# Signal expression manipulation

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Hi guys!

I need help on manipulating a signal expression that I can't go through it. I have the result but I'm struggling hard to get there!

The original expression is this:

$$\displaystyle {S_{AM}\left ( t \right )=4\cdot \cos \left ( 2\cdot \pi \cdot 720\cdot 10^{3}\cdot t \right )+\cos \left ( 2\cdot \pi \cdot 710\cdot 10^{3}\cdot t \right )+\cos \left ( 2\cdot \pi \cdot 730\cdot 10^{3}\cdot t \right )}$$

The final result I'm trying to achieve is this:

$$\displaystyle {S_{AM}\left ( t \right )=4\cdot \left ( 1+0.5\cdot \cos \left ( 2\cdot \pi \cdot 10^{4}\cdot t \right ) \right )\cdot \cos \left ( 2\cdot \pi \cdot 72\cdot 10^{4}\cdot t\right )}$$

I already identified that the carrier is the term with 720 kHz with amplitude of 4 and that the other 2 terms are something like 720 + 10 and 720 - 10 and that the 10 is a common term but I can't move on!

Any help is appreciated!

#### WBahn

Joined Mar 31, 2012
25,902
Try working it backwards starting from the final result and using trig identities to express it as the sum of sinusoids.