Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Chapters

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Sinc Impulse

Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Sinc Impulse

The preceding Fourier pair can be used to show that

$\displaystyle \zbox {\lim_{\tau\to\infty} \tau\,\mbox{sinc}(f\tau) = \delta(f).} $



Proof: The inverse Fourier transform of $ \tau\,$sinc$ (f\tau)$ is

\begin{eqnarray*} p_\tau(t) &=& \ensuremath{\int_{-\infty}^{\infty}}\tau\,\mbo... ...eq 1/2 \\ [5pt] 0, & \mbox{otherwise}. \\ \end{array} \right. \end{eqnarray*}

In particular, in the middle of the rectangular pulse at $ t=0$, we have

$\displaystyle p_\tau(0)=\ensuremath{\int_{-\infty}^{\infty}}\tau\,$sinc$\displaystyle (f\tau) df = 1, \quad \forall \tau>0. $

This establishes that the algebraic area under $ \tau\,$sinc$ (\tau f)$ is 1 for every $ \tau>0$. Every delta function (impulse) must have this property.

We now show that $ \tau\,$sinc$ (f\tau)$ also satisfies the sifting property in the limit as $ \tau\to\infty$. This property fully establishes the limit as a valid impulse. That is, an impulse $ \delta(t)$ is any function having the property that

$\displaystyle \ensuremath{\int_{-\infty}^{\infty}}g(t)\delta(t)dt = \left<g,\delta\right> = g(0) $

for every continuous function $ g(t)$. In the present case, we need to show, specifically, that

$\displaystyle \lim_{\tau\to\infty}\ensuremath{\int_{-\infty}^{\infty}}G(f)\tau\,$sinc$\displaystyle (\tau f)\,df = G(0). $

Define $ P_\tau(f)\isdef \tau\,$sinc$ (f\tau)$. Then by the power theorem (§2.4.8),

$\displaystyle \left<G,P_\tau\right> = \left<g,p_\tau\right> = \ensuremath{\int_{-\infty}^{\infty}}g(t) p_\tau(t)\,dt = \int_{-\tau/2}^{\tau/2} g(t)\,dt. $

Then as $ \tau\to\infty$, the limit converges to the algebraic area under $ g$, which is $ G(0)$ as desired:

$\displaystyle \lim_{\tau\to\infty}\int_{-\tau/2}^{\tau/2} g(t)\,dt = \ensurema... ...int_{-\infty}^{\infty}}e^{-j\omega t} g(t)\,dt \right\vert _{\omega=0} = G(0). $

We have thus established that

$\displaystyle {\lim_{\tau\to\infty}\tau\,\mbox{sinc}(f\tau) = \delta(f),} $

where

   sinc$\displaystyle (f)\isdef \frac{\sin(\pi f)}{\pi f}. $

For related discussion, see [31, p. 127].

Order a Hardcopy of Spectral Audio Signal Processing

Previous: Rectangular Pulse
Next: Impulse Trains
written by 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.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )