# Taking inverse Fourier transform of complicated multipart equation

Define

$\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} \right)\omega } d\tau '} \right]$

$\Lambda (\tau ,\omega ) = \frac{{\beta (\tau ,\omega ) + {\sigma ^2}}}{{{{\left( {\beta (\tau ,\omega )} \right)}^2} + {\sigma ^2}}}$

$\beta (\tau ,\omega ) = \exp \left[ { - \int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{{ - 1}}{{\pi Q(\tau ')}}}}} d\tau '} \right]$

In the above, $\tau$ is the time, and $\omega$ is the frequency, and $Q(\tau)$ is another function of time $\tau$.

The equations are for a filter applied to seismic data. The equation can be found on pg. 128 of this monograph.

I would like to take the inverse continuous Fourier transform of $\tilde U(\tau ,\omega )$ so that I get a function $\tilde U(\tau ,\tau )$ which is expressed only in terms of $\tau$.

This gets rid of $\omega$ so that the expression is only in the time domain.

I have tried using both a CAS and numerous attempts on paper, but I am uncertain as to whether this can be done using continuous mathematics. I am inclined to believe that it would be simpler to approximate $\tilde U(\tau ,\tau )$ using numerical methods. How to approach this problem?

-
just out of curiosity, do these equations represent anything such as a physical system? – Navin Oct 29 '12 at 0:40
Yes, the equations are a signal processing filter applied in the spectral domain (STFT spectrogram). However, most filter kernel equations are generated in the time domain (and then taken into the frequency domain via the FFT) due to circular convolution effects associated with direct multiplication in the frequency domain. I will update my question above to add a little more information. – Nicholas Kinar Oct 29 '12 at 0:49