Re: Harmonic Analysis (Harmonic Matching) (Symbolic) in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg25598] Re: Harmonic Analysis (Harmonic Matching) (Symbolic) in Mathematica
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 11 Oct 2000 03:50:38 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <8rtsoq\$dpd@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

insert your definitions and expand the polynom

expr = a00 + a10 f[t] + a01 g[t] + a11 f[t] g[t] +
a20*f[t]^2 + a21 f[t]^2 g[t] +
andSoOn /. {f[t_] :> f1*Exp[I*w*t] + f2*Exp[-I*w*t],
g[t_] :> g1*Exp[I*w*t] + g2*Exp[-I*w*t]} // Expand

and here are the first harmonic

Plus @@ Cases[expr, a_.*Exp[I*w*t]]

Regards
Jens

AES wrote:
>
> 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

```

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