Resources and Help The chirplet transform: physical considerations Abstract: We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions which we will call q-chirps for short , giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a "time-frequency-scale volume" and thus encompasses both the short-time Fourier transform as a slice along the time and frequency axes and the wavelet transform as a slice along the time and scale axes. In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time obtained through convolution with a q-chirp and shear in frequency obtained through multiplication by a q-chirp.

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Published24 Dec Abstract An instantaneous frequency identification method of vibration signal based on linear chirplet transform and Wigner-Ville distribution is presented. This method has an obvious advantage in identifying closely spaced and time-varying frequencies.

The matching pursuit algorithm is employed to select optimal chirplets, and a modified version of chirplet transform is presented to estimate nonlinear varying frequencies.

Because of the high time resolution, the modified chirplet transform is superior to the original method. The proposed method is applied to time-varying systems with both linear and nonlinear varying stiffness and systems with closely spaced modes. A wavelet-based identification method is simulated to compare with the proposed method, with the comparison results showing that the chirplet-based method is effective and accurate in identifying both time-varying and closely spaced frequencies.

A bat echolocation signal is used to verify the effectiveness of the modified chirplet transform. The result shows that it will significantly increase the accuracy of nonlinear frequency trajectory identification. Introduction Natural frequency is a crucial parameter in dynamic systems, and most of the frequency identification methods are based on the Fourier transform.

However, these methods are effective only when the frequency contents of the vibration signals are time-invariant. Linear time-varying LTV systems are often used in engineering, such as large flexible structures for outer space exploration, spacecraft, launch vehicles, and vehicle-bridge systems [ 1 , 2 ].

Parameter identification methods for LTV systems have been widely studied in the last two decades, but most of them are based on inaccurate and computationally sectional time invariance [ 3 , 4 ]. Because responses of LTV systems are nonstationary, time-frequency analysis becomes a necessary and unique approach for identification of time-varying parameters of LTV systems.

Shi et al. Unfortunately, EMD is not capable of decomposing a signal having multiple components of close frequencies. Chen and Wang [ 7 ] proposed a signal decomposition theorem called analytical mode decomposition to address the challenges in closely spaced modes. Due to its multiresolution analysis ability, wavelet transform has been widely studied for linear and nonlinear system identification.

Ruzzene et al. A comparative study of the Morlet wavelet, Cauchy wavelet, and harmonic wavelet for system identification was performed by Le and Argoul [ 9 ], and a criterion for choosing mother wavelets was also introduced, with this technique being extended to the response from ambient excitation tests [ 10 ]. Based on the orthogonality of wavelet scaling functions, Ghanem and Romeo [ 11 ] proposed a method to transform the differential equation of an LTV system into simple linear equations and hence the parameter identification problem became a simple problem of solving linear equations.

Xu et al. Recently an exploratory study of wavelet-based frequency response functions for time-variant systems was presented by Staszewski and Wallace [ 13 ]. It is a new way to describe time-variant systems in the time-frequency domain using CWT. However, according to the Heisenberg uncertainty principle, wavelet transform is incapable of simultaneously providing high resolutions in both time and frequency domains.

Wigner-Ville distribution WVD is a quadratic time-frequency representation of a dynamical signal, and its time-frequency resolution is higher than that of CWT [ 14 ]. However, cross-terms occur in WVD if the processed signal contains multiple components.

Many studies have been done to reduce these cross-terms [ 15 ]. Chen et al. The matching pursuit MP algorithm [ 17 ] was employed in the chirplet signal decomposition, and the vibration signal is decomposed adaptively into a series of linear chirplets.

The method has been applied in fault diagnosis [ 18 ] and earthquake signal analysis [ 19 ]. Wang and Jiang [ 21 ] introduced a parameter called curvature to extend the original chirplet atom to cubic form, and results show that the accuracy of time-frequency representation is improved. Yin et al. In this paper, an instantaneous frequency identification method based on chirplet transform CT and MP is proposed.

The effectiveness of the method in identifying close modes and time-varying systems is investigated in simulated mass-stiffness-damping systems. A modified CT is presented to reduce the errors occurring in identifying nonlinear varying frequencies. An echolocation chirp signal is utilized to assess the method for practical signals. The results are compared with the CWT-based method.

Conclusions are drawn in Section 5. Continuous Wavelet Transform is a square integral function which has the following admissibility condition: where is the Fourier transform of. This kind of function is fast attenuated and supports a short duration in the time domain.

A whole family of these functions can be generated by shifts in the time domain and scaling in the scale domain: where is the scale parameter which is related to frequency and is the translation parameter related to time. The local characteristic and time-frequency representation of a signal can be illustrated by decomposing into wavelet coefficients using the basis of wavelet functions.

This can be expressed by the following equations: where is the complex conjugate of and are the wavelet coefficients. The resolution of the wavelet transform in time and frequency domains is defined by the following equations: where are the duration and bandwidth of the wavelet function, respectively. The work presented in this paper utilizes the Morlet wavelet, defined as The Morlet wavelet is a complex wavelet as shown in Figure 1. Each wavelet basis supports a time-frequency window restricted by.

It is noted that an increase of time resolution will lead to a decrease of frequency resolution, which meets the requirements of the Heisenberg uncertainty principle. Figure 1 Morlet wavelet. Most of the real signals are asymptotic and can be given in the form of the sum of single components: where are the amplitudes and are the phases. It is assumed that, in an asymptotic signal, the variation of phase is much faster than the variation of amplitude.

The analytical signal of a single component can be given by The ridge and skeleton of the CWT contain most of the information of the original signal, with the ridge being defined as where is a constant depending on the mother wavelet.

Several algorithms can be found to extract the ridge. In this paper, the local maximum algorithm is performed. Once the ridge is obtained, the instantaneous angle frequency can be obtained by is the center angle frequency of the wavelet. For example, the WVD of signal.

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## Chirplet transform

Resources and Help Velocity Synchronous Linear Chirplet Transform Abstract: Linear transform has been widely used in time-frequency analysis of rotational machine vibration. However, the linear transform and its variants in current forms cannot be used to reliably analyze rotational machinery vibration signals under nonstationary conditions because of their smear effect and limited time variability in time-frequency resolution. As such, this paper proposes a new time-frequency method, named velocity synchronous linear chirplet transform VSLCT. It can effectively alleviate the smear effect and can dynamically provide desirable time-frequency resolution in response to condition variations. The smearing problem is resolved by using linear chirplet bases with frequencies synchronous with shaft rotational velocity, and the time-frequency resolution is made responsive to signal condition changes using time-varying window lengths.

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## P-Chirps and P-Chirplets

CA Abstract We propose a novel transform, an expansion of an arbitrary function onto a basis of multi-scale chirps swept frequency wave packets. We apply this new transform to a practical problem in marine radar: the detection of floating objects by their "acceleration signature" the "chirpyness" of their radar backscatter , and obtain results far better than those previously obtained by other current Doppler radar methods. Each of the chirplets essentially models the underlying physics of motion of a floating object. Because it so closely captures the essence of the physical phenomena, the transform is near optimal for the problem of detecting floating objects. Besides applying it to our radar image processing interests, we also found the transform provided a very good analysis of actual sampled sounds, such as bird chirps and police sirens, which have a chirplike nonstationarity, as well as Doppler sounds from people entering a room, and from swimmers amid sea clutter. We then extended that generalization further to include what we call "chirplets". We have coined the term "chirplet transform" to denote this overall generalization.

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## Select a Web Site

Published24 Dec Abstract An instantaneous frequency identification method of vibration signal based on linear chirplet transform and Wigner-Ville distribution is presented. This method has an obvious advantage in identifying closely spaced and time-varying frequencies. The matching pursuit algorithm is employed to select optimal chirplets, and a modified version of chirplet transform is presented to estimate nonlinear varying frequencies. Because of the high time resolution, the modified chirplet transform is superior to the original method. The proposed method is applied to time-varying systems with both linear and nonlinear varying stiffness and systems with closely spaced modes. A wavelet-based identification method is simulated to compare with the proposed method, with the comparison results showing that the chirplet-based method is effective and accurate in identifying both time-varying and closely spaced frequencies.