Linear systems in Gabor time-frequency space

A new approach for the description and operation of linear time-varying systems is presented. The proposed formalism implements the superposition integral describing a general linear operation in Gabor time-frequency space, enabling the use of its joint properties. The method is based on transforming both the input and the time-varying impulse response function into the Gabor space. The 2- and 4-D arrays, corresponding to the Gabor spectra of the input signal and the impulse response, respectively, are multiplied yielding a 2-D array corresponding to the output Gabor spectrum, from which the output time signal is subsequently reconstructed. Unlike other time-varying operation methods carried out in different mixed time-frequency spaces, e.g., Wigner space, the suggested approach, when employing the critically sampled Gabor expansion, generates a legal output spectrum, and eliminates the need to resort to cumbersome and numerically taxing approximate synthesis techniques. The proposed formalism can be implemented in the areas of time-variant filtering, speech scrambling, and any other linear operations that vary their characterizations in time