Joint Time-frequency Distribution Research Articles

Interpretation of Transient Electromagnetic Field Data Using Joint Time-Frequency Analysis

Using the joint time-frequency (TF) analysis method, we interpret transient electromagnetic (TEM) data so as to fully employ information contained in TEM signals. We choose joint TF distribution (or representation) of Wigner-Will distribution and Gabor expansion to transform the one-dimensional time-domain signal to the two-dimensional TF plane, which not only distinguishes responses of different geo-electric sections from each other successfully, but also provides clear physical meaning between exhibited signal feature and electrical structure of the stratum. This describes the transient eddy field excitation and underground attenuation process in time and frequency respects. The fitting difference is reduced and the precision of quantitative interpretation is improved by means of joint TF inversion with time and frequency window data.

Time–frequency signal analysis based on the windowed fractional Fourier transform

A new signal-adaptive joint time–frequency distribution for the analysis of nonstationary signals is proposed. It is based on a fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms. Improvement over the standard time–frequency representations is achieved when the principal axes of a signal (defined as mutually orthogonal directions in the time–frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum) do not correspond to time and frequency. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the windowed Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the knowledge of three second-order fractional Fourier transform moments. Numerical simulations confirm a qualitative advantage in the time–frequency representation, when the calculation is done in the optimal fractional domain. The approach can be generalized to the time–frequency distributions from the Cohen class.

Measurement of the Page function of an ultrashort laser pulse

We demonstrate an experimental setup based on an ultrafast optical plasma shutter which measures the Page function of an ultrashort laser pulse. The Page function is a joint time frequency distribution function that allows for a direct reconstruction of the phase and the amplitude of the electric field of an ultrashort optical pulse without employing iterative algorithms.

Modified Cohen-Lee time-frequency distributions and instantaneous bandwidth of multicomponent signals

Cohen (1989, 1995) has introduced and extensively studied and developed the concept of the instantaneous bandwidth of a signal. Specifically, instantaneous bandwidth is interpreted as the spread in frequency about the instantaneous frequency, which is itself interpreted as the average frequency at each time. This view stems from a joint time-frequency distribution (TFD) analysis of the signal, where instantaneous frequency and instantaneous bandwidth are taken to be the first two conditional spectral moments, respectively, of the distribution. However, the traditional definition of instantaneous frequency, namely, as the derivative of the phase of the signal, is not consistent with this interpretation, and new definitions have therefore been proposed previously. We show that similar problems arise with the Cohen-Lee (1988, 1989) instantaneous bandwidth of a signal and propose a new formulation for the instantaneous bandwidth that is consistent with its interpretation as the conditional standard deviation in frequency of a TFD. We give the kernel constraints for a distribution to yield this new result, which is a modification of the kernel proposed by Cohen and Lee. These new kernel constraints yield a modified Cohen-Lee TFD whose first two conditional moments are interpretable as the average frequency and bandwidth at each time, respectively.

New Gear-Fault-Detection Parameter by Use of Joint Time-Frequency Distribution

Anewgear-fault-detectionparametercalledNP4 isintroduced.Thisfault-detectionparameterutilizestheproperties of the joint time ‐frequency analysis given by the Wigner ‐Ville distribution (WVD) and kurtosis. With the WVD, the instantaneous power of the gear-vibration signature for one complete rotor revolution can be obtained. The presence of single-gear-tooth damage can be manifested by the existence of an instantaneous power distribution with a peakedness larger than the normal distribution. The normalized kurtosis, a fourth-order statistical parameter calculated for the instantaneous power distribution, provides the gear-fault-detection parameter NP4. The developed fault-detection parameter NP4 is sensitive to gear-tooth damage, especially for damage in a single tooth. The application of this NP4 fault-detection parameter was demonstrated by experimental data obtained from a gear test rig. The results showed that the NP4 parameter, used with the WVD, can provide an accurate fault identie cation of gear-tooth damage. The parameter NP4 would be of help to the practitioners in the e eld of machine health monitoring.

On complex demodulation and time-frequency distributions for broadband reflectometry signals

Complex demodulation is used to generate a positive joint time-frequency distribution for broadband reflectometry signals, the so-called demodulogram, which is actually the dual of the well-known spectrogram and which, for a given signal, is obtained by squaring the modulus of the inverse Fourier transform of the signal frequency-windowed Fourier spectrum. The demodulogram and its properties are discussed, and the former is applied to the analysis of typical reflectometry data, being shown, in particular, that it can indeed yield what is strictly defined as the instantaneous frequency of discrete-time reflectometry signals.

Response of thin semi-infinite cylindrical elastic shell to axisymmetric impulsive radial excitation

The transient response of a semi-infinite, thin cylindrical elastic shell to an impulsive, axisymmetric radial excitation applied at the origin is investigated in the time and frequency domains using Fourier and Laplace transform methods. Solutions for the radial and axial displacements as functions of time and axial position show the dispersive nature of the wave propagation. High-frequency components propagating at the plate wave speed are followed by slower components occurring near the shell ring frequency. The temporal variation of the spectral components of the transient displacement is examined using a joint time–frequency distribution function. Analytical results are confirmed by experimental measurements on a brass tube.