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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:

$\displaystyle \Delta t$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\left\Vert\,x\,\right\Vert _2} \sqrt{\int_{-\infty}^\infty t^2 \left\vert x(t)\right\vert^2 dt}
\quad\isdef \quad \frac{\left\Vert\,tx\,\right\Vert _2}{\left\Vert\,x\,\right\Vert _2}$  
$\displaystyle \Delta \omega$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\left\Vert\,X\,\right\Vert _2} \sqrt{\int_{-\infty}^\infty \omega^2 \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}}
\quad\isdef \quad \frac{\left\Vert\,\omega X\,\right\Vert _2}{\left\Vert\,X\,\right\Vert _2},
\protect$ (C.1)

$\displaystyle \left\Vert\,x\,\right\Vert _2^2$ $\displaystyle \isdef$ $\displaystyle \int_{-\infty}^\infty \left\vert x(t)\right\vert^2 dt$  
$\displaystyle \left\Vert\,X\,\right\Vert _2^2$ $\displaystyle \isdef$ $\displaystyle \int_{-\infty}^\infty \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}.$ (C.2)

By the DTFT power theorem, which is proved in a manner analogous to the DFT case in §7.4.8, we have $ \left\Vert\,x\,\right\Vert _2=\left\Vert\,X\,\right\Vert _2$ . Note that writing `` $ \left\Vert\,tx\,\right\Vert _2$ '' and `` $ \left\Vert\,\omega X\,\right\Vert _2$ '' 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 [54, p. 273-274]:

Theorem: If $ x(t)
0$ and $ \sqrt{\vert t\vert}\,x(t) \to 0$ as $ \left\vert t\right\vert\to\infty$ , then

$\displaystyle \zbox {\Delta t\cdot \Delta \omega \geq \frac{1}{2}} \protect$ (C.3)

with equality if and only if

$\displaystyle x(t) = Ae^{-\alpha t^2}, \quad \alpha>0, \quad A\ne 0.

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 $ x(t)$ to be real and normalized to have unit $ \ensuremath{L_2}$ norm ( $ \left\Vert\,x\,\right\Vert _2=1$ ). From the Schwarz inequality (see §5.9.3 for the discrete-time case),

$\displaystyle \left\vert\int_{-\infty}^\infty t x(t) \left[\frac{d}{dt}x(t)\right] dt\right\vert^2 \leq \int_{-\infty}^\infty t^2 x^2(t) dt \int_{-\infty}^\infty \left\vert\frac{d}{dt}x(t)\right\vert^2 dt. \protect$ (C.4)

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)}{2} \right\vert _{-\infty}^{\infty} - \frac{1}{2}
\int_{-\infty}^\infty x^2(t) dt \isdef -\frac{1}{2}\left\Vert\,x\,\right\Vert _2^2 = -\frac{1}{2}

where we used the assumption that $ \sqrt{\vert t\vert}\,x(t) \to 0$ as $ \left\vert t\right\vert\to\infty$ .

The second term on the right-hand side of Eq.(C.4) can be evaluated using the power theorem (§7.4.8 proves the discrete-time case) and differentiation theoremC.1 above):

$\displaystyle \int_{-\infty}^\infty \left\vert\frac{dx(t)}{dt}\right\vert^2 dt
= \int_{-\infty}^\infty \left\vert j\omega X(\omega)\right\vert^2 \frac{d\omega}{2\pi}
= \int_{-\infty}^\infty \omega^2 \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}

Substituting these evaluations into Eq.(C.4) gives

$\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.

Taking the square root of both sides gives the uncertainty relation sought.

If equality holds in the uncertainty relation Eq.(C.3), then Eq.(C.4) implies

$\displaystyle \frac{d}{dt}x(t) = c t x(t)

for some constant $ c$ , which implies $ x(t)=A e^{\frac{c}{2} t^2}$ for some constants $ A$ and $ c$ . Since $ x(\pm\infty)=0$ by hypothesis, we have $ c<0$ while $ A\ne 0$ remains arbitrary.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University