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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.

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1 Answer 1

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The notation in this monograph is strange. After some investigation, I would suggest using another paper [1] to compute the Gabor transform, which is nothing more than a Short Time Fourier Transform (STFT) with a Gaussian window.

[1] J. B. Allen and L. R. Rabiner, “A unified approach to short-time Fourier analysis and synthesis,” Proceedings of the IEEE, vol. 65, no. 11, pp. 1558 – 1564, Nov. 1977.

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Did you implement the discrete Gabor transform by some programming language? i.e. Matlab? –  freak_warrior Nov 12 '13 at 8:36
    
Yes I did; you can look here for the inverse and forward transforms: dsp.stackexchange.com/questions/2757/… –  Nicholas Kinar Nov 13 '13 at 0:54
    
Ok thanks! I am trying to understand how the Windowed Fourier transform works, so I used Matlab to create an artificial signal, and plotted the Windowed Fourier transform using a gaussian window. ![plot][1] However, sometimes,(i.e. $\frac{1}{2}\leq{t}<\frac{3}{4}$, and $\frac{3}{4}\leq{t}\leq1$), we can see some form of bifurcating effect (vertically). The effect will gradually decrease and disappear, if we use a larger window. Do you know the reason? [1]: i.stack.imgur.com/DqM4z.png –  freak_warrior Nov 13 '13 at 1:16
    
This might be due to band-limiting of the input signal by the window size. I believe that the Allen and Rabiner paper discusses this for the OLA method. –  Nicholas Kinar Nov 13 '13 at 16:13

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