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MSU EELE 578 - Time-Frequency Distributions-A Review

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Time-Frequency Distributions-A Review LEON COHEN lnvited Paper 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 is changing in time, and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic goal 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 presented to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. I. INTRODUCTION The power of standard Fourier analysis is that it allows the decomposition of a signal into individual frequency components and establishes the relative intensity of each component. The energy spectrum does not, however, tell us when those frequencies occurred. During a dramatic sunset, for example, it is clear that the composition of the light reaching us isverydifferentthan what it isduring most of the day. If we Fourier analyze the light from sunrise to sunset, the energy density spectrum would not tell us that the spectral composition was significantly different in the last 5 minutes. In this situation, where the changes are rel- atively slow, we may Fourier analyze 5-minute samples of the signal and get a pretty good idea of how the spectrum during sunset differed from a 5-minute strip during noon. This may be refined by sliding the 5-minute intervals along time, that is, by taking the spectrum with a 5-minute time window at each instant in time and getting an energy spec- trum as acontinuous function of time. As long as the 5-min- Ute intervals themselves do not contain rapid changes, this will give an excellent idea of how the spectral composition of the light has changed during the course of the day. If significant changes occurred considerably faster than over 5 minutes, we may shorten the time window appropriately. This is the basic idea of the short-time Fourier transform, or spectrogram, which is currently the standard method for Manuscript received March21,1988; revised March30,1989,This workwas supported in part by the City University Research Award Program. The author is with the Department of Physics and Astronomy, Hunter Collegeand Graduatecenter, City Universityof NewYork, New York, NY 10021, USA. IEEE Log Number 8928443. the study of time-varying signals. However, there exist nat- ural and man-made signals whose spectral content is changing so rapidly that finding an appropriate short-time window is problematic since there may not be any time interval for which the signal is more or less stationary. Also, decreasing the time window so that one may locate events in time reduces the frequency resolution. Hence there is an inherent tradeoff between time and frequency resolu- tion. Perhaps the prime example of signals whose fre- quency content is changing rapidly and in a complex man- ner is human speech. Indeed it was the motivation to analyze speech that led to the invention of the sound spec- trogram [113], [I601 during the 1940s and which, along with subsequent developments, became a standard and pow- erful tool for the analysis of nonstationary signals [5], [6], [MI, [751, [1171, U121, F261, [1501, [1511, [1581,[1631, [1641, [1741. Its possible shortcomings not withstanding, the short-time Fourier transform and its variations remain the prime meth- ods for the analysis of signals whose spectral content is varying. Starting with the classical works of Gabor [BO], Ville [194], and Page [152], there has been an alternative development for the study of time-varying spectra. Although it is now fashionable to say that the motivation for this approach is to improve upon the spectrogram, it is historically clear that the main motivation was for a fundamental analysis and a clarification of the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic idea is to devise a joint function of time and frequency, a distribution, that will describe the energy density or inten- sityof a signal simultaneously in time and frequency. In the ideal case such a joint distribution would be used and manipulated in the same manner as any density function of more than one variable. For example, if we had a joint density for the height and weight of humans, we could obtain the distribution of height by integrating out weight. Wecould obtain the fraction of peopleweighing more than 150 Ib but less than 160 Ib with heights between 5 and 6 ft. Similarly, we could obtain the distribution of weight at a particular height, the correlation between height and weight, and so on. The motivation for devising a joint time- frequencydistribution is to be able to use it and manipulate it in the same way. If we had such a distribution, we could ask what fraction of the energy is in a certain frequency and time range, we could calculate the distribution of fre- 0018-9219/89/0700-0941501.00 0 1989 IEEE PROCEEDINGS OF THE IEEE, VOL. 77, NO. 7, IULY 1989 941quency at a particular time, we could calculate the global and local moments of the distribution such as the mean frequency and its local spread, and so on. In addition, if we did have a method of relating a joint time-frequency dis- tribution to a signal, it would be a powerful tool for the con- struction of signals with desirable properties. This would be done by first constructing a joint time-frequency func- tion with the desired attributes and then obtaining the sig- nal that produces that distribution. That is, we could synthesize signals having desirable time-frequency char- acteristics. Of course, time-frequency analysis has unique features, such astheuncertaintyprinciple,which add tothe richness and challenge of the field. From standard Fourier analysis, recall that the instanta- neous energy of a signal s(t) is the absolute value of the sig- nal squared, (s(t)I2 = intensity per unit time at time t or (s(t)12At = fractional energy in time interval At at time t. (1.1) The intensity per unit frequency,’ the energy density spec- trum, is the absolutevalue of the Fourier transform squared, (S(O)(~ = intensity per

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