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 .
Geometric Signal Theory