Rectangular Pulse

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

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

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

$$\zbox {p_\tau(t) \longleftrightarrow \tau\,\mbox{sinc}(f\tau)} \protect$$ (B.6)

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

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

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

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