Gabor transform

It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time.

[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: 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.

The window function width can also be varied to optimize the time-frequency resolution tradeoff for a particular application by replacing the

Because it is over-complete, the original signal can be recovered in a variety of ways.

The main application of the Gabor transform is used in time–frequency analysis.

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.

Note that for each point in time there is both a negative (upper white part) and a positive (lower white part) frequency component.

can be derived easily by discretizing the Gabor-basis-function in these equations.

Hereby the continuous parameter t is replaced by the discrete time k. Furthermore, the now finite summation limit in Gabor representation has to be considered.

In this way, the sampled signal y(k) is split into M time frames of length N. According to

For overall M time windows with N sample values, each signal y(k) contains K = N

M sample values: (the discrete Gabor representation) with

correspond to the number of sample values K of the signal.

with N′ > N, which results in N′ > N summation coefficients in the second sum of the discrete Gabor representation.

N′ > K. Hence, more coefficients than sample values are available and therefore a redundant representation would be achieved.

As in short time Fourier transform, the resolution in time and frequency domain can be adjusted by choosing different window function width.

In Gabor transform cases, by adding variance

, the window function will be narrow, causing higher resolution in time domain but lower resolution in frequency domain.

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 [2] 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 [2] for further details.