Early on, our interest in chirping analysis functions was motivated by a different kind of chirping phenomenon: chirping due to perspective. Our urban or indoor world contains a plethora of periodicity, repeating rows of bricks, tiles, windows, or the like abound, yet pictures of these structures fail to capture the true essence of this periodicity. When photographed at an oblique angle (where the film plane is not necessarily parallel to the planar surface), these surfaces give rise to an image whose spatial frequency changes as we move across the image plane. The distant bricks will appear smaller and smaller as we move toward the vanishing point which may be defined to be the point of infinite spatial frequency. Our first generalization of the wavelet transform was to take the ``zooming-in'' property of wavelets and extend it to panning and tilting, to model the movements of a camera. Our interest in radar, however, drew us toward processes that are more accurately analyzed by linear-FM chirplets. We realized that, listening to radar sounds from marine radar, automobile traffic radar, and the like, that in many cases there was a strong ``chirping'', and so the usual Fourier Doppler methods seemed inappropriate in these cases. In particular, the warbling sound of small iceberg fragments suggested that we should consider alternatives to windowed harmonic oscillations and the like (e.g. alternatives to waves and wavelets).
Of the many different kinds of chirping analysis primitives possible, we may distinguish two families of analysis primitives that are of particular interest in practice: the ``projective chirplet'' (p-chirplet), and the ``quadratic chirplet'' (q-chirplet), the latter being the one described in this paper. These two forms have been presented in a combined fashion with the ``time-frequency perspectives''mannicassp, which is a more general chirplet that has eight parameters. The resulting eight-parameter signal representation includes the ``projective chirplet transform'' as one five-parameter subspace, and the ``quadratic chirplet transform'' (e.g. the one presented in this paper) as another five-parameter subspace with the time, frequency, and dilation axes being common to both of these two subspaces. Computational issues have yet to be addressed, although special-purpose hardware has been proposedmannDSPworld with an emphasis on use of FFT-based hardware.
We have also constructed other chirplet transforms, such as a three-parameter Doppler chirplet representation that models a source producing a sinusoidal wave, while moving along a straight line (e.g. a train whistle). The three parameters are center-frequency, maximum rate of change of frequency, and frequency swing. Also, a log-frequency chirplet has been formulated where the underlying chirps appear as straight lines in the time-scale plane.
Generalizations of the STFT and wavelet transform, that make use of chirping analyzing functions, have been previously suggestedmannVI91,mihovilovic,mannspieconference,manneleclett,mannws,mannicassp,baraniukicassp,mihovilovic2. Comparisons between traditional TF methods and chirplets have also been made, in the context of practical applications in both radarmannVI91,cunningham, and geophysicsmihovilovic2.