       Re: Fitting to complex values

• To: mathgroup at smc.vnet.net
• Subject: [mg30720] Re: Fitting to complex values
• From: "David M. Wood" <dmwood at slate.Mines.EDU>
• Date: Sat, 8 Sep 2001 02:55:58 -0400 (EDT)
• Organization: Colorado School of Mines
• References: <9nalpb\$npj\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Max Ulbrich <ulbrich at biochem.mpg.de> wrote:
> I have complex data (re+im) from a lock-in amplifier and want to fit
> them
> to a complex function. Though, the NonlinearFit function doesn't work
> with complex data. Has anyone a solution?
> mailto:ulbrich at biochem.mpg.de

Such 'response functions' are complex functions of a real variable.
They should obey the Kramers-Kronig relations (i.e., the real and
imaginary parts are essentially Hilbert transforms of one another).
In principle you can fit the real part and then use the K-K relations
to find the imaginary part, but this would provide a check of the
imaginary part data rather than a fit.  So I'd guess you'll need
to do coupled real fits, as below.

At first glance, the squared deviation function (what you minimize
in a least squares fit) I think would look like
W= \Sum_{all data pts n}[(fR(x_n)-fitR(x_n))^2 + (fI(x_n)-fitI(x_n))^2]
(where fR and fI are the real and imaginary parts of your data at
real argument x, and fitR and fitI are the corresponding fit
functions).  So you'd need to minimize W with respect to the parameters
of the two independent fitting functions fitR(x) and fitI(x).

Hope this helps.
--
David M. Wood
Department of Physics, Colorado School of Mines, Golden, CO 80401
Phone: (303) 273-3853; Fax: (303) 273-3840
e-mail: dmwood at physics.Mines.EDU ; NeXTMail welcome

```

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