I am trying to perform the following operation:
Let us assume I have a two-tone input to my system in the form
x = A Sin[Ω1] + A Sin[Ω2]
and our system is nonlinear so it looks like this
y = x + x2 + x3
Now to substitute x (the input) into y (the system) I use the expand function as follows
Out= ASin[Ω1] + A2Sin[Ω1] 2 + A3 Sin[Ω1]3 + A Sin[Ω2] + 2A2 Sin[Ω1] Sin[Ω2] + 3A3 Sin[Ω1]2 Sin[Ω2] + A2 Sin[Ω2]2 + 3A3 Sin[Ω1] Sin[Ω2]2 + A3 Sin[Ω2]3
Now that is good as it is substituted properly but this is just direct substitution and it is not what I am looking for, I want to break all the high power terms on the Sin functions so I try using simplify as follows:
Out= A (Sin[Ω1] + Sin[Ω2]) (1 + A2 Sin[Ω1]2 + A Sin[Ω2] + A2 Sin[Ω2]2 + A Sin[Ω1] (1 + 2A Sin[Ω2]))
Now it looks better but unfortunately it doesn’t get a lot further than that.
The question is, how can I persuade Mathematica to use some of the well known trigonometric functions for example:
Sin[Ω1+ Ω2]=Sin[Ω1] Cos[Ω2] + Cos[Ω1] Sin[Ω2],
Sin[Ω1- Ω2]=Sin[Ω1] Cos[Ω2] - Cos[Ω1] Sin[Ω2],
Cos[Ω1+ Ω2]= Cos[Ω1] Cos[Ω2] - Sin[Ω1] Sin[Ω2],
Cos[Ω1- Ω2]= Cos[Ω1] Cos[Ω2] + Sin[Ω1] Sin[Ω2],
Cos[Ω1]2=.5 + .5 Cos[2Ω1]
Cos[Ω1]2=.5 - .5 Cos[2Ω1]
To produce the terms that has [Ω1+ Ω2], [Ω1+ Ω2], [2Ω1] or [2Ω2]. These are the terms that I am most interested it which do come out of that non linear system when fed with a two-tone input.
Which functions can I use to give me that, give me these Sin or Cos [Ω1+ Ω2], [Ω1+ Ω2], [2Ω1] or [2Ω2] terms?
Thanks all in advance for reading and for your help.