An analysis of some time-frequency and time-scale distributions

This paper presents an analysis of the representation of instantaneous frequency and group delay using time-frequency transforms or distributions of energy density domain. The time-frequency distributions which ideally represent the instantaneous frequency or group delay (itfd) are defined. Closeness to the itfd is chosen as a criterion for comparison of various commonly used distributions. It is shown that the Wigner distribution is the best among them, with respect to this criterion. The wavelet and scaled forms of the Wigner distribution are defined and analyzed. In the second part of the paper we extended the analysis to the multicomponent signals and cross terms effects. On the basis of that analysis, an efficient method, derived from the analysis of the Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution. The theory is illustrated on numerical examples.     

Unified approach to noise analysis in the Wigner distribution and spectrogram

An analysis of time-frequency representations of noisy signals is performed. Using the method for time-frequency signal analysis which was recently defined by Stankovic (the S-method), the influence of noise on the two most important distributions (spectrogram and Wigner distribution) is analyzed in unified manner. It is also shown that, for signals whose instantaneous frequency is not constant, an improvement over the spectrogram and the Wigner distribution performances in a noisy environment may be achieved using the S-method. The expressions for mean and variance are derived. Results are given for several illustrative and numerical examples.     

A new time-frequency transform for non-stationary signals with any nonlinear instantaneous phase

A new generic adaptive time-frequency transform based on the Wigner distribution is proposed for amplitude estimation of transient signals with any nonlinear non-polynomial variation of the instantaneous frequency. It is for use in situations where independent synchronous measurements of the instantaneous frequency are available. It shows that the new transform tracks the instantaneous frequency and estimates signal amplitude without errors along the curve of the instantaneous frequency. The paper also applies the new transform to signals with the non-linear sinusoidal and exponential variations in the instantaneous phase and determines formulae for transform in these cases. The paper compares the new transform with the Wigner distribution in several cases and demonstrates that the new transform is more effective at amplitude estimation of signals with nonlinear variation of the instantaneous frequency. New analytic formulae are obtained for the new transform and the Wigner distribution in both the sinusoidal and the exponential cases. An analytic formula is obtained which relates the new transform to the Wigner distribution in the sinusoidal case. This formula is inverted to obtain a previously unknown formula for the Wigner distribution of any signal. It is shown that the new transform could be used for adaptive processing of transient signals with instantaneous frequency variations. The paper studies the performance of the new transform with no adaptation, partial adaptation and complete adaptation.

A method for improved distribution concentration in thetime-frequency analysis of multicomponent signals using the L-Wignerdistribution

The energy location in the Cohen class of time-frequency distributions is analyzed. If the instantaneous frequency is linear, then only the Wigner distribution produces the ideal energy concentration. The scaled version of the Wigner distribution (L-Wigner distribution), is used to improve the time-frequency representation of signals with nonlinear instantaneous frequencies. In the case of multicomponent signals, the cross terms, appearing in the Wigner distribution and in the L-Wigner distribution, can be easily removed or reduced in a computationally very efficient way. The theory is illustrated on the numerical examples with multicomponent noisy signals     

Estimation of instantaneous frequency using the discrete Wigner distribution

Analytical expressions for the performance of the discrete Wigner distribution (DWD) in estimating the instantaneous frequency of linear frequency modulated signals in additive white noise are derived and verified using simulation. It is shown that the DWD peak provides an optimal estimate at high input signal-to-noise ratios. The applicability of these results to the general case of nonlinear FM signals is discussed.<>     

Signal reconstruction from two close fractional Fourier power spectra

Based on the definition of the instantaneous frequency (signal phase derivative) as a local moment of the Wigner distribution, we derive the relationship between the instantaneous frequency and the derivative of the squared modulus of the fractional Fourier transform (fractional Fourier transform power spectrum) with respect to the angle parameter. We show that the angular derivative of the fractional power spectrum can be found from the knowledge of two close fractional power spectra. It permits us to find the instantaneous frequency and to solve the phase retrieval problem up to a constant phase term, if only two close fractional power spectra are known. The proposed technique is noniterative and noninterferometric. The efficiency of the method is demonstrated on several examples including monocomponent, multicomponent, and noisy signals. It is shown that the proposed method works well for signal-to-noise ratios (SNRs) higher than about 3 dB. The appropriate angular difference of the fractional power spectra used for phase retrieval depends on the complexity of the signal and can usually reach several degrees. Other applications of the angular derivative of the fractional power spectra for signal analysis are discussed. The proposed technique can be applied for phase retrieval in optics, where only the fractional power spectra associated with intensity distributions can be easily measured.     

On the local frequency, group shift, and cross-terms in somemultidimensional time-frequency distributions: a method formultidimensional time-frequency analysis

Presents an analysis of the representation of local frequency and group shift using multidimensional time-frequency distributions. In the second part of the correspondence, the authors extend the analysis to the multicomponent signals and cross-terms effects. On the basis of that analysis, an efficient method, derived from the analysis of the multidimensional Wigner distribution defined in the frequency domain, is proposed. This method provides some substantial advantages over the Wigner distribution: the well known cross-terms effects are reduced or completely removed; the oversampling of signals is shown to be unnecessary; and the computation time can be significantly reduced, as well. The theory is illustrated by a two-dimensional numerical example     

Performance of quadratic time-frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic time-frequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed.     

二维信号Wigner分布的数值计算

Wigner分布函数作为一个同时表达空间 频率变量的函数 ,在光学领域中有着广泛的应用。提出了二维信号Wigner分布函数的数值计算方法 ,并模拟了一些二维实信号 (圆孔 ,圆环和方孔 )的Wigner分布函数     

Multiwindow time-varying spectrum with instantaneous bandwidth andfrequency constraints

We build on Cohen's work (Cohen and Lee 1988, 1989; Cohen 1990, 1995) on instantaneous bandwidth and frequency by extending it to a multiwindow framework for polynomial phase signals. Unlike the case with a single spectrogram, which Cohen considered, our multiwindow framework allows one to obtain a time-varying spectral estimate that simultaneously satisfies instantaneous bandwidth and frequency constraints. We then develop a method utilizing this new multiwindow time-varying spectral technique for estimating the instantaneous frequency of a signal. The method is computationally simple, asymptotically unbiased for noise-free signals, and provides a signal-to-noise ratio (SNR) improvement of more than 3 dB over other estimators, including the cross-polynomial Wigner distribution method, for quadratic and cubic FM signals     

Tomographic time-frequency analysis and its application towardtime-varying filtering and adaptive kernel design for multicomponentlinear-FM signals

Since line integrals through the Wigner spectrum can be calculated by dechirping, calculation of the Wigner spectrum may be viewed as a tomographic reconstruction problem. In the paper, the authors show that all time-frequency transforms of Cohen's class may be achieved by simple changes in backprojection reconstruction filtering. The resolution/cross-term tradeoff that occurs in time-frequency kernel selection is shown to be analogous to the resolution-ringing tradeoff that occurs in computed tomography (CT). “Ideal” reconstruction using a purely differentiating backprojection filter yields the Wigner distribution, whereas low-pass differentiating filters produce cross-term suppressing distributions such as the spectrogram or the Born-Jordan distribution. It is also demonstrated how this analogy can be exploited to “tune” the reconstruction filtering (or time-frequency kernel) to improve the ringing/resolution tradeoff. Some properties of the projection domain, which is also known as the Radon-Wigner transform, are characterized, including the response to signal delays or frequency shifts and projection masking or convolution. Last, time-varying filtering by shift-varying convolution in the Radon-Wigner domain is shown to yield superior results to its analogous Cohen's class adaptive transform (shift-invariant convolution) for the multicomponent, linear-FM signals that are investigated     

Applied bias slewing in transient Wigner function simulation ofresonant tunneling diodes

The Wigner function formulation of quantum mechanics has shown much promise as a basis for accurately modeling quantum electronic devices, especially under transient conditions. In this work, we demonstrate the importance of using a finite applied bias slew rate (as opposed to instantaneous switching) to better approximate experimental device conditions, and thus to produce more accurate transient Wigner function simulation results. We show that the use of instantaneous (and thus unphysical) switching can significantly impact simulation results and lead to incorrect conclusions about device operation. We also find that slewed switching can reduce the high computational demands of transient simulations. The resonant tunneling diode (RTD) is used as a test device, and simulation results are produced with SQUADS (Stanford QUAntum Device Simulator)     

Virtues and vices of quartic time-frequency distributions

We present results concerning three different types of quartic (fourth order) time-frequency distributions (TFDs). First, we present new results on the previously introduced local ambiguity function and show that it provides more reliable estimates of instantaneous chirp rate than the Wigner distribution. Second, we introduce the class of quartic, shift-covariant, time-frequency distributions and investigate distributions that localize quadratic chirps. Finally, we present a shift covariant distribution of time and chirp rate     

Evolutionary time?frequency distributions using Bayesian regularised neural network model

Time-frequency distributions (TFDs) that are highly concentrated in the time-frequency plane are computed using a Bayesian regularised neural network model. The degree of regularisation is automatically controlled in the Bayesian inference framework and produces networks with better generalised performance and lower susceptibility to over-fitting. Spectrograms and Wigner transforms of various known signals form the training set. Simulation results show that regularisation, with input training under Mackay's evidence framework, produces results that are highly concentrated along the instantaneous frequencies of the individual components present in the test TFDs. Various parameters are compared to establish the effectiveness of the approach.     

A resolution comparison of several time-frequency representations

Two signal components are considered resolved in a time-frequency representation when two distinct peaks can be observed. The time-frequency resolution limit of two Gaussian components, alike except for their time and frequency centers, is determined for the Wigner distribution, the pseudo-Wigner distribution, the smoother Wigner distribution, the squared magnitude of the short-time Fourier transform, and the Choi-Williams distribution. The relative performance of the various distributions depends on the signal. The pseudo-Wigner distribution is best for signals of this class with only one frequency component at any one time, the Choi-Williams distribution is most attractive for signals in which all components have constant frequency content, and the matched filter short-time Fourier transform is best for signal components with significant frequency modulation. A relationship between the short-time Fourier transform and the cross-Wigner distribution is used to argue that, with a properly chosen window, the short-time Fourier transform of the cross-Wigner distribution must provide better signal component separation that the Wigner distribution     

Time-frequency distributions with complex argument

A distribution highly concentrated along the group delay or the instantaneous frequency (IF) is presented. It has been defined by introducing a signal with a complex argument in time-frequency (TF) analysis. Realization of a signal with a complex argument, using a signal with a real argument, is described. The reduced interference realization of the complex argument distribution, in the case of multicomponent signals, is presented. The proposed distribution is used for the IF estimation. It is shown that the estimation can be improved with respect to the Wigner distribution based one since the bias can be significantly reduced with only a slight increase of the variance. The theory is illustrated by examples     

Generalized instantaneous parameters and window matching in thetime-frequency plane

The concept of instantaneous parameters, which has previously been associated exclusively with 1-D measures like the instantaneous frequency and the group delay, are extended to the 2-D time-frequency plane. Such generalized instantaneous parameters are associated with the short-time Fourier transform. They may also be interpreted as local moments of certain time-frequency distributions. It is shown that these measures enable local signal behavior to be characterized in the time-frequency plane for nonstationary deterministic signals. The usefulness of the generalized instantaneous parameters is demonstrated in their application to optimal selection of windows for spectrograms. This is achieved through window matching in the time-frequency plane. An algorithm is provided that illustrates the performance of this window matching. Results based on simulated and real data are presented     

Dynamic properties of an all solid-state frequency-shifted feedbacklaser

Frequency-shifted feedback (FSF) laser exhibits outstanding features in its oscillation spectrum. We analyze build-up dynamic properties of the FSF laser by means of rate equation and analyze steady-state dynamic properties of the FSF laser by means of Wigner-Ville distribution of intracavity electric field. Furthermore, we analyze instantaneous oscillation frequency at peak spectral intensity and oscillation bandwidth of its spectrum using the formula of instantaneous spectral intensity derived from Wigner-Ville distribution of intracavity electric field. These analytical results are in good agreement with the experimental ones which have been observed by a diode-pumped Nd:YVO₄ FSF laser. It becomes clear that the FSF laser supports many frequency components simultaneously even though the gain medium is homogeneously broadened and has a continuously chirped frequency components of comb in which the creation of chirped frequency components are strongly correlated in phase because of a replica of the preceding components. Also, the instantaneous oscillation frequency is closely related to the detuning frequency which depends on the total net gain in the cavity and the gain bandwidth of atomic transition. The oscillation bandwidth is defined as the product of the saturation-broadened bandwidth and the total resonant modes contributing to FSF operation