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 . Inverse Gabor transform 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 : Alternatively, all of the time components can be combined together: Properties of the Gabor transform The Gabor transform has many properties like those of the Fourier transform. These properties are listed in the following tables. Application and example 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