Duration and Bandwidth as Second Moments
More interesting definitions of duration and bandwidth are obtained
for nonzero
signals using the normalized
second moments of the
squared magnitude:
where

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
[
52, p. 273-274]:
Theorem: If

and

as

, then
 |
(C.2) |
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 (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.
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