We derive the$s$- and$z$-plane pole regions for continuous-time and discrete-time LTI systems to yield resonance. In this way, resonance can be identified by visual inspection of the pole-zero plot, without the need for calculations.

Galleani and Cohen have developed a new approach to the study of random differential equations. Recently, they applied their method to the interesting case that they called the generalized Wiener process. We show that while the phase space equation they derived is correct, their solution to the equation is not the full solution but holds only under certain conditions. We obtain the general solution and discuss under what circumstances their solution is exact, and under what circumstances their solution is a good approximation to the exact solution. In addition, we pinpoint where in their method of solution they neglected a term, and give the correction. In many cases, their approach may be accurate enough, and hence preferable, as it is a simpler calculation than the exact solution. However, for processes with long correlation times (i.e., autocorrelation functions that do not decay rapidly to zero), the complete method of solution presented here may be required.

An approximation of the time-varying (Wigner) spectrum of a filtered signal is presented. The approximation is simple to apply, yet insightful, in that it shows the effects of the magnitude and phase of the frequency response of the filter on the time-frequency properties of the signal. In addition, the approximation yields the correct Fourier spectrum of the filtered signal, to within a complex constant

We present methods for obtaining basic signal parameters and apply them to the problem of detection, estimation, and enhancement of biological signals. We apply these methods to human speech and whale sounds. In speech, the methods are used to obtain statistical properties of the signal so that the derived features may be used to cluster speakers. For whale sounds we show that the methods may be effectively used not only to obtain signal components and parameters but to also adaptively filter off the noise.

We introduce the concept of the instantaneous kurtosis of a signal. Using the general approach of Cohen (1993), we obtain an explicit expression in terms of the amplitude and phase of the signal. The idea and results are analogous to those of the instantaneous frequency and the instantaneous bandwidth of a signal, which are low-order conditional moments of the time-varying spectral density of the signal.

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.

A non-linear multi-window method for generating a time-varying spectrum of non-stationary signals in noise is presented. The time-varying spectrum is computed from an optimally weighted average of multiple Hermite windowed spectrograms. The weights are determined using linear least squares estimation with respect to a reference time-frequency distribution. A masking operation is also used to reduce extraneous side lobes introduced by higher order Hermite windows. Several examples are provided, with performance criteria measures, to demonstrate and quantify the effectiveness of this new method.

We apply time-frequency analysis to various acoustic transients. Estimates of the conditional mean frequency and the conditional bandwidth exhibit characteristic trends for different transients. This information may be helpful in distinguishing between different classes of transients, and in understanding the underlying mechanisms generating the transient.

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.

A method for generating an informative prior when constructing a positive time-frequency distribution (TFD) by the method of the minimum cross-entropy (MCE) is developed. The prior is obtained from a combination of the Wigner distribution (WD) and the evolutionary periodogram, and results in a more informative MCE-TFD, as quantified via the mutual information of the distribution. The procedure allows any of the bilinear distributions to be used in the prior. Examples illustrate the performance of the new technique.

To study potential time-varying dynamics of postural sway as measured via center-of-pressure (COP), the authors applied time-frequency analysis to COP data from ten vestibularly impaired subjects and thirteen unimpaired controls, during quiet stance and in response to visual perturbation. This analysis revealed that: (1) the spectral characteristics of the COP change over time; (2) there are time-dependent and frequency-dependent differences in the COP between impaired and unimpaired populations during visual perturbation; and (3) there is no difference in the COP during quiet stance (eyes open) between impaired and unimpaired 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.

We investigate the question of what the joint moments of a signal are by considering the joint moments of the spectrogram for limiting cases of the window. Operator methods are also explored. Expressions for the joint moments are derived, which reveal the distorting effects of the spectrogram window. Knowledge of the joint moments of a signal may be useful in estimating positive time-frequency distributions, or in signal classification of nonstationary signals.

We present a general approach to approximating positive time-frequency distributions (TFDs) through nonlinear combinations of spectrograms. Closed-form solutions for the combinations are obtained via optimization of entropy functionals subject to an energy constraint. We apply two such combinations to generating approximate TFDs for whale sounds and speech. Through these applications, it can be seen that these methods give results superior to that achieved with individual spectrograms, and remarkably close to the positive TFDs obtained via computationally-intensive methods.

It is generally stated that the conditional mean frequency of a time-frequency distribution (TFD) should equal the instantaneous frequency of the signal. The commonly accepted definition of instantaneous frequency as the derivative of the phase of the analytic signal sometimes leads to curious results. Although it is commonly held that positivity of the TFD and satisfaction of the so-called "instantaneous frequency constraint" are generally incompatible, we show that one can always find a complex signal, the real part of which is the given signal, for which the derivative of the phase is consistent with the marginals and positivity. Furthermore, for the cases considered, the derivative of the phase of this signal, which by design equals the conditional mean frequency of a positive TFD, is a reasonable, readily interpretable choice for instantaneous frequency.

Two results concerning time-frequency distributions (TFD's) are presented. First, it is shown that the Cohen-Posch class of distributions (or "positive distributions") are scale invariant. Second, it is shown that the commonly stated condition for scale invariance of a TFD, namely, that the kernel be a product kernel, is not a necessary condition for signal-dependent kernels.

We demonstrate that two previously proposed methods for combining the information content from multiple spectrograms into a single, positive time-frequency function are optimal in a cross-entropy sense. The goal in combining the spectrograms is to obtain an improved approximation of the joint time-frequency signal density by overcoming limitations of any single spectrogram. An example of each method is provided, and results are compared with spectrograms and a Cohen-Posch (1985) time-frequency density (TFD) of a nonstationary pulsed tone signal. The proposed combinations are effective and can be efficiently computed.>

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.>

The principle of minimum cross entropy (MCE) is used to generate positive distributions in the Cohen-Posch (1985) class of proper time-frequency distributions (TFDs). The MCE-TFDs are not only intuitively satisfying, but they also yield the correct marginals, have strong finite support, and are everywhere nonnegative. The usual cross-term artifacts that hinder interpretation of other time-frequency representations (most notably, the Wigner-Ville) are not a problem with the MCE-TFDs. Examples of speech and chirps are given and compared to the spectrogram. An interesting observation in the case of speech is that spectrograms more closely resemble time-conditional-frequency distributions (narrowband) or frequency-conditional-time distributions (wideband) than they do joint time-frequency distributions.>

The assumption of quasi-stationarity in the analysis of speech is questioned from the standpoint of the best resolution of global periodicity and vocal tract formant frequency locations. The generalized class of time-frequency representations is discussed from the standpoint of speech analysis and is integrated with a description of the spectrogram. The spectrogram is known to have minimal interference artifacts, and the truly nonstationary cone-kernel time-frequency representation (CK-TFR) is shown to be similarly free of interference. The CK-TFR is observed to give higher resolution of the time point of glottal closure, and the clarity of some formant frequencies, especially for a nasal consonant, is shown to be better than the spectrogram. It is shown that interference-free representations of speech with a higher simultaneous resolution in time and frequency than the spectrogram are possible, and that these new representations may be applicable to the better analysis and understanding of speech.>

Two properties are presented that constrain the cross-terms of Cohen-class time-frequency representations (TFRs) to appear only at signal frequencies, and only when the signal is nonzero. These properties thus guarantee strong finite support, i.e. the TFR is zero everywhere the signal or its spectrum is zero. When combined with cross-term attenuation, one can obtain TFRs with spectrogram-like interference suppression, but without the inherent time-frequency resolution tradeoff of the spectrogram.>

A nonstationary analysis technique is applied to speech, with encouraging results. This technique makes use of a generalized time-frequency representation (GTFR) which uses a kernel that allows finite-time support while suppressing interference terms. This kernel has a cone shape in the (t, tau ) plane, where t is time of a signal and tau is an autocorrelationlike lag. The processing thus allows time and frequency resolution equivalent to the Wigner distribution but does not have significant interference terms. It is shown that the effect of this distribution on the long-term (1.5-s) display of speech is a visible enhancement of formant tracks. Potentially more important shorter-term (30-ms or less) advantages are illustrated and quantified as more accurate locations of bursts in the time-frequency distribution. It is shown that the GTFR technique offers a representation which is less sensitive to threshold settings and is thus more robust.>