Given a signal is angle modulate and is:
Xc(t)=10cos(2 pi 10^8 t + 200cos 2 pi 10^3 t)
Find what is the bandwith?
The analytic representation of this signal is:
$\displaystyle s(t)=10\exp(\text{i}[2\pi f_0 t + 200 \cos(2 \pi f_1 t)])$
so the instantaneous phase is:
$\displaystyle
\phi(t)=2\pi f_0 t + 200 \cos(2 \pi f_1 t)
$
and the instantaneous frequency is:
$\displaystyle
f(t)=\frac{1}{2\pi}~\frac{\partial}{\partial t}\phi(t)= f_0 + 200 f_1 \sin(2 \pi f_1 t)
$
and the bandwidth is therefor the difference between the higest and lowest instataneous frequencies:
$\displaystyle
{\bf{B}}=400 \pi f_1
$
CB