In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the chirplet transform is an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of an input signal with a family of analysis primitives called chirplets.
[S. Mann and S. Haykin,]
The Chirplet transform: A generalization of Gabor's logon transform
, ''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).]
Similar to the
wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
, 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).
Definitions
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:
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
Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
, 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.]
Applications
The first practical application of the chirplet transform was in water-human-computer interaction (WaterHCI) for marine safety, to assist vessels in navigating through ice-infested waters, using marine radar to detect growlers (small iceberg fragments too small to be visible on conventional radar, yet large enough to damage a vessel).
Other applications of the chirplet transform in WaterHCI include the SWIM (Sequential Wave Imprinting Machine).
More recently other practical applications have been developed, including image processing (e.g. where there is periodic structure imaged through projective geometry),
as well as to excise chirp-like interference in spread spectrum communications, in EEG processing, and Chirplet Time Domain Reflectometry.
Extensions
The warblet transform[
Zelinsky, N. R., & Kleimenova, N. G. Chirplet transform as the useful tool for study the time-frequency structure of geomagnetic pulsations.] is a particular example of the chirplet transform introduced by Mann and Haykin in 1992 and now widely used. It provides a signal representation based on cyclically varying frequency modulated signals (warbling signals).
See also
* Time–frequency representation
A time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved b ...
;Other time–frequency transforms
* Short-time Fourier transform
The short-time Fourier transform (STFT), 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. In practice, the procedure for computing STFTs is to divi ...
* Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
* Fractional Fourier transform
References
* LEM, Logon Expectation Maximization
* {{cite journal
, first1=S.
, last1= Mann
, first2=S.
, last2=Haykin
, url=http://wearcam.org/chirplet/adaptive_chirplet1992/
, title= Adaptive chirplet transform
, journal=Optical Engineering
, volume=31
, issue=6
, pages=1243–1256
, doi=10.1117/12.57676
, bibcode=1992OptEn..31.1243M
, year=1992 introduces Logon Expectation Maximization (LEM) and Radial Basis Functions (RBF) in Time–Frequency space.
* Osaka Kyoiku
Gabor, wavelet and chirplet transforms...(PDF)
* J. "Richard" Cui, etal
Time–frequency analysis of visual evoked potentials using chirplet transform
IEE Electronics Letters, vol. 41, no. 4, pp. 217–218, 2005.
Florian Bossmann, Jianwei Ma, Asymmetric chirplet transform--Part 2: phase, frequency, and chirp rate, Geophysics, 2016, 81 (6), V425-V439.
Florian Bossmann, Jianwei Ma, Asymmetric chirplet transform for sparse representation of seismic data, Geophysics, 2015, 80 (6), WD89-WD100.
External links
DiscreteTFDs - software for computing chirplet decompositions and time–frequency distributions
(web tutorial and info).
Transforms
Fourier analysis
Time–frequency analysis
Image processing
Radar signal processing