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:

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.

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

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 ${\displaystyle \alpha }$.

The Gabor transform is invertible. Because it is over-complete, the original signal can be recovered in a variety of ways. For example, the "unwindowing" approach can be used for any ${\displaystyle \tau _{0}\in (-\infty ,\infty )}$:

Alternatively, all of the time components can be combined:

The Gabor transform has many properties like those of the Fourier transform. These properties are listed in the following tables.