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The Uncertainty Principle

The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem (§B.4). It may be loosely stated as

Time Duration $ \times$ Frequency Bandwidth $ \geq$ c
where $ c$ 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 useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse (§2.2).



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