Gabor transform

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.^[1] 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:

${\displaystyle G_{x}(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)e^{-\pi (t-\tau )^{2}}e^{-j\omega t}\,dt}$

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.

${\displaystyle {\begin{cases}e^{-{\pi }a^{2}}\geq 0.00001;&\left|a\right|\leq 1.9143\\e^{-{\pi }a^{2}}<0.00001;&\left|a\right|>1.9143\end{cases}}}$

Outside these limits of integration (${\displaystyle \left|a\right|>1.9143}$) the Gaussian function is small enough to be ignored. Thus the Gabor transform can be satisfactorily approximated as

${\displaystyle G_{x}(\tau ,\omega )=\int _{-1.9143+\tau }^{1.9143+\tau }x(t)e^{-\pi (t-\tau )^{2}}e^{-j\omega t}\,dt}$

This simplification makes the Gabor transform practical and realizable.

The window function width can also be varied to optimize the time-frequency resolution tradeoff for a particular application by replacing the ${\displaystyle {-{\pi }(t-\tau )^{2}}}$ with ${\displaystyle {-{\pi }\alpha (t-\tau )^{2}}}$ for some chosen ${\displaystyle \alpha }$.