More interesting definitions of duration and bandwidth are obtained
for nonzero signals using the normalized second moments of the
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.4) 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.4) gives
Taking the square root of both sides gives the uncertainty relation sought.
If equality holds in the uncertainty relation Eq.(C.3), then Eq.(C.4) implies
for some constant , which implies for some constants and . Since by hypothesis, we have while remains arbitrary.