Rectangular Pulse

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Rectangular Pulse

The rectangular pulse of width $\tau$ centered on time 0 may be defined by

$\displaystyle p_\tau(t) \isdef \left\{\begin{array}{ll} 1, & \left\vert t\right... ...} \\ [5pt] 0, & \left\vert t\right\vert>\frac{\tau}{2}. \\ \end{array}\right.$

Its Fourier transform is easily evaluated:

$\begin{eqnarray*} P_\tau(\omega) &\isdef & \hbox{\sc FT}_\omega(p_\tau) \isdef \... ...sin(\pi f\tau)}{\pi f\tau}\\ &\isdef & \tau\,\mbox{sinc}(f\tau) \end{eqnarray*}$

Thus, we have derived the Fourier pair

$\displaystyle \zbox {p_\tau(t) \longleftrightarrow \tau\,\mbox{sinc}(f\tau)} \protect$

(3.8)

Note that sinc

is the Fourier transform of the one-second rectangular pulse:

$\displaystyle p_1(t) \longleftrightarrow$ sinc $\displaystyle (f)$

From this, the scaling theorem implies the more general case:

$\displaystyle p_1\left(\frac{t}{\tau}\right) \longleftrightarrow \tau\,$ sinc $\displaystyle (f\tau)$

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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.

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Chapters

Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Rectangular Pulse