Gabor transform

Article summary

The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula: : G_x(\tau,\omega) = \int_{-\infty}^\infty x(t)e^{-\pi(t-\tau)^2}e^{-j \omega t}\,dt The Gaussian function has infinite range and it is impractical for implementation. However, a level of significance can be chosen (for instance 0.00001) for the distribution of the Gaussian function. : \begin{cases} e^{-{\pi}a^2} \ge 0.00001; & \left| a \right| \le 1.9143 \\ e^{-{\pi}a^2} 1.9143 \end{cases} Outside these limits of integration ( \left| a \right| > 1.9143 ) the Gaussian function is small enough to be ignored. Thus the Gabor transform can be satisfactorily approximated as : G_x(\tau,\omega) = \int_{-1.9143+\tau}^{1.9143+\tau} x(t) e^{-\pi(t-\tau)^2} e^{-j \omega t} \, dt This simplification makes the Gabor transform practical and realizable.…

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