Inferential estimation of high frequency LNA gain performance usingmachine learning techniques

Peter C. Hung, Seain F. McLoone, Magdalena Sainchez, Ronan FarrellInstitute ofMicroelectronics and Wireless Systems, Department ofElectronic Engineering,

National University ofIreland Maynooth, Maynooth, Co. Kildare, Irelandtphung, sean.mcloone, msanchez, rfarrell}geeng.nuim.ie

Abstract

Functional testing of radio frequency integratedcircuits is a challenging task and one that is becomingan increasingly expensive aspect of circuitmanufacture. Due to the difficulties with bringing highfrequency signals off-chip, current automated testequipment (ATE) technologies are approaching thelimits of their operating capabilities as circuits are

pushed to operate at higher and higher frequencies.This paper explores the possibility of extending theoperating range of existing ATEs by using machinelearning techniques to infer high frequency circuitperformance from more accessible lower frequencyand DC measurements. Results from a simulationstudy conducted on a low noise amplifier (LNA) circuitoperating at 2.4 GHz demonstrate that the proposedapproach has the potential to substantially increasethe operating bandwidth ofATE.

1. Introduction

Reliable high frequency testing of radio frequencyintegrated circuits (RFIC) has become a significantfactor in the cost and time-to-market of novel wirelessproducts [1]. Testing of RFICs is generally performedusing dedicated automatic test equipment (ATE) thatrecord a set of measurements with the circuitsoperating in their functional mode so that performancecan be compared against design specifications. Thistraditional testing solution is becoming prohibitivelyexpensive as RFIC designs push into multi-gigahertzoperating frequencies due to the increased cost ofATEat these frequencies and the difficulty with bringingmulti-gigahertz RF measurements off-chip in a factoryenvironment. Indeed, it is predicted that this laterconstraint may become a limiting factor in extendingwireless technologies to even higher frequencies.

Manufacturers are therefore increasingly looking atalternative approaches to functional testing whichavoid taking measurements at very high frequencies.Significant advances have been made in a variety ofapplication domains in recent years through thedevelopment of Design for Testability (DfT) and Built-In-Test (BIT) methodologies [2-5]. In BIT, forexample, on-chip testing circuitry allows evaluation ofICs using lower frequency or DC external testers.However, these methodologies come with significantoverheads in terms of area and power consumption andare not readily applicable to RF mixed signal circuits.

In [6] the authors propose a different strategy. It ishypothesised that for many RFICs knowledge ofresponses at DC and lower frequencies may providesufficient information to allow inference ofperformance at higher frequencies. Training of suchinferential performance estimators can be achievedusing data generated from detailed circuit simulationscovering the space of process-dependent parametervariations [6]. When successful, this approacheffectively extends the operating range of existingATEs, allowing them to be employed to test nextgeneration devices.

In this paper a case study is presented on theapplication of the proposed test methodology toclassification of the gain performance of a low noiseamplifier (LNA), a key component in modemtelecommunication systems. Inferential classificationof the amplifier gain at its operating frequency isperformed using feature vectors consisting of DCmeasurements (quiescent currents and voltages) andgain measurements at other more accessible (lower)frequencies.

Since RF circuit interactions are inherentlynonlinear and complex, it is anticipated that machinelearning modelling and classification will generally berequired to adequately capture the relationshipsbetween variables. This is demonstrated in the case

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study through comparative results for linear andnonlinear classifiers. In each case, both direct andindirect classifier paradigms are considered. In thelatter, linear and neural network models are trained topredict the value of the amplifier gain and a thresholdrule applied to the resulting prediction to perform thecircuit classification, while in the former, lineardiscriminant analysis, support vector machine and k-nearest neighbour classifiers are trained to directlyclassify the circuit performance on the basis of thefeature vector provided.

The remainder of the paper is structured as follows.The case study and test methodology are described inSection 2 while the classical and machine learninginferential classifiers considered are briefly presentedin Section 3. The results of a comprehensive MonteCarlo simulation study are presented in Section 4together with a discussion of the implications for theproposed test methodology. Finally, the conclusionsare given in Section 5.

The following notation is used to distinguishbetween the different feature vectors considered in thestudy.

Xf = vector of the amplifier gains at allsampling frequencies from 0.1 tofGHz

XDC = vector of the amplifier DC bias voltageand current measurements

Xf+DC = [Xf, XDC], a concatenation of Xf and XDC

The goal of testing is to classify amplifiers as beingwithin specification ('good') or not withinspecification ('bad'). In the proposed test methodologythis is achieved without direct measurement of g2.4 byinferring the classification from other circuitmeasurements, specifically XDC and/or Xf, withfideallymuch less than 2.4 GHz. This is achieved by building aclassifier, h, to map from a circuit feature vector, x, tothe appropriate classification

h(x) - Z2.4 (g2.4 ) = {+1} - (1)

2. Case Study and Test Methodology

In this study, the low noise amplifier (LNA) usedwas a standard low-voltage 2.4 GHz MOSFET design,simulated in ADS® using UMC's 0.18 ptm siliconprocess technology [7]. The LNA circuit consisted of 2bias transistors, 4 RF transistors (0.18 pim channellength and 0.5 pim channel width), 4 resistors, 3capacitors and 4 inductors. The gain of the LNA at itsoperating frequency (2.4 GHz), g2.4, was considered tobe the critical performance parameter and was deemedto be within specification if 14.7 dB< 9g24 < 17.2 dB.

To simulate LNA manufacturing process variationszero mean uniform random variations were introducedinto the 38 most significant model parameters, asdetermined by a sensitivity analysis [6]. While inpractice circuit parameters might be expected to varynormally around their nominal values, uniformdistributions were chosen to give an even coverage ofthe LNA parameter space. Catastrophic failures suchas short-circuits were excluded as these faults can bedetected relatively easily using existing IC testingtechniques.

Using this approach, 10,000 circuit simulationswere performed. For each circuit the amplifier gain, gfwas recorded at sampling frequencies, f, of 0.1, 0.3,0.6, 1.2, 1.4, 1.7 and 2.0 GHz and also at the operatingfrequency (2.4 GHz). In addition, 8 key DC biasvoltages and currents were recorded. This data wasthen normalised to have zero mean and unit varianceand divided into training and test data sets, eachcontaining 5,000 samples.

Here, Z2.4 is a threshold function defined as

{+1 if 14.7<x<17.2Z2.4 (X) = l otherwise (2)

The mapping h(x) is determined through training onsimulated data generated using circuit simulationsoftware such as ADS®, where g2.4 measurements andtarget circuit classifications can easily be generated.Once trained, the classifier can then be used to testproduction circuits, with only the measurementsneeded to generate the feature vector, x, required fortesting. In the case study the 5,000 sample test data setis used to represent production circuits.

3. LNA Inferential Classifiers

The inferential LNA classifier can be implementeddirectly as in (1) using either linear classifiers such aslinear discriminant analysis (LDA) or machinelearning classifiers such as support vector machines(SVM) and k-nearest neighbour (kNN) classifiers.Alternatively it can be estimated indirectly by firstlearning a model that predicts the value of 2.4 from thefeature vector,

g(x) - 2.4 (3)

and then using a threshold function to perform theclassification, that is

h(x)- Z2.4 (g(X)) - (4)

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A simple linear model or an artificial neural networksuch as a Multilayer Perceptron can be used to learnthe mapping defined by (3). A brief overview of eachof these models and classifiers will now be given.

3.1. Linear Model (LM)

A linear model (LM) for predicting g24 can beconstructed as

g2.4 = LM (X) = W'X, (5)where w, the model parameters, are estimated fromtraining data using least squares optimisation. Definingthe matrix of training feature vectors as

X = [x1 x2 ...x], (6)

where n is the number of training samples, and thecorresponding vector of targets as

Y = [g2.4(l) 92.4(2) ... 92.4(n)]T , (7)

the least squares solution is given by

w =[XTX] 1IXTy = XtY,

the optimum weights can be determined using gradientbased optimisation techniques. Here, training wasperformed using the hybrid BFGS training algorithmwith stopped minimisation used to prevent over-fitting[9]. The optimum number of neurons (M) wasdetermined for each model by cross-validation on thetest data set.

3.3. Linear Discriminant Analysis (LDA)

In linear discriminant analysis (LDA) the featurevectors are projected onto directions that havemaximum discriminatory power, P, as measured by theratio of between and within class variances. Here, theclassical two-class Fisher's LDA [10] is used, in whichthe feature data is projected onto a singlediscriminatory direction

d = W'X

and w is chosen to maximise

W'SBWP(w) =

wTSWW

(12)

(13)(8)

where Xt is the Moore-Penrose pseudo-inverse of X.

3.2. Multilayer Perceptron (MLP)

The between-class and within-class scatter matrices(SB and Sw), defined as

SB = (XG -XB)(XG XB) (14)

The multilayer perceptron (MLP) is one of the bestknown and most widely used neural networkarchitectures because of its universal functionapproximation capabilities, good generalisationproperties and the availability of robust efficienttraining algorithms [8]. In this application a singlehidden layer MLP is used to capture a nonlinear modelbetween the LNA feature vector, x, and gain, g24:

g2.4 gMLP(x) (9)

where

M h

gmLp(x) = bh + W11w x 0)+ep(w' .x+b')

Parameters wh w b ,(i 2,-,M) and bh areweights and biases which collectively form thenetwork weights vector, w, and M is the number ofhidden layer neurons. Defining a Mean Squared Error(MSE) cost function over the training data,

and

SW E ~(Xi -xj)(Xi _Xj)TjeG,B ieC.j

(15)

are computed over the training data. In (14) and (15)vector Xi is the mean of the feature vectors in the jh

class, Cj. P(w), which is in the form of a generalizedRayleigh quotient, is maximised when w is chosen asthe eigenvector corresponding to the largest eigenvalueof the generalised eigenvector problem

SBW = /SWW . (16)

Having determined d, the optimum projection of thefeature vector, circuit classification is performed byapplying a threshold function to d, that is:

hLDA (X) = z (d) = z, (w'x) , (17)

where

z +1 if x2 aZzx (x) =dl-I ifx<a

(18)

The value of the threshold, a, is chosen to minimisethe misclassification rate (MCR) over the training data.

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(1 1)pn

E(w=-E (9MLp (X,p w) g2.4(,p))p=l

Linear discriminant analysis assumes classes areapproximately normally distributed. However, theperformance specification for the amplifier gain in thecase study is such that the class of out-of-spec ('bad')circuits is bi-modal. To take this into account, circuitclassification is implemented as a three-class problemwith classes defined as bad-lower (BL), good (G) andbad-upper (BU). Classification is then performed intwo-stages. Firstly, two one-versus-all classifiers aretrained, one to classify LNAs as either 'BL' or 'not-BL', and the other to classify them as either 'BU' or'not-BU'. The overall classification is then obtained bytaking a logical AND of the output of these classifiers,where 'not-BL' and 'not-BU' are considered to be'good' and all other combinations 'bad'.

3.4. Support Vector Machines (SVM)

Support Vector machines (SVM), proposed byVladimir Vapnik in 1963, classify data sets bydetermining the margin which achieves the maximumseparation between classes [11]. Consider a separatinghyperplane, with orientation vector w and locationparameter b, that divides two classes of data:

and v, are the corresponding weighting factors. Thelinear SVM (LSVM) decision function is then given by

hLsvM(x) = Za(wX -b) (23)

where a = 0.Substituting (22) into (23) allows the SVM

classifier to be expressed as a linear combination ofsupport vector dot products, that is,

hLsvM(x) zo K V,(x.xs) b . (24)

This important property allows SVMs to be extendedto nonlinear classification problems using the kerneltrick. This involves replacing the dot products by akernel function which meets Mercer's condition [12]:

(25)

so that the data is mapped into a higher dimensionfeature space, x -* D(x), where classification can beperformed using a linear SVM. Here the Gaussianradial basis function (RBF) kernel,

k(xi, x .) = exp! - xij

and two additional hyperplanes that are parallel to theseparating hyperplane:

WTX - b = 1(20)

W'X -b = 1

is used. The parameter, o-, controls the kernel widthand is determined as part of the classifier trainingprocess. Note, in the original data space, the resultingdecision function will be nonlinear and takes the form

The perpendicular distance between the parallelhyperplanes, 2/llwll, is referred to as the classifiermargin and the optimal hyperplane is the one whichresults in the maximum margin. The problem ofdetermining the maximum margin can be expressedmathematically as

2maxw llwll min(wTw),

w

subject to / 1)

Z2.4 (g24(i) )(WTXi -b) >1, i =1,2, ... ., n.

This is a constrained quadratic optimisation problemwhose solution w has an expansion

W ,vS , (22)

s

where xs are the subset of the training data, referred toas support vectors, located on the parallel hyperplanes,

hsVM(x) = zo VS k(x,xs) - bJ. (27)

In non-separable problems where different classesof data overlap, slack variables are introduced so that a

certain amount of misclassification of data is allowed.A smoothing parameter C controls the amount ofmisclassification permitted with large values resultingin a larger penalty on errors. This 'soft margin'classifier is the most general form of SVM [12].SVM training is generally formulated as a quadratic

programming problem and, as such, is computationallycomplex for large training sets. Consequently muchresearch effort has gone into developing efficienttraining algorithms for SVMs [13]. Here, training was

performed using the Matlabg package simpleSVM[14], a fast SVM solver based on active constraint sets.The kernel width parameter, o, and smoothingparameter C were optimised on the basis ofclassification performance on the test data set. The a

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W'X - b = 0 (19) (26)

k(xi, x.) = (D(xi) - (D(xj) ,i

priori knowledge of the bi-modal distribution of 'bad'circuits was taken into account by implementingclassification as a three-class problem using the one-versus-all approach, as outlined in Section 3.3.

3.5. k-Nearest Neighbours (kNN)

The third classifier considered in the case study isthe k-nearest neighbour (kNN) classifier [10], anonparametric method in which a test feature vector, x,is assigned the most frequently occurring class amongthe k most similar feature vectors in the training set.Mathematically, the kNN classifier can be expressed as

classifiers (with optimised k values) perform verypoorly for this problem, but this is to be expectedgiven the high level of overlap between classes in thefeature space, particularly when the DC and lowestfrequency measurements are included. Moresurprisingly, the SVM classifiers do not yield goodresults. This may be a consequence of the nature of theproblem and the fact that the indirect classificationapproach is more suited to the form of the data. It isalso suspected that the simpleSVM algorithm used totrain the SVM may have deficiencies for this type ofproblem. This is currently under investigation.

Table 1. MCRs for various LNA classifier paradigm-featurevector combinations

(28)

where K(x) is the index set of the k nearest training setfeature vectors. The similarity measure used was theEuclidian distance between vectors,

di(x) = llx-xi 1- (29)

and the optimum value of k for each classifier wasdetermined by cross-validation on the test data set.

4. Simulation Study

To evaluate the performance of the variousinferential performance classifiers under considerationand assess the value of different choices of featurevector, a Monte Carlo simulation was performed, asdescribed in Section 2. The first 5,000 LNAs wereused to train the various classifiers, while the second5,000 formed the test set. Performance was measuredin terms of the misclassification rate (MCR), the'good' circuit pass rate (GPR) and the 'bad' circuit failrate (BFR) estimated over the test LNAs. To providerobust estimates, these metrics were computed byaveraging over 100 batches of 500 'good' and 500'bad' LNAs selected from the test data set usingsampling with replacement.

Table 1 shows the mean MCR obtained with eachof the classifiers for five specific feature vectorcombinations, namely XDC, x14, X1.4+DC, x2.0 and X2.0+DCThe standard deviation of the MCR estimates is givenin parenthesises. For comparison purposes the MCRsobtained for single frequency measurements at 1.4(g1.4) and 2.0 GHz (g2.o) are also included.

The results clearly show that the indirect classifiersoutperform the direct classifiers with the MLPconsistently giving the lowest MCR. The kNN

MCR mean (standard deviation)Input Indirect Direct

Linear NN LDA SVM kNN36.92 35.96 36.95 37.04 39.01

XDC (1.57) (1.49) (1.44) (1.47) (1.60)35.96

91.4 (1.36)22.16 20.84 22.78 23.28 33.84

X1.4 (1.33) (1.31) (1.22) (1.46) (1.44)20.06

g2.0 (1.20)20.58 18.53 22.50 23.26 34.10

XI.4 DC (1.21) (1.25) (1.19) (1.29) (1.62)7.12 3.69 8.75 7.55 21.71

X20 (0.74) (0.55) (0.94) (0.88) (1.35)6.88 2.46 12.31 7.55 26.18

X2.0+DC (0.77) (0.83) (1.08) (0.84) (1.44)

The feature vectors in Table 1 are in the order ofreducing MCR. It is clear that the DC measurementshave only very limited value for classification, whilegain measurements at high sample frequencies havethe most value. The best results are obtained when thegain measurements from several sample frequenciesare combined. For example, x14 has similardiscriminatory power to g2.0 and x1.4+DC is marginallysuperior to g2.0. This is further highlighted in Figure 1,which shows a plot of the mean MCR as a function ofsample frequency, f, for MLP classifiers with featurevectors Xf, xf+DC, gf and g+DC. Note that the DCmeasurements contribute significantly to classifierperformance at lower frequencies (f<1.2 GHz) but theirvalue decreases rapidly thereafter.

Since the good pass rate (GPR) and bad fail rate(BFR) of a classifier vary as a function theclassification threshold, with one increasing as the

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hkNN (x) = zo Y, Z2.4 (92.4(i))\,iEK(x) I/

other decreases, the threshold can be adjusted tocontrol one or other of these metrics. The relationshipbetween GPR and BFR for the MLP LNA classifier iscaptured in the operating curves plotted in Figure 2 forvarious choices of feature vector. Again this shows therelative merits of the different feature vectors, butmore importantly, from a manufacturing perspective itshows the trade-off that can be obtained with eachclassifier. For example, for a target BFR of 90°0 aGPR of 55% is obtained with x1.4, and 65% withxI.4+DC. This increases to over 99%0 with x2.0+DC.Alternatively, a 100% BFR can be obtained whenusing X2 +DC if the GPR is dropped to 80%. The levelof wastage at x1.4 may well be acceptable in amanufacturing context where the cost of deliveringfaulty product to consumers may be substantial.

5. Conclusions

In this paper machine learning based inferentialclassifiers are used to predict the high-frequency gainperfornance of a 2.4 GHz LNA. Of the variousclassifiers investigated, the indirect MLP classifiergives the best overall results, effectively extending theoperating frequency range of ATE for LNA gainclassification by 20% at 2 GHz and 42% at 1.4 GHz.

6. Acknowledgments

The authors gratefully acknowledge the financialsupport of Enterprise Ireland.

[7] P.E. Allen, DR. Holberg, CMOS analog circuit design,2nd Edn. Oxford University Press, Oxford, 2002.[8] S. Haykin, Neural Networks: A comprehensivefoundation, 2nd Edn. Prentice Hall, New Jersey, 1998.[9] S. McLoone, M. Brown, G. Irwin, G. Lightbody, "Ahybrid linear/nonlinear training algorithm for feedforwardneural networks", IEEE Trans. on Neural Networks, 9, 1998,pp. 669-684.[10] R.O. Duda, P.E. Hart, D.G. Stork, PatternClassification, Wiley Inc., 2001.[11] V. Vapnik, A. Lerner, "Pattern recognition usinggeneralised portrait method", Automation and RemoteControl, 24, 1963, pp. 774-780.[12] C.J. Burges, "A tutorial on support vector machines forpattern recognition", Data Mining and KnowledgeDiscovery, 2, 1998, pp. 273-297.[13] I.W. Tsang, J.T. Kwok, P.M. Cheung. "Core vectormachines: Fast SVM training on very large data sets",Journal ofAMachine Learning Research, 6:363-392, 2005.[14] S.V.N. Vishwanathan, A.J. Smola, M.N. Murty,"SimpleSVM", Proc. 20th Int. Conf Machine Learning,2003, pp. 760-767.

45

40

35

30

a 25

- 20

15

Xf10 + gf

5+ gf+DC.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Frequency (GHz)

Fig. 1: MCR as a function of frequency for the MLP LNAclassifier for feature vectors xf, X+DC,, gf and gf+DC-

7. References

[1] J. Ferrario, R. Wolf, H. Ding, "Moving from mixed signalto RF test hardware development", IEEE Int. Test Conf,2001, pp. 948-956.[2] W.Y. Lau, "Measurement challenges for on-wafer RF-SOC test", Annual IEEEISEMI Int. Elect. Manufact. Tech.Symp., 2002, pp. 353-359.[3] M. Negreiros, L. Carro, A. Susin, "Low cost on-linetesting of RF circuits", 10th IEEE Int. On-Line TestingSymp., 2004, pp. 73-78.[4] D.C. Doskocil, "Advanced RF built in test",AUTOTESTCON '92 IEEE Sys. Readiness Tech. Conf, 1992,pp. 213-217.[5] M.E. Goff, C.A. Barratt, "DC to 40 GHz MMIC powersensor", Gallium Arsenide IC Symp., 1990, pp. 105-108.[6] M. Sanchez, P.C. Hung, R. Farrell, S.F. McLoone and G.Zhang, "A Structural Test Methodology Utilising IndirectPerformance Estimation For Low-Cost RF Test", TechnicalReport, IMWS, NUIMaynooth..

100 *

95

90

85

80

m 75 -

65 14X1.4. X2.0

60 3 x1.4+DC

55 -

xDC

50 55 60 65 70 75 80 85 90 95 100%GPR

Fig. 2: MLP LNA classifier BFR v GPR operating curves forvarious feature vector choices.

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