In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. [1][2]
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Much as in the wavelet transform, the chirplets are usually generated from (or can be expressed as being from) a single mother chirplet (analogous to the so-called mother wavelet of wavelet theory).
The term chirplet transform was coined by Steve Mann, as the title of the first published paper on chirplets. The term chirplet itself (apart from chirplet transform) was also used by Steve Mann, Domingo Mihovilovic, and Ronald Bracewell to describe a windowed portion of a chirp function. In Mann's words:
A wavelet is a piece of a wave, and a chirplet, similarly, is a piece of a chirp. More precisely, a chirplet is a windowed portion of a chirp function, where the window provides some time localization property. In terms of time–frequency space, chirplets exist as rotated, sheared, or other structures that move from the traditional parallelism with the time and frequency axes that are typical for waves (Fourier and short-time Fourier transforms) or wavelets.
The chirplet transform thus represents a rotated, sheared, or otherwise transformed tiling of the time–frequency plane. Although chirp signals have been known for many years in radar, pulse compression, and the like, the first published reference to the chirplet transform described specific signal representations based on families of functions related to one another by time–varying frequency modulation or frequency varying time modulation, in addition to time and frequency shifting, and scale changes. In that paper, the Gaussian chirplet transform was presented as one such example, together with a successful application to ice fragment detection in radar (improving target detection results over previous approaches). The term chirplet (but not the term chirplet transform) was also proposed for a similar transform, apparently independently, by Mihovilovic and Bracewell later that same year.[2]
The chirplet transform is a useful signal analysis and representation framework that is widely used in
There are two broad categories of chirplet transform:
These categories may be further subdivided by:
In either the fixed or adaptive case, the chirplets may be:
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.
The chirplet transform is a generalized representation that includes as special cases:
Josef Segman proposed the idea of incorporating scale into the Heisenberg group (position, momentum, phase, or equivalently any canonical conjugate variables taken together with phase, such as, for example, time, frequency, and phase). This gave rise to a four parameter space of time, frequency, phase, and scale. Segman introduced this idea of phase scale. (Personal communication with Mann, from Josef Segman, at Harvard University and at Massachusetts Institute of Technology). Further personal communication between Irving Segal (the principal behind the Segal, Shale Weil representation, known also as the metaplectic representation—a double covering of the symplectic group) and Mann led to additional insight into the chirplet transform, in particular, to the variation of the chirplet transform that is based on q-chirplets.
Other time–frequency transforms: