22 January 2016

Revision as of 11:18, 23 January 2016

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To obtain that relationship suppose that there are two distributions, <math>C_1</math> and <math>C_2</math>, with corresponding kernels, <math>\phi_1</math> and <math>\phi_2</math>. Their characteristic functions are

: <math>M_1(\phi,\tau) = \phi_1(\theta,\tau)\int s^*\left(u-\dfractfrac{1\tau}{2}\tau\right)s\left(u+\dfractfrac{1\tau}{2}\tau\right)e^{j\theta u}\, du</math> (4)

: <math>M_2(\phi,\tau) = \phi_2(\theta,\tau)\int s^*\left(u-\dfractfrac{1\tau}{2}\tau\right)s\left(u+\dfractfrac{1\tau}{2}\tau\right)e^{j\theta u}\, du</math> (5)

Divide one equation by the other to obtain

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Now express <math>M_2</math> in terms of <math>C_2</math> to obtain

: <math>C_1(t,\omega) = \dfrac{1}{4\pi^2}\iiiint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}C_2(t,\omega^')e^{j\theta(t^'-t)+j\tau(\omega^'-\omega)}\, d\theta\,d\tau\,dt^'\,d\omega^'</math> (8)

This relationship can be written as

: <math>C_1(t,\omega) = \iint g_{12}(t^'-t,\omega'-\omega)C_2(t,\omega')\,dt^'\,d\omega'</math> (9)

with