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Chirplet transform
Chirplet transform
Chirplet transform
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Comparison of wave, wavelet, chirp, and chirplet[1]
Chirplet in a computer-mediated reality environment.

In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.[2][3]

Similar to the wavelet transform, 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

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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.[2] In that paper,[2] 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.[3]

Applications

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(a) In image processing, periodicity is often subject to projective geometry (i.e. chirping that arises from projection). (b) In this image, repeating structures like the alternating dark space inside the windows, and light space of the white concrete, chirp (increase in frequency) towards the right. (c) The chirplet transform is able to represent this modulated variation compactly.

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).[4][5]

Other applications of the chirplet transform in WaterHCI include the SWIM (Sequential Wave Imprinting Machine).[6][7]

More recently other practical applications have been developed, including image processing (e.g. where there is periodic structure imaged through projective geometry),[6][8] as well as to excise chirp-like interference in spread spectrum communications,[9] in EEG processing,[10] and Chirplet Time Domain Reflectometry.[11]

Extensions

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The warblet transform[12][13][14][15][16][17] 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

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Chirplet transform

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from Grokipedia
The chirplet transform is a signal processing method for time-frequency analysis of non-stationary signals, defined as the inner product between the input signal and a parameterized family of chirplet functions, which are Gaussian-windowed linear frequency-modulated chirps with instantaneous frequency varying linearly over time. Introduced by Steve Mann and Simon Haykin in 1995, the chirplet transform extends the short-time Fourier transform and continuous wavelet transform by incorporating an additional chirp rate parameter (μ), enabling representation in a multidimensional space that includes time (t₀), frequency (f₀), scale (σ), and chirp rate, thus providing superior resolution for signals exhibiting linear frequency sweeps, such as those arising from Doppler effects or accelerating sources. The mathematical formulation involves chirplets of the form ψ(t) = exp(- (t-t₀)² / (2σ²)) exp(i (2π f₀ (t-t₀) + π μ (t-t₀)²)), where the quadratic phase term captures the chirp modulation. This generalization allows the transform to encompass the time-frequency plane of the STFT and the time-scale plane of wavelets as special cases (μ=0), while introducing "shear" deformations for more accurate modeling of physical phenomena like perspective distortion in imaging or gravitational acceleration in radar returns. Key advantages include enhanced energy concentration in the time-frequency domain for chirp-like components compared to fixed-window methods, making it robust for low environments, and its ability to decompose multicomponent signals via iterative algorithms like . Applications span and systems for detecting accelerating targets, ultrasonic non-destructive testing for flaw characterization in materials, biomedical such as ECG QRS delineation and EEG recognition, speech feature extraction for improved recognition accuracy, and mechanical fault diagnosis in rotating machinery like wind turbines and bearings under variable speeds. Subsequent developments, including synchrosqueezed and high-resolution variants, have further refined its performance for nonlinear chirps and real-time implementations.

Overview

Definition

The chirplet transform is a time-frequency technique that computes the inner product of an input signal with a of analysis primitives known as chirplets, which are essentially windowed functions featuring linear . This approach allows for the representation of signals in a multidimensional parameter space that captures variations in time, , and chirp rate, making it particularly suited for analyzing non-stationary signals with evolving frequency content. Chirplets serve as a of by incorporating an additional chirp rate parameter, which accounts for linear frequency sweeps or modulations over time, thereby extending the flexibility of traditional wavelet decompositions to handle more complex, time-varying frequency structures. Unlike , which rely on one-dimensional scaling and , chirplets are generated through two-dimensional affine transformations—including translations, dilations, rotations, and shears—in the time-frequency plane, enabling a more adaptive matching to signal components. Conceptually, the chirplet transform builds on the (STFT) and by introducing the chirp parameter to address curved instantaneous frequency trajectories that these methods struggle with, such as those appearing as non-parallel ridges in spectrograms. This sheared or rotated perspective in the time-frequency domain provides enhanced resolution for signals exhibiting rapid frequency changes, positioning the chirplet transform as a versatile tool within the broader framework of time-frequency analysis.

History

The chirplet transform was coined by Steve Mann in 1991 during his collaboration with Simon Haykin at McMaster University's Communications Research Laboratory, initially developed to address physical considerations in , such as modeling the acceleration and motion of objects in data. This work built on the need for analysis tools that could capture frequency-modulated signals, or chirps, more effectively than existing methods. In the same year, Mann and Haykin presented the first publication on the topic, titled "The Chirplet Transform: A Generalization of Gabor's Logon Transform," at the Vision Interface '91 conference, where they introduced the transform as an expansion onto multi-scale chirps and applied it to practical problems like detecting floating objects in marine environments. Independently, D. Mihovilovic and R. N. Bracewell also invented a form of the in , focusing on time-frequency representations to adaptively decompose signals on the time-frequency plane, as detailed in their Electronics Letters paper "Adaptive Chirplet Representation of Signals on Time-Frequency Plane." This parallel development highlighted the transform's potential for handling dynamic spectra with linear , complementing Mann and Haykin's physical modeling approach. Mihovilovic and Bracewell's contribution emphasized adaptive parameter selection to represent signal components as chirplets, enabling separation of overlapping features in the time-frequency domain. The chirplet transform evolved from earlier time-frequency analysis techniques, particularly the short-time Fourier transform (STFT) introduced by Dennis Gabor in 1946, which provided fixed-resolution analysis but struggled with non-stationary chirp signals due to its constant window size. It also extended wavelet transforms, pioneered by Alexandre Grossmann and Jean Morlet in 1984, by incorporating chirp modulation to better resolve signals with varying instantaneous frequencies, such as those in seismic or radar data, while addressing limitations in scale and frequency adaptability. Early applications of the chirplet transform, as explored in Mann's work, included systems for detecting small ice fragments, or , by analyzing their acceleration signatures in Doppler returns, outperforming traditional methods in cluttered ocean environments. This initial use in of demonstrated the transform's utility for real-world challenges involving moving objects.

Mathematical Foundations

Chirplet Functions

The chirplet functions serve as the fundamental basis elements for the chirplet transform, generalizing traditional fixed-frequency atoms by incorporating a linear frequency modulation. In their general form, a chirplet function is expressed as ψ(t)=g(t)exp⁡(j(ω0t+μ2t2))\psi(t) = g(t) \exp\left(j \left( \omega_0 t + \frac{\mu}{2} t^2 \right)\right), where g(t)g(t) denotes a window function, typically of finite duration, ω0\omega_0 represents the central angular frequency, and μ\mu is the chirp rate parameterizing the quadratic phase that induces frequency sweeping. This structure was introduced by Mann and Haykin as an extension of Gabor logons, enabling representation of signals with time-varying frequencies. The standard implementation employs a Gaussian window for its optimal localization in the time-frequency plane, yielding the Gaussian chirplet ψt0,f0,μ,s(t)=1sg(t−t0s)exp⁡(j(2πf0(t−t0)+πμ(t−t0)2))\psi_{t_0, f_0, \mu, s}(t) = \frac{1}{\sqrt{s}} g\left(\frac{t - t_0}{s}\right) \exp\left(j \left(2\pi f_0 (t - t_0) + \pi \mu (t - t_0)^2 \right)\right)
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