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Harmonic Analysis (Harmonic Matching) (Symbolic) in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg25591] Harmonic Analysis (Harmonic Matching) (Symbolic) in Mathematica
• From: AES <siegman at stanford.edu>
• Date: Mon, 9 Oct 2000 21:43:31 -0400 (EDT)
• Organization: Stanford University
• Sender: owner-wri-mathgroup at wolfram.com

How can one do symbolic harmonic analysis (aka "harmonic matching") in 
Mathematica?

That is, I'd like to insert sinusoidal functions f[t] and g[t] written as

   f[t] = f1 Exp[+j w t] + f1Star  Exp[-j w t]

   g[t] = g1 Exp[+j w t] + g1Star  Exp[-j w t]

into a polynomial in f[t] and g[t], e.g.

   expr = a00 + a10 f[t] + a01 g[t] + a11 f[t] g[t] 
                    + a20 [f[t]^2 + a21 f[t]^2 g[t] + and so on

and then pull out the individual harmonic components, i.e. the 
Exp[+j w t], Exp[-j w t], Exp[+2 j w t], Exp[-j w t], etc., components.

(This may not be what some people call harmonic analysis, but it's a 
common approach in engineering analysis of nonlinear systems.)

A simple way to do this?

(Note: the coefficients may themselves be expressions that contain 
factors like Exp[constant] -- though not t explicitly.]

Thanks,   AES

• Follow-Ups:
  • Re: Harmonic Analysis (Harmonic Matching) (Symbolic) in Mathematica
    • From: "Carl K. Woll" <carlw@u.washington.edu>