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. Discrete Gabor-transformation A discrete version of Gabor representation with 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 , the factor Ω for critical sampling is . Similar to the DFT (discrete Fourier transformation) a frequency domain split into N discrete partitions is obtained. An inverse transformation of these N spectral partitions then leads to N values y(k) for the time window, which consists of N sample values. For overall M time windows with N sample values, each signal y(k) contains K = N M sample values: (the discrete Gabor representation) with According to the equation above, the N M coefficients correspond to the number of sample values K of the signal. For over-sampling is set to with N′ > N, which results in N′ > N summation coefficients in the second sum of the discrete Gabor representation. In this case, the number of obtained Gabor-coefficients would be MN′ > K. Hence, more coefficients than sample values are available and therefore a redundant representation would be achieved. Scaled Gabor transform 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 , as following equation: The scaled (normalized) Gaussian window denotes as: So the Scaled Gabor transform can be written as: With a large , the window function will be narrow, causing higher resolution in time domain but lower resolution in frequency domain. Similarly, a small will lead to a wide window, with higher resolution in frequency domain but lower resolution in time domain. Time-causal analogue of the Gabor transform 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