Patrick J. Loughlin

Construction of positive time-frequency distributions

A general method for constructing nonnegative definite, joint time-frequency distributions (TFDs) satisfying the marginals of time |s(t)|/sup 2/ and frequency |S(f)|/sup 2/ is presented. As nonnegative-definite distributions with the correct marginals, these TFDs are members of the Cohen-Posch class. Several examples illustrating properties of these TFDs are presented for both synthetic and real signals. >

ON THE AMPLITUDE- AND FREQUENCY-MODULATION DECOMPOSITION OF SIGNALS

In general, the problem of determining the amplitude and frequency modulations (AM and FM) of a signal is ill posed because there is an unlimited number of combinations of AM and FM that will generate a given signal. Although Gabor proposed a method for uniquely defining the AM and FM of a signal, namely via the analytic signal, the results obtained are sometimes physically paradoxical. In this paper, four reasonable physical conditions that the calculated AM and FM of a signal should satisfy are proposed. The analytic signal method generally fails to satisfy two of the four conditions. A method utilizing the positive (Cohen–Posch) time‐frequency distribution and time‐varying coherent demodulation of the signal is given for obtaining an AM and FM that satisfy the four proposed conditions. Contrary to the accepted definition, the instantaneous frequency (i.e., the FM) that satisfies these conditions is generally not the derivative of the phase of the signal. Rather, the phase is separated into two parts, o...

Comments on the interpretation of instantaneous frequency

Instantaneous frequency, taken as the derivative of the phase of the signal, is interpreted in the time-frequency literature as the average frequency of the signal at each time. We point out some difficulties with this interpretation, and show that for a generic two-component AM-FM signal, the interpretation holds only when the components are of equal strength. We conclude that instantaneous frequency and the average frequency at each time are generally two different quantities. One possible interpretation of the difference between these two quantities is suggested.

Bilinear time-frequency representations: new insights and properties

An analysis of the interference terms of Cohen-class bilinear time-frequency representations (TFR) of multicomponent signals is presented. Constraints for achieving new interference properties are derived. Imposing these interference time and interference frequency concentration constraints on a TFR guarantees that the TFR will be zero everywhere the signal s(t) is zero, and the TFR will contain only those frequencies that occur in the signal. Thus, these new constraints guarantee strong finite support in a TFR. When these interference concentration properties are combined with interference attenuation, tradeoffs between finite support, the marginals, and the interference properties are shown to be unavoidable. The useful class of product kernels are considered and generalized further to allow TFR with potentially superior interference properties. The interference frequency concentration and attenuation properties allow TFR with spectrogram-like interference suppression, but without the spectrograms inherent time/frequency resolution tradeoff. Other useful combinations of properties are discussed and analyzed, and properties and tradeoffs are illustrated by examples. >

When is instantaneous frequency the average frequency at each time

The interpretation of instantaneous frequency has been a subject of interest for many years. One interpretation is that it is the average frequency at each time in the signal. We prove that instantaneous frequency equals the average frequency at each time only when there is symmetry in the instantaneous spectrum, as previous empirical evidence has suggested. Also, when there is such symmetry, the average frequency at each time equals the median frequency at each time.

The uncertainty principle: global, local, or both?

We address the issue of the relation between local quantities and the uncertainty principle. We approach the problem by defining local quantities as conditional standard deviations, and we relate these to the uncertainty product appearing in the standard uncertainty principle. We show that the uncertainty product for the average local standard deviations is always less than or equal to the standard uncertainty product and that it can be arbitrarily small. We apply these results to the short-time Fourier transform/spectrogram to explore the commonly held notion that the uncertainty principle somehow limits local quantities. We show that, indeed, for the spectrogram, there is a lower bound on the local uncertainty product of the spectrogram due to the windowing operation of this method. This limitation is an inherent property of the spectrogram and is not a property of the signal or a fundamental limit. We also examine the local uncertainty product for a large class of time-frequency distributions that satisfy the usual uncertainty principle, including the Wigner distribution, the Choi-Williams distribution, and many other commonly used distributions. We obtain an expression for the local uncertainty product in terms of the signal and show that for these distributions, the local uncertainty product is less than that of the spectrogram and can be arbitrarily small. Extension of our approach to an entropy formulation of the uncertainty principle is also considered.

Stiffness and Damping in Postural Control Increase With Age

Upright balance is believed to be maintained through active and passive mechanisms, both of which have been shown to be impacted by aging. A compensatory balance response often observed in older adults is increased co-contraction, which is generally assumed to enhance stability by increasing joint stiffness. We investigated the effect of aging on standing balance by fitting body sway data to a previously developed postural control model that includes active and passive stiffness and damping parameters. Ten young (24 ± 3 years) and seven older (75 ± 5 years) adults were exposed during eyes-closed stance to perturbations consisting of lateral pseudorandom floor tilts. A least-square fit of the measured body sway data to the postural control model found significantly larger active stiffness and damping model parameters in the older adults. These differences remained significant even after normalizing to account for different body sizes between the young and older adult groups. An age effect was also found for the normalized passive stiffness, but not for the normalized passive damping parameter. This concurrent increase in active stiffness and damping was shown to be more stabilizing than an increase in stiffness alone, as assessed by oscillations in the postural control model impulse response.

Applications of positive time-frequency distributions to speech processing

Much of our current knowledge and intuition of speech is derived from analyses involving assumptions of short-time stationarity (e.g., the speech spectrogram). Such methods are, by their very nature, incapable of revealing the true nonstationary nature of speech. A careful consideration of the theory of time-frequency distributions (TFDs), however, allows the construction of methods that reveal far more of the nonstationarities of speech, thereby highlighting just what it is that conventional approaches miss. We apply two iterative methods for generating positive time-frequency distributions (TFDs) to speech analysis. Both methods make use of multiple sources of information (e.g., multiple spectrograms) to yield a high-resolution estimate of the joint time-frequency energy density of speech. Plosive events and formant harmonic structure are simultaneously preserved in these TFDs. Rapidly time-varying formants are also resolved by these TFDs, and harmonic structure is revealed, independent of sweep rate; this result is quite different from that seen with conventional speech spectrograms. The speech features observed in these distributions demonstrate that conventional sliding window techniques lose or distort much of the rich nonstationary structure of speech. Examples for synthetic formants and real speech are provided. The differences between joint distributions and conditional distributions are also illustrated. >

Time-varying characteristics of visually induced postural sway

To study potential time-varying dynamics of postural sway as measured via center-of-pressure (COP) under the feet, we applied time-frequency analysis to COP data from ten vestibularly impaired subjects and 13 nonimpaired controls, during quiet stance and in response to visual perturbation. This analysis revealed that 1) the spectral characteristics of COP change over time; 2) there are time-dependent and frequency-dependent differences in COP between impaired and nonimpaired populations during visual perturbation, and 3) there is no difference in COP during quiet stance (eyes opne) between impaired and nonimpaired populations for the parameters investigated. A novel finding of this research is that controls appear to adapt to constant frequency visual perturbation, while vestibularly impaired subjects do not. This difference could not have been observed with conventional Fourier analysis, which is commonly used in COP data analysis, because time is not a parameter of the spectrum and adaptation is, by nature, a time-varying process. These results suggest that time-frequency analysis of COP data is useful for studying temporal dynamics of postural control, and in particular the differences between vestibularly impaired subjects and healthy controls during visual perturbation.

A Wigner approximation method for wave propagation

An approximation method for pulse propagation based on the Wigner position-wavenumber representation is presented. The method is very easy to apply and moreover is physically illuminating. One obtains the evolved approximate Wigner distribution from the initial Wigner distribution by a simple linear translation in phase space. Each phase space point propagates at constant velocity given by the group velocity at the phase space point. Dissipative propagation (damping) is also taken into account. From the approximate Wigner distribution, one can obtain the approximate magnitude of the evolved pulse and the approximate local wavenumber, that is, the spatial derivative of the phase of the pulse. Examples are given to illustrate the method.