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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    More Fourier Theorems
       Selected Continuous Fourier Theorems
          The Uncertainty Principle
             Duration and Bandwidth as Second Moments

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Duration and Bandwidth as Second Moments

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

$\displaystyle \Delta t$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\left\Vert\,x\,\right\Vert _2} \sqrt{\int_{-\infty}^\inf... ...sdef \quad \frac{\left\Vert\,tx\,\right\Vert _2}{\left\Vert\,x\,\right\Vert _2}$  
$\displaystyle \Delta \omega$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\left\Vert\,X\,\right\Vert _2} \sqrt{\int_{-\infty}^\inf... ...{\left\Vert\,\omega X\,\right\Vert _2}{\left\Vert\,X\,\right\Vert _2}, \protect$ (B.9)

    where

\begin{eqnarray*} \nonumber \\ [10pt] \left\Vert\,x\,\right\Vert _2^2 &\isdef &... ...y}^\infty \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi} \end{eqnarray*}

By the DTFT power theorem (§2.3.8), 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.B.2Under these definitions, we have the following theorem [192, p. 273-274]:



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

$\displaystyle \zbox {\Delta t\cdot \Delta \omega \geq \frac{1}{2}} \protect$ (B.10)

with equality if and only if

$\displaystyle x(t) = Ae^{\alpha t^2}, \quad \alpha>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 consider $ x(t)$ to be real and normalized to have unit $ L2$ norm ( $ \left\Vert\,x\,\right\Vert _2=1$). From the Schwarz inequality [248],B.3

$\displaystyle \left\vert\int_{-\infty}^\infty t x(t) \left[\frac{d}{dt}x(t)\rig... ...) dt \int_{-\infty}^\infty \left\vert\frac{d}{dt}x(t)\right\vert^2 dt. \protect$ (B.11)

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} $

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 (B.11) can be evaluated using the power theorem and differentiation theorem (§B.1.2):

$\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} $

Substituting these evaluations into (B.11) 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 (B.10), then (B.11) 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$.

Previous: The Uncertainty Principle
Next: Time-Limited Signals

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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