I'm reading a monograph where the forward and inverse Gabor transforms are being computed. (See pg. 122 of Seismic Inverse Q Filtering for further details). For a time $t$ in a time series $u(t)$, and a window time $\tau$ at the center of a window (a sub-array of the time series with $N$ elements), a Gabor window (with $N$ elements) is defined as the time-domain product:
$u(\tau ,t) = u(t)w(t - \tau )$
The window function is from my other post:
$ w(t) = \left\{ {\begin{array}{*{20}{c}} {\frac{2}{{T\sqrt \pi }}\exp \left[ { - {{\left( {\frac{{2t}}{T}} \right)}^2}} \right],{\rm{ for }} - T \le t \le T}\\ {0,{\rm{ otherwise}}} \end{array}} \right. $
The Gabor transform is defined as the Fourier Transform of $u(\tau,t)$. I think this could also be computed using the FFT:
$U(\tau ,\omega ) = {\rm{FFT}}({\rm{ }}u(\tau ,t){\rm{ }})$
Once the Gabor transform has been computed, the monograph shows how to compute the inverse Gabor transform. First, using the Inverse Fourier Transform (or IFFT):
${\rm{ }}u(\tau ,t) = {\rm{IFFT(}}U(\tau ,\omega ){\rm{)}}$
Now the monograph refers to a very curious "data synthesis integral" that is used to recompose $u(t)$, the sequence in the time domain:
$u(t) = h(t)\int\limits_{ - \infty }^\infty {u(\tau ,t)d} \tau $
The $h(t)$ function is given as the following:
$h(t) = {\left[ {\int\limits_{ - \infty }^\infty w (t - \tau )d\tau } \right]^{ - 1}}$
I am finding it challenging to understand what is meant by the "data synthesis integral", and how the integral allows for the re-computation of the original time series $u(t)$.
How do I numerically compute this integral (listed below)?
$\int\limits_{ - \infty }^\infty {u(\tau ,t)d} \tau $
Note that the operation of the inverse Gabor transform must be able to reverse the operation of the forward Gabor transform. I can't understand why the integration is occurring over $\tau$, which should be the window center.