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We have presented the chirplet transform, which may be viewed as a generalization of both the shorttime Fourier transform (STFT) and the wavelet transform (WT). These generalizations are based on the fact that both the STFT and WT can be written as inner products of the signal under analysis with versions of a single analysis primitive (window/wavelet) acted on by various operators. In the case of the wavelet, these operators result in 1D affine coordinate tranformations of the timeaxis. In the case of the chirplet, these operators result in 2D affine coordinate transformations of the TF plane (of the timedomain function that they operate on, if one prefers to regard the operators as acting in the timedomain). The family of chirplets is the result of a family of TFaffine coordinate transformation operators acting on a single window/wavelet (the ``mother chirplet''). The chirplet transform is the resulting signal representation on this family of chirplets.
The chirplet transform allows for a unified framework for comparison of various timefrequency methods, because it embodies many other such methods as lowerdimensional subspaces in the chirplet analysis space. For example, the wavelet transform, the shorttime Fourier transform (STFT), the ``frequencyfrequency'' transform, and the scalefrequency transform are planar slices through the proposed multidimensional chirplet parameter space, while many adaptive methodsmannlem,baraniuktsp,baraniuksp are either collections of arbitrary points or twoparameter curved surfaces (manifolds) taken from the multidimensional chirplet parameter space. In addition to unifying some of the existing methods, the chirplet transform provides us with a framework for both formulating and evaluating entirely new subspace transforms.
As pointed out in 1.1, many others have contributed directly or indirectly to the development of the chirplet transform. In many ways, however, we have taken its development further toward becoming a useful signal processing tool for practical engineering problems, as evidenced by the material presented in this paper.
ACKNOWLEDGMENT
The authors wish to express their gratitude to the following individuals
for their valuable assistance:
Rosalind Picard, Irving Segal,
Shawn Becker, and Kris Popat of the Massachusetts Institute of Technology;
Douglas Jones
of the University of Illinois at UrbanaChampaign;
Richard Baraniuk of Rice University;
and the anonymous reviewers, whose careful efforts resulted
in a substantially improved presentation.