### Duration and Bandwidth as Second Moments

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

where

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

with equality if and only if

(B.63) |

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 [264],^{B.2}

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

(B.65) |

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):

(B.66) |

Substituting these evaluations into (B.65) gives

(B.67) |

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

(B.68) |

for some constant , implying for some constants and .

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Time-Limited Signals

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Geometric Signal Theory