Rectangular Pulse

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

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

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

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

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

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

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