Chirplet Transform - Taxonomy of Chirplet Transforms

Taxonomy of Chirplet Transforms

There are two broad categories of chirplet transform:

• Fixed
• Adaptive

These categories may be further subdivided by:

• choice of chirp
• choice of window

In either the fixed or adaptive case, the chirplets may be:

• q-chirplets (quadratic chirplets) of the form exp or, in general, some kind of quadratically varying exponent, linear swept wave packet, or the like. These are sometimes called linear FM chirplets (linear frequency-modulated chirplets, since quadratic phase is linear frequency). Commonly used families of q-chirplets are metaplectomorphisms of one another (i.e., the energy distribution of any member of the family of q-chirplets can be generated from any other member by shear-in-time, shear-in-frequency, dilation, translation-in-time, and translation-in-frequency).
• w-chirplets, also known as warblets. A family of warblets are like the sound made by birds called warblers. Unwindowed warblets have a sinusoidally varying time–frequency distribution, or similar cyclostationary or periodically varying time–frequency plot. The sound of a police siren is an example, in which the pitch goes up and down periodically. Of course, the warblet is a "piece of" a warble (i.e., a windowed section of something that has a time–frequency periodicity).
• d-chirplets, also known as Doppler chirplets. These are analysis functions that mimic the Doppler shift of a passing tone (e.g., the sound you hear from a train whistle as it moves past).
• p-chirplets, in which the scale varies projectively. Whereas the wavelet transform is based on wavelets of the form g(ax + b), the p-type chirplet transform is based on chirplets of the form g((ax + b)/(cx + 1)), where a is the scale, b is the translation, and c is the chirpiness (chirp-rate, as defined by the degree of perspective, or projection).

The choice of window is also another matter of decision. A Gaussian window is one possible choice, leading to a four parameter chirplet transform (for which time–shear and frequency–shear only give one degree of freedom that may thus be encapsulated as rotation angle—Radon transform of the Wigner distribution may, for example, be used, as may the fractional Fourier transform).

Another possible choice is the rectangular window, and discrete prolate spheroidal sequences ( also called Multitaper#The_Slepian_sequences ) may be used, by way of the method of multiple mother chirplets. This method gives a total chirplet transform as the sum of energies in various contributory chirplet transforms made from multiple windows, akin to the way in which DPSSs are used to get a perfect rectangular tiling of the time–frequency plane. Thus it is now possible to get perfect parallelogram tiling of the time–frequency plane, using the method of multiple mother chirplets.