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 $c$ 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 very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §E.2.