### 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.1}Under these definitions, we have the following theorem
[52, p. 273-274]:

**Theorem: **If
and
as
, then

with equality if and only if

*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),

The left-hand side can be evaluated using integration by parts:

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):

If equality holds in the uncertainty relation Eq.(C.2), then Eq.(C.3) implies

**Next Section:**

Time-Limited Signals

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Relation of the DFT to Fourier Series