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MathGroup Archive 2006

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General--Trigonometric functions manipulations in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71504] General--Trigonometric functions manipulations in Mathematica
  • From: ali.abuelmaatti at elec.gla.ac.uk
  • Date: Mon, 20 Nov 2006 18:12:07 -0500 (EST)

Hi all,

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

Expand [y]

Out[1]= 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:

Simplify[Out[1]]

Out[2]= 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 doesnÂ?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]

And

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.

Ali


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