Comparison of wave, wavelet, chirp, and chirplet

In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.

Similarity to other transforms

Much as like 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).

What is a chirplet?

The term "chirplet" was reportedly coined by Steve Mann in the 1980s to describe a windowed portion of a chirp function. Loosely speaking, a wavelet is a piece of a wave, and similarly a chirplet 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" as such was in Mann and Haykin (1991), describing the Gaussian chirplet transform together with a successful application to ice fragment detection in radar (improving target detection results over previous approaches). The term was also proposed for a similar transform, apparently independently, by Mihovilovic and Bracewell in the same year.

Applications

(a) In image processing, we seldom have direct periodicity, but, rather, we encounter periodicity-in-perspective. (b) Repeating structures like the alternating dark space inside the windows, and light space of the white cement, "chirp" (increase in frequency) to towards the right. (c) Thus the p-chirplet transform is often more appropriate than either the Fourier or wavelet transforms.

The chirplet transform is a useful signal analysis and representation framework that is widely used in radar, biomedical signal processing, and image processing.

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(i 2pi (at^2 +bt +c)), 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 sound 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 of course, Discrete Prolate Spheroidal 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 contributant 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.

Related work

The chirplet transform is a generalized representation that includes as special cases:

• The Fourier transform
• The Short Time Fourier Transform (STFT), also known as the spectrogram
• The Wigner Ville Distribution
• The wavelet transform.

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.

Further ongoing work

Work on the chirplet transform is ongoing. One of the most exciting developments is that of Richard Cui, who has developed a chirplet-based Brain Computer Interaction system that allows a person wearing eyetap eyeglasses to interact with a computer by way of Visual Evoked Potentials. Chirplet-based VEP is the subject of Richard Cui's PhD thesis.

Various companies, such as Andromed, National Instruments, etc., use and support the chirplet transform in a wide range of product offerings.

References

• S. Mann and S. Haykin, "The Chirplet transform: A generalization of Gabor's logon transform (http://wearcam.org/chirplet/vi91/index.htm)", Proc. Vision Interface 1991, 205–212 (3–7 June 1991).
• D. Mihovilovic and R. N. Bracewell, "Adaptive chirplet representation of signals in the time-frequency plane," Electronics Letters 27 (13), 1159–1161 (20 June 1991).
• S. Mann and S. Haykin, "The adaptive chirplet: An adaptive wavelet like transform (http://wearcam.org/chirplet/adaptive_chirplet1991/)", Proc. SPIE 36th Intl. Symp. Optical and Optoelectronic Appl. Sci. Eng. (21–26 July 1991).
• S. Mann, The Chirplet Transform (http://wearcam.org/chirplet.htm) (web tutorial and info).