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 useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse (§2.2).
Duration and Bandwidth as Second Moments
More interesting definitions of duration and bandwidth are obtained
using the normalized second moments of the squared magnitude:
By the DTFT power theorem (§2.3.8), 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 .Under these definitions, we have the following theorem [202, p. 273-274]:
Theorem: If as , then
with equality if and only if
That is, only the Gaussian function (also known as the ``bell curve'' or ``normal curve'') achieves the lower bound on the time-bandwidth product.
The left-hand side can be evaluated using integration by parts:
where we used the assumption that as .
Substituting these evaluations into (B.65) gives
Taking the square root of both sides gives the uncertainty relation sought.
for some constant , implying for some constants and .
If for , then
where is as defined above in (B.62).
Proof: See [202, pp. 274-5].
Time-Bandwidth Products 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.B.3 The allpass filter cannot affect bandwidth , but the duration can be arbitrarily extended by successive applications of the allpass filter.
Relation of Smoothness to Roll-Off Rate