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 [54, p. 273-274]:
Theorem: If and as , then
That is, only the Gaussian function (also known as the ``bell curve'' or ``normal curve'') achieves the lower bound on the time-bandwidth product.
Proof: Without loss of generality, we may take to be real and normalized to have unit norm ( ). From the Schwarz inequality (see §5.9.3 for the discrete-time case),
where we used the assumption that as .
The second term on the right-hand side of Eq. (C.3) can be evaluated using the power theorem (§7.4.8 proves the discrete-time case) and differentiation theorem (§C.1 above):
Substituting these evaluations into Eq. (C.3) gives
Taking the square root of both sides gives the uncertainty relation sought.
If equality holds in the uncertainty relation Eq. (C.2), then Eq. (C.3) implies
for some constant , which implies for some constants and . Since by hypothesis, we have while remains arbitrary.