Improved Cole-Cole Parameter Extraction from
Frequency Response Using Least Squares Fitting
Todd J. Freeborn, Brent Maundy, and Ahmed Elwakil*
Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada.
*Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, Emirates (elwakil@ieee.org).
Abstract—In this paper we propose using a method of non-linear
least squares to extract the Cole-Cole impedance parameters (Ro,
R∞, α, τ ) from collected frequency response datasets. We show
how this method reduces the error of the extracted parameters
when compared to direct extraction methods whose accuracy are
limited by the measured low and high frequency gains. Parameters
extracted using the non-linear least squares show up to 4 orders
of magnitude lower relative error from ideal simulated datasets
compared to direct methods. Experimentally extracted parameters
from fruit tissue frequency responses verify the accuracy of the
proposed non-linear least squares method over the direct extraction
method.
I. I NTRODUCTION
Fractional calculus, the branch of mathematics dealing with
differentiation and integration to non-integer order, has tradition-
ally been the domain of theoretical pursuits by mathematicians.
With a fractional derivative of order α given by the Grünwald-
Letnikov approximation [1] below
a
D
α
f (x) lim
h→0
1
h
α
[
x-a
h
]
m=0
Γ(α + 1)
m!Γ (α − m + 1)
f (x − mh) (1)
While there are no physical analogies to these derivatives,
like slope or area under a curve, the concepts of fractional
calculus and fractional-order systems show many useful appli-
cations in diverse fields of engineering. A few of these include
materials theory [2], control theory [3], electronic filters [4],
bioimpedance measurements [5], [6], [7] and more emerging
regularly.
In the field of bioimpedance measurements, the Cole-Cole
impedance model [8] is widely used for characterizing biologi-
cal tissues and biochemical materials [9] because of its simplic-
ity and good fit with measured data illustrating the behaviour
of tissue impedance as a function of frequency. The Cole-Cole
model is composed of three hypothetical circuit elements: a
high-frequency resistor R
∞
, a low-frequency resistor R
o
, and a
constant phase element (CPE). The CPE’s impedance is given
as Z
CPE
=1/(jω)
α
C ; C is the capacitance and α is its order,
where 0 <α ≤ 1. The expression for the impedance of the
Cole-Cole model is given as
Z = R
∞
+
R
o
− R
∞
1+(jωτ )
α
= Z
′
+ jZ
′′
(2)
where τ is the time constant given by τ = [(R
o
− R
∞
) C ]
1/α
.
Research has shown that the impedance parameters of healthy
and cancerous tissues differ [10]. Therefore, the accurate mea-
surement of these characteristics has the potential to aid doctors
+
-
+
-
+
-
+
-
- -
+
+
(a) (b)
Figure 1. Circuits for (a) filter and (b) integrator extraction of the Cole-Cole
model parameters.
and researchers in the diagnosis and study of cancerous tissues.
To characterize a particular tissue requires the determination of
the four parameters R
o
,R
∞
, α, and τ . The traditional method
to extract these parameters is to measure the tissue impedance
and construct a plot relating the imaginary impedance Z
′′
to
the real impedance Z
′
from which the four parameters can be
graphically extracted. Recent work has shown two techniques
to extract the parameters requiring only the frequency response
without direct measurement of the impedance and has been
presented in [6] and [7].
In this paper, we propose using a method of non-linear
least squares fitting (NLSF) to extract the Cole-Cole impedance
parameters from collected frequency response datasets. We show
how the methods presented in [6] and [7] are sensitive to
the accuracy of select measurements from a collected dataset,
while the NLSF method reduces the error in the extracted
parameters. The Cole-Cole impedance parameters are extracted
from MATLAB simulated datasets and experimentally collected
fruit tissue datasets verifying the improved accuracy of this
proposed method.
II. ANALYSIS OF DIRECT EXTRACTION METHOD
In [6] and [7] a Cole-Cole impedance is used as a component
in the extraction circuits; shown in Figs. 1(a) and (b), respec-
tively, exhibiting the frequency responses described by
T
1
(s) =
V
o
(s)
V
in
(s)
=
1
G
1
1+(τs)
α
1+
G2
G1
(τs)
α
(3)
T
2
(s) =
V
o
(s)
V
in
(s)
= −
G
1
+ G
2
(τs)
α
1+(τs)
α
(4)
where T
1
(s) and T
2
(s) are the filter and integrator transfer
functions, respectively. G
1,2
are the low and high frequency
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