# Amplitude modulated

yoyo
An AM cosine wave is represented by x(t)=12+7*sin(pi*t-(1/3)*pi)]*cos(13*pi*t). Use phasors to show that x(t) can be expressed in form of:

A1cos(w1*t+phi1) + A2cos(w2*t+phi2)+A3cos(w3*t+phi3) where w1<w2<w3.

I am really stuck with this. don't know where to start. can someone please help me out?

Homework Helper
yoyo said:
An AM cosine wave is represented by x(t)=12+7*sin(pi*t-(1/3)*pi)]*cos(13*pi*t). Use phasors to show that x(t) can be expressed in form of:

A1cos(w1*t+phi1) + A2cos(w2*t+phi2)+A3cos(w3*t+phi3) where w1<w2<w3.

I am really stuck with this. don't know where to start. can someone please help me out?

Use that

$$cos(\omega t +\phi)= \frac{e^{i(\omega t + \phi)}+e^{-i(\omega t + \phi)}}{2}$$

and

$$sin(\omega t +\phi)= \frac{e^{i(\omega t + \phi)}-e^{-i(\omega t + \phi)}}{2i}$$

Replace the sin and cos functions in x(t) with the exponential expressions, do all the multiplications, collect the terms with the same angular frequency and see what you get.

ehild