CONCLUSION

We have presented the chirplet transform, which may be viewed as a generalization of both the short-time 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 1-D affine coordinate tranformations of the time-axis. In the case of the chirplet, these operators result in 2-D affine coordinate transformations of the TF plane (of the time-domain function that they operate on, if one prefers to regard the operators as acting in the time-domain). The family of chirplets is the result of a family of TF-affine 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.

1. As is well-known, taking the Fourier transform of a one dimensional function results in a complex-valued function of a single variable.
2. As is also well-known, the STFT results in a function of two variables: time and frequency. The wavelet transform results in a complex function of two variables: time and scale.
3. The combined TFS transform results in a complex function of three variables: time, frequency and scale.
4. The Gaussian chirplet transform (GCT) results in a complex function of four variables: time, frequency, scale, and ``chirprate''.
5. Another complex-valued four-dimensional parameter space is given by: time, frequency, ``chirprate'' and ``dispersionrate''. This space has the interesting property that it does not require dilation of the mother chirplet, and may therefore be applied to discrete mother chirplets that do not have a mathematical description (e.g. no need for interpolation or antialiasing).
6. The full continuous chirplet transform (CCT) that can be obtained using only a single mother chirplet results in a complex function of five variables: time, frequency, scale, chirprate, and dispersionrate.
7. The multiple-mother-chirplet transform (e.g. using the prolate family) results in a real function of six variables: time, frequency, scale, chirprate, dispersionrate, and TF tile size. The coordinate axes of this six-dimensional parameter space correspond to the six affine coordinate transformations in the TF plane: translation along each of the time and frequency axes, change in aspect ratio, shear along each of the time and frequency axes, and change in area occupied in the TF plane. The last of these six dimensions is discretized, while the other five are continuous.

The chirplet transform allows for a unified framework for comparison of various time-frequency methods, because it embodies many other such methods as lower-dimensional subspaces in the chirplet analysis space. For example, the wavelet transform, the short-time Fourier transform (STFT), the ``frequency-frequency'' transform, and the scale-frequency transform are planar slices through the proposed multi-dimensional chirplet parameter space, while many adaptive methodsmannlem,baraniuktsp,baraniuksp are either collections of arbitrary points or two-parameter curved surfaces (manifolds) taken from the multi-dimensional 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 Urbana-Champaign; Richard Baraniuk of Rice University; and the anonymous reviewers, whose careful efforts resulted in a substantially improved presentation.