The Uncertainty Principle

The Uncertainty Principle

The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem. It may be loosely stated as

Time Duration $\times$ Frequency Bandwidth $\geq$ c

where

is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.

If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain $c=\infty$ , which is not very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §B.2.

Subsections

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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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