### 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:

 (C.1)

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 theoremC.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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Relation of the DFT to Fourier Series