Rectangular Pulse

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

$\displaystyle p_\tau(t) \isdef \left\{\begin{array}{ll} 1, & \left\vert t\right\vert\leq\frac{\tau}{2} \\ [5pt] 0, & \left\vert t\right\vert>\frac{\tau}{2}. \\ \end{array} \right.$ (B.31)

Its Fourier transform is easily evaluated:

\begin{eqnarray*}
P_\tau(\omega) &\isdef & \hbox{\sc FT}_\omega(p_\tau) \isdef \int_{-\infty}^\infty p_\tau(t) e^{-j\omega t}dt\\
&=& \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} e^{-j\omega t}dt
= \left.-\frac{1}{j\omega} e^{-j\omega t}\right\vert _{t=-\frac{\tau}{2}}^{\frac{\tau}{2}}\\
&=& \frac{e^{j\omega \frac{\tau}{2}} - e^{-j\omega \frac{\tau}{2}}}{j\omega}
= \frac{2j\sin\left(\omega \frac{\tau}{2}\right)}{j\omega}\\
&=& \tau\frac{\sin\left(\omega \frac{\tau}{2}\right)}{\omega\frac{\tau}{2}}
= \tau\frac{\sin(\pi f\tau)}{\pi f\tau}\\
&\isdef & \tau\,\mbox{sinc}(f\tau)
\end{eqnarray*}

Thus, we have derived the Fourier pair

$\displaystyle \zbox {p_\tau(t) \;\longleftrightarrow\;\tau\,\mbox{sinc}(f\tau)} \protect$ (B.32)

Note that sinc$ (f)$ is the Fourier transform of the one-second rectangular pulse:

$\displaystyle p_1(t) \;\longleftrightarrow\;$sinc$\displaystyle (f)$ (B.33)

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

$\displaystyle p_1\left(\frac{t}{\tau}\right) \;\longleftrightarrow\;\tau\,$sinc$\displaystyle (f\tau)$ (B.34)


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Sinc Impulse
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Gaussian Pulse