Re: fitting functions which contain 'i': more details

[Harbinson, Jeremy]

Hi,
The function I need to fit is one that describes the frequency dependence of 
the electrical impedance (Z) of biological tissue in terms of four parameters:
capacitance, 
two limiting resistances, and 
alpha, which basically compensates for the fact that biological materials do 
not behave as ideal Resistor-Capacitor systems. 

It is difficult to clearly write the function here as it has lots of subscripts 
etc, but here is the function in Latex:

Z\left(freq\right)=R_{\infty}+\frac{R_{0}-R_{\infty}}{1+i\omega 
C\left(R_{0}-R_{\infty}\right)^{\alpha}}

C is capacitance, and R0 and Rinfinity are the limiting resistances. 

If C is non-zero the function returns an complex number that contains the real 
and imaginary  parts of the impedance. If the real and imaginary parts of the 
impedance are plotted against each other (real on x, imaginary on y) the result 
is a so-called Cole-Cole plot. I think it is also sometimes called a Nyquist 
diagram.

The data I collect is the complex impedance (Z) as a function of frequency 
(omega). I would like to estimate the parameters R0, Rinfinity, C and alpha 
from this data by the fitting the above function to it. So I need some fitting 
tool that is happy working with complex numbers. I have a recollection that 
this is/was possible with a function in the basic Matlab package, but I do not 
know which.

An alternative approach to fitting the function above is to fit the Cole-Cole 
plot (the real and imaginary components plotted against each other) with a 
circle, as the locus of the points on a Cole-Cole plot is a chord or semicircle 
whose centre is shifted away from the origin of the graph. Problems with this 
approach are that the least-squares fitting routine does not (so far as I know) 
fit parametric equations (maybe fsolve does?) and real data often deviates from 
the chord/semicircle of the ideal Cole-Cole plot (typical biology - we are not 
ideal). These distorted responses are relatively easy to deal with by tweaking 
the basic function shown above, but not so easy to work with via the graphical 
approach of the Cole-Cole plot. 
Hope this clarifies things a bit,
all the best,
Jeremy