2) Redo example 3.1.1 considering a periodic signal with 3 different time
durations
(Page 26) e.g. $\tau$ = (1/2, 1¼, ½).
Example 3.1.1 Consider the following signal
$x(t) = \begin{cases} 0, & -T/2 < t < -\tau/2 \\ K, & -\tau/2 < t < \tau/2 \\ 0, & \tau/2 < t < T/2 \end{cases}$
$x(t) = x(t+T)$
Signals of this type can be produced by a pulse generator and are used extensively in radar and
sonar systems. Calculate the Fourier coefficients of $x(t)$.
$X_n = \frac{1}{T} \int_{-T/2}^{T/2} x(t) \exp\left(-j\frac{2\pi nt}{T}\right) dt$
$= \frac{1}{T} \int_{-\tau/2}^{\tau/2} K \exp\left(-j\frac{2\pi nt}{T}\right) dt$
$= \frac{K}{T} \frac{T}{2\pi n} \left[ \exp\left(-j\frac{2\pi n\tau}{2T}\right) - \exp\left(+j\frac{2\pi n\tau}{2T}\right) \right]$
$= \frac{K}{2\pi n} K2j \sin\left(\frac{2\pi n\tau}{2T}\right)$
$= \frac{K\tau}{\pi} \frac{\sin\left(\frac{\pi n\tau}{T}\right)}{\frac{\pi n\tau}{T}} = \frac{K\tau}{T} \frac{\sin\left(\frac{\pi n\tau}{T}\right)}{\frac{\pi n\tau}{T}}$
$= \frac{K\tau}{T} \text{sinc}\left(\frac{n\tau}{T}\right)$
$\text{where } \quad \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$
