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. 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 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 with for some chosen .

The main application of the Gabor transform is used in time–frequency analysis. Take the following function as an example. The input signal has 1 Hz frequency component when t ≤ 0 and has 2 Hz frequency component when t > 0

But if the total bandwidth available is 5 Hz, other frequency bands except x(t) are wasted. Through time–frequency analysis by applying the Gabor transform, the available bandwidth can be known and those frequency bands can be used for other applications and bandwidth is saved. The right side picture shows the input signal x(t) and the output of the Gabor transform. As was our expectation, the frequency distribution can be separated into two parts. One is t ≤ 0 and the other is t > 0. The white part is the frequency band occupied by x(t) and the black part is not used. Note that for each point in time there is both a negative (upper white part) and a positive (lower white part) frequency component.

When processing temporal signals, data from the future cannot be accessed, which leads to problems if attempting to use Gabor functions for processing real time signals. A time causal analogue of the Gabor filter has been developed in based on replacing the Gaussian kernel in the Gabor function with a time causal and time recursive kernel referred to as the time causal limit kernel. In this way, time frequency analysis based on the resulting complex valued extension of the time causal limit kernel makes it possible to capture essentially similar transformations of a temporal signal as the Gabor function can, and corresponding to the Heisenberg group, see for further details.