### Duration and Bandwidth as Second Moments

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

 (B.61)

where

By the DTFT power theorem2.3.8), 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.62)

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

 (B.64)

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 theoremB.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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