This paper presents an evolutionary algorithm for symbolic macromodeling of parameterized frequency responses. The method does not require an a priori specification of the multivariate functional form or complexity of the model. Numerical results are shown to illustrate the performance of the technique

Deschrijver, Dirk

Dhaene, Tom

English

Ghent University Academic Bibliography

Symbolic Macromodeling of ParameterizedS-Parameter Frequency ResponsesDirk DeschrijverDept. of Information TechnologyGhent University - IBBT9000 Ghent, BelgiumEmail: dirk.deschrijver@intec.ugent.beTom DhaeneDept. of Information TechnologyGhent University - IBBT9000 Ghent, BelgiumEmail: tom.dhaene@intec.ugent.beAbstract—This paper presents an evolutionary algorithm forsymbolic macromodeling of parameterized frequency responses.The method does not require an a priori specification of the mul-tivariate functional form or complexity of the model. Numericalresults are shown to illustrate the performance of the technique.I. INTRODUCTIONEfficient design space exploration requires parametricmacromodels that describe the dynamic behavior of scalablesystems in terms of the frequency and several design variables,e.g. geometrical layout or substrate characteristics. The calcu-lation of compact and accurate models is not a trivial task,and the development of novel macromodeling techniques hasreceived a lot of attention over the past few years. An overviewof recent work is found in [1]-[6] and the references therein.This paper presents an alternative macromodeling procedurethat is based on symbolic regression. The goal of the algorithmis to find a mathematical multivariate expression that providesan accurate approximation of the parameterized frequencyresponse. The new approach is more generic than conventionalmacromodeling techniques, in a sense that it does not requirean a priori specification of the functional form of the model(e.g. rational, polynomial, neural network, etc) or the modelcomplexity (e.g. number of basis functions or the number ofhidden layers). Note that this method provides a compact andscalable macromodel, but it does not take physical propertiesinto account (such as causality, stability and passivity).The usefulness of the symbolic macromodeling approach isillustrated by applying it to a scalable waveguide example.II. SYMBOLIC MACROMODELINGSymbolic macromodeling is a biologically inspired methodthat mimics the process of Darwin’s evolution theory andthe mechanisms of genetic variation and natural selection[7]. It is based on the concept of genetic programming, andcomputes a multi-gene regression model that approximates aset of tabulated, parameterized frequency samples. These datasamples can be S-parameters S(~x) that depend on several inputparameters ~x = {x1, ..., xn} (e.g. frequency, length and widthsof metallizations in an EM-circuit or substrate parameters).The multivariate model expression is represented as an α-weighted sum of parse trees Tm(~x) (also called genes) thatFig. 1. Example : Multi-gene regression modeldepend on some (or all) of the n input parameters ~x.T (~x) = α0 + α1T1(~x) + ...+ αMTM (~x) (1)Each parse tree Tm(~x) is composed of primitive functions (e.g.+,−, /, ∗, log, sin, exp, ...) and terminals (i.e. input variables~x and constants). The optimal values of the scalar weights αmare determined by a least-squares procedure that regresses thegenes against the output data. The structure of the genes isevolved automatically as described in the following section.Fig. 1 shows an example of a multi-gene regression model.III. EVOLUTIONARY ALGORITHMThe algorithm starts by generating an initial population ofrandomly generated individuals T (~x), which all represent apotential solution to the approximation problem. Each indi-vidual has a random number of genes Tm(~x) with m ≤ M .In order to measure the quality of a particular solution, thefitness function F assigns a fitness value to each individual.This fitness function is the objective function that the algorithmaims to optimize. As some individuals will be more fit thanothers, the best performing individuals are selected and threegenetic operators are applied to create new offspring [8] :− Direct Reproduction, Crossovers and Mutations −1) The direct reproduction operator simply copies the entireindividual to the next generation without modification.2) The crossover operator selects two parent individualsfrom the population and combines them to create newoffspring by swapping genes or subtrees of the genes3) The mutation operator selects a single gene from theindividual and makes a random alteration to its subtree.Fig. 2. Cross-sectional view and top view of inductive posts [9]This evolutionary process is repeated over many genera-tions, in order to produce individuals with increasing fitness.Once the algorithm is terminated, the “best” individual fromthe population is selected and designated as the final result.More details about genetic programming can be found in [7].IV. EXAMPLE : RECTANGULAR WAVEGUIDEA. Calculation of Frequency ResponseIn this example, the method is applied to build a multivariatemacromodel of a parameterized frequency response. A mo-ment method is used to compute the transmission coefficientS21 of two perfectly conducting round posts, centered inthe E-plane of a rectangular waveguide [10]. The system isparameterized by 2 variables : frequency f and post-spacingw. A symbolic macromodel is calculated for a standard WR90rectangular waveguide, where the frequency f varies between[7-10] GHz and the spacing w between [15-17] mm. Thelayout of the inductive posts is shown in Fig. 2 and thefrequency response is shown in Fig. 3 and Fig. 4.B. Symbolic MacromodelingBoth the real part <e(S21) and the imaginary part =m(S21)of the calculated frequency response are modeled indepen-dently by two multi-gene regression models, see (1). [11]The maximum number of genes (M ) is set to 12, and themaximum tree depth of each gene is set to 3. The primitiveoperations are chosen to be ∗,−,+, /,√|...|, (...)2. Note thatother primitive operations can be included, although this wasnot necessary to obtain a sufficiently accurate macromodel.First an initial population of 300 randomly generated in-dividuals is generated, and the fitness of each individual iscomputed as the root-mean-squared prediction error over thedata set. Based on the fitness values, tournament selectionchooses the individuals that are used to create offspring. Ifone or more individuals have the same fitness value, then theone with the fewest nodes is selected. Elitism ensures thatindividuals with the best fitness values (2% of the population)are copied directly to the next generation without applyinggenetic operators. This process is repeated for 200 generations.15161777.588.599.5100.10.20.30.40.50.60.70.80.9w [mm]Frequency [GHz]Real(S12)Fig. 3. Real part of transmission coefficient S2115161777.588.599.5100.340.360.380.40.420.440.460.480.50.52w [mm]Frequency [GHz]Imag(S12)Fig. 4. Imaginary part of transmission coefficient S21As illustrated by Fig. 5, the accuracy of the fittest individualin each population improves over the course of the run.C. Selection of the Best (Pareto-Optimal) ModelOnce the evolutionary algorithm has converged, one of theindividuals in the final population is selected and designatedas the “best” solution. This individual is preferably the onewith the highest accuracy (i.e. the model with the smallestfitness value) and the smallest model complexity (i.e. fewestnumber of nodes in the parse tree). Since these two designobjectives are usually conflicting, a pareto front is formedthat presents a compromise among these two criteria. Thefitness and model complexity of each individual in the finalpopulation is shown by a dark (blue) circle in Fig. 6 and Fig.7. The light (green) circles correspond to models that are not0 50 100 150 20010−410−310−2GenerationFitness  RealImagModels of real partModels of imaginary partFig. 5. Fitness of most accurate individual over course of the run5 10 15 20 25 30 35 40 45 5010−410−310−210−1FitnessTotal number of nodesComparable Fitness Fewer nodesFittest individualFig. 6. Pareto front : Individuals final population (models of real part)outperformed by any other model in terms of both fitnessand model complexity. These are so-called “pareto-optimal”models. The most accurate individual, i.e. the one with thesmallest fitness value, is marked by a bold (red) circle. It isnoted that some of the pareto-optimal models have a fitnessvalue that is comparable to the optimal solution, but a lowermodel complexity. This information can be exploited by thedesigner to select a compact model that also meets the desiredaccuracy specifications.D. Accuracy of MacromodelAs mentioned above, the best models for the real andimaginary part of S21 are selected from the final populationsby investigating the Pareto front. Fig. 8 shows the absolutefitting error over the design space for both models, and it isfound that the maximum absolute error is approximately 10−30 10 20 30 40 50 60 7010−410−310−210−1FitnessTotal number of nodesFittest individualComparable Fitness Fewer nodesFig. 7. Pareto front : Individuals final population (models of imaginary part)1515.51616.5177891010−610−410−2100 w [mm]Frequency [GHz] Absolute errorRealImagFig. 8. Absolute fitting error of the best modelsin both cases, which is sufficiently accurate. The parse trees ofthese models are concatenated as shown in Fig. 9. This yields alarger, complex-valued parse tree that characterizes the overallfrequency response. Based on this compound expression, themagnitude and phase of the model can also be calculated andcompared to the reference data, as shown in Fig. 10 and Fig.11. It is clear that a good agreement is observed in both cases.As a final test, a dense set of 2500 validation samples arecomputed and compared to the response of the model. It isfound that the maximum absolute error over all data samplesis -55.68 dB. The distribution of the absolute error over thereference samples is visualized by a histogram in Fig. 12.V. CONCLUSIONThis paper proposes an evolutionary algorithm for the sym-bolic macromodeling of parameterized frequency responses.Fig. 9. Concatenation of 2 parse trees (models of real and imaginary part)7 7.5 8 8.5 9 9.5 10−9−8−7−6−5−4−3−2−10Frequency [GHz]Magnitude [dB]  ModelDatawFig. 10. Magnitude of complex model and reference data7 7.5 8 8.5 9 9.5 100.40.50.60.70.80.911.11.21.3Frequency [GHz]Phase [rad]  ModelDatawFig. 11. Phase of complex model and reference data−100 −95 −90 −85 −80 −75 −70 −65 −60 −55050100150200250Absolute error [dB]Number of samplesFig. 12. Histogram : absolute error distribution over 2500 validation samplesThe method is applied to a scalable waveguide example, andits performance is analysed by numerical results. It is foundthat the method yields compact models with a fair accuracy.ACKNOWLEDGMENTThis work was supported by the Fund for Scientific Re-search in Flanders (FWO-Vlaanderen). Dirk Deschrijver isa post-doctoral research fellow of FWO-Vlaanderen. Theauthors thank Dr. Robert Lehmensiek for provision of the data.REFERENCES[1] S. Grivet-Talocia, S. Acquadro, S. Bandinu et al., “A ParameterizationScheme for Lossy Transmission Line Macromodels with Application toHigh Speed Interconnects in Mobile Devices”, IEEE Trans. on Electro-magnetic Compatibility, pp. 18-24, vol. 49, no. 1, Feb 2007.[2] D. Deschrijver, T. Dhaene and D. De Zutter, “Robust Parametric Macro-modeling Using Multivariate Orthonormal Vector Fitting”, IEEE Trans.on Microw. Theory and Techn., vol. 56, no. 7, pp. 1661–1667, Jul 2008.[3] D. Deschrijver and T. Dhaene, “Stability and Passivity Enforcement ofParametric Macromodels in Time and Frequency Domain”, IEEE Trans.on Microw. Theory and Techn., vol. 56, no. 11, pp. 2435-2441, Nov 2008.[4] T. Dhaene, D. Deschrijver, “Stable Parametric Macromodeling usinga Recursive Implementation of the Vector Fitting Algorithm”, IEEEMicrow. and Wireless Comp. Lett., vol. 19, no. 2, pp. 59-61, Feb 2009.[5] P. Triverio, S. Grivet-Talocia and M. S. Nakhla, “A ParameterizedMacromodeling Strategy with Uniform Stability Test”, IEEE Trans. onAdv. 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Symbolic macromodeling of parameterized S-parameter frequency responses

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