Duration and Bandwidth as Second MomentsMore interesting definitions of duration and bandwidth are obtained using the normalized second moments of the squared magnitude:
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.
Proof: Without loss of generality, we may take consider to be real and normalized to have unit norm ( ). From the Schwarz inequality ,B.2
The left-hand side can be evaluated using integration by parts:
where we used the assumption that as . The second term on the right-hand side of (B.65) can be evaluated using the power theorem and differentiation theorem (§B.2):
Substituting these evaluations into (B.65) gives
Taking the square root of both sides gives the uncertainty relation sought. If equality holds in the uncertainty relation (B.63), then (B.65) implies
for some constant , implying for some constants and .
Geometric Signal Theory