A multivariate orthonormal vector fitting based estimation technique

A Multivariate Orthonormal Vector Fitting

based estimation technique

Francesco Ferranti∗

Yves Rolain∗∗

Koen Vandermot∗∗

Luc Knockaert∗

Tom Dhaene∗

∗ Department of Information Technology (INTEC), Ghent University -IBBT, Ghent, Belgium, (e-mail: [francesco.ferranti, knokaert,

Tom.Dhaene]@Ugent.be)∗∗ Department ELEC, Vrije Universiteit Brussel, Brussel, Belgium,

(e-mail: [yrolain, koen.vandermot]@vub.ac.be)

Abstract: This paper modifies a recent robust parametric macromodeling technique calledMultivariate Orthonormal Vector Fitting (MOVF), to handle noisy data in an output errorestimation framework. The new method provides accurate and compact rational parametricmacromodels based on measurements in the frequency domain. The performance of themultivariate method is shown on simulation as well as on real measurements.

Keywords: Modelling, Identification algorithms, Parametrization, Multivariable Systems,Frequency measurements.

1. INTRODUCTION

The increasing demand for performance of telecommuni-cation devices pushes operation to higher signal band-widths and lower power consumption. To be able to as-sist microwave designers, accurate modeling of previouslyneglected second order effects becomes increasingly impor-tant during circuit and system simulations. These effectsare very different and can range from the dynamics ofinterconnection effects, to the inclusion of the nonlinearbehavior of devices or subsystems. The accurate predictionof these effects is fundamental for a successful designand involves the solution of large systems of equationswhich are often prohibitively expensive to solve. For real-time design space exploration and fast optimization, thereis a significant need for accurate broadband parametricmacromodels that approximate the frequency domain be-havior of a system in terms of several design variables(for example the power level for a nonlinear system orthe dimension of an interconnect structure) by a rationalanalytic function.

A frequency domain technique called Multivariate Or-thonormal Vector Fitting (MOVF) was presented in (De-schrijver et al. [2008]), to compute accurate parametricmacromodels from Laplace domain data samples. A gener-alization of (Deschrijver et al. [2008]), including parameterderivatives in the macromodeling process, was recentlyproposed by (Ferranti et al. [2008]). The aim of thispaper is to extend the MOVF technique to deal with noisy,measured data and estimate a parametric macromodel.As for microwave systems the measurement of the sys-tem response commonly uses S-parameters, which are inessence a multiple-input multiple-output transfer matrixof the system, it seems evident to use an output error

⋆ This work was supported by the Research Foundation Flanders(FWO).

framework. The proposed method is an approximation ofthe maximum-likelihood estimator. The validity and theperformance of the proposed approach are first shownon a simulation example. Next, the frequency and powerdependent model for the behavior of a microwave amplifierconfirms the ability of the modified algorithm to buildparametric macromodels of dynamic systems with a goodaccuracy, starting from measurement data.

2. THE PARAMETRIC MACROMODEL

2.1 The Multivariate Rational Model

To simplify the notation, the algorithm is only describedfor bivariate systems. The extension to the full multivari-ate formulation is straightforward, however. As in (De-schrijver et al. [2008]), the MOVF method proposes torepresent the parametric macromodel as the ratio of abivariate numerator and denominator

R(θ, s, g) =N(θN , s, g)

D(θD, s, g)=

∑P

p=0

∑V

v=0cpvφp(s)ϕv(g)

∑P

p=0

∑V

v=0c̃pvφp(s)ϕv(g)

(1)

where P and V represent the maximum order of the corre-sponding basis functions φp(s) and ϕv(g) in the complexfrequency variable s and the real design variable g, respec-tively. θ represents the vector of the model parameterscpv and c̃pv. The multivariate model in (1) is general,flexible and resulted to be able to model parameterizedfrequency responses with a highly dynamic behavior accu-rately (Deschrijver et al. [2008]). The choice of the basisfunctions has an influence on the numerical stability of themultivariate modeling technique. One can decide to useorthonormal basis functions to obtain a better numericalstability of the estimator. The numerical stability is a keyissue for multivariate modeling as the number of parame-ters contained in θ to be estimated increases exponentially

Preprints of the

15th IFAC Symposium on System Identification

Saint-Malo, France, July 6-8, 2009

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with the number of dimensions that are present in themultivariate model. The numerator and denominator in(1) have the same order, but, of course, selecting differentorders is possible.

2.2 Frequency-Dependent Basis Functions φp(s)

In the context of MOVF, the basis functions φp(s) are builtup using a set of Muntz-Laguerre orthonormal rationalbasis functions φp(s,−→a ), which are based on a prescribedfixed set of stable poles −→a = {−ap}P

p=1, provided thatφ0(s) = 1. These poles are grouped as complex conjugatepole pairs, and are selected such that they have smallnegative real parts and the imaginary parts linearly spacedover the frequency range of interest (Gustavsen et al.[1999]).

To make sure that the transfer function has real-valued co-efficients, a linear combination of φp(s,−→a ) and φp+1(s,−→a )is formed as follows

φp(s,−→a ) =

√

2ℜe(ap) (s − |ap|)(s + ap)(s + ap+1)

(

p−1∏

i=1

s − a∗

i

s + ai

)

(2)

φp+1(s,−→a ) =

√

2ℜe(ap) (s + |ap|)(s + ap)(s + ap+1)

(

p−1∏

i=1

s − a∗

i

s + ai

)

(3)

It was shown in (Deschrijver et al. [2007]) that the or-thonormal rational basis functions can improve the con-ditioning of the system equations and are less sensitiveto the choice of the initial poles. Their use ensures amore numerically robust macromodeling procedure. Notethat identical rational basis functions are chosen to modelthe numerator and the denominator of R(θ, s, g). Hence,both are approximated by a rational expression that hasan identical denominator, and this common denominatorcancels out in R(θ, s, g). The choice of the poles of the basisfunctions has an influence on the convergence and accuracyof the proposed method. The choice proposed here for thepoles of N(θN , s, g) and D(θD, s, g) has the advantage tokeep the numerical conditioning to an acceptable level. Weremark that being the model in (1) general, any kind ofbasis functions can be selected, e.g. polynomial basis.

2.3 Design Variable-Dependent Basis Functions ϕv(g)

The second set of basis functions ϕv(g,−→b ) depends on

the design variable. If the model response varies smoothlywith the design variable, the corresponding basis functionsusually have relatively low orders and a suitable choice isa set of polynomial basis functions with real coefficients.If the model response variation is highly dynamic withthe design variable, a set of rational basis functions is abetter choice to deal with high model orders and avoid ill-conditioning. In this section we discuss the choice of thebasis functions in rational form being more general andflexible. They are chosen in partial fraction form and arefunctions of jg instead of g. These basis functions are built

up using a prescribed set of poles−→b = {−bv}V

v=1, whichare chosen as complex pole pairs with small real parts ofopposite sign and imaginary parts that are linearly spacedover the parameter range of interest (Deschrijver et al.[2007]).

The first basis function ϕ0(g) is chosen to be equal to one.A linear combination of two partial fractions is formed

to ensure that ϕv(g,−→b ) and ϕv+1(g,

−→b ) are strictly real

functions by construction

ϕv(g,−→b ) = (jg + bv)

−1 − (jg − (bv)∗)−1 (4)

ϕv+1(g,−→b ) = j(jg + bv)

−1 + j(jg − (bv)∗)−1 (5)

The choice of the poles for numerator and denominator,as in the previous section, helps to keep the numericalconditioning under control for this set of basis functions.

3. SIGNAL AND NOISE MODEL

In this work, we consider that the transfer function of theDevice Under Test (DUT) is a measured quantity. As themain applications of the modeling algorithm will lie in thedomain of high-frequency electronics and telecommunica-tions, this can safely be assumed. The parametric transferfunction Sm(s, g) is measured at a set of discrete frequen-cies and discrete values of the design variable, overall K

measurement data {(s, g)k, Sm(s, g)k}K

k=1. Each individual

measurement of the parametric transfer function is as-sumed to be perturbed by colored additive noise nS(s, g)k

with circular Normal distribution N(0, σS(s, g)k). Themeasured quantity at each sample (s, g)k is then obtainedas

Sm(s, g)k = Se(s, g)k + nS(s, g)k (6)

where Se(s, g) is the exact parametric transfer functionwithout the influence of the noise. One of the majoradvantages of frequency domain based transfer functionmeasurements is that it becomes very easy and inexpensiveto measure not only the transfer function, but also theestimate of the variance of the transfer function. Tothis end, it is sufficient to repeat the measurements andcalculate the sample variance on a point-by-point basis. Itcan be shown that even for a modest number of repetitions(minimum 8), one can safely replace the exact variance bythe sample variance without impairing the properties ofthe estimators defined below (Pintelon et al. [2001]).

4. THE MAXIMUM LIKELIHOOD ESTIMATOR

For ease of notation (s, g)k is replaced by −→α k. The maxi-mum likelihood cost function LK(θ) for this output errorframework is given by

LMLK (θ) =

K∑

k=1

|Sm(−→α k) − R(θ,−→α k)|2σ2

S(−→α k)= e

H(θ)e(θ) (7)

where the error vector e(θ) ∈ CKx1. The maximum

likelihood estimator θ̂ML is the minimizer of LMLK (θ),

defined as

θ̂ML = minθ

LMLK (θ) (8)

Note that the calculation of θ̂ML boils down to a minimiza-tion problem that is nonlinear in the parameter vector θ.An iterative minimization is required to solve this prob-lem. As the dimension of the parameter vector increasesexponentially with the number of independent variables

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in the rational model, the numerical cost of a nonlinearminimization problem becomes extremely high, even for amodest number of dimensions. To get around this issue,an approximation of the maximum likelihood cost thatremains linear in the parameters is proposed next.

4.1 A Linear Approximation for the Maximum LikelihoodCost Function

The estimation procedure proposed here uses a minimiza-tion method that is similar to the method that is used tofit the parameter vector θ in the noise-free case proposedin the MOVF algorithm (Deschrijver et al. [2008]). Thefitting procedure used there calculates an initial parameterestimate using Levi’s cost function (Levi [1959])

LLeviK (θ) =

K∑

k=1

|Sm(−→α k)D(θD,−→α k) − N(θN ,−→α k)|2 (9)

This initial value is then used to bootstrap the iterativeseries of T linear minimization problems as is proposed bySanathanan-Koerner (SK) (Sanathanan et al. [1963])

LSKK (θ t) =

K∑

k=1

|Sm(−→α k)D(θ tD,−→α k) − N(θ t

N ,−→α k)|2|D(θt−1

D ,−→α k)|2(10)

In the following steps (t = 1, .., T ) of the SK itera-tion, the inverse of the previously estimated denominatorD(θt−1

D ,−→α k) is used as an explicit least-squares weightingfactor. A non-triviality constraint similar to (Gustavsen[2006]) is added as an additional row in the system ma-trix to avoid the trivial null solution and improve theconvergence of the algorithm. Each equation is split inits real and imaginary parts, to ensure that the modelcoefficients ct

pv, c̃tpv are real. Scaling each column to unity

length (Gustavsen et al. [1999]) is suitable to improve thenumerical accuracy of the results.

To extend this approach, the output error weighting asis used in the maximum likelihood is introduced in theweighted SK cost as follows

LSKWK (θt) =

K∑

k=1

|Sm(−→α k)D(θ tD,−→α k) − N(θ t

N ,−→α k)|2σ2

S(−→α k)|D(θt−1

D ,−→α k)|2(11)

Even if the series of weighted cost functions is very similarto the maximum likelihood cost, it can be shown that theseries of cost functions does not always converge to themaximum likelihood cost, as has been shown in (Pintelonet al. [2001]). So a bias error may occur in the estimatedparameters.

5. SIMULATION RESULTS

The goal of the simulation example is to show the im-portance of the bias error that is present in the previousestimation framework. To this end, we consider a systemthat is perfectly known in advance. The system is chosensuch that it exhibits dynamic behavior in both frequencyand design variable dimensions. In this case, a second orderChebyshev filter is chosen in the frequency direction whilea second order Butterworth filter is used for the designvariable dependence. The 3-dB bandwidth of both filters

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−14

−12

−10

−8

−6

−4

−2

0

Design variable normalizedFrequency normalized

|Se(j

ω,g

)| [

dB

]

Fig. 1. Magnitude of Se(jω, g) in the simulation example.

00.2

0.40.6

0.81

00.2

0.40.6

0.81

−5

−4

−3

−2

−1

0

Design variable normalizedFrequency normalized

Ph

ase

(Se(j

ω,g

)) [

rad

]

Fig. 2. Phase of Se(jω, g) in the simulation example.

is chosen to correspond to 80% of the full span of thecorresponding variable, so ensuring a persistence of theexcitation. The parametric trasfer function to be modeledis:

Se(jω, g) = S2nde,Chebyshev(jω)S2nd

e,Butterworth(jg) (12)

where S2nde,Chebyshev(jω), S2nd

e,Butterworth(jg) are the transferfunctions of second order Chebyshev and Butterworthfilters written in a polynomial form. The magnitude andphase of the system Se(jω, g) under study are shown inFig. 1 and Fig. 2.

A first estimation is done with the noise-free data, toestimate the “true” model and relative coefficients. Themodel used in the identification process is:

R(jω, g) =c00

∑2

p=0

∑2

v=0c̃pv(jω)p(jg)v

(13)

The maximum absolute error model is equal to −277 dBand we can consider that the “true” coefficients are found.After that, a large number P = 1000 of estimations isperformed under circular complex Gaussian perturbations.The variance of the noise source is chosen such as to yielda signal-to-noise ratio of 10 dB in the band of the multi-variate system. The analysis of the sample mean µ̂P andstandard deviation σ̂P of the estimated coefficients allows

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Bias/(2 σ̂P ) NUM Bias/(2 σ̂P ) DEN

(c00) 5.0613 (c̃00) 6.3799(c̃10) 5.7594(c̃20) 6.3527(c̃01) 6.1920(c̃11) 4.9447(c̃21) 5.5521(c̃02) 6.0303(c̃12) 5.0670(c̃22) 5.3891

Table 1. Normalized bias term of the coeffi-cients

0

0.5

1

0

0.5

1

−70

−60

−50

−40

−30

−20

−10

0

Design variable normalizedFrequency normalized

Am

plit

ud

e [

dB

]

residuals

95% confidence bound

Fig. 3. Comparison between the residuals and the 95%confidence bound (

√3σS).

to detect the presence of a bias term. The results of thisanalysis are shown in Table 1. The bias term normalizedby the 95% confidence bound (2 σ̂P ) is shown for thenumerator and denominator coefficients, respectively.

By selecting a model in the previous estimation processwith noisy data, the residuals (difference between the data

and the model) and the 95% confidence bound (√

3σS) arecompared and it was found that 95% of the residuals areunder the defined uncertainty bound. This comparison isshown in Fig. 3.

To conclude, the simulation results indicate that the modelcreated by MOVF explains very well the data, even if abias term is present in the numerator and denominatorcoefficients.

6. EXPERIMENTAL VALIDATION

The goal of this experiment is to show that the method isalso usable in a practical context. To allow to easily obtaina large number of experiments in the frequency and thedesign variable, the design variable has been chosen to bean electrical quantity. This should however not be seen asa limitation to the method: it is perfectly possible to usea dimension or a value of some component instead.

6.1 The measurement setup

In this particular example, the dynamic non-linearity ofa microwave amplifier will be modeled as a function of

Fig. 4. Measurement setup used for the experimentalvalidation.

�� ���

����

���

�����

����

Fig. 5. Supply voltages of the Motorola MRFIC2006.

both the frequency and the power of the excitation signal.The power is considered to be the design variable. Themeasurement made here is closely related to the conceptof the power dependent Best Linear Approximation (BLA)(Pintelon et al. [2001]), (Schoukens et al. [1998]). TheBLA is extended to a 2 dimensional rational model inthis paper. The difference between the full blown BLAmeasurement setup and the setup used here and shownin Fig. 4 is that the large excitation signal that is usedto set the operating point of the non-linearity is a sine-wave rather than a multisine signal. This choice has beenimposed by limitations of the current measurement setup.

Note that the measurement of the frequency responseof the DUT is obtained injecting a probing sine-wavethrough port 1 of the Performance Network Analyzer(PNA). The power level of this signal is to be selectedsuch that its influence on the operating point of thenonlinear device is negligible. It is easy to verify thatthis hypothesis is indeed fulfilled. Therefore, the measuredfrequency response that is obtained at 2 small differentprobing levels for the same large signal amplitude shouldbe equal up to the uncertainty of the measurements. Thishas been checked for the power levels that were appliedduring the experiments.

6.2 Description of the experiment

The MRFIC2006 power amplifier from Motorola is used asa DUT for this experiment (Motorola [1998]). The supplyvoltages of this amplifier are 1V for Vcc1 that biases thefirst amplification stage and 4V for Vcc2 that biases thesecond and output stage (see Fig. 5).

The amplifier has been excited by a large sine-wave signalwith a frequency of 1GHz to set the operating point ofthe non-linearity. The power of this signal is the designvariable and has been swept in order to measure thefrequency response of the amplifier over a power rangestarting at -14 dBm and ramping up to 4 dBm in powersteps of 0.6 dBm. This sine-wave pump signal is generatedby a HP83650B signal generator and fed to the amplifierthrough a power splitter.

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An E8364B PNA has been used to measure the transferfunction S21 of the device under test. The device injectsa small probing signal in a frequency range that is setfrom 300MHz to 1700MHz with a frequency resolutionof 1MHz. Due to the rather modest frequency selectivityof the detectors of the PNA, the measured small signalresponse cannot be measured accurately for frequencies inthe vicinity of the frequency of the imposed large pumpsignal. For this reason, the behavior of the amplifier isnot measured in a frequency band of 160MHz centeredaround the frequency of the pump signal. One of theadvantages of the frequency domain approach is that thisunequally spaced frequency grid poses little or no signif-icant problems to the estimation procedure. Of course, ifsharp resonances were to be expected in the frequencyband that is left unmeasured, under-modeling may result.

In order to estimate the sample variance of the noise on themeasurements, 10 repeated measurements were performed.The sample variance is then plugged instead of the realvariance to start the estimation process.

6.3 Calibration of the experiment

The proposed measurement setup has the disadvantagethat the power splitter and the sine source are located inbetween the wave reflectometers of the PNA (see Fig. 4)together with the DUT. This implies that even a minimalchange in the setup of the source will automaticallyrequire a recalibration of the complete measurement setup.Instead of one single SOLT calibration (Hewlett-Packard[1986]), as much as 31 calibrations were needed to obtaincorrect measurements. This disadvantage can be avoidedin future measurements using a different Vector NetworkAnalyzer (VNA), where the additional source can be putoutside the wave reflectometers.

6.4 The identified model

To model the transfer function S21, whose magnitude andphase are shown in Fig. 6 and Fig. 7, a model, as in(1), with P = 14 and V = 8 poles, respectively for thefrequency and the power of the excitation signal, is chosen.These orders have been selected by computing the costfunction of the minimization problem in (11) for differentnumber of poles and looking at the zone where it beginsdecreasing slowly in frequency and power directions.

As in the simulation example a test on the residualsis realized, to compare the model error with the 95%confidence bound (

√3σS21

). The test shows that 51% ofthe residuals are under the defined uncertainty bound. Thecomparison is shown in Fig. 8.

This result reveals that the model error is larger thanthe considered uncertainty bound and some unmodeleddynamics are present. As the perturbations on the mea-surements caused by the calibration are often an order ofmagnitude larger than the noise (Van Moer [2001]), a newuncertainty equal to 10σS21

is chosen to take this effect intoaccount. The model obtained by MOVF technique explainswell the measurements, because 98% of the residuals areunder the new uncertainty bound, as shown in Fig. 9.

−15−10

−50

5

0500

10001500

2000

−15

−10

−5

0

5

10

15

20

Power [dBm]Frequency [MHz]

|S2

1| [d

B]

Fig. 6. Magnitude of S21.

−15−10

−50

5

0500

10001500

2000

−10

−8

−6

−4

−2

0

2

4

Power [dBm]Frequency [MHz]

Ph

ase

(S2

1)

[ra

d]

Fig. 7. Phase of S21.

−15−10

−50

5

0500

10001500

2000−70

−60

−50

−40

−30

−20

−10

Power [dBm]Frequency [MHz]

Am

plit

ud

e [

dB

]

residuals

95% confidence bound

Fig. 8. Comparison between the residuals and the 95%confidence bound (

√3σS21

).

7. CONCLUSION

This paper presents an extension of the recent MultivariateOrthonormal Vector Fitting (MOVF) technique, to han-dle noisy data in an output error estimation framework.The ability of the modified algorithm to build parametric

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−15−10

−50

5

0500

10001500

2000−70

−60

−50

−40

−30

−20

−10

0

10

Power [dBm]Frequency [MHz]

Am

plit

ud

e [

dB

]

residuals

10σS

21

uncertainty bound

Fig. 9. Comparison between the residuals and the uncer-tainty bound 10σS21

.

macromodels of dynamic systems with a good accuracy,starting from measurement data was tested on simulationand real measurements examples. Both tests show that thepresented algorithm is able to compute accurate and com-pact rational parametric macromodels based on parame-terized frequency responses obtained from measurementsin the frequency domain.

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