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Historical Notes

  In 1946, in his seminal paper on communication theory [7], Gabor (who later won the Nobel prize for his work on holography), provided a new interpretation of the one-dimensional Gaussian-windowed STFT and examined the time-frequency plane in terms of a two-dimensional tiling. Although Gabor's development was not completely rigorous (and, in fact his representation was later shown to be unstable[1]), his notion of a time-frequency tiling was a very significant contribution. Gabor referred to the elements of his tiling as logons.

Beginning around 1956, Siebert began to formulate a radar detection philosophy with some particularly useful insights in terms of time-frequency[8][9]. Much of his insight was obtained through the use of Woodward's uncertainty function[10], also known as the radar ambiguity function[11] or the Fourier-Wigner transform[12]. Siebert also considered chirping functions for pulse compression radar, and studied these in detail, observing that chirping in the time domain gives rise to a shearing in the time-frequency plane (or equivalently, a shearing in the 2D Fourier transform of the time-frequency plane).

In 1985, Grossman and Paulgrossmann-paul rigorously formulated some of these important ideas in terms of affine canonical coordinate transformations to a coherent space representation. They also considered two-parameter subgroups of these affine coordinate transformations.

Papoulis, in his book[14] described the use of a linear frequency-modulated (chirped) signal as the basis of an ordinary Fourier analyzer, and also presented the chirped signals as shearing operators in the time-frequency plane, foreshadowing the development of the chirplet transform.

In 1987, Jones and Parksjonesparks2 formulated the problem of window selection in terms of time-frequency leakage. They made an important connection between the work of Szu and Blodgettszu who showed that frequency shearing is accomplished through multiplication by a chirp, and the work of Janssenjanssen who proved that any area-preserving affine coordinate transformation of the time-frequency plane yields a valid time-frequency plane of some other signal, though they were unaware of Siebert's earlier unpublished work. In a simple and insightful example, Jones and Parks showed the time-frequency distribution of both a Hamming window and a chirped Hamming window, one being a sheared version of the other.

Berthon[18] proposed a generalization of the radar ambiguity function based on the semi-direct product of two important groups:

In 1989 and early 1990, we formulated the chirplet transform, a multidimensional parameter space whose coordinate axes correspond to the pure parameters of planar affine coordinate transformations in the time-frequency plane. (This formulation was motivated by a discovery made by the senior author and his research associates, namely, that the Doppler radar return from a small piece of ice floating in an ocean environment is chirp-like[19].) We also formulated a variety of new and useful transforms that were two-dimensional subspaces of this multidimensional parameter space. Furthermore, we suggested using the work of Landaudps1[21][22][23][24][25] who introduced prolate spheroidal functions, and we noted their significance in the context of the shearing phenomenon in the time-frequency plane, as they form idealized parallelogram tilings of this plane.

Later, we applied the chirplet transform and some of the new two-dimensional subspace transforms to problems in marine radar and obtained results that were better than previous methods, so we published these findings[26]. Independently, at around the same time (ironically, only a few days later) Mihovilovic and Bracewell also presented a related idea[27] (ironically, using the same name, ``chirplets''), though not in the same level of generality of the multidimensional parameter space. Later they also presented a practical application of chirplets[28].

A point that needs to be emphasized here is that there is more to the chirplet transform than just the shear phenomenon. In particular, time-shear and frequency-shear are examples of affine coordinate transformations - mappings from the TF-plane to the TF-plane - while the chirplet transform is a mapping from a continuous function of one real variable to a continuous function of five (or six) real variables.

In 1991, Torresani[29] examined some relations that were intermediate between the affine and the Weyl-Heisenberg groups. The work of Segman and Schemppsegman incorporates scale into the Heisenberg group, and the work of Wilson kn:IT38[32] examines the use of a TFS representation that they call the multiresolution Fourier transform.

Baraniuk and Jones studied several ``chirplet transform subspaces'' and made precise some of the mathematical details of the two-dimensional chirplet transform subspaces[33]. They also provided an alternative derivationbaraniukthesis of the chirplet transform, based on the Wigner distribution. This derivation involved noting, as we did, that each point in the analysis space of the chirplet transform corresponds to a particular operator in the time domain. This time-domain operator acting on the analysis primitive (`mother chirplet') also has, associated with it, a 2-D area-preserving affine coordinate transformation in the TF plane. Baraniuk and Jones also addressed discretization issues[33][34].

Recently, researchers have considered fractional Fourier domains and their relation to chirp and wavelet transforms[35].

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Steve Mann
Thu Jan 8 19:50:27 EST 1998