Ingrid Daubechies and Fabrice Planchon
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Adaptive Gabor transforms Ingrid Daubechies and Fabrice Planchon 1 Program in Applied and Computational Mathematics, Princeton University, Princeton NJ 08544-1000, USA Abstract We aim to provide time-frequency representations of a 1D signal where the window is locally adapted to the signal, thus providing a better readability of the representation. Introduction All the linear transforms which allow to depict a signal in phase space ( = time-frequency plane) have a blurring eect, because they typically introduce an auxiliary function which can be chosen arbitrarily and which serves as a \template: the window for a Gabor transform, the wavelet for a wavelet transform. Various attempts have been made to correct such blurring arti- facts while retaining the interesting properties of such transforms. Let us cite reallocation methods ([11]), or squeezing ([6]). On the other hand, bi- linear transform such as the Wigner transform don't introduce extraneous \templates, and have less blurring for some classes of signals. In partic- ular, the Wigner transform is well localized for linear chirps. However, for more complicated signals, interferences appear due to the quadratic nature of the transform; often these are diÆcult to separate from the interesting components in the representation.
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