Leon Cohen

Time‐Frequency Analysis

Conventionally, time series have been studied either in the time domain or the frequency domain. The representation of a signal in the time domain is localized in time, i.e. the value of the signal at each instant in time is well defined. However, the time representation of a signal is poorly localized in frequency, i.e. little information about the frequency content of the signal at a certain frequency can be known by looking at the signal in the time domain. On the other hand, the representation of a signal in the frequency domain is well localized in frequency, but is poorly localized in time, and as a consequence it is impossible to tell when certain events occurred in time.

Time-frequency distributions-a review

A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

Generalized Phase-Space Distribution Functions

A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will, of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.

Positive time-frequency distribution functions

We demonstrate the existence of positive joint distributions of time and frequency for arbitrary signals. A method is given to readily generate an infinite number of them for any signal. General properties of these distribution functions are derived and specific examples for some common signals are presented.

Positive quantum joint distributions

We demonstrate the existence of positive phase space density functions which yield the quantum mechanical marginal distributions of position and momentum.

Local kinetic energy in quantum mechanics

The question of local kinetic energy in quantum mechanics is considered by using the phase space formulation of quantum mechanics. We derive the totality of possible quantum mechanical expressions for local kinetic energy and give a subset which satisfy Bader’s criterion. The different definitions that various authors have used are shown to be directly related to particular quasiprobability distributions and correspondence rules.

Scale transform in speech analysis

In this paper, we study the scale transform of the spectral-envelope of speech utterances by different speakers. This study is motivated by the hypothesis that the formant frequencies between different speakers are approximately related by a scaling constant for a given vowel. The scale transform has the fundamental property that the magnitude of the scale-transform of a function X(f) and its scaled version /spl radic//spl alpha/X(/spl alpha/f) are same. The methods presented here are useful in reducing variations in acoustic features. We show that the F-ratio tests indicate better separability of vowels by using scale-transform based features than mel-transform based features. The data used in the comparison of the different features consist of 200 utterances of four vowels that are extracted from the TIMIT database.

What is a multicomponent signal

Multicomponent signals, which produce delineated concentrations in the time-frequency plane, are common in nature, human speech being a prime example. An explanation of what types of signals are multicomponent is presented. The idea is generalized to time-scale. The recognition of multicomponent signals is considered.<<ETX>>

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.

Representable local kinetic energy

An infinite set of expressions for the local kinetic energy are derived using proper joint distributions of position and momentum. The joint distributions yield the correct quantum marginal distributions and are positive. This assures that the local kinetic energy is positive.