Identification of a Block-Structured Nonlinear Feedback System

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Identification of a Block-Structured Nonlinear Feedback System, Applied to a Microwave Crystal Detector Johan Schoukens, Fellow, IEEE, Liesbeth Gommé, Student Member, IEEE, Wendy Van Moer, Senior Member, IEEE, and Yves Rolain, Fellow, IEEE

Abstract—This paper studies the identification of a blockstructured nonlinear Wiener–Hammerstein system that is captured in the feedforward or the feedback path of a closed-loop system. First, nonparametric initial estimates are generated for the three dynamic blocks, which are modeled by their frequency response function, and the static nonlinear system, which is described using an input–output table. Next, a parametric model is identified. Random or periodic excitations can be used in the experiments. The method is illustrated on the identification of an Agilent HP420C crystal detector. Index Terms—Block structure, crystal detector, feedback (FB), nonlinear (NL) systems, system identification, Wiener– Hammerstein.

Fig. 1.

Basic block-structured models.

I. I NTRODUCTION

M

EASURING the frequency response function (FRF) of a linear dynamic system is one of the main topics of the measurement society, because it is a generic problem that appears in many application fields, for example, vibrating mechanical structures, the FRF of electrical systems like filters and amplifiers, and the dynamic behavior of sensors. In the second step, the nonparametric response is modeled by a parametric transfer function model, as explained, for example, in [11]. These models are widely used, among others, in simulations, design, and calibration processes. In many applications, the system is prone to nonlinear (NL) distortions, requiring the generalization of the linear transfer function model. This leads to NL models and measuring techniques. Compared to measuring and modeling linear systems, the characterization of NL dynamic systems is a very difficult problem because no universal valid model structure is available, as it is for linear systems. An attempt to solve that problem is to extend the class of linear systems by adding static NL blocks, which leads to block-structured models that describe the NL system as a connection of linear dynamic and static NL blocks [1], [9], [10]. Initially, attention was focused on Wiener and Hammerstein systems, followed more recently by Manuscript received July 2, 2007; revised May 5, 2008. This work was supported in part by the Research Foundation–Flanders (FWO–Vlaanderen), by the Flemish Government (Methusalem METH1), and by the Belgian Federal Government (IUAP VI/4). The authors are with the Department of Fundamental Electricity and Instrumentation (ELEC), Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail: [email protected]). Digital Object Identifier 10.1109/TIM.2008.925721

Fig. 2. Examples of a block-structured FB model. (Top) Nonlinearity in the FF path. (Bottom) Nonlinearity in the feedback path.

Wiener–Hammerstein systems, which are natural extensions of the linear theory, as shown in Fig. 1. In this figure, G0 , R0 , and S0 are linear dynamic systems, and f0 is a static NL system. For these systems, extended identification methods are developed (see [3] and references therein). All these systems are open-loop systems; there is no NL feedback (FB) present. Such systems cannot describe many of the phenomena that are observed in practice for NL systems: a shifting resonance frequency or a changing damping as a function of the input amplitude of the excitation. To also include these phenomena, an FB around the nonlinearity should be added. This leads to the system class as shown in Fig. 2. It is the responsibility of the measurement society to also provide for this generalized class of systems good measuring/modeling tools to the user. It turns out that the most difficult step in the identification process is finding good initial estimates for the individual building blocks G0 , R0 , and S0 (the FRFs of the linear dynamic blocks) and for f0 (the static nonlinearity); see Fig. 2. The aim of this paper is to solve this problem,

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SCHOUKENS et al.: IDENTIFICATION OF A BLOCK-STRUCTURED NONLINEAR FEEDBACK SYSTEM

TABLE I EQUIVALENCIES OF THE STRUCTURES IN FIG . 2

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B. Nonparametric Identification of the Dynamics of the FF and the FB Path The first step of the initialization procedure is to generate initial estimates for the dynamics G0 (FF) and G2 = R0 S0 (FB). In the second step, G2 is split over R0 and S0 , and simultaneously, an initial estimate for f0 is estimated. Therefore, the nonparametric initialization procedure consists of the following steps. — Estimate the FRF of the FF and FB dynamics: G0 and G2 . — Split G2 in R0 and S0 , and estimate f0 .

providing nonparametric initial estimates, starting from a set of measured input and output data r(k), y(k), k = 1, . . . , N . Once these initial values are available, an NL optimization problem is solved to tune the model to the data. This is a nontrivial extension of the existing identification methods for blockstructured NL systems. Until now, no methods are available to solve this problem. This NL FB modeling technique will be applied to a coaxial HP420C crystal detector. This device is used in microwave applications to measure the envelope of a signal. In this paper, the model of the detector will be identified. An extended validation of the model is discussed in [7]. This paper is an extended version of the work presented in [14]. In the first part of the paper, it will be verified if an NL FB structure can be uniquely identified from input–output data only. Next, an initialization procedure to identify the linear dynamic blocks and the static NL blocks in the FB structure is explained, followed by a brief discussion of the parametric estimation step. Finally, the case study is made. II. I DENTIFICATION OF A B LOCK -S TRUCTURED NL FB S YSTEM A. Structure Selection: An Indistinguishability Problem A detailed study [8] reveals that it is impossible to distinguish between the two structures in Fig. 2 from input–output data only. It is always possible to replace the upper model with the lower one (or the other way around), without changing the input–output relations. Even if one of the two structures is chosen, it is still impossible to uniquely identify G0 . An arbitrary part of the dynamics of the FB branch can be shifted to the feedforward (FF) branch by changing at the same time the static NL characteristic f0 (x) → f0 (x) − α(x) (see also Appendix A). Table I gives the relations between the different structures and their linear dynamic and static NL blocks. From the identification point of view, it is impossible to make a physical interpretation from input–output measurements only; additional assumptions or prior information is needed. On the basis of this discussion, we select in this paper, without any loss of generality, the bottom structure with the static nonlinearity in the FB as the default block structure.

Both steps are discussed below in more detail. 1) Estimate the FRF of the FF and FB Dynamics—G0 and G2 : The method starts from the observation that the best linear approximation (BLA) GCL of an NL closed-loop system varies with the amplitude of the excitation: The resonance frequencies can shift, and the damping can vary. By making experiments at different excitation levels, a series of varying FRFs is obtained, and these variations will be used to split the input and output dynamics. BLA of a Wiener–Hammerstein system: It is shown that a wide class of NL systems can be approximated by its BLA plus an NL noise source [2], [12], [13]. For the Wiener– Hammerstein system in the FB loop of the system at the bottom of Fig. 2, which is symbolically denoted as q = S0 f0 (R0 p)

(1)

the BLA for Gaussian noise excitations is the product αS0 R0 = αG2

(2)

with α constant [11]. The constant α depends on the power spectrum of the input. BLA of a Wiener–Hammerstein system captured in an FB loop (Fig. 2): In the first step, we replace the NL FB branch (the Wiener–Hammerstein system) by its BLA (2). This leads to the following linearized approximation for the closed-loop system: GCL ≈

G0 . 1 + αG0 G2

(3)

Note that even if the reference signal r(t) is Gaussian distributed, it is not guaranteed to be the case for the driving signal p(t) of the FB. However, to generate initial estimates, all approximations are allowed as long as they result in reasonable estimates. Separating the FF and FB dynamics: Apply R experiments with a changing amplitude of the excitation. Measure for each experiment the BLA GCL (i, j), where i = 1, . . . , F is the frequency index ωi , and j = 1, . . . , R is the experiment index. The inverted measurements are used as entries for the −1 matrix gCL 

−1 gCL

· = · ·

·

1 GCL (i,j)

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·

 · · ·

(4)

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where it should be noted that 1 1 = + αj G2 (i). GCL (i, j) G0 (i)
G0


G2

sample frequency). The static nonlinearity is written as a linear combination of a set of basis functions (5) f0 (x, θf ) =

Vinit

  2     1 1

 −
= + α j G2 (i)  . (6)   GCL (i, j)  G (i) i

j

V (Y, θ) =

ˆ −1 G FF

1 = + β1 G2 , G0

ˆ 2 = β2 G2 G

(7)

with β1 and β2 being arbitrary parameters. The presence of these parameters directly follows from the indistinguishability study: β1 is discussed in Table I in Section II-A. A change of β2 leads to an appropriate variation of the scaling of fˆ such that the total gain of the FB path remains constant. 2) Separating the Dynamics and the Static Nonlinearity in the NL Branch: Once initial estimates are available for the FF and FB dynamics, it is possible to calculate the initial estimates pˆlin and qˆlin from r and y pˆlin = y,

ˆ −1 (y). qˆlin = r − G FF

(8)

Next, a nonparametric initial estimate for the Wiener– Hammerstein FB branch can be generated [15]. A detailed discussion of this method is out of the scope of this paper. The final result of this procedure is a nonparametric estimate of the FRF of the dynamic blocks at a user-specified fast-Fouriertransform frequency grid. In addition, a nonparametric model of the static nonlinearity is obtained in the form of an input output table. This characteristic can be further smoothed using, for example, spline functions. Remark: Since we use only nonparametric frequency response representations of the dynamic blocks, the stability of the estimates is not an issue here. C. Estimation of a Parametric Model Parametric initialization: Once the nonparametric initial estimates are available for the static and dynamic blocks, it is possible to identify a parametric model. All dynamic blocks are individually modeled [11] with a transfer function model (discrete time or continuous time) of the form F (Ω, θF ) =

nb  k=1

bk Ωk /

na 

ak Ωk

(9)

k=1

with F being G0 , R0 , and S0 , and Ω = jω for continuous-time systems or Ω = e−j2πf /fs for discrete-time systems (fs is the

(10)

Possible choices are, for example, Bk (x) = xk or piecewise linear representations. Parametric optimization: In the final optimization step, a (weighted) least squares problem is solved

0

This eventually results in the following estimates for the FF and FB dynamics:

βk Bk (x).

k=0




The FRFs 1/ (i) and (i) and the scale factors α j are estimated by minimizing the following cost function (see Appendix B):

nf 

N1 1  wk |Y (Ωk ) − M (Ωk , θ)|2 N1

(11)

k=1

with M (Ωk , θ) being the block-structured FB model evaluated at frequency Ωk , and θ = {θG , θR , θS , θf }. The weighting wk can be used to select the frequency band of interest, putting wk = 0 at those frequencies that should not be considered. It also allows us to weight with the noise variances as a function of the frequency [11]. This should only be done if it turns out that the model errors are (much) smaller than the noise errors. In this paper, we choose wk equal to one or zero to select the considered frequency band. D. Some Practical Remarks — During the measurement and processing of the data, the sample frequency should be much larger than the maximum signal frequency to avoid aliasing during the calculations. The nonlinearity f (p) can create higher harmonics, far above the maximum input frequency. — The FRF measurements GCL (i, j) (see Section II-B) are smoothed using a parametric transfer function estimate for each excitation level j. — During the parametric minimization of V (Y, θ), the pseudoinverse [6] is used to solve the normal equations to deal with the intrinsic overparameterization of the blockstructured model. Indeed, it is easy to see that it is possible to push a gain factor or a delay from one block to another so that the number of independent parameters is smaller than the total number of free parameters in the model, which results in normal equations that are not of full rank. Furthermore, the presence of the “indistinguishability parameters” β1 and β2 reduces the rank. — The parametric linear dynamic models might be unstable. This is not really an issue, because all linear relations are calculated in the frequency domain. The steadystate solution of these systems is obtained. During the calculation of the closed-loop solution, an iterative calculation scheme is used to retrieve the steadystate solution, assuming that the whole experiment is periodically repeated. It might happen that this process does not converge. In that case, more powerful NL equation solvers should be used. See the literature. When selecting a method, it is important to realize that the equations should be solved for long records (e.g., 10 000 to 50 000 points) in each iteration step.

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SCHOUKENS et al.: IDENTIFICATION OF A BLOCK-STRUCTURED NONLINEAR FEEDBACK SYSTEM

Fig. 3. Output measurement, segmented in five blocks for the separation of the FF and FB dynamics.

Fig. 4. (Left) FRF of the BLA. (Right) Its evolution over the different blocks of the experiment.

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ˆ Fig. 5. (Top left and right) Amplitude of the identified transfer functions G ˆ and R.

Fig. 6. Estimated nonparametric static nonlinearity. Note that the input and output are scaled with an arbitrary factor.

III. C ASE S TUDY : I DENTIFICATION OF A C RYSTAL D ETECTOR In this section, the method is illustrated on the identification of an Agilent HP420C crystal detector. This device is excited with a Gaussian distributed excitation with a slowly growing amplitude. The bandwidth of the excitation is 800 kHz. This signal is generated with an arbitrary waveform generator (Agilent HP-E1445A). The input and output voltage is measured using Agilent HP-E1430A data-acquisition cards. Since these have a 50-Ω input resistance, a high-impedance homemade buffer is used to avoid a loading of the device. All measurements are alias protected with antialias filters. The sampling frequency is 10 MHz. The noise sequence has a length of 50 000 samples. This sequence is repeated five times, and the final measurements are obtained by averaging these repeated measurements. This allows us to also estimate the standard deviation of the noise on the measurements. It turns out that the standard deviation of the input and output noise is 0.23 mV. The measured output is shown in Fig. 3. As described in the Section II, the input and output is segmented in five subrecords, and for each of these, the best approximating linear system is identified. The amplitude of the FRF of these estimates for the five successive blocks is shown in Fig. 4. From these results, an estimate of the FF and FB dynamics is made. Next, the static nonlinearity is identified, and

Fig. 7. Physical representation of the identified crystal detector model.

finally, these results are used to identify a parametric model. A discrete-time model is identified with the following blocks: −1

ˆ −1 , θG ) = 0.1700 + 0.1753z G(z 0.8345 − 0.4952z −1

(12)

−1 ˆ −1 , θG ) = 5.034 − 5.052z R(z −7.634 − 5.367z −1

(13)

ˆ −1 , θG ) = 0.7675. S(z

(14)

The static nonlinearity is modeled as a ninth-order polynomial. These models are shown in Figs. 5 and 6. A physical interpretation of the NL block structure is given in Fig. 7. The static nonlinearity shown in this scheme is that of the “diode” after removing a linear part (see Section II-A). Using this model, the output of the crystal detector is simulated and compared to the measurements. As can be seen in Fig. 8, the difference between measurement and simulation is very small (mean error: 0.18 mV, standard deviation: 0.30 mV), particularly when this is compared with the standard deviation

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A PPENDIX A I NDISTINGUISHABILITY OF THE B LOCK -S TRUCTURED NL FB S YSTEM Shifting Dynamics From the NL FB to the FF Branch: Consider the structure with the nonlinearity in the FB (Fig. 2, bottom). We will show that it is possible to change the dynamics of the FF branch by subtracting a linear term of the static nonlinearity in the FB y = G0 (r − S0 f0 (R0 y)) .

(15)

Define the modified NL function Fig. 8. (Black, top) Comparison of the measured and simulated output. (Black, bottom) Error. (Gray) Nonlinear contribution to the output.

Then

or

fˆ0 (x) = f0 (x) − αx.

(16)



y = G0 r − S0 fˆ0 (R0 y) − αS0 R0 y

(17)

 −1 y = G−1 r − fˆ0 (R0 y) 0 + αS0 R0

ˆ −1 r − fˆ0 (R0 y) =G 0

(18)

ˆ −1 = G−1 + αS0 R0 G 0 0

(19)

G0 . 1 + αG0 S0 R0

(20)

with

Fig. 9. Comparison of the measured and simulated output spectrum. (Black, top) Measured output. (Gray) Error linear model. (Black, bottom) Error nonlinear model.

of the noise on the input and output measurements, which equals 0.23 mV. Note that the mean error is also influenced by the offset errors of the acquisition channels. The contribution of the NL FB to the output is also shown in the figure. It can be seen that it is significantly larger than the remaining error, and hence, a NL model is indeed needed to model the crystal detector. In practice, it is exactly this NL part of the output that will be used in the normal operation of the crystal detector. The conversion gain (NL/linear contributions) is about −16 dB. In Fig. 9, the same results are given in the frequency domain. This clearly shows that the simulation error is reduced with 20 dB by using the NL FB structure compared to the BLA.

or ˆ0 = G

Switching the Static Nonlinearity From the FF to the FB Branch: Consider the structure with the nonlinearity in the FF (Fig. 2, top). We will show that it is possible to switch the static NL function f˜0 from the FF to the FB. Assume that the inverses exist where needed

˜ 0 f˜0 S˜0 (r − G ˜ 0 y) y=R (21) or



˜ −1 (y) = r − G ˜ 0 y. S˜0−1 f˜0−1 R 0

This leads, after some rearrangements, to y = G0 (r − S0 f0 (R0 (y)))

IV. C ONCLUSION In this paper, we have studied the identification of NL FB systems using block-structured models. First, it is shown that such a system cannot be uniquely identified from input–output measurements only. Next, a system identification method is proposed to identify a block-structured NL FB system. The method can be used with periodic or random excitations. With minimum user interaction, initial nonparametric estimates are obtained for the three linear dynamic blocks and the static nonlinearity. Starting from these results, a parametric NL optimization is initialized. The technique is successfully applied to the identification of a crystal detector.

(22)

(23)

with ˜ −1 , R0 = R ˜ −1 , S0 = S˜−1 . f0 (x) = f˜0−1 (x), and G0 = G 0 0 0 (24) A PPENDIX B G ENERATING I NITIAL E STIMATES Consider the quadratic cost function   2 F  R    1 1

  −
+ α j G2 (i)  Vinit =   GCL (i, j)  G (i) i j

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0

(25)

SCHOUKENS et al.: IDENTIFICATION OF A BLOCK-STRUCTURED NONLINEAR FEEDBACK SYSTEM

that minimizes the difference between the inverse measured values 1/GCL (i, j) and the modeled inverse closed-loop transfer function 1
G0





+ α j G2 (i)

(26)

(i)

with respect to the parameters 1







and α j , G2 (i).




(27)

G0 (i)


In the first step, Vinit is minimized with respect to 1/ G0 (i) 1  R j=1 R

1 ˆ ∗ (i) G 0

= argmin Vinit = 1
G 0 (i)

 1

− α j G2 (i) . GCL (i, j) (28)

This result is substituted in the cost function (25). Define 1 1 1  1 = − ∗ GCL (i, j) GCL (i, j) R j=1 GCL (i, j) R




αj∗ = α j −

1 
αj . R j=1

(29)

R

(30)

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[4] S. Duym, “An alternative force state map for shock absorbers,” Proc. Inst. Mech. Eng., D: J. Automobile Eng., vol. 211, no. 3, pp. 175–179, May 1997. [5] M. Enqvist and L. Ljung, “Linear approximations of nonlinear FIR systems for separable input processes,” Automatica, vol. 41, no. 3, pp. 459–473, Mar. 2005. [6] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: John Hopkins Univ. Press, 1996. [7] L. Gomme, J. Schoukens, Y. Rolain, and W. Van Moer, “Validation of a crystal detector model for the calibration of the Large Signal Network Analyzer,” in Proc. IEEE Instrum. Meas. Technol. Conf., Warsaw, Poland, May 1–3, 2007, pp. 948–953. [8] K. R. Godfrey and M. J. Chapman, “Identifiability and indistinguishability of linear compartmental models,” Math. Comput. Simul., vol. 32, no. 2, pp. 273–295, Jun. 1990. [9] M. J. Korenberg and I. W. Hunter, “The identification of nonlinear biological systems: LNL cascade models,” Biol. Cybern., vol. 55, no. 2/3, pp. 125–134, Nov. 1986. [10] M. J. Korenberg, “Parallel cascade identification and kernel estimation for nonlinear systems,” Ann. Biomed. Eng., vol. 19, no. 4, pp. 429–455, Jul. 1991. [11] R. Pintelon and J. Schoukens, System Identification: A Frequency Domain Approach. Piscataway, NJ: IEEE Press, 2001. [12] J. Schoukens, T. Dobrowiecki, and R. Pintelon, “Identification of linear systems in the presence of nonlinear distortions—A frequency domain approach,” IEEE Trans. Autom. Control, vol. 43, no. 2, pp. 176–190, Feb. 1998. [13] J. Schoukens, R. Pintelon, T. Dobrowiecki, and Y. Rolain, “Identification of linear systems with nonlinear distortions,” Automatica, vol. 41, no. 3, pp. 451–504, Mar. 2005. [14] J. Schoukens, L. Gomme, W. Van Moer, and Y. Rolain, “Identification of a crystal detector using a block structured nonlinear feedback model,” in Proc. IEEE IMTC, Warsaw, Poland, 2007, pp. 68–72. [15] J. Schoukens, R. Pintelon, J. Paduart, and G. Vandersteen, “Nonparametric initial estimates for Wiener–Hammerstein systems,” in Proc. 14th IFAC Symp. Syst. Identification, Newcastle, Australia, 2006, pp. 778–783.

Then, we get that the reduced cost function becomes

Vinit

2 F  R  
  1 ∗   =  G∗ (i, j) − αj G2 (i) . CL i j

(31)

The solution of this problem immediately follows from a singular value decomposition, because (31) can be written as  ∗   −1
∗  − G2 [α1∗ , . . . , αR ] Vinit = gCL

Frobenius

(32)

∗

−1 with gCL (i, j) = 1/G∗CL (i, j). Hence, the solution is retrieved as

Johan Schoukens (F’97) received the degree of engineer and the Ph.D. degree in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1980 and 1985. He is currently a Professor with the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. The prime factors of his research are system identification for linear and nonlinear systems. Dr. Schoukens is the recipient of 2002 Best Paper Award and the 2003 Society Distinguished Service Award from the IEEE Instrumentation and Measurement Society.

ˆ 2 is the first left singular vector of g ∗ −1 . • G CL ∗ −1 −1 equals gCL where the mean of each row is removed • gCL row by row. ˆ 1 (i) = 1/(1/R) R (1/gCL (i, j)). ˆ 1 is given by G • G j=1 R EFERENCES [1] S. A. Billings and S. Y. Fakhouri, “Identification of systems containing linear dynamic and static nonlinear elements,” Automatica, vol. 18, no. 1, pp. 15–26, Jan. 1982. [2] J. J. Bussgang, “Crosscorrelation functions of amplitude-distorted Gaussian signals,” Res. Lab. Electron., Mass. Inst. Technol., Cambridge, MA, MIT Tech. Rep. P. No 216, 1952. [3] P. Crama and J. Schoukens, “Computing an initial estimate of a Wiener–Hammerstein system with a random phase multisine,” IEEE Trans. Intrum. Meas., vol. 54, no. 1, pp. 117–122, Feb. 2005.

Liesbeth Gommé (S’06) received the M.Eng. degree in electrical engineering, with an option in photonics, from Vrije Universiteit Brussel, Brussels, Belgium, in July 2005. She is currently a Ph.D. Researcher with the Department of Fundamental Electricity and Instrumentation (ELEC), Vrije Universiteit Brussel, sponsored by a Grant from the Research Foundation–Flanders (FWO–Vlaanderen). Her main interests are in the field of microwave measurements, particularly advanced RF calibration and instrumentation setups.

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Wendy Van Moer (SM’07) received the M.Eng. degree in telecommunication and the Ph.D. degree in applied sciences from Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1997 and 2001, respectively. She is currently a Postdoctoral Researcher with the Department of Fundamental Electricity and Instrumentation (ELEC), VUB. Her main research interests include nonlinear microwave measurements and modeling techniques. Dr. Van Moer is the recipient of the 2007 IEEE Instrumentation and Measurement Outstanding Young Engineer Award.

Yves Rolain (F’06) received the M.Eng. degree, the degree of computer sciences, and the Ph.D. degree in applied sciences from the Vrije Universiteit Brussel, Brussels, Belgium, in 1984, 1986, and 1993, respectively. He is currently a Professor with the Department of Fundamental Electricity and Instrumentation, Vrije Universiteit Brussel. His main research interests include nonlinear microwave measurement techniques, applied digital signal processing, parameter estimation/system identification, and biological agriculture. Dr. Rolain received the Society Award from the IEEE Instrumentation and Measurement Society in 2004.

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