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Chirplet transform

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.^[1] 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.^[1] 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.^[1] 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.^[1] 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 signal-to-noise ratio environments, and its ability to decompose multicomponent signals via iterative algorithms like matching pursuit.^[2] Applications span radar and sonar systems for detecting accelerating targets, ultrasonic non-destructive testing for flaw characterization in materials, biomedical signal processing such as ECG QRS delineation and EEG motor imagery recognition, speech feature extraction for improved recognition accuracy, and mechanical fault diagnosis in rotating machinery like wind turbines and bearings under variable speeds.^[3]^[4] Subsequent developments, including synchrosqueezed and high-resolution variants, have further refined its performance for nonlinear chirps and real-time implementations.^[5]

Overview

Definition

The chirplet transform is a time-frequency analysis technique that computes the inner product of an input signal with a family of analysis primitives known as chirplets, which are essentially windowed chirp functions featuring linear frequency modulation.^[6] This approach allows for the representation of signals in a multidimensional parameter space that captures variations in time, frequency, and chirp rate, making it particularly suited for analyzing non-stationary signals with evolving frequency content.^[6] Chirplets serve as a generalization of wavelets 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.^[6] Unlike wavelets, which rely on one-dimensional scaling and translation, 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.^[6] Conceptually, the chirplet transform builds on the short-time Fourier transform (STFT) and wavelet transform 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.^[6] 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.^[6]

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 signal processing, such as modeling the acceleration and motion of objects in radar data.^[7] 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.^[7] Independently, D. Mihovilovic and R. N. Bracewell also invented a form of the chirplet transform in 1991, 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 frequency modulation, 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 marine radar systems for detecting small ice fragments, or growlers, by analyzing their acceleration signatures in Doppler returns, outperforming traditional methods in cluttered ocean environments.^[7] This initial use in remote sensing of sea ice demonstrated the transform's utility for real-world signal processing challenges involving moving objects.^[8]

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 \psi(t) = g(t) \exp\left(j \left( \omega_0 t + \frac{\mu}{2} t^2 \right)\right), where g(t) denotes a window function, typically of finite duration, \omega_0 represents the central angular frequency, and \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.^[9] The standard implementation employs a Gaussian window for its optimal localization in the time-frequency plane, yielding the Gaussian chirplet \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), where t_0 specifies the time shift, f_0 the central frequency, \mu the chirp rate, and s > 0 the scale parameter controlling the window's width. Here, g(u) = \exp\left(-\frac{u^2}{2}\right) is the unit-variance Gaussian, ensuring the function's energy concentration. This parameterized form allows flexible adaptation to signal characteristics, with the exponential term capturing linear frequency modulation (LFM) akin to chirp signals in radar and sonar.^[9]^[2] The key distinguishing parameter is the chirp rate \mu, which governs the instantaneous frequency sweep: the instantaneous frequency evolves as f(t) = f_0 + \mu (t - t_0), enabling chirplets to model signals where frequency changes linearly over time, in contrast to constant-frequency basis functions in the short-time Fourier transform or wavelets. Nonzero \mu introduces a shear in the time-frequency plane, providing enhanced resolution for chirp-like components without fixed-frequency assumptions.^[9] Normalization in chirplet functions ensures unit energy in the L^2 norm, typically achieved through the factor $1/\sqrt{s} for scale invariance and the Gaussian window's inherent properties, which minimize the Heisenberg uncertainty product for time-frequency concentration. Alternative windows, such as rectangular or Hamming, may be used but Gaussian is preferred for its mathematical tractability and asymptotic optimality in localization, as it yields the minimum time-bandwidth product.^[9]

The Transform

The continuous chirplet transform (CCT) provides a time-frequency-scale-chirp rate representation of a signal by computing inner products with a family of chirplet atoms. It is formally defined as

C(t_0, f_0, \mu, s) = \int_{-\infty}^{\infty} x(\tau) \psi_{t_0, f_0, \mu, s}^{*}(\tau) \, d\tau,

where x(\tau) is the input signal, \psi_{t_0, f_0, \mu, s}(\tau) denotes the chirplet function parameterized by center time t_0, center frequency f_0, chirp rate \mu, and scale s, and ^* indicates the complex conjugate.^[6] This integral formulation generalizes traditional time-frequency transforms by incorporating chirp modulation to capture linear frequency sweeps inherent in nonstationary signals.^[1] The output of the CCT resides in a four-dimensional parameter space spanned by t_0, f_0, \mu, and s, which enables the decomposition of signals with quadratic phase structures that elude fixed-window analyses like the short-time Fourier transform.^[6] Sampling this space densely allows for adaptive resolution in both time-frequency localization and chirp characterization, with the scale parameter s controlling the temporal extent of the analysis window and \mu quantifying the instantaneous frequency variation.^[1] Discrete implementations approximate the CCT by discretizing the continuous integral over a finite parameter grid, often using numerical quadrature or basis pursuit methods to evaluate coefficients at sampled points. Fast algorithms, leveraging FFT-based convolutions and chirp modulations, reduce the computational complexity from O(N^2) to O(N \log N) for an N-point signal, facilitating practical applications in real-time processing.^[10] Under complete coverage of the parameter space, the original signal can be reconstructed via an inverse transform that integrates the chirplet coefficients weighted by the dual basis functions, akin to frame expansions in overcomplete dictionaries, ensuring perfect recovery for bandlimited signals.^[11]

Properties and Comparisons

Key Properties

The chirplet transform exhibits superior time-frequency resolution for chirp signals, as its basis functions incorporate a linear frequency modulation parameter that aligns with the instantaneous frequency trajectory of the signal, thereby concentrating energy along curved ridges in the time-frequency plane more effectively than fixed-window methods.^[6] The inclusion of a scale parameter s imparts a multiresolution capability to the chirplet transform, allowing analysis of signals at varying levels of detail by adjusting the temporal support of the basis functions, akin to the dilation in wavelet analysis.^[6] A notable limitation of the chirplet transform arises from its high computational cost, stemming from the need to search a multidimensional parameter space (often four or more dimensions) to identify optimal chirp parameters, which can render full implementations impractical for real-time applications without dimensionality reduction to subspaces.^[6]

Comparison to Other Transforms

The chirplet transform extends the Fourier transform by incorporating localized chirp functions, enabling analysis of non-stationary signals with linear frequency modulation (FM), whereas the Fourier transform assumes global stationarity and struggles with time-varying frequencies.^[6] This addition of time, scale, and chirp-rate parameters allows the chirplet transform to capture hyperbolic trajectories in the time-frequency plane, providing a more adaptive representation for signals like radar pulses.^[12] Compared to the short-time Fourier transform (STFT), which uses a fixed window size that compromises either time or frequency resolution for chirp signals, the chirplet transform employs variable chirp rates to align basis functions with the signal's instantaneous frequency, yielding superior resolution for linear FM components.^[12] In applications involving dispersive waves, such as ultrasonics, the chirplet's flexibility results in sparser decompositions without the Heisenberg uncertainty limitations inherent in STFT.^[13] The wavelet transform relies on scale and translation for multiresolution analysis but lacks an explicit chirp parameter, making it less effective for signals with pronounced linear FM trajectories; the chirplet transform addresses this by integrating chirp modulation into its affine framework, encompassing wavelets as a subset while adding rotation and shear for enhanced adaptability.^[6] This parametric extension facilitates better matching to hyperbolic signal paths, as seen in sonar processing where wavelets may dilute energy across scales.^[12] In contrast to the Wigner-Ville distribution (WVD), a quadratic time-frequency method that suffers from cross-term interference in multicomponent signals, the chirplet transform is linear and additive, avoiding such artifacts at the cost of requiring parameter optimization.^[4] For chirp-dominated environments like radar and sonar, this linearity promotes sparse representations, concentrating signal energy efficiently without the bilinear distortions of WVD.^[12]

Applications

Signal Processing

The chirplet transform finds significant application in radar and sonar systems for detecting linear frequency-modulated (FM) signals, especially non-stationary ones exhibiting chirp-like behavior. In the 1990s, Steve Mann and Simon Haykin applied the transform to marine radar data for identifying growlers—small iceberg fragments hazardous to shipping—in ice-infested waters. By decomposing radar returns into a basis of multiscale chirps, the method captures the acceleration signatures (or "chirpyness") of bobbing objects, outperforming conventional Doppler processing with detection accuracies up to 98% through successive time-frequency snapshots.^[14]^[7] In time domain reflectometry (TDR), chirplet-based analysis enhances fault detection and localization in cables by processing reflections from chirp-excited signals. Unlike Fourier methods, which suffer from limited resolution for dispersive reflections, the chirplet transform provides sharper time-frequency representations, enabling precise identification of fault positions and types in branched network cables. For instance, polynomial chirplet transform applied to multiple chirp reflectometry signals improves directionality and accuracy in live systems, mitigating noise and dispersion effects.^[15] For harmonics detection in power systems, improved variants of the chirplet transform isolate chirp-like distortions in voltage and current signals under noisy conditions. Research from the 2010s introduced an enhanced chirplet approach that constructs inverse transforms for reconstruction, demonstrating superior contour separation of harmonic components compared to wavelet methods in simulated power quality assessments. This enables effective monitoring and mitigation of non-stationary harmonics in electrical grids.^[16] In speech and audio analysis, the chirplet transform models dynamic spectral features to improve recognition accuracy. It extracts robust time-frequency features for tasks like emotion classification, outperforming short-time Fourier transform in non-stationary signal analysis.^[17] A notable example in marine safety employs the chirplet transform on radar imagery to enhance edge detection amid sea clutter and noise. In growler detection scenarios, it sharpens boundaries of floating hazards by emphasizing chirp-modulated echoes, thereby supporting real-time vessel navigation and collision avoidance in polar regions.^[7]

Other Fields

In biomedical engineering, the chirplet transform has been applied to analyze electroencephalogram (EEG) signals, specifically for characterizing visual evoked potentials (VEPs). Cui et al. (2005) demonstrated its utility in detecting chirp-like neural responses by providing a time-frequency representation that captures the transient and steady-state behaviors of VEPs more sharply than traditional methods like the short-time Fourier transform.^[18] This approach reveals the time-dependent frequency modulations in neural activity evoked by visual stimuli, aiding in the diagnosis of visual pathway disorders. In communications systems, the chirplet transform supports spread spectrum techniques by enabling the despreading and processing of linear frequency modulated (FM) signals, which exhibit chirp characteristics. It has been particularly effective in suppressing nonstationary, chirp-like interference in direct sequence spread spectrum environments, where adaptive chirplet-based excisers correlate the received signal with chirplet atoms to isolate and remove distortions while preserving the desired spread signal. This enhances signal recovery in noisy channels, as shown in applications involving alpha-stable noise interference.^[19] For image processing, the chirplet transform leverages projective geometry to extract features from images subjected to rotations, shears, or other affine distortions, extending its time-frequency framework to two-dimensional domains. Mann's seminal work (1995) introduced this capability by parameterizing chirplets to model affine transformations in the time-frequency plane, which translates to robust detection of oriented or warped patterns in visual data, such as edges or textures in radar imagery. This makes it suitable for tasks like object recognition under viewpoint variations.^[6] In geophysics, the chirplet transform facilitates seismic signal analysis by modeling the chirp-like dispersion in wave propagation, which is crucial for fault detection. It excels at extracting high-resolution time-frequency components from seismograms, such as velocity pulses in near-fault ground motions, allowing geophysicists to identify fault rupture characteristics and predict seismic hazards more accurately than with wavelet-based methods alone. Studies have applied it to sparse representation of seismic data, enhancing the detection of subtle fault-related anomalies.^[20]^[21] Emerging applications since 2020 have incorporated the chirplet transform into machine learning workflows for feature extraction from non-stationary time series, where it generates concentrated time-frequency maps to feed into classifiers. For instance, it has been used to derive discriminative features from speech signals for Parkinson's disease detection, achieving higher classification accuracy by capturing instantaneous frequency variations that linear models overlook.^[22] As of 2025, further applications include the dynamic masking chirplet transform for fault diagnosis in rotating machinery and optical chirplet transform for observing ultrafast pulse structures.^[23]^[24]

Extensions and Variants

Warblet Transform

The warblet transform, introduced by Mann and Haykin in 1992, is a specific variant of the chirplet transform, featuring cyclically varying frequency modulation referred to as the warble rate to better capture quasi-periodic phenomena. This predates the detailed 1995 formulation of the chirplet transform.^[1] The mathematical form of the warblet basis function builds on the chirplet structure but incorporates a periodic chirp parameter, expressed as

\psi(t) = g(t) \exp\left(j \left( \omega_0 t + \frac{\mu(t) t^2}{2} \right) \right),

where g(t) is a window function, \omega_0 is the central frequency, and \mu(t) is a periodic function representing the warble rate. This transform finds applications in modeling quasi-periodic signals, such as vibrations from rotating machinery, where frequency modulation exhibits oscillatory patterns.^[25] Compared to the standard chirplet transform, the warblet offers advantages in handling nonlinear and oscillatory frequency changes through its cyclic modulation, avoiding the need for multiple linear chirp approximations to represent periodic deviations.

Recent Developments

Since the 2010s, research on the chirplet transform has advanced toward greater flexibility in handling complex frequency-modulated signals, with key innovations focusing on linear and nonlinear extensions, improved resolution in noisy environments, and enhanced computational efficiency. These developments address limitations in earlier formulations, such as restricted chirp paths and sensitivity to noise, enabling more robust applications in time-frequency analysis.^[26] The General Linear Chirplet Transform (GLCT), introduced in 2016, extends the traditional chirplet transform to accommodate arbitrary linear frequency-modulated (FM) paths by incorporating a more general kernel that unifies various time-frequency methods like the short-time Fourier transform and linear chirp transform. This allows for optimized parameter estimation through adaptive kernel design, improving energy concentration for signals with varying chirp rates. The GLCT has been applied in mechanical fault diagnosis, demonstrating superior performance over standard chirplet methods in resolving multicomponent signals.^[27] In 2021, the High-Resolution Chirplet Transform was proposed to enhance parameter analysis for noisy signals, employing an iterative refinement approach that combines multiple chirplet transforms with varying parameter settings to achieve finer time-frequency resolution without excessive computational overhead. This method mitigates cross-term interference and improves accuracy in estimating chirp rates and instantaneous frequencies. It addresses post-2010 gaps in handling nonlinear chirps by enabling parameter combinations that provide improved time-frequency resolution compared to single-parameter chirplet transforms, particularly useful in noisy environments, such as biomedical signal processing.^[26] More recently, the Synchrosqueezed Chirplet Transform, developed in 2024, integrates synchrosqueezing techniques with the chirplet framework to enable sharper time-frequency reassignment, significantly reducing the spread of energy distributions for multicomponent linear FM signals. By refining instantaneous frequency and chirp rate estimates through high-order corrections, it achieves higher concentration in the time-chirp-rate plane, outperforming unsynchrosqueezed versions in resolving closely spaced chirps with minimal artifacts. This advancement enhances computational efficiency for real-time processing while maintaining theoretical guarantees for signal recovery.^[28] Adaptive variants of the chirplet transform emerged in the 2010s for power quality analysis, particularly in detecting harmonics through improved formulations that adjust window parameters to better isolate time-varying distortions in electrical signals. The Improved Chirplet Transform (ICT), for instance, refines the kernel to suppress sidelobes and enhance harmonic separation, enabling precise reconstruction of distorted voltages in power systems for simulated harmonics. These adaptations often incorporate machine learning for automated parameter selection, such as optimizing chirp rates via supervised algorithms to adapt to varying disturbance patterns, thereby filling gaps in efficiency for nonlinear and noisy power quality scenarios.^[16]

References

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[PDF] On the Chirp Function, the Chirplet Transform and the ... - IAENG
Mar 25, 2020 · Abstract—The purpose of this extended paper is to provide a review of the chirp function and the chirplet transform and.<|separator|>
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Speech feature extraction using linear Chirplet transform and its ...
This work aims to exploit a new approach to design a speech signal representation in the time-frequency domain via Linear Chirplet Transform (LCT).
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Chirplet Transform in Ultrasonic Non-Destructive Testing and ...
In this paper, the principles of the chirplet transform are explained with a simplified presentation and the studies that used the transform in ultrasonic non- ...
[5]
[PDF] High-resolution chirplet transform: from parameters analysis to ...
The chirplet transform (CT) uses parameters in its window for time-frequency analysis. Multi-resolution CT (MrCT) combines multiple CTs with different ...
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[PDF] The Chirplet Transform, Physical Considerations - WearCam
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PDF | We propose a novel transform, an expansion of an arbitrary function onto a basis of multi-scale chirps (swept frequency wave packets). We apply.
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[PDF] The Chirplet Transform : A Generalization of Gabor ' s Logon ...
We propose a novel transform, an expansion of an arbitrary function onto a basis of multi-scale chirps (swept frequency wave packets).
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Multiple Chirp Reflectometry for Determination of Fault Direction and ...
Polynomial chirplet transform (PCT) is one of the PTFTs that can produce an ... Besides cable fault detection, the potential applications of the ...
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An Improved Chirplet Transform and Its Application for Harmonics ...
In this paper, we proposed the improved chirplet transform (ICT) and constructed the inverse ICT. Finally, by simulating the harmonic voltages, The power of the ...
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Chirplet transform based time frequency analysis of speech signal ...
The chirplet transform is more robust to noise than the traditional TF methods. This is because the chirplet basis functions can adapt to the signal's frequency ...
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Modulation Parameter Estimation of LFM Interference for Direct ...
Aug 6, 2025 · Thus, we formulate a new generalized extended linear chirplet transform to suppress the alpha-stable noise, for a robust time-frequency, ...
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Detection and extraction of velocity pulses of near-fault ground ...
The chirplet transform is a powerful signal processing method that provides more precise insights into the complex structure of a seismic signal. By this ...
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Time-frequency analysis of speech signal using Chirplet transform ...
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The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform.