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Extracting coefficients for sinusoidals

• To: mathgroup at smc.vnet.net
• Subject: [mg60477] Extracting coefficients for sinusoidals
• From: Helge Stenstrom <helge.stenstrom at ericsson.com>
• Date: Sat, 17 Sep 2005 02:32:01 -0400 (EDT)
• Organization: Ericsson
• Sender: owner-wri-mathgroup at wolfram.com

Suppose I have a signal 
 signal = Sin[p]

and a non-linear function
 f[x_] := x - x^3/3

then
  f[signal] = Sin[p] - Sin[p]^3/3

but I don't want this form (with powers of the Sin function), but
rather the form with multiples of p. TrigReduce can solve that:

  signal // f // TrigReduce

  (9*Sin[p] + Sin[3*p])/12

It's easy to extract the coefficients by visual inspection, 

  9/12, 1/12

but how can it be done programmaticaly? With a more complex non-linear
function, for example 

  a1*x + a1*x^2 + a1*x^3 + a4*x^4 + a5*x^5
using 
  signal // f // TrigReduce // InputForm
resulting in 
  (8*a1 + 6*a4 - 8*a1*Cos[2*p] - 8*a4*Cos[2*p] + 2*a4*Cos[4*p] + 28*a1*Sin[p] + 
  10*a5*Sin[p] - 4*a1*Sin[3*p] - 5*a5*Sin[3*p] + a5*Sin[5*p])/16

visual ispection is no longer simple enough. So how can I get the
coefficients, individually or as a list?

I'm not sure how I want the Cos and Sin terms to be treated; probably
a*Cos[x] should correspond to a*I*Sin[x] when the coefficients are
collected. 

-- 
Helge Stenström

• Follow-Ups:
  • Re: Extracting coefficients for sinusoidals
    • From: "W. Craig Carter" <ccarter@mit.edu>