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A K Peters/CRC Press eBooks

The Heisenberg Uncertainty Principle and Time-Frequency Decompositions

Barbara Burke Hubbard

Published May 30, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781439864555-28

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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.
It is not extremely difficult to accept the fact that a signal can't be "localized" simultaneously in time and in frequency. Speaking of a frequency \nat a given instant doesn't really have a meaning; a frequency has to have \nthe time to oscillate. It is much harder to come to terms with the fact that \none cannot speak of an elementary particle's precise position and precise \nmomentum. One may be tempted to think that Heisenberg imposes a \nlimit on what one can know (lacking the necessary intelligence or tools). \nThat is not the case. As in signal processing, the Heisenberg principle \ndescribes a reality: an elementary particle does not have simultaneously \na precise position and a precise momentum.
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.
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Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.
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