The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem. It may be loosely stated as
Time Duration Frequency Bandwidth cwhere 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 , which is not very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §B.2.
Duration and Bandwidth as Second Moments
More interesting definitions of duration and bandwidth are obtained
for nonzero signals using the normalized second moments of the
By the DTFT power theorem, which is proved in a manner analogous to the DFT case in §7.4.8, we have . Note that writing `` '' and `` '' is an abuse of notation, but a convenient one. These duration/bandwidth definitions are routinely used in physics, e.g., in connection with the Heisenberg uncertainty principle.C.1Under these definitions, we have the following theorem [52, p. 273-274]:
Theorem: If and as , then
with equality if and only if
The left-hand side can be evaluated using integration by parts:
If for , then
Proof: See [52, pp. 274-5].
Time-Bandwidth Products are Unbounded Above
We have considered two lower bounds for the time-bandwidth product based on two different definitions of duration in time. In the opposite direction, there is no upper bound on time-bandwidth product. To see this, imagine filtering an arbitrary signal with an allpass filter.C.2 The allpass filter cannot affect bandwidth , but the duration can be arbitrarily extended by successive applications of the allpass filter.
Introduction to Sampling