Least squares estimation technique of Cole-Cole parameters from step response
times very near [vo(t ¼ 0)] and very far [vo(t ¼ 1)] from the applied voltage step. In particular, R1 R2 = vo (0) (4) Vcc − vo (0)
T.J. Freeborn, B. Maundy and A.S. Elwakil A method of nonlinear least squares to estimate the Cole-Cole impedance parameters (Ro , R1 , a, C) from collected time-domain step response datasets is reported. Experimentally estimated parameters from fruit tissue step responses are given and compared with numerical simulations of the acquired model.
Introduction: In the field of bioimpedance measurements, the ColeCole impedance model [1] is widely used for characterising biological tissues and biochemical materials [2] because of its simplicity and good fit with measured data. The Cole-Cole model is composed of three hypothetical circuit elements: a high-frequency resistor R1, a low-frequency resistor Ro , and a constant phase element (CPE). The CPE’s impedance is given as ZCPE ¼ 1/( jv)aC where C is the capacitance and 0 , a ≤ 1 is its order. The expression for the impedance of the Cole-Cole model is given as Z = R1 +
Ro − R1 = Z ′ + jZ ′′ 1 + ( jvt)a
(1)
where t is the time constant given by t ¼ [(Ro 2 R1) C ]1/a. To characterise a particular tissue requires the determination of the four parameters Ro , R1 , a, and t. Multiple techniques have been used to extract the Cole-Cole impedance parameters from the measured impedance data [3, 4] and measured frequency response [5, 6]. A recent step response method was proposed in [7]. The detailed analysis in [7] outlines how the Cole-Cole parameters can be extracted from the time-domain dataset measurements of the step response. In this Letter, we propose using the alternative method of nonlinear least squares fitting (NLSF) to extract the Cole-Cole impedance parameters from collected step response datasets. We show that the method presented in [7] is sensitive to the accuracy of measurements from a collected dataset, while the NLSF method has much lower sensitivity towards the accuracy of the collected time-domain data. The ColeCole impedance parameters are extracted from MATLAB simulated datasets and experimentally collected fruit tissue datasets verifying this NLSF method.
uin
R1
uo required Cole-Cole model
R2 1 s aC
R3
Fig. 1 Circuit for step response estimation of Cole-Cole model parameters
Direct extraction analysis: Fig. 1 shows the simple circuit used to obtain the step response of the Cole-Cole impedance model with an applied voltage step-input vin , where R2 ¼ R1 and R3 ¼ Ro 2 R1. It has been shown in [7] that the time domain expression of the output voltage, vo(t), of the circuit in Fig. 1, assuming zero initial conditions, is given by R2 R1 + R2 + R3 Ea,1 −t a vot = Vcc R1 + R2 CR3 (R1 + R2 ) (2) R2 + R3 a a R1 + R2 + R3 t Ea,a+1 −t + CR3 (R1 + R2 ) CR3 (R1 + R2 ) where Ea,b(.) is the two-term Mittag-Leffler function defined as [8] Ea,b =
1
zk G(ak + b) k=0
(3)
Using the step response described by (2), both R2 and R3 can be extracted, when R1 is known, by measuring the output voltage at
vo (1)(R1 + R2 ) − Vcc R2 (5) Vcc − vo (1) Then a and C can be extracted knowing R2,3 and the slope of the rise time and time of the midpoint voltage from the step response [7]. It is clear that the measurement of vo(0) and vo(1) has a significant impact on the accuracy of the extracted parameters. Since their exact values cannot be measured from a response that does not extend from t ¼ 0 to t ¼ 1, their measured values will only ever be an approximation. The error in these approximate values introduces errors into the extracted R2,3 that propagate into the extracted a and C, which require accurate R2,3 values for their calculation. For example, the effect of these errors on the extracted parameters from (2) simulated with R1,2,3 ¼ 1 kV, a ¼ 0.75, C ¼ 100 nF, Vcc ¼ 5 V, vo (1) ≃ vo(1 s), and vo(0) ≃ vo(1 ps), vo(1 ns), and vo(1 ms), respectively, are given in Table 1. As the error in the vo(0) value increases, the errors of the extracted parameters get significantly larger. This highlights how sensitive this extraction process is to the accuracy of the vo(0) approximation; which serves as the foundation for the extraction of all the impedance parameters. The problem becomes worse with lower a and C values, which decrease both the rise time and location of the midpoint voltage [7], which increases the hardware complexity by requiring a high precision step input generator. R3 =
Table 1: Relative errors of parameters extracted from simulated step responses for different measured values of vo(0) vo(0) measured after 1 ps 1 ns 1 ms
Relative error (%) R2 R3 a C 0.00109 0.000383 0.00140 20.00114 0.193 21.92 0.456 22.71 29.1 229.1 33.3 278.6
Least squares extraction: The limitation of the direct extraction method can be overcome by using an NLSF method. This numerical method attempts to solve the problem: n (6) min T (x) − ydata22 = min (T (x)i − ydatai )2 x
x
i
where x is the vector of impedance parameters (R2 , R3 , a, C ), T(x) is the time domain step re"sponse (2) calculated using x, ydata is the collected step response to fit to (2), T(x)i and ydatai are the simulated response and collected response at time ti , and n is the total number of data points in the collected step response. This routine aims to find the impedance parameters that would ideally reduce the least squares error to zero. The accuracy of the parameters extracted from the collected dataset using the NLSF method is not limited by the accuracy of R2,3 , determined by two datapoints in the direct method, because every collected datapoint is used to find the parameters that result in the best fit. Ideal step response: Both the direct and NLSF methods were applied to the ideal step responses of the circuit in Fig. 1, given by (2), using fruit impedance parameters from [5] with R1 ¼ 1 kV. These MATLAB simulations were carried out for a 1 ms to 1 s time set. The two term Mittag- Leffler function was calculated using [9] with each element calculated to an accuracy 1026. From the results in Table 2 the NLSF shows up to five orders of magnitude less error than the direct method. Therefore, using the NLSF method we can maintain a high level of accuracy while relaxing the hardware requirements since it is enough to gather data over 6 decades (from 1 ms to 1 s). Recall from Table 1 that this is not possible with the direct extraction method.
Table 2: Absolute relative errors from simulated step responses of fruit tissues from 1 ms to 1 s Fruit
Absolute relative error (%) [NLSF method/Direct method]
R2 R3 a 1.93 × 1023 1.23 × 1024 1.60 × 1023 Apple 1.78 × 102 5.77 × 100 3.38 × 101 1.38 × 1023 3.40 × 1024 3.54 × 1023 Apricot 5.74 × 101 5.71 × 100 3.18 × 101 Kiwi Potato
ELECTRONICS LETTERS 21st June 2012 Vol. 48 No. 13
1.82 × 100 8.02 × 102 7.08 × 1023 3.42 × 102
9.98 × 1022 4.38 × 101 2.93 × 1024 1.62 × 100
9.00 × 1023 4.77 × 101 8.80 × 1024 3.68 × 101
C 1.15 × 1022 8.76 × 101 1.87 × 1022 7.52 × 101 1.00 × 1021 8.18 × 101 6.97 × 1023 8.57 × 101
Experimental step responses: The Cole-Cole impedance of Fig. 1 was realised using components R1 ¼ 1.007 kV, R2 ¼ 1.753 kV, R3 ¼ 32 kV, C ¼ 1 mF, and a ¼ 1. Using this setup, a voltage step input was applied using an Agilent 33250A waveform generator. A pulse with 15 ns edge time, 500 ms pulse width, and period of 1 s was generated and applied to the circuit under test with the resulting step response measured using a Tektronix 745D digital oscilloscope. The collected step response is shown in Fig. 2 as the solid line. The MATLAB algorithm was applied to the experimentally collected dataset with relative errors of (4.91, 7.11, 3.46 × 1027, 2.22 × 1026)% for (R2 , R3 , a, C), respectively. The response simulated using the estimated parameters (dashed line in Fig. 2) shows very good agreement with the experimentally collected response.
Acknowledgment: This work has been supported by the National Sciences and Engineering Research Council (NSERC) and Alberta Innovates – Technology Futures. # The Institution of Engineering and Technology 2012 2 February 2012 doi: 10.1049/el.2012.0360 T.J. Freeborn and B. Maundy (Department of Electrical and Computer Engineering, University of Calgary, Alberta, Canada)
4.8 4.6
E-mail:
[email protected]
4.4 voltage, V
Conclusion: We have presented a method to improve significantly the accuracy of the extracted Cole-Cole impedance parameters from a collected time-domain step response using a NLSF method. The method also reduces the size of the acquired dataset by reducing the sampling time of the step response.
A.S. Elwakil (Department of Electrical and Computer Engineering, University of Sharjah, United Arab Emirates)
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References
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1 Cole, K., and Cole, R.: ‘Dispersion and absorption in dielectrics’, J. Chem. Phys., 1941, 9, pp. 341– 351 2 Grimnesand, S., and Martinsen, O.: ‘Bioimpedance and bioelectricity basics’ (Academic Press, 2000) 3 Ionescu, C., and Keyser, R.D.: ‘Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease’, IEEE Trans. Biomed. Eng., 2009, 56, pp. 978–987 4 Tang, C., You, F., Cheng, G., Gao, D., Fu, F., and Dong, X.: ‘Modeling the frequency dependence of the electrical properties of the live human skull’, Physiol. Meas., 2009, 30, pp. 1293– 1301 5 Elwakil, A.S., and Maundy, B.: ‘Extracting the Cole-Cole impedance model parameters without direct impedance measurement’, Electron. Lett., 2010, 46, pp. 1367–1368 6 Maundy, B., and Elwakil, A.S.: ‘Extracting single dispersion Cole-Cole impedance model parameters using an integrator setup’, Analog Integr. Circuits Signal Process., 2011, DOI: http://dx.doi.org/10.1007/s10470011-9751-1 (in press) 7 Freeborn, T.J., Maundy, B., and Elwakil, A.S.: ‘Numerical extraction of Cole-Cole impedance parameters from step response’, IEICE Nonlinear Theory Appl., 2011, 2, pp. 548–561 8 Podlubny, I.: ‘Fractional differential equations’ (Academic Press, 1999) 9 Podlubny, I.: ‘Mittage-Leffler function’, 2005, http://www.mathworks. com/matlabcentral/fileexchange/8738
3.4 3.2 10−4
10−3
10−2
10−1
time, s
Fig. 2 Experimental (solid) and MATLAB simulated (dashed) step responses of Fig. 1 when (R2 , R3 , a, C) ¼ (1.753 kV, 32 kV, 1, 1 pF)
To verify further the NLSF method, the circuit of Fig. 1 was implemented using an apple and a tomato as the Cole-Cole impedances with R1 ¼ 1.007 kV. The values extracted from the step response, collected from 1 ms to 500 ms, from both fruits are given in Table 3. The collected step response of the circuit realised with the tomato is shown in Fig. 3 as a solid line, with the MATLAB simulated response using the NLSF estimated parameters shown as a dashed line.
Table 3: NLSF extracted Cole-Cole impedance parameters from experimentally collected step responses of fruits Impedance parameters
Fruit
R2 (kV) R3 (kV) a C (nF) Apple 2.206 29.88 0.8814 4.769 Tomato 0.0246 19.64 0.6145 372.5 5.0
voltage, V
4.5 4.0 3.5 3.0 2.5 2 10−6
10−6
10−4
10−3 time, s
10−2
10−1
Fig. 3 Experimental (solid) and MATLAB simulated (dashed) step responses using a tomato in circuit of Fig. 1
ELECTRONICS LETTERS 21st June 2012 Vol. 48 No. 13
