Smoothed Wigner Distribution Research Articles

Time–Frequency Analysis of the Impulse Response

The impulse response is a key function for the analysis and design of systems, therefore understanding its properties is a fundamental problem. To investigate the nonstationary structure of the impulse response, we derive its time–frequency representation both in the Wigner and smoothed Wigner distribution domains. First, we consider the class of single-input single-output (SISO) systems, which models a wide variety of physical phenomena. The obtained Wigner distribution for the class of SISO systems is made by frequency-translated versions of the Wigner distribution of the impulse response for a first-order system, a T-shaped function of time and frequency, plus the interference terms, which are effectively filtered out in the smoothed Wigner domain. Then, we consider the case of multiple-input multiple-output (MIMO) systems, whose impulse response is the collection of the individual responses obtained by applying an impulse to every input separately. Also in this case, the Wigner distribution of the impulse response depends on the T function. We apply our results to a two-dimensional MIMO harmonic oscillator, and to a SISO system with three resonances. Finally, we propose the key idea of a system identification method that operates directly in the time–frequency domain by fitting the smoothed T function. We show that the method works for a nonlinear oscillator with a large cubic term, as well as for a time-varying harmonic oscillator, and that it can tolerate the presence of both stationary and nonstationary noise.

Nonstationary vibration analysis in the smoothed Wigner domain

Dynamical systems represent fundamental models for vibration analysis. When the input of such dynamical systems is nonstationary also the output is nonstationary, and its frequency content changes with time. Time-frequency analysis provides an effective representation of this time-varying spectrum. Even more effective is the direct time-frequency representation of dynamical systems. We first show how to transform a dynamical system in the domain of the smoothed Wigner distribution. The result is a time-frequency dynamical system whose input and output are the smoothed Wigner distributions of the input and output signals in time, respectively. Then, we illustrate how to compute the smoothed Wigner output when the input to the dynamical system in time belongs to a class of common nonstationary inputs, including a delta function and a short duration sinusoid. Finally, we show how to obtain the smoothed Wigner output when the input is a linear combination of these nonstationary signals. We provide a series of examples that show how the time-frequency representation of dynamical systems can unveil the spectral structure of nonstationary vibrations.

Measurement-induced decoherence and Gaussian smoothing of the Wigner distribution function

We study the problem of measurement-induced decoherence using the phase-space approach employing the Gaussian-smoothed Wigner distribution function. Our investigation is based on the notion that measurement-induced decoherence is represented by the transition from the Wigner distribution to the Gaussian-smoothed Wigner distribution with the widths of the smoothing function identified as measurement errors. We also compare the smoothed Wigner distribution with the corresponding distribution resulting from the classical analysis. The distributions we computed are the phase–space distributions for simple one-dimensional dynamical systems such as a particle in a square-well potential and a particle moving under the influence of a step potential, and the time–frequency distributions for high-harmonic radiation emitted from an atom irradiated by short, intense laser pulses.

Application of time-frequency analysis to landmine detection using a laser Doppler vibrometer

The feasibility of detecting landmines acoustically was recently demonstrated by measuring insonified ground surface velocity using a laser Doppler vibrometer (LDV), since the presence of a mine enhances ground matrix velocity [J. M. Sabatier and N. Xiang, J. Acoust. Soc. Am. 106, No. 4, Pt. 2 (1999)]. Although mines were initially detected using discrete LDV measurements, they were detected for this work by spatially continuous measurement, obtained by sweeping the beam onto the ground surface with a constant velocity in order to hasten the detection process. The ground was excited by single-tone acoustic waves. After LDV signal demodulation, a sine wave was obtained which was amplitude-modulated due to mine presence and corrupted by speckle noise ‘‘bursts’’ whose energy grew with the sweeping beam velocity. The signal was time-frequency analyzed by means of the smoothed Wigner distribution that provides straightforward physical interpretation since time is proportional to beam position. The mine was detected when the signal energy was enhanced at the excitation frequency. The wideband nature of speckle ‘‘bursts’’ allowed corrupted signal sections to be discarded. Results are provided for different mine depths, types of ground, and sweeping velocities. The limits of the analysis are investigated.

Comparison of techniques for time–frequency analysis of the ECG during human ventricular fibrillation

Ventricular fibrillation (VF) is a catastrophic and potentially lethal breakdown of heart rhythm. Despite its importance, it remains poorly understood; there are no effective techniques for predicting or preventing VF, and it can only be reliably terminated by a large current pulse: defibrillation. There is currently great interest in using linear and nonlinear signal processing techniques to characterise recordings of VF. Five episodes of VF were recorded from patients in a Coronary Care Unit. Time–frequency distributions (TFDs) of the first 10 s of these recordings were estimated with the short-time Fourier transform, Wigner–Ville, smoothed Wigner–Ville and Choi–Williams algorithms. The smoothed Wigner distribution was found to give a TFD reasonably free of cross-terms, without the additional computational cost of the Choi–Williams algorithm. Used in tandem with the predictable short-time Fourier transform, the smoothed Wigner TFD is a valuable tool for characterising the TFD of human VF.

Comparison of certain methods of analysis of multicomponent nonstationary signals

Metrological aspects are discussed of the application of time-frequency representations such as Fourier spectrograms, Wigner distributions, and smoothed Wigner distributions to the analysis of multicomponent nonstationary signals encountered, in particular, in vibration diagnosis.

Pseudo-Wigner distribution for analysis of pulsed Doppler ultrasound

Existing pulsed Doppler ultrasound systems apply the spectrogram as a tool for analysis and display of signals scattered from the blood. The spectrogram is a time-frequency representation (TFR) of a signal that maps a one-dimensional signal of time into a two-dimensional function of time and frequency. The analysis of Doppler ultrasound signals requires application of a two-dimensional TFR rather than one-dimensional spectral representations due to the nonstationary nature of the signals scattered from blood. The classical spectrogram is a smoothed Wigner distribution (SWD) with a specific smoothing function. For this smoothing function, the smoothing, and hence the resolution in time and frequency, cannot be controlled independently. The purpose of this study is to examine the application of other SWD's to analysis and display of Doppler ultrasound signals. The present paper concentrates on the pseudo-Wigner distribution (PWD). The PWD and the spectrogram are examined and compared as analysis tools for nonstationary Doppler ultrasound signals. The performance of these two TFR's as a function of Doppler bandwidth is evaluated and compared for time-varying flow. >

Analysis and filtering using the optimally smoothed Wigner distribution

The authors consider the analysis and filtering of a deterministic signal with slowly time-varying spectra using the optimally smoothed Wigner distribution (OSWD). They compare this mixed time-frequency representation (MTFR) to other MTFRs such as the spectrogram, the short-time Fourier transform (STFT), and the Wigner and pseudo-Wigner distributions. The authors propose an approach to designing linear time-varying filters for slowly time-varying signals which is based on the concept of local nonstationarity cancellation and show that it is equivalent to masking the optimal STFT. The performance of the filter in suppressing white noise and in decomposing a slowly time-varying signal into its components is studied and compared to the performance of the techniques based on the STFT. >

Time-Frequency Design and Processing of Signals Via Smoothed Wigner Distributions

The Wigner distribution (WD) and WD-based sig- nal synthesis can be used for designing and processing signals in a joint time-frequency domain. Unfortunately, the perfor- mance of this method is adversely affected by the occurrence of interference terms (IT'S) in the WD. This paper provides an analysis of IT effects in WD-based signal synthesis, and shows that IT effects can be substantially reduced by using a smoothed Wigner distribution (SWD) instead of the WD. An iterative al- gorithm for SWD-based signal synthesis is presented, and the improvement over WD-based signal synthesis is verified via computer simulation.

Impulse response for backscattering by a thin spherical shell displayed in time and frequency.

Time-frequency resolutions of the acoustical scattering from submerged targets utilizing exact form functions have been given by Yen etal. [J. Acoust. Soc. Am. 87, 2359–2370 (1990)]. Spectrograms of backscattered signals from a thin spherical are computed in the present research as smoothed Wigner distributions employing an algorithm given by Nuttall [NUSC Technical Report 8785 (1990)]. For the response to an impulse, the underlying dynamics are discussed in light of recent investigations by several authors working from different but complementary viewpoints. A deficiency in the spectrogram’s ability to simultaneously resolve the instantaneous time-frequency content of backscattered signals is exploited to display either the underlying resonant structure of steady-state scattering or the transient response to wideband impulses or tone bursts. The spectrograms identify features associated with the midfrequency enhancement present in corresponding time-domain plots given by L. G. Zhang etal. [J. Acoust. Soc. Am. 91, 1862–1874 (1992)]. Prompt features associated with the high-frequency enhancements are also evident. [Work supported by ONR.]

Formant onset duration: A secondary cue for identifying the place of articulation in stop consonants

Formant onset duration (FOD) is suggested as a possible alternative to voice onset time (VOT), to characterize the place of articulation in stop consonants. The FOD is defined as the duration from the start of the burst (denoting consonantal release) to the start of the formant structure (denoting vowel onset). In many languages, VOT is the most salient cue distinguishing between voiced and voiceless stop phonemes. Although VOT has been shown to increase in duration from bilabials to alveolars to velars [Lisker and Abramson, Word 20; Zue, Tech. Rep. 523, MIT], attempts to use VOT as a cue to place of articulation [Kewley‐Port, J. Acoust. Soc. Am. 73, 322–335 (1983)] were not successful. In a recent study of stop consonants using the partially smoothed Wigner distribution [Garudadri et al., J. Acoust. Soc. Am. Suppl. 1 82, S55 (1987)], FOD and F2 of the following vowel were used along with the spectral shape during the burst, to resolve context dependencies due to coarticulation effects. The FOD increases from bilabials to alveolars to velars and appears to be related to the inertia of the articulators. Unlike VOT, FOD is always positive and corresponds to the vocal tract rather than the source of excitation. Evidence from English, Telugu, and French stop consonants supporting these ideas is presented. [Work supported in part by NSERC, Canada.]

Application of smoothed Wigner distribution (WD) to speech signals

The WD is a powerful tool for the simultaneous time‐frequency analysis of time‐varying signals [Martin and Flandrin, IEEE Trans. Acoust. Speech Signal Process. ASSP‐33 (Dec. 1985)]. For signals with simple time‐frequency structures (e.g., frequency‐modulated signals), the WD provides excellent resolution in the time‐frequency (t − ω) plane, as compared with conventional short‐time spectral analysis techniques. The interpretation of the WD becomes more difficult as the time‐frequency relationship of the signal gets more complex (e.g., in speech), because of the presence of cross terms and of regions below (i.e., negative) the t − ω plane. This disadvantage can be overcome by smoothing the WD in the t and ω directions over a region TΩ > 1 (T and Ω are the “smear” values in time and frequency) [Janssen and Claasen, IEEE Trans. Acoust. Speech Signal Process. ASSP‐33 (Aug. 1985)]. Such a smoothing process, however, makes the WD and the spectrogram equivalent in terms of the time and frequency resolutions. This presentation is directed towards comparing the performances of the WD (with partial smoothing, i.e., T < 1) and the spectrogram. Nonspeech signals as well as speech signals will be used to demonstrate the application of WD to the analysis of speech signals. [Work supported by NSERC, Canada.]

Correspondence rules and properties of smoothed phase space distribution functions

According to the link between phase space distribution functions and correspondence rules via an arbitrary weighting function, we show that a necessary and sufficient condition for obtaining a direct correspondence between a real phase space distribution function and the density operator of a pure state imposes the weighting function to be unimodular. The same condition is also shown to a particular case of interest for deriving a general formula from which previous known results such as orthonormality of phase space eigenfunctions, non-negativity of smoothed Wigner distributions or phase space interpretation of the scalar product are recovered as special cases.

A measure of phase space delocalization for the eigenfunctions of non-linear vibrational systems

A measure of phase space delocalization for the eigenfunctions of regular and irregular eigenstates of a bound hamiltonian is introduced on the basis of a smoothed Wigner distribution function and a phase space graining. This measure, which is an entropy-like quantity, shows a large dispersion as a function of energy above and below the critical energy E C for classically chaotic motion. It is conjectured that many chaotic states also show properties of regular states too.