A.E. Ruehli

B. Gustavsen

C. Sanathanan

D. Deschrijver

E.C. Levi

F. Ferranti

J. Ureel

J.M. Jin

J.W. Bandler

K.C. Sou

N. Wong

N.K. Nikolova

P. Liu

P.S.C. Heuberger

R.F. Harrington

T. Dhaene

U.M.  Al-Saggaf

English

Ferranti, Francesco

Deschrijver, Dirk

Knockaert, Luc

Dhaene, Tom

Ghent University Academic Bibliography

Data-driven Parameterized Model OrderReduction Using z-domain MultivariateOrthonormal Vector Fitting TechniqueFrancesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom DhaeneAbstract Efficient real-time design space exploration, design optimization and sen-sitivity analysis call for Parameterized Model Order Reduction (PMOR) techniquesto take into account several design parameters, such as geometrical layout or sub-strate characteristics, in addition to time or frequency. This chapter presents a robustmultivariate extension of the z-domain Orthonormal Vector Fitting technique. Thenew method provides accurate and compact rational parametric macromodels basedon numerical electromagnetic simulations or measurements in either frequency-domain or time-domain. The technique can be seen as a data-driven PMOR method.1 IntroductionNowadays, full-wave electromagnetic methods [9],[17],[11] are widely used to sim-ulate a variety of complex electromagnetic systems and are considered to be essen-tial for efficient design. The use of these methods usually results in the computationof a huge number of field (E,H) or circuit (i,v) unknowns, in the frequency-domainor time-domain, although users are usually only interested in a few of them at theinput and output ports. These methods provide high accuracy, often at a significantcost in terms of memory storage and computing time. Therefore, Model Order Re-duction (MOR) techniques are crucial to reduce the complexity of the model definedby the full-wave numerical method and the computational cost required by simula-tions, while retaining the important physical features of the original system [7],[3].F. Ferranti, L. Knockaert, and T. Dhaene are with the Department of Information Technology (IN-TEC), Ghent University-IBBT, Ghent 9000, Belgium (e-mail: francesco.ferranti@intec.ugent.be;luc.knockaert@intec.ugent.be; tom.dhaene@intec.ugent.be).D. Deschrijver is with the Department of Information Technology (INTEC), Ghent University-IBBT, Ghent 9000, Belgium. He is also a post-doctoral research fellow of FWO Vlaanderen (e-mail: dirk.deschrijver@intec.ugent.be).12 Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom DhaeneEfficient real-time design space exploration, design optimization and sensitivityanalysis require the development of accurate parametric broadband macromodelsthat approximate the dynamic behavior of a system characterized by several designparameters, such as geometrical layout or substrate characteristics, in addition totime or frequency. These applications call for Parameterized Model Order Reduction(PMOR) techniques.A frequency-domain technique called Multivariate Orthonormal Vector Fitting(MOVF) was presented in [4], to compute accurate rational parametric macromod-els, based on parameterized frequency responses with a highly dynamic behavior.This technique can be seen as a data-driven PMOR method. Instead of reducingthe size of the matrices of a parameterized state-space model directly (model-basedPMOR), MOVF builds rational parametric macromodels with a reduced model com-plexity based on a set of input-output data samples. The goal of the macromodelingalgorithm is to find a multivariate rational function which approximates a large set ofK + 1 data samples {(s,g)k,H(s,g)k}Kk=0 in a least-squares sense. These data sam-ples depend on the complex frequency s = jω and several additional parametersg = (g(n))Nn=1 as design variables which describe e.g. the metallizations in an EMcircuit (lengths, widths,...) or the substrate features (thickness, dielectric constant,losses, ...). The proposed approach results in accurate and compact rational paramet-ric macromodels of complex electromagnetic systems. A generalization of MOVFto include parameter derivatives in the modeling process was proposed in [6]. Pa-rameter derivatives provide additional information about the underlying system andcan often be simulated at a significantly lower computational cost than additionalsamples [13],[5],[20],[15]. The inclusion of derivatives can be useful to reduce therequired amount of data samples, while preserving the accuracy of the results. Inthis chapter a new technique, the z-domain Multivariate Orthonormal Vector Fitting(ZD-MOVF) is described, representing the z-domain counterpart of [4]. It is a robustmultivariate extension of the z-domain Orthonormal Vector Fitting technique (ZD-OVF) proposed in [16],[14],[21]. A microstrip example confirms the ability of thenew algorithm to build parametric macromodels of dynamic systems with a goodaccuracy.2 BackgroundIn this section we explain the generation of z-domain data starting from frequency-domain or time-domain data and the choice of the λ parameter of the Tustin trans-form.Title Suppressed Due to Excessive Length 32.1 Generation of z-domain dataMicrowave circuits and components can be characterized in frequency-domain ortime-domain by numerical electromagnetic simulations or measurements. To obtainthe corresponding parameterized z-domain response, Hd(z,g), where z is the com-plex discrete frequency variable and g is a real design variable, a Tustin (bilinear)transform:s−→ z = λ + sλ − s , λ ∈ R+ (1)can be used starting from frequency-domain data Hc(s,g):Hc(s,g)−→ Hd(z =λ + sλ − s ,g)(2)where c stands for continuous and d for discrete. If time-domain data is available,under the hypothesis of a negligible or absent aliasing in the sampling process, thefrequency response of a continuous-time system can be computed by applying stan-dard techniques, such as Fast Fourier Transform (FFT) algorithms on the data sam-ples:hd([n],g) = hc(nTs,g) (3)where the real sequence hd([n],g) is equal to the signal in the time domain hc(t,g)at the equally spaced time samples nTs and Ts is the sampling period. Once theparameterized frequency response is computed, the Tustin transform (1) is used asbefore. The obtained z-domain data can be normalized by discrete frequency z [14].Once the parametric macromodel is computed in the z-domain, it can be convertedback to the s-domain by using inverse Tustin tranform.2.2 Choice of λ of the Tustin transformThe λ parameter of the Tustin transform can be freely chosen [19] under the con-straint that it is not a real pole of the continuous-time system [1]. The numericalexample in this letter shows that the algorithm is robust with respect to an arbi-trary choice of λ , since its value does not affect the accuracy of the results over awide range. To avoid harmful numerical conditions, extreme values of λ have to bediscarded, such as very low (near zero) or very high (near infinity).4 Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom Dhaene3 Parametric MacromodelingTo simplify the notation, the algorithm is only described for bivariate systems. Theextension to the full multivariate formulation is straightforward. As in [4], the ZD-MOVF algorithm proposes to represent the parametric macromodel as the ratio of abivariate numerator and denominatorR(z,g) =N(z,g)D(z,g)=∑Pp=0 ∑Vv=0 cpvχp(z)ψv(g)∑Pp=0 ∑Vv=0 c˜pvχp(z)ψv(g)(4)where P and V represent the maximum order of the corresponding basis functionsχp(z) and ψv(g) in the complex discrete frequency variable z and the real designvariable g, respectively. To establish the coefficients cpv and c˜pv of numerator anddenominator in (4), the ZD-MOVF algorithm minimizes the Sanathanan-Koerner(SK) cost function [18] on a set of K + 1 data samples {(z,g)k,Hd(z,g)k}Kk=0. SKis an iterative procedure, in the first iteration step of the algorithm (t = 0) Levi’scost function [12] is minimized to obtain an initial estimate of the coefficients cpvand c˜pv. In the following steps (t = 1, ..,T ) of the SK iteration, the inverse of thepreviously estimated denominator D(t−1)(z,g) is used as an explicit least-squaresweighting factor. A relaxed non-triviality constraint is added as an additional row inthe system matrix [8], to avoid the trivial null solution and improve the convergenceof the algorithm. Each equation is split in its real and imaginary parts, to ensure thatthe model coefficients c(t)pv , c˜(t)pv are real. Scaling each column to unity length [7] issuitable to improve the numerical accuracy of the results.4 Choice of basis functionsIn this section we describe the choice of the basis functions for the discrete fre-quency and other parameters.4.1 Discrete Frequency-Dependent Basis FunctionsBased on a prescribed set of stable poles a = {−ap}Pp=1, a set of partial fractionsχp(z,a) is chosen, with χ0(z) = 1. To select the poles two steps are followed: first,they are chosen in the s-domain as complex conjugate pairs with small real partsand the imaginary parts linearly spaced over the frequency range of interest [7] andafter that, the Tustin transform (1) is applied to map them from s- to z-domain. Alinear combination of two fractions is chosen to ensure that the residues of χp(z,a)and χp+1(z,a) come in perfect conjugate pairs leading to real-valued time domainresponses, i.e. :Title Suppressed Due to Excessive Length 5χp(z,a) = z(z + ap)−1 + z(z + ap+1)−1 (5)χp+1(z,a) = jz(z + ap)−1− jz(z + ap+1)−1 (6)To improve the numerical stability of the modeling algorithm, the Takenaka-Malmquistbasis functions [10] can be used, as shown in [16]:χp(z,a) = kp(p−1∏i=11 + a∗i zz + ai)(1− z)|1−ap|(z + ap)(z + ap+1)(7)χp+1(z,a) = kp+1(p−1∏i=11 + a∗i zz + ai)(1 + z)|1 + ap|(z + ap)(z + ap+1)(8)wherekp = kp+1 =√22√1−|ap|2 (9)The orthonormal basis functions can improve the conditioning of the systemequations and are less sensitive to the choice of the initial poles. Their use ensures amore numerically robust macromodeling procedure [3].4.2 Parameter-Dependent Basis FunctionsThe parameter-dependent basis functions ψv(g,b) are also chosen in partial frac-tion form as a function of jg, hence in rational form. The set of starting polesb = {−bv}Vv=1 is composed by complex pairs with small real parts of opposite signand imaginary parts linearly spaced over the parameter range of interest, providedthat ψ0(g) = 1. A linear combination of two fractions is used to ensure that ψv(g,b)and ψv+1(g,b) are real functions [4]:ψv(g,b) = ( jg + bv)−1− ( jg− (bv)∗)−1 (10)ψv+1(g,b) = j( jg + bv)−1 + j( jg− (bv)∗)−1 (11)5 Example: Double folded stub microstrip bandstop filterThe double folded stub microstrip bandstop filter [2] under study is shown in Figure1. The substrate is 0.1270 mm thick with a relative dielectric constant εr equal to 9.9.The scattering parameters of the system are simulated by ADS-Momentum1 in thes-domain and subjected to (2), to obtain the corresponding parametrized z-domainresponse.1 Momentum EEsof EDA, Agilent Technologies, Santa Rosa, CA.6 Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom DhaeneP1 P2SSLLFig. 1 Geometry of the double folded stub microstrip bandstop filter [2].The parametric macromodels of scattering parameters S11 and S21 are built asfunctions of the varying length of each folded segment L ∈ [1.98 mm - 2.40 mm]and varying spacing between a folded stub and the main line S ∈ [0.061 mm - 0.243mm] over the frequency range [5 GHz - 20 GHz]. The desired model accuracy is setto −60 dB, which corresponds to 3 significant digits. The initial data grid for S11and S21 is of size 14× 10× 22 samples (L,S, f req). The corresponding number ofpoles is chosen 6, 4 and 10 for both scattering parameters. Figure 2 and Figure 3show the magnitude of the trivariate macromodels of S11 and S21 for the minimumand maximum values of the spacing variable S.1.822.22.4510152000.20.40.60.81Frequency [GHz]Length [mm]|S 11|,|S 21|Fig. 2 Magnitude of the trivariate macromodels of S11 (light grey surface) and S21 (dark greysurface) for S = 0.061 mm.Title Suppressed Due to Excessive Length 71.822.22.4510152000.20.40.60.81Frequency [GHz]Length [mm]|S 11|,|S 21|Fig. 3 Magnitude of the trivariate macromodels of S11 (light grey surface) and S21 (dark greysurface) for S = 0.243 mm.−120 −110 −100 −90 −80 −70 −60 −500100020003000400050006000700080009000Absolute error [dB]Number of samplesFig. 4 Histogram : error distributions of the trivariate macromodels of S11 (light grey) and S21(dark grey) over 226500 validation samples.To compute the macromodels only 4 and 3 iterations of SK method discussedin Section 3 are needed and the maximum absolute error in the initial data gridcorresponds to −62.84 dB and −67.54 dB, respectively. To confirm the quality ofbuilt macromodels a set of validation data samples is computed on a very densegrid of size 50× 30× 151 samples. The histogram in Figure 4 shows the numberof validation samples that corresponds to a certain absolute error for both trivariatemacromodels. Figure 4 shows that they have a good overall accuracy and the max-8 Francesco Ferranti, Dirk Deschrijver, Luc Knockaert and Tom Dhaeneimum absolute error over all the validation samples is bounded by −60.17 dB and−61.06 dB for S11 and S21 respectively. The choice of the λ parameter in the Tustintransform (1) does not influence the model accuracy over a broad range of values[10−3− 1023]. It confirms that λ is free to choose and illustrates the robustness ofthe algorithm.6 ConclusionsThis chapter presents a robust multivariate extension of the z-domain Vector Fit-ting technique [16],[14],[21], for the calculation of accurate and compact paramet-ric macromodels of high-speed components. By combining rational basis functionsand the Sanathanan-Koerner least-squares estimator, the robustness of the methodis ensured. An example illustrates the capability of the algorithm to model dynamicparameterized frequency responses with a good accuracy. Once the multivariatemacromodeling process is completed, the resulting scalable behavior model can ef-ficiently be employed in real-time design space exploration, fast optimization andsensitivity analysis.Acknowledgements This work was supported by a grant of the Research Foundation-Flanders(FWO-Vlaanderen).References1. U. M. Al-Saggaf and G. F. Franklin, “Model Reduction Via Balanced Realizations: An Ex-tension and Frequency Weighting Techniques”, IEEE Trans. on Automatic Control, vol. 33,no. 7, pp. 687-692, July 1988.2. J. W. Bandler, R. M. Biernacki, S. H. Chen, P. A. Grobelny, and R. H. Hemmers, “Spacemapping technique for electromagnetic optimization”, IEEE Trans. on Microwave Theoryand Techniques, vol. 42, no. 12, pp. 2536-2544, December 1994.3. D. Deschrijver, B. Haegeman and T. Dhaene, “Orthonormal Vector Fitting: A Robust Macro-modeling Tool for Rational Approximation of Frequency Domain Responses”, IEEE Trans.on Advanced Packaging, vol. 30, no. 2, pp. 216-225, May 2007.4. D. Deschrijver, T. Dhaene and D. De Zutter, “Robust Parametric Macromodeling using Mul-tivariate Orthonormal Vector Fitting”, IEEE Trans. on Microwave Theory and Techniques,vol. 56, no. 7, pp. 1661-1667, July 2008.5. T. Dhaene and D. Deschrijver, “Generalised Vector Fitting Algorithm for Macromodelling ofPassive Electronic Components ”, IEE Electronics Letters, vol. 41, no. 6, pp. 299-300, March2005.6. F. Ferranti, D. Deschrijver, L. Knockaert and T. 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Liu, Z-F. Li and G-B. Han, “Application of Asymptotic Waveform Evaluation to Eigen-mode Expansion Method for Analysis of Simultaneous Switching Noise in Printed CircuitBoards (PCBs)”, IEEE Transactions on Electromagnetic Compatibility, vol. 48, no. 3, pp.485-492, August 2006.14. Y. S. Mekonnen and J. E. Schutt-Aine´, “Broadband Macromodeling of Sampled FrequencyData Using z-domain Vector Fitting”, IEEE Workshop on Signal Propagation on Intercon-nects, SPI 2007, Camogli Genova (Italy), pp. 45-48, May 2007.15. N. K. Nikolova, Y. Li, Y. Li and M. H. Bakr, “Sensitivity Analysis of Scattering Parameterswith Electromagnetic Time-Domain Simulators”, IEEE Transactions on Microwave Theoryand Techniques, vol. 54, no. 4, pp. 1598-1610, April 2006.16. B. Nouri, R. Achar, M. Nakhla and D. 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De Zutter, “A New Method for Obtaining the Shape Sensitivities of PlanarMicrostrip Structures by a Full-Wave Analysis ”, IEEE Transactions on Microwave Theoryand Techniques, vol. 44, no. 2, pp. 249-260, February 1996.21. N. Wong and C. Lei, “IIR Approximation of FIR Filters Via Discrete-Time Vector Fitting”,IEEE Trans. on Signal Processing, vol. 56, no. 3, pp. 1296-1302, March 2008.

Data-driven parameterized model order reduction using z-domain multivariate orthonormal vector fitting technique

Francesco Ferranti

Dirk Deschrijver

Luc Knockaert

Tom Dhaene

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Data-Driven Parameterized Model Order Reduction Using z-Domain Multivariate Orthonormal Vector Fitting Technique

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Data-driven parameterized model order reduction using z-domain multivariate orthonormal vector fitting technique

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