More interesting definitions of duration and bandwidth are obtained
using the normalized second moments of the squared magnitude:
By the DTFT power theorem (§2.3.8), 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 [59].Under these definitions, we have the following theorem [202, p. 273-274]:
Theorem: If
as
, then
(B.63) |
Proof: Without loss of generality, we may take consider
to be real
and normalized to have unit
norm (
). From the
Schwarz inequality [263],^{B.2}
(B.65) |
The second term on the right-hand side of (B.65) can be evaluated using the power theorem and differentiation theorem (§B.2):
(B.66) |
(B.67) |
If equality holds in the uncertainty relation (B.63), then (B.65) implies
(B.68) |