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
![\begin{eqnarray*}
\nonumber \\ [10pt]
\left\Vert\,x\,\right\Vert _2^2 &\isdef &...
...}^\infty \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1742.png)
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
with equality if and only if
![$\displaystyle x(t) = Ae^{-\alpha t^2}, \quad \alpha>0, \quad A\ne 0.
$](http://www.dsprelated.com/josimages_new/mdft/img1750.png)
Proof: Without loss of generality, we may take consider to be real
and normalized to have unit
norm (
). From the
Schwarz inequality (see §5.9.3 for the discrete-time case),
The left-hand side can be evaluated using integration by parts:
![$\displaystyle \int_{-\infty}^\infty tx \frac{dx}{dt} dt
= \left . t \frac{x^2(t...
...ty x^2(t) dt \isdef -\frac{1}{2}\left\Vert\,x\,\right\Vert _2^2 = -\frac{1}{2}
$](http://www.dsprelated.com/josimages_new/mdft/img1753.png)
![$ \sqrt{\vert t\vert}\,x(t) \to 0$](http://www.dsprelated.com/josimages_new/mdft/img1747.png)
![$ \left\vert t\right\vert\to\infty$](http://www.dsprelated.com/josimages_new/mdft/img1748.png)
The second term on the right-hand side of Eq.(C.3) can be
evaluated using the power theorem
(§7.4.8 proves the discrete-time case)
and differentiation theorem (§C.1 above):
![$\displaystyle \int_{-\infty}^\infty \left\vert\frac{dx(t)}{dt}\right\vert^2 dt
...
...\infty}^\infty \omega^2 \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}
$](http://www.dsprelated.com/josimages_new/mdft/img1754.png)
![$ \,$](http://www.dsprelated.com/josimages_new/mdft/img131.png)
![$\displaystyle \left\vert-\frac{1}{2}\right\vert^2 \leq \left\Vert\,tx\,\right\Vert _2^2 \left\Vert\,\omega X\,\right\Vert _2^2.
$](http://www.dsprelated.com/josimages_new/mdft/img1755.png)
If equality holds in the uncertainty relation Eq.(C.2), then
Eq.
(C.3) implies
![$\displaystyle \frac{d}{dt}x(t) = c t x(t)
$](http://www.dsprelated.com/josimages_new/mdft/img1756.png)
![$ c$](http://www.dsprelated.com/josimages_new/mdft/img163.png)
![$ x(t)=A e^{\frac{c}{2} t^2}$](http://www.dsprelated.com/josimages_new/mdft/img1757.png)
![$ A$](http://www.dsprelated.com/josimages_new/mdft/img367.png)
![$ c$](http://www.dsprelated.com/josimages_new/mdft/img163.png)
![$ x(\pm\infty)=0$](http://www.dsprelated.com/josimages_new/mdft/img1724.png)
![$ c<0$](http://www.dsprelated.com/josimages_new/mdft/img148.png)
![$ A\ne 0$](http://www.dsprelated.com/josimages_new/mdft/img1758.png)
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Time-Limited Signals
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Relation of the DFT to Fourier Series