Signal expression manipulation

Discussion in 'Wireless & RF Design' started by PsySc0rpi0n, Jan 23, 2016.

  1. PsySc0rpi0n

    Thread Starter Well-Known Member

    Mar 4, 2014
    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:

    <br />
\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 )}<br />

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

    <br />
\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 )}<br />

    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!
  2. WBahn


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