Duration and Bandwidth as Second Moments

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

$$\Delta t \isdef \frac{1}{\left\Vert\,x\,\right\Vert _2} \sqrt{\int_{-\infty}^\infty t^2 \left\vert x(t)\right\vert^2 \,dt} \quad\isdef \quad \frac{\left\Vert\,tx\,\right\Vert _2}{\left\Vert\,x\,\right\Vert _2}$$

$$\Delta \omega \isdef \frac{1}{\left\Vert\,X\,\right\Vert _2} \sqrt{\int_{-\infty}^\infty \omega^2 X(\omega) \frac{d\omega}{2\pi}} \quad\isdef \quad \frac{\left\Vert\,\omega X\,\right\Vert _2}{\left\Vert\,X\,\right\Vert _2}, \protect$$ (B.61)

where

$$\begin{eqnarray*} \left\Vert\,x\,\right\Vert _2^2 &\isdef & \int_{-\infty}^\infty \left\vert x(t)\right\vert^2 dt\nonumber \\ \left\Vert\,X\,\right\Vert _2^2 &\isdef & \int_{-\infty}^\infty \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi} \end{eqnarray*}$$

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 [59].Under these definitions, we have the following theorem [202, p. 273-274]:

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

$$\zbox {\Delta t\cdot \Delta \omega \geq \frac{1}{2}} \protect$$ (B.62)

with equality if and only if

$$x(t) = Ae^{\alpha t^2}, \quad \alpha>0.$$ (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 $x(t)$ to be real and normalized to have unit $\ensuremath{L_2}$ norm ( $\left\Vert\,x\,\right\Vert _2=1$ ). From the Schwarz inequality [264],^B.2

$$\left\vert\int_{-\infty}^\infty t x(t) \left[\frac{d}{dt}x(t)\right] dt\right\vert^2 \leq \int_{-\infty}^\infty t^2 x^2(t) dt \int_{-\infty}^\infty \left\vert\frac{d}{dt}x(t)\right\vert^2 dt. \protect$$ (B.64)

The left-hand side can be evaluated using integration by parts:

$$\int_{-\infty}^\infty tx \frac{dx}{dt} dt = \left . t \frac{x^2(t)}{2} \right\vert _{-\infty}^{\infty} - \frac{1}{2} \int_{-\infty}^\infty x^2(t) dt \isdef -\frac{1}{2}\left\Vert\,x\,\right\Vert _2^2 = -\frac{1}{2}$$ (B.65)

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.65) can be evaluated using the power theorem and differentiation theorem (§B.2):

$$\int_{-\infty}^\infty \left\vert\frac{dx(t)}{dt}\right\vert^2 dt = \int_{-\infty}^\infty \left\vert j\omega X(\omega)\right\vert^2 \frac{d\omega}{2\pi} = \int_{-\infty}^\infty \omega^2 \left\vert X(\omega)\right\vert^2 \frac{d\omega}{2\pi}$$ (B.66)

Substituting these evaluations into (B.65) gives

$$\left\vert-\frac{1}{2}\right\vert^2 \leq \left\Vert\,tx\,\right\Vert _2^2 \left\Vert\,\omega X\,\right\Vert _2^2.$$ (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

$$\frac{d}{dt}x(t) = c t x(t)$$ (B.68)

for some constant $c$ , implying $x(t)=A e^{\frac{c}{2} t^2}$ for some constants $A$ and $c$ .