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

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==Relation of the spectrogram to other bilinear representations==

Now we specialize to the case where one transform from an arbitrary representation to the spectrogram. In Eq. (9), both <math>C_1</math> to be the spectrogram and <math>C_2</math> to be arbitrary are set. In addition, to simplify notation, <math>\phi_{SP} = \phi_1</math>, <math>\phi = \phi_2</math>, and <math>g_{SP} = g_{12}</math> are set and written as

: <math>C_{SP}(t,\omega) = \iint g_{SP} \left (t^'-t,\omega^'-\omega \right)C \left (t,\omega^' \right )\,dt^'\,d\omega^'</math> (11)

The kernel for the spectrogram with window, <math>h(t)</math>, is <math>A_h(-\theta,\tau)</math> and therefore

: <math>\begin{alignat}{3align}

g_{SP}(t,\omega) & = \dfrac{1}{4\pi^2}\iint \dfrac{A_h(-\theta,\tau)}{\phi(\theta,\tau)}e^{j\theta t+j\tau\omega}\, d\theta\,d\tau \\

& = \dfrac{1}{4\pi^2}\iiint \dfrac{1}{\phi(\theta,\tau)}h^*(u-\dfractfrac{1\tau}{2}\tau)h(u+\dfractfrac{1\tau}{2}\tau)e^{j\theta t+j\tau\omega-j\theta u}\, du\,d\tau\,d\theta \\

& = \dfrac{1}{4\pi^2}\iiint h^*(u-\dfractfrac{1\tau}{2}\tau)h(u+\dfractfrac{1\tau}{2}\tau)\dfrac{\phi(\theta,\tau)}{\phi(\theta,\tau)\phi(-\theta,\tau)}e^{-j\theta t+j\tau\omega+j\theta u}\, du\,d\tau\,d\theta \\

\end{alignatalign}</math> (12)

If takingwe theonly consider kernels for which <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>, <math>g_{SP}(t,\omega)</math>holds isthen just the distribution of the window function, except that it is evaluated at <math>-\omega</math>. Therefore,

: <math>g_{SP}(t,\omega) = \dfrac{1}{4\pi^2}\iiint h^*(u-\tfrac{\tau}{2})h(u+\tfrac{\tau}{2}) \phi(\theta,\tau) e^{-j\theta t+j\tau\omega+j\theta u}\, du\,d\tau\,d\theta = C_h(t,-\omega)</math> (13)

and therefore

for kernels that satisfy <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>

: <math>C_{SP}(t,\omega) = \iint C_s(t^',\omega^')C_h(t^'-t,\omega^'-\omega)\,dt^'\,d\omega^'</math> (14)

and

forThis kernelswas thatshown satisfyby Janssen[4]. When <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math> does not equal one, then

: <math>C_{SP}(t,\omega) = \iint C_s(t^',\omega^')C_h(t^'-t,\omega^'-\omega)\,dt^'\,d\omega^'</math> (14)

: <math>C_{SP}(t,\omega) = \iiiint G(t^{''},\omega^{''})C_s(t^',\omega^')C_h(t^{''}+t^'-t,-\omega^{''}+\omega-\omega^')\,dt^'\,dt^{''}\,d\omega^'\,d\omega^{''}</math> (15)

for kernels that satisfy <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>

This was shown by Janssen[4]. For the case where <math>\phi(-\theta,\tau)\phi(\theta,\tau)</math> does not equal one, then

: <math>C_{SP}(t,\omega) = \iiiint G(t^{''},\omega^{''})C_s(t^',\omega^')C_h(t^{''}+t^'-t,-\omega^{''}+\omega-\omega^')\,dt^'\,dt^{''}\,d\omega^\,d\omega^{''}</math> (15)

where

: <math>G(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{e^{-j\theta t-j\tau\omega}}{\phi(\theta,\tau)\phi(-\theta,\tau)}\, d\theta\,d\tau</math> (16)

==References==