Question

Consider the message signal shown in the figure below:
If this message signal m(t) is applied to an FM modulator whose frequency sensitivity is $k_f=0.5Hz/V$, then the maximum pluse deviation of the resultant FM signal will be ____radians.

Answer 1

For an FM signal
$\Delta {\varphi }_{max}=2{\pi }_y\left|\underset{-\infty }{\overset{t}{\int }}m\left(t\right)dt\right|$

$m\left(t\right)=2\left[u\right(t)-(t-1\left)u\right(t-1)-(t-3\left)u\right(t-3\left)u\right(t-3)+(t-4\left)u\right(t-4\left)\right]$

$\underset{-\infty }{\overset{t}{\int }}m\left(t\right)dt=\left[tu\right(t)-{(t-1)}^{2}u(t-1)-{(t-3)}^{2}u(t-3)+{(t-4)}^{2}u(t-4\left)\right]$

By plotting $\underset{-\infty }{\overset{t}{\int }}m\left(t\right)dt$, we get,

So,$\left|\underset{-\infty }{\overset{t}{\int }}m\left(t\right)dt\right|=6$

$\Delta {\varphi }_{max}=2\pi \left(0.5\right)\left(6\right)=6\pi rad=18.85rad$