More interesting definitions of duration and bandwidth are obtained
for nonzero signals using the normalized second moments of the
squared magnitude:

(C.1)

where

By the DTFTpower 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.1}Under these definitions, we have the following theorem
[51, p. 273-274]:

Proof: Without loss of generality, we may take consider to be real
and normalized to have unit norm (
). From the
Schwarz inequality (see §5.9.3 for the discrete-time case),

(C.3)

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 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.