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Birkhäuser Boston eBooks

The Uncertainty Principle for the Short-Time Fourier Transform and Wavelet Transform

Leon W. Cohen

Published Jan 1, 2001• 29 citations
Publisher

Abstract

We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0137-3_8

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We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
The method of wavelet transforms , which provides a decomposition of functions in terms of a fixed family of functions of constant shape but varying scales and locations, recently acquired broad significance in the analysis of signals and of experimental data from various physical phenomena. It is clear that the potential of this method has not yet been fully tapped. Nevertheless, its value for the whole spectrum of problems in many areas of science and engineering, including the study of electromagnetic and turbulent hydrodynamic fields, image reconstruction algorithms, prediction of earthquakes and tsunami waves, and statistical analysis of economic data, is by now quite obvious. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
It is sometimes said that one can't localize a signal simultaneously in time and frequency. This can be misleading. It's not that we can't localize the signal in time and frequency. The signal itself can't be concentrated simultaneously in time and in frequency. The Heisenberg uncertainty principle does not limit what we can know about reality; it describes that reality. The shorter-lived a function, the wider the band of frequencies given by its Fourier transform; the narrower the band of frequencies of its Fourier transform, the more the function is spread out in time.
We present a general method for calculating relevant moments of the wavelet transform.Explicit results are given for the time, frequency, and scale moments.Using the results obtained, we show that the wavelet transform has unique characteristics that are not possessed by other methods.We discuss whether these unusual characteristics are physically or mathematically important.Exactly solvable examples are given, and the results are contrasted to those of the standard methods such as the spectrogram and the Wigner distribution.