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.

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 [

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. [

Due to its multiresolution analysis ability, wavelet transform has been widely studied for linear and nonlinear system identification. Ruzzene et al. [

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 [

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. For completeness, the basic theory of CWT and CT is introduced in Section

A mother wavelet

The local characteristic and time-frequency representation of a signal

The resolution of the wavelet transform in time and frequency domains is defined by the following equations:

The work presented in this paper utilizes the Morlet wavelet, defined as

The Morlet wavelet is a complex wavelet as shown in Figure

Morlet wavelet.

Most of the real signals are asymptotic and can be given in the form of the sum of single components:

The ridge and skeleton of the CWT contain most of the information of the original signal, with the ridge being defined as

For an arbitrarily signal

WVD has the highest time-frequency resolution, and it is an ideal approach to represent mono component nonstationary signals. As shown in (

The linear chirplet transform is defined as the inner product of the signal

The Gauss chirplet is generated by time shifting, frequency shifting, scaling, and rotating of the original Gauss function. A typical Gauss mother chirplet and its WVD are shown in Figure

Chirplet and its Wigner-Ville distribution.

Since the Gauss chirplets do not form an orthogonal basis, matching pursuit is a promising solution to realize adaptive decomposition. The vital idea of the method is to select optimal chirplets

The object of MP is to find optimal parameters of chirplets and generate

The cross-term does not exist in the chirplet spectrogram, and the resolution of time-frequency depends on the durations of the optimal chirplets. Once the chirplet spectrum is obtained, a local maximum algorithm [

In the continuous wavelet transform-based (CWT) method, the vibration signal is decomposed using wavelet functions. The resolution is dependent on the time and frequency windows of each wavelet. Instantaneous frequencies can be estimated in different sizes of frequency windows by controlling translation and dilatation. However, as mentioned before, an increase of time resolution will lead to a decrease of frequency resolution, and hence different resolutions will be used in different modes. In the time-varying systems, the frequency resolution varies with time. The identification procedure using CWT is as follows:

Calculate the CWT of the vibration signal using the Morlet wavelet.

Extract the ridges of the CWT using local maximum algorithm.

Estimate the instantaneous frequencies by (

In the CT-based method, the cross-term of the classical WVD is eliminated. The resolution is dependent on the time-frequency property of optimal chirplets. The MP algorithm ensures the sparsity of the signal decomposition. The noise is departed in the residual signal while most of the time-frequency characteristics of the original response are retained in the selected chirplets. It is also noticed that CWT is a special case of CT where only time shifting and scaling parameters are available and other parameters are equal to zero. The identification procedure using CT is as follows:

Obtain the parameters of optimal chirplets via MP algorithm.

Calculate the WVD of each optimal chirplet basis. The chirplet spectrum is obtained by the linear weighted sum of the WVDs.

Estimate the instantaneous frequencies using local maximum algorithm.

Since the basis functions of CT are linear chirplet functions, the method is effective in identifying systems with linear varying frequencies; however, in case of nonlinear varying frequencies, the nonlinear trajectories are approached by several linear segments, and thus significant errors occur. In order to solve this problem, a modified version of CT is utilized. In the matching pursuit process, the key step is to solve the optimization problem presented in (

In the modified CT, the duration of optimal chirplets is restricted to

The procedure of the modified CT based method is shown in Figure

Identification procedure.

It should also be noted that the bandwidth of each chirplet gets wider because of the restriction of duration, and the frequency resolution is reduced. Nevertheless, it is still much better than that of the CWT-based method. The effectiveness of the modified CT is verified in the following section.

In this section, several examples of time-varying systems and systems with closely spaced modes are investigated to illustrate the effectiveness and accuracy of the algorithms presented in Figure

The single-degree-of-freedom linear time-varying system is governed by the general equation:

Displacement response of case 1.

Displacement response with noise of case 1.

The FFT spectrum of the response is shown in Figure

FFT of displacement response of case 1.

CWT scalogram of displacement response of case 1.

CT spectrogram of case 1.

From the comparison, it can be seen that both of the methods present the true natural frequency of the system. However, a difference is clear: the frequency resolution of CT is much higher than that of CWT. In the CWT scalogram, higher frequency has lower resolution, while the CT spectrogram has uniform resolution for all frequencies. The identification results obtained by local maximum algorithm of both CWT and CT are shown in Figure

Identification results of case 1.

The methods are applied to another kind of time-varying system: the parameters vary nonlinearly. In this case,

The CWT scalogram and the CT spectrogram are shown in Figures

CWT scalogram of displacement response of case 2.

CT spectrogram of displacement response of case 2.

Modified CT spectrogram of case 2.

Identification results of case 2.

In order to evaluate the accuracy of the modified CT, the residual sum of squares (RSS) is introduced as follows:

Table

RSS of different identification results of case 2.

Algorithms | CWT | CT | Modified CT |
---|---|---|---|

RSS | 24.332 | 0.442 | 0.261 |

In order to find the optimal value of the restriction parameters

RSS with the increase of

A two-degree-of-freedom time-varying system with closely spaced modes is developed to evaluate the effectiveness of the proposed identification methods. The spring-mass-damper model is shown in Figure

Spring-mass-damper model of closely spaced modes.

Both CWT- and CT-based methods are performed to estimate the natural frequencies of the two modes. Figure

CWT scalogram of displacement response of case 3.

CT spectrogram of displacement response of case 3.

In order to verify the effectiveness of the proposed method in the application to real data, the

CT spectrogram of

Modified CT spectrogram of

From Figures

Time-frequency resolution is a key factor in instantaneous frequency identification. As mentioned in Section

In cases of nonlinear varying frequencies, computation cost remains a challenge. Due to the restriction in time support, the number of optimal chirplets increases in the proposed modified approach; in other words, more iterations are required in MP and the computation cost is significantly increased. Introducing a nonlinear basis chirplet is also a promising approach to improve accuracy; see reference [

Instantaneous frequency identification based on CT and WVD is proposed. The adaptive chirplet transform using MP is employed to represent the vibration signal in the time-frequency plane. Complete frequency identification procedure is established, and local maximum algorithm is utilized to extract the instantaneous frequency from the CT spectrogram.

A system with closely spaced modes and two kinds of time-varying systems are investigated to validate the proposed method. The results are compared with an instantaneous frequency identification method based on CWT. Results show that both the CWT- and CT-based methods present the true frequencies. However, the frequency resolution of the CT-based method is much higher than that of the CWT-based method; CT clearly separated the close modes while CWT failed. The antinoise performances of the two methods are compared by adding a noise of 10 dB signal-to-noise ratio. Results show that the CT-based identification method has an excellent antinoise performance.

A modified version of CT is proposed to identify nonlinear varying frequencies, and its effectiveness is verified using both simulated and practical signals. Results show that the modified CT successfully improves the accuracy in identifying nonlinear varying frequencies.

The authors declare that there is no conflict of interests regarding the publication of this paper.