In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, the
chirplet transform is an inner product of an input signal with a family of analysis primitives called
chirplets.
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 MannSteve Mann , is a tenured professor at the Department of Electrical and Computer Engineering at the University of Toronto.-Education:...
, as the title of the first published paper on chirplets.
In
signal processingSignal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
, 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 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 the chirplet and chirplet transform?
The term "
chirplet transform" was coined by
Steve MannSteve Mann , is a tenured professor at the Department of Electrical and Computer Engineering at the University of Toronto.-Education:...
, 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
chirpA chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
function. In Mann's words:
http://wearcam.org/chirplet.htm
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 transformThe short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
s) or waveletLoosely speaking, a wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully...
s.
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
radarRadar is an object detection system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR was coined in 1941 as an acronym for RAdio Detection And...
, 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
GaussianGAUSSIAN is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70...
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
BracewellRonald Newbold Bracewell AO was the Lewis M. Terman Professor of Electrical Engineering, Emeritus of the Space, Telecommunications and Radioscience Laboratory at Stanford University until his death.- Education :...
later that same year.
Applications
The chirplet transform is a useful signal analysis and representation framework that is widely used in
- radar
Radar is an object detection system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The term RADAR was coined in 1941 as an acronym for RAdio Detection And...
- biomedical, the most commonly and widely used chirplet applications in biomedical being:
- heart sound processing by companies such as Andromed;
- EEG processing, as for example, outlined by Cui, et al..
- signal processing
Signal processing is an area of electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time to perform useful operations on those signals...
- image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as photographs or frames of video; the output of image processing can be either an image or a set of characteristics or parameters related to the image...
- SETI@home
SETI@home is a distributed computing project using Internet-connected computers, hosted by the Space Sciences Laboratory, at the University of California, Berkeley, in the United States. SETI is an acronym for the Search for Extra-Terrestrial Intelligence...
uses chirp functions to compensate for Doppler drift.
- Chirplet Time Domain Reflectometry (from National Instruments website)
Taxonomy of chirplet transforms
There are two broad categories of chirplet transform:
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 2π (a t² + b t + 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 metaplectomorphism
In mathematics and signal processing, a metaplectomorphism is a transformation by way of an operator from a metaplectic group of operators.Examples of metaplectomorphisms, in time-frequency, include:*translation in time;*translation in frequency;...
s 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 warbler
There are a number of Passeriformes called "warblers". They are not particularly closely related, but share some characteristics, such as being fairly small, vocal and insectivorous....
s. 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 transformthumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.thumb|right|Original function is equal to one on the white region and zero on the dark region....
of the
Wigner distributionThe Wigner distribution is either of two things:* Wigner semicircle distribution - A probability function used in mathematics * Wigner quasi-probability distribution - A distribution in phase space encoding, in a convenient representation, properties of quantum-mechanical wave-functions...
may, for example, be used, as may the
fractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...
).
Another possible choice is the rectangular window, and 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
In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
- The Short-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
(STFT), also known as the spectrogram
- The Wigner-Ville distribution
- The wavelet transform
- Canonical conjugate variables
- Segal-Shale-Weil distribution
The Segal-Shale-Weil distribution is a distribution based on metaplectomorphisms of a plane of two canonical conjugate variables; such as time and frequency.This work was pioneered by Irving Segal, a mathematician at MIT....
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 UniversityHarvard University is a private university located in Cambridge, Massachusetts and a member of the Ivy League. Founded in 1636 by the colonial Massachusetts legislature, Harvard is the oldest institution of higher learning in the United States and currently comprises ten separate academic units...
and at
Massachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological research...
). Further personal communication between
Irving SegalIrving Ezra Segal was a mathematician known for work on theoretical quantum mechanics.He was at the Massachusetts Institute of Technology...
(the principal behind the Segal, Shale Weil representation, known also as the
metaplecticIn mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers...
representation --- a double covering of the
symplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. In this article, we shall denote these two groups Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former...
) 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. Recent developments include the
chirplet-based Brain Computer Interaction system that allows a person wearing
eyetapAn EyeTap is a device that is worn in front of the eye that acts as a camera to record the scene available to the eye as well as a display to superimpose a computer-generated imagery on the original scene available to the eye....
eyeglasses to interact with a computer.
Andromed and National Instruments use and support the chirplet transform in a wide range of product offerings.
See also
- Time-frequency representation
A time–frequency representation is a view of a signal represented over both time and frequency. Time–frequency analysis means analysis into a TFR....
Other time-frequency transforms:
- short-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
- wavelet transform
- fractional Fourier transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...
External links