Analysis and compact behavioral modeling of nonlinear distortion in analog communication circuits

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 22, NO. 9, SEPTEMBER 2003 1215

Analysis and Compact Behavioral Modeling ofNonlinear Distortion in Analog

Communication CircuitsPetr Dobrovolný, Member, IEEE, Gerd Vandersteen, Member, IEEE, Piet Wambacq, Member, IEEE, and

Stéphane Donnay, Member, IEEE

Abstract—The design of analog front-ends of digital telecom-munication transceivers requires mixed-signal simulations at thearchitectural level. The nonlinear nature of the analog front-endblocks is a complication for their modeling at the architecturallevel, especially when the nonlinear behavior is frequency de-pendent. This paper describes an analysis and modeling methodbased on Volterra theory [1], [2]. The method derives bottom-upmodels of nonlinear analog continuous-time circuits. Thesebehavioral models predict the dominant nonlinear effects using acomposition of linear transfer functions and multiplications. Thismakes it possible to accurately model frequency dependenciesand to gain insight into the dominant nonlinear sources of thecircuit. The basic models are afterwards described using its mul-ticarrier complex low-pass representation to enable their efficientcosimulation with the digital circuits in a dataflow simulationenvironment.The multicarrier representation is a direct extensionof the classically used complex low-pass equivalent models, whichconsiders the modulation of a single carrier only. The accuracy ofthe multicarrier representation is higher than classical complexlow-pass equivalent models since out-of-band nonlinear distortionis taken into account. The main advantage of the proposedtechnique is that it yields both insight in the nonlinear behaviorat the circuit level and that it provides an important gain insimulation efficiency of RF integrated circuits at the system level.Both aspects are demonstrated on a 5-GHz WLAN design.

Index Terms—Circuit, cosimulation, device models, modeling,simulation.

I. INTRODUCTION

T HE DESIGN of today’s digital wireless communicationsystems is driven by requirements for miniaturization,

lower power consumption, and high flexibility. Meeting thesedemands means a serious challenge for the design of the analogfront-ends in the transmitters and the receivers.

The signal degradation caused by the analog front-ends(noise, nonlinear distortion, etc.) must be kept as low aspossible. The total effect of signal degradations of the completesystem, including both the analog and the digital part, can bequantified by means of the bit-error-rate (BER). A predictionof the BER requires lengthy simulations since a large numberof symbols must be transmitted from the transmitter to thereceiver. Therefore, it is very important that the inclusion of

Manuscript received May 31, 2002; revised September 25, 2002 and March15, 2003. This paper was recommended by Guest Editor H. A. Mantooth.

The authors are with the Interuniversity MicroElectronics Center, B-3001Leuven, Belgium (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCAD.2003.816220

Fig. 1. Design flow for the analog front-ends that links the architectural,circuit, and layout levels of description.

the RF front-end nonidealities in BER simulations do not slowdown the computations too much. A cosimulation of the digitalparts, described at a high abstraction level, with the analogparts described at the circuit-level, is, therefore, not feasible.

The possible strategy of a mixed-signal design flow thatcovers various abstraction levels was suggested in [3]. The partof the mixed-signal flow, related to the analog front-end blocks,is shown in Fig. 1.

The design flow starts with a high-level simulation of afront-end architecture. At this stage, analog–digital tradeoffscan be made. Afterwards, the architecture is split into analogand digital parts according to the chosen tradeoff. The digitalparts are designed in a separate design flow (not shown inFig. 1) [4]. The models for the analog blocks are very roughduring these first high-level simulations. These simulationsprovide an initial set of specifications for the individualanalog/RF circuits that are used as input for the circuit-leveldesign. This circuit-level design can be performed in a manualway or by using analog synthesis tools [5]. After the circuitdesign, the layout can be generated. Again, CAD tools foranalog placement and routing could be used here to speed upthe process [6].

0278-0070/03$17.00 © 2003 IEEE

1216 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 22, NO. 9, SEPTEMBER 2003

The design flow as described up until now is atop-downprocess. The reliability of the results depends highly on theaccuracy of the high-level models used. The flow, therefore,also contains abottom-uppath that extracts system-level modelsfrom a circuit netlist in an automated way.

Commercial simulators contain libraries with high-levelmodels of analog blocks that can be used during the initialsimulation at the architectural level. The available modelsare often very general and take only the nominal behaviorinto account. This can lead to inaccurate simulations and,consequently, to wrong conclusions at the system level. Thedifficulty in modeling analog blocks at the system level istheir nonlinear behavior that requires a careful study and evenadvanced mathematical methods. Moreover, this nonlinearbehavior is generally frequency dependent, especially at highfrequencies.

This paper presents an analysis and modeling approach forthe extraction of reliable high-level models that are consistentwith the designed circuit. The described modeling method isbased on Volterra theory and its application to weakly nonlinearsystems. The extracted models are more accurate than the initialhigh-level models, leading to more realistic specifications forthe analog circuits. Moreover, the accuracy of the models is userdefinable, such that the user can control the consistency betweena circuit and the corresponding high-level model.

The modeling of the frequency-dependent, weakly nonlinearbehavior is done using the Fourier transforms of Volterra kernels[7], [8]. The first-order kernel transform describes the linear be-havior of the circuit. The second- and third-order kernel trans-forms describe the second- and third-order nonlinear behavior,respectively. The discussion here is limited to second- and third-order nonlinear behavior only, which is a reasonable assumptionin most practical cases. It should be noted, however, that the de-scribed approach is also valid for nonlinear behavior of orderfour and higher.

A Volterra series development of a circuit can both predictthe response of the circuit and can reveal the dominant non-linear sources. The latter is important for both potential modelreduction—and, hence, simulation efficiency—and to providevital feedback to the circuit designers. Feedback to the circuitdesigner is feasible since the Volterra series can be interpretedphysically. This gives the circuit designers the insight on whereand how they can reduce the nonlinear distortion.

Revealing the dominant nonlinear sources requires that thedifferent nonlinear contributions be separated since it mustbe possible to determine the impact of the different nonlinearcontributions individually. This differentiates the proposedmethod with existing Volterra-based simulation techniques[9]–[11]. These techniques determine the overall nonlinearresponse by computing of the successive orders of nonlinearity.All the different contributions of the same order are combinedduring the computation of that one order of nonlinearity. Thisimplies that the different nonlinear sources will be combinedduring the computation, which makes it impossible to separatethe different nonlinear sources.

The proposed method also generates high-level models thatcan be used in conjunction with digitally modulated signals[12]. Such signals are often difficult to approximate with a small

Fig. 2. Different steps of the modeling approach in view of various levels ofsimulation abstraction and domains.

number of sinusoidal signals. The modeling approach firstcomputes the multidimensional Fourier transforms of second-and third-order Volterra kernels of the output of interest. Thesekernel transforms contain many contributions. The modelreduction determines the dominant contributions according to auser-defined error. This method is an extension to a widebandmultitone excitation of the algorithm described in [8] thatonly holds for a single-tone excitation. Finally, a translationof the dominant contributions into a block diagram yieldsthe final model. This model can be efficiently and accuratelycosimulated with a digital system using acomplex low-passequivalentwith amulticarrier signal representation[13]. Fig. 2shows the different steps of our modeling approach in view ofvarious levels of a simulation abstraction and domains.

The rest of the paper is organized as follows. Section II re-views the power series description of nonlinearities. Section IIIdiscusses the application of the Volterra theory to weakly non-linear circuits with a multitone excitation. Section IV describesthe modeling approach together with the model reduction andtransformation toward its multicarrier complex low-pass equiv-alent representation. Section V illustrates the proposed tech-niques on an integrated low-noise amplifier design for 5-GHzWLAN receiver. Several aspects related to the simulation accu-racy, efficiency, and performance are highlighted.

II. DESCRIPTION OFNONLINEARITIES

The nonlinear devices most commonly used in silicon in-tegrated circuits (ICs) are bipolar and MOS transistors, resis-tors, capacitors, and diodes. The large-signal characteristics ofthese devices are usually described by a set of nonlinear model

DOBROVOLNÝ et al.: ANALYSIS AND COMPACT BEHAVIORAL MODELING OF NONLINEAR DISTORTION 1217

Fig. 3. Nonlinear equivalent circuit of a bipolar transistor.

equations together with an equivalent circuit. Some of the ele-ments of the nonlinear equivalent circuit are so weakly nonlinearthat they can be taken linear. The remaining nonlinear elementsare called “basic nonlinearities” and are described by nonlinear

or relationships. For most applications one can re-strict this set to:

• nonlinear resistances;• nonlinear conductances;• nonlinear transconductances;• nonlinear capacitances;• multidimensional transconductances (i.e., current sources

that depend upon several controlling voltages, where oneof the controlling voltages can be the voltage over the el-ement itself).

These basic nonlinearities are characterized using a power se-ries expansion of the nonlinear modeling equations in the quies-cent point. Resulting formulas, hence, contain the higher-orderderivatives of the output quantity (current or voltage) with re-spect to the controlling quantities.

An example of an equivalent nonlinear circuit for a bipolartransistor is shown in Fig. 3. It is based upon the Gummel–Poonmodel and consists of lumped elements only. The collectorcurrent, which flows between the collector and the emitter, is anonlinear function of two voltages, the base-emitter voltageand the base-collector voltage . In the linearized equivalentcircuit of a transistor, the early effect is modeled by takinginto account a resistance between collector and emitter. Tofollow this approach, the is substituted with - inthe Gummel–Poon equations, so the collector currentcanbe expressed as a function of controlling voltages and

. We assume these voltages, expressed as a sum of constantcomponents and and time-varying small-signalcomponents and . After subtracting the constantcomponents from the complete Taylor series expansion of thecollector current in the close vicinity of a quiescent point, we

can write the following relation small-signal quantitiesand

(1)

where coefficients , , and are referred to as thefirst-, second- and third-ordernonlinearity coefficients. They areproportional to the first-, second-, and third-order derivatives ofthe collector current with respect to , whereas coefficients

, , and are proportional to the first-, second-, andthird-order derivatives of the collector current with respect to

. The coefficients that correspond to derivatives with respectto one voltage are called one-dimensional (1-D) nonlinearitycoefficients. The ones proportional to cross-derivatives, such as

(2)

are two-dimensional (2-D) nonlinearity coefficients.Through the similar procedure based on the Taylor series ex-

pansion, any basic nonlinearity of a circuit can be characterizedby its nonlinearity coefficients and

III. A PPLICATION OFVOLTERRA SERIES TO THEANALYSIS OF

WEAKLY -NONLINEAR ANALOG CIRCUITS

The response of weakly nonlinear circuits can be describedby a functional series, referred to as the Volterra series, where

represents an output signal and represents an inputsignal in the time domain. The multidimensional function

is called the th-order Volterra Kernel(VK) or the th-order nonlinear impulse response. The linearbehavior represented by is the well-known impulseresponse of the system.

The functional series [(3) see equation at the bottom of thepage] can be expressed in a more compact form as

(4)

(3)


Fig. 4. Nonlinear system described by Volterra operators.

where —referred to as the th-order Volterra oper-ator—stands for the th-order convolution integral from (3).The convolution can be computed in the frequency domain bymultiplying the Fourier transform of theth-order impulse re-sponse with the Fourier transform of theinput signals. The

th-dimensional Fourier transformation of theth-order time-domain Volterra kernel leads to the Volterrakernel expressed in the frequency domain ,i.e., the so-callednonlinear transfer functionor Volterra KernelTransform(VKT)

The characterization of a system by (4) leads to the schematicrepresentation shown in Fig. 4, where each operatorde-scribes the th-order nonlinear behavior either in the time do-main or in the frequency domain.

The efficient application of Volterra series analysis for non-linear systems is limited by a condition of convergence. A non-linear system, whose response for a given excitation can be rep-resented by a converging Volterra series, behaves weakly non-linear. The question remains on how to determine the radius ofconvergence of Volterra series. However, [8] shows that for mostpractical ICs under normal signal conditions (except, e.g., lim-iters, comparators, and unstable circuits, which are outside ofthe scope of this paper), the following more restrictive definitionfor weakly nonlinear behavior can be used.A circuit behavesweakly nonlinearly if, for the applied input signal, it can be ac-curately described by the first three terms of its (converging)Volterra series.

A. Calculation of Volterra Kernels

The calculation of the VKs adopts the known analysismethod called themethod of nonlinear currents[8], [15]. Thisapproach exploits the power series description of basic non-linear elements discussed in detail in Section II and calculatescurrent components from VKTs of a lower order. Let us sup-pose a nonlinear circuit displayed in Fig. 5 with only one basicnonlinear element—the nonlinear conductance—described byits nonlinear characteristic and its power series expansionin the close vicinity of a quiescent point (see(1)). According to this power series expansion, the nonlinearconductance could be represented as a parallel connection of itslinearized model and controlled current sources correspondingto the higher-order terms in the power series.

An excitation signal described by a frequency bandaround a carrier frequency is usually mod-

eled by a properly chosen finite set of sinusoidal signals that

Fig. 5. Simple nonlinear circuit and its equivalent circuit according to a powerseries expansion of the nonlinear conductance.

covers the frequency band. Hence, we assume the circuit ex-citation as a sum of unitary sinusoidal signals

(5)

where for each frequency index. To simplify the in-

dexing notation, we define the frequency setthat covers

positive and negative frequencies of all of the components ofthe original circuit excitation. Then, the controlling voltage

could be expressed in the following form:

(6.1)

(6.2)

(6.3)

(6.4)

when limiting the analysis to the third-order mixing prod-ucts. The first-order component depends on the lineartransfer function from the original input excitationto the controlling nodes. The second- and third-order product

and are functions of the second- and third-orderVKTs and , respectively, whichrepresent the second- and third-order nonlinear transfer fromthe original input to the controlling nodes.


Fig. 6. Rearrangement of the circuit in Fig. 5 with the current sourcesrepresenting the appropriate single order of mixing products.

From Fig. 5 we see that the square and third power of thecontrolling voltage are required. Using (6.1) these powersare given by

(7.1)

(7.2)

where only the term from (7.1) produces only second-order mixing products. The terms from (7.1) and

from (7.2) produce the third-order mixing product.Now we can rearrange the circuit as shown in Fig. 6.Inserting (6.1), (6.2), (6.3), (6.4), (7.1), and (7.2) into the

power series expansion of the nonlinear characteristics ofthe conductance (1) can be found that the nonlinear currentsources and satisfy

This clearly shows that theth-order nonlinear current sourceassociated with a basic nonlinearity depends on the type of thenonlinearity, on its nonlinearity coefficients of order equal toor less than and on the VKTs of its controlling voltage(s) ofan order equal or less than ( ). Using this property, it ispossible to compute the VKTs in the frequency domain usingthe same linear network with different excitations generated bythe different orders of nonlinearity.

• The first-order VKT at the output and the first-orderVKT at the controlling nodes are equal to the cor-

responding transfer function of the linearized circuit (seeFig. 6). They are obtained by an ac analysis for all fre-quencies from the frequency set .

• The second-order VKTs and are derivedfrom the same linear network. The original excitation isset to zero and the new excitation is established by thenonlinear current source that corresponds to thesecond-order nonlinearity coefficient. The amplitude ofthe second-order nonlinear current source is derivedfrom (8.1) for a given set of two frequency arguments

using

(9)

The second-order signal generated by propagatesthrough the linear network and contributes to both thesecond-order response at the output (see the linear transferfunction in Fig. 6) and also to the second-order response at the controlling nodes of the nonlinearity(linear transfer function that equals to unity forour simple example and not displayed in Fig. 6)

(10.1)

(10.2)

is evaluated for all possible frequency argumentcombinations from the frequency set andcharacterizes the second-order nonlinear behavior of thecircuit at the output. represents the nonlinear transferfunctions toward the controlling nodes of the nonlinearityfor the determination of higher order nonlinear contribu-tions.

• The third-order response at the output are derived fromthe same linear network but now excited by the third-ordercurrent source , namely

(11)

where denotes the transfer functionfrom the applied current source to the output. is

(8.1)

(8.2)


the amplitude of a third-order nonlinear current source[see (8.2)] for a given set of three frequency arguments

, namely

(12)

It can be seen that is a function of the nonlinearitycoefficients and and and (the first-and second-order VKTs at the controlling nodes of thenonlinearity) computed in the previous steps.

Evaluating (11) at all possible combinations of frequency ar-guments from the frequency set completelycharacterizes the third-order nonlinear behavior of the circuit fora given input excitation. Then, the third-order circuit responseat a chosen frequency is determined by

(13)

which means that the third-order output at frequencyis a sum of all third-order VKTs whose frequency argumentssatisfy the condition .

Indeed, a practical circuit contains many nonlinear elementsand the procedure must be repeated for each of the basic non-linearities in the circuit. Then, the VKTs at the observed outputcomprise all contributions from all basic nonlinear elements.This implies that (11) can be generalized toward

(14)

where is the amplitude of the third-order nonlinear cur-rent source corresponding to theth nonlinearity of the circuitand denotes the transfer function from appliedcurrent source to the output. The sum is taken over all nonlin-earities in the network. It should be noted that the sum over thedifferent nonlineairities can be calculated during the calculationof each order of the Volterra series [10]. However, summing upthe contributions at an early stage will remove the ability to pin-point the dominant nonlinear contributions.

It can be easily proven that anth-order VKT’s functionis symmetric with respect to its frequency arguments whenits time-domain equivalent is symmetric. This is very natural,because a nonlinear operator cannot distinguish the differentapplied frequencies. An interesting property for symmetric

th-order transfer functions is that its complex conjugate can

be obtained by simply changing the sign of the frequencyarguments

(15)

Both mentioned properties can be employed during thecomputation of VKTs and considerably decrease the numberof computations.

The estimate of the time required to perform the above-de-scribed analysis, contains several contributions that roughly cor-respond to the outlined steps of the procedure. At first, we haveto evaluate all first-order VKTs at all frequencies from the fre-quency set of an input excitation [see (5)]. This requires an acanalysis of the linearized circuit at different frequencies fromthe set . It has computational complexity ex-pressed by means of a so-called-notation, where

and is the number of nodes in the circuit. Next,the determination of ath-order VKT requires determining all

th-order contributions from all nonlinearities of a circuit. Theth-order contribution of a nonlinearity is represented by a set

of nonlinear current sources, each corresponding to a nonlin-earity coefficient of an order equal toor less. Let be theoverall number of nonlinearity coefficients of a circuit that in-fluence the th-order VKT. Then, the number of operations re-quired to evaluate ath-order VKT can be approximated by

, where the functionexpresses the

number of all possible combinations of different fre-quency arguments when taking into account the symmetry prop-erty of a VKT [see (15)]. The factor in correspondsto the number of operations required for the evaluation of allpossible transfer functions induced by nonlinear currentsources. The member of the bounds the number ofoperations for evaluating values of nonlinear currents sources

[see (9) and (12)]. Finally, the functionexpresses the number of operations re-

quired to sort all contributions. This step is necessary for furtherapproximation of a complete VKT by the dominant contribu-tions (according to some error criterion). The combination of

, , , and leads to the overall computational complexity. This limit shows that

high-order behavior (i.e., higher than three) yields an excessiverunning time limit if the frequency resolution is very high.

IV. M ODELING APPROACH

Our primary motivation for the analysis described in the pre-vious section is the need of a white box model for weakly non-linear system under analysis. The analysis approach providesboth numerical/symbolic results and provides information aboutorigin of the dominant nonlinear behavior. It will be utilized inthe construction of a block diagram of a model.

This section describes in detail the particular steps of themodeling approach according to the concise description given inthe introduction section and the modeling flow shown in Fig. 2.

A. Setting up the Volterra Model

A block diagram for a VKT of a voltage or current in a non-linear circuit can be constructed by an adoption of the technique


Fig. 7. Block diagram for the computation of the second-order nonlinearbehavior of a nonlinear circuit.

described in [7]. To understand this construction, let us imi-tate the analysis procedure. Consider a nonlinear behavior ofa 1-D nonlinear admittance described by its second-order non-linearity coefficient. A second-order nonlinearity squares thevoltage over its controlling terminals, then multiplies the re-sult with its second-order nonlinearity coefficient to produce asecond-order “nonlinear” current. This current propagates to theoutput. When the second-order response is to be computed, thenonly the linear propagation of this current to the output mustbe considered. The effect of other nonlinearities on this cur-rent can indeed be neglected, since this yields a nonlinear be-havior of an order higher than two. This reasoning holds for allsecond-order nonlinearity coefficients in the circuit. Hence, thetotal second-order output is the sum of the contributions of allsecond-order nonlinearity coefficients. Each contribution corre-sponds to one path in the block diagram representation as shownin Fig. 7.

A block diagram representation of a second-order VKcontains three different types of paths, corresponding to 1-Dsecond-order nonlinearity coefficients , 2-D coefficients

, and to coefficients that correspond tononlinear capacitors. The symbols denote transfer func-tions from the original input to the controlling voltages of thenonlinearity. Transfer functions from a second-order current tothe output of the circuit are denoted by . The andblocks are computed in the linearized circuit in the frequencydomain and they are primarily characterized by tabulated valuesof their transfer functions. Each path consists of a second-ordernonlinearity coefficient, linear transfer functions, and a staticnonlinearity. For a 1-D nonlinearity, this static nonlinearityblock is a squarer whereas, for a 2-D nonlinearity, this is amultiplier that takes the product of the two parallel paths thatcorrespond to the two controlling voltages.

The block diagrams used throughout this paper presume atime-domain representation for the signals and the involvedbuilding blocks. The linear transfer function blocks must,therefore, be characterized by their impulse response functionsderived from the original frequency-domain transfer functions.Nevertheless, we will keep the original notation or

Fig. 8. Path from the input to the output of a weakly nonlinear circuitcorresponding to the contribution of a 1-D third-order coefficientK3 to theoverall third-order nonlinear behavior.

Fig. 9. Path from the input to the output of a weakly nonlinear circuitcorresponding to the contribution of a 1-D second-order coefficientK2 tothe overall third-order nonlinear behavior.

to reflect the primarily derived frequency character-istics.

With a block diagram such as in Fig. 7, the output of thecircuit due to second-order nonlinear behavior can be computedfor any input excitation, provided that this excitation is not toolarge such that the assumption of weakly nonlinear behavior isnot violated.

The construction of a block diagram for third-order nonlinearbehavior caused by third-order nonlinearity coefficients (seeFig. 8) proceeds in a similar way as the second-order behaviorcaused by second-order nonlinearities.

It should be noted that second-order nonlinearities also pro-duce a third-order nonlinear response by combining a second-order signal with a first-order signal at their controlling termi-nals to produce a third-order “nonlinear” current [see Fig. 9].The propagation of the third-order nonlinear current through thelinearized circuit yields the contribution of the second-order co-efficient to the overall third-order output signal. The total third-order kernel is then modeled as the parallel connection of thedifferent paths as the ones shown in Figs. 8 and 9. Clearly, blockdiagrams that describe third-order nonlinear behavior containonly scale factors, linear transfer functions, and static nonlin-earities, just as in the second-order case.

B. Simplification of the Block Diagram

Due to the large number of nonlinearities in a circuit of apractical size, the complete high-level models of VKTs arequite complex. Remember that the number of nonlinearitycoefficients roughly corresponds to the number of parallelpaths in a block diagram. The large number of nonlinearitycoefficients describing nonlinear dependencies of a MOS or abipolar junction transistor currents leads to the block diagramsthat consist of many parallel paths. The use of such bulkydiagrams for system-level simulations is not practical.

Fortunately, in most practical circuits many nonlinearity co-efficients give a negligible contribution to the overall second- orthird-order nonlinear behavior. This is exploited in an algorithmthat eliminates all negligible contributions up to a user-definable


error in a given frequency band. The algorithm is an extension tomultiple dimensions of the method described in [8] to determinethe most important contributions to the second- and third-orderharmonics.

A translation of the dominant contributions into a block di-agram yields the approximated model in the form of relativelysimple block diagram that comprises only a limited number ofstatic nonlinearities, scale factorsandtransfer functions.

It leads to compact high-level models in the frequency do-main and, moreover, knowledge of the dominant contributionsyields insight in the nonlinear circuit behavior. Because manyblocks are identical in different parallel paths, the approximatedhigh-level models can be further simplified in its structure by ap-plying various graph-rewriting rules. These graph operations re-sult in a simplified model with less complexity while preservingaccuracy of the original approximated one.

Finally, we will end up with high-level models containingonly a few linear blocks—transfer functions—described in thefrequency domain and static nonlinear blocks. Their accuracyis easily controlled and the simulation performance, in conjunc-tion with digitally modulated excitations, is superior in compar-ison with circuit-level simulations. The impact of described ap-proximations and simplifications rules on a model complexityand evaluation time will be illustrated in Section V-A dealingwith realistic example.

C. Multicarrier Low-Pass Model

The signals that we deal with in communication systems areoften carrier-modulated signals that are processed in bandpasssystems. In a simulation, these signals have to be sampled ata rate that is at least twice the highest frequency in the signalspectrum. In the case of RF analog blocks, this could cause highsimulation inefficiency, especially if the cosimulation with dig-ital parts is required. This can be overcome with the complexenvelope method [16], which uses complex low-pass equivalentsignals and systems. If a signal spectrum around a carrier at fre-quency is contained in the band ,then with a complex low-pass representation the sampling fre-quency only needs to be on the order of B.

The procedure of the nonlinear analysis and modeling de-scribed in the previous Sections IV-A and B leads directly tothe approximated bandpass high-level model in the frequencydomain. The complex envelope simulation techniques requirethe complex low-pass equivalent transformation of all systemblocks and signals.

Classical low-pass equivalent transformation approach dealsonly with a frequency band around the main carrier frequency.Hence, it only takes in-band distortion into account. The mod-eling out-band nonlinear distortions is necessary when each cir-cuit block is modeled individually. In contrast to the approach,where the front-end system is modeled as one, this approach al-lows the designers to play with the architectures of each circuitblock independently. However, it may lose certain system-levelsimulation efficiency since the resulting model becomes morecomplicated.

Highly integrated RF frond-end architectures are character-ized by cascade connections of active RF blocks without any in-termediate frequency-selective blocks often occurring. A prac-

tical signal often consists of different bandpass signals, e.g., themodulated carrier of interest and its (modulated) harmonics, oran interfering signal. Due to the nonlinear behavior of a subse-quent front-end block, these out-of-band signals can be foldedback inside the band of interest and cause in-band distortion. Itimplies that out-of-band distortion has to be taken into accountin a modeling procedure of each individual circuit block. Forexample, in a WLAN zero-IF receiver, both the- and -sec-tions each often contain a baseband variable-gain amplifier, fol-lowed by an active antialias filter and an analog-to-digital con-verter. The signals that are applied have a bandwidth of 10 MHz.Second-order nonlinear behavior of the variable-gain amplifiergenerates both in-band distortion (e.g., the part of the signal at1 MHz yields a signal at 2 MHz) and out-of-band distortion(e.g., the part of the spectrum around 5.5 MHz yields a signal at11 MHz). The in-band distortion is passed to the output of theanalog filter (since it is inside the passband) and the out-of-banddistortion at can intermodulate with any tonein the input signal between 1 and 10 MHz to produce a signal at

inside the passband of the filter.Another example of a nonlinear RF circuit where the

out-of-band distortion analysis can significantly improve accu-racy is a class A power amplifier. The nonlinear distortion onthe input signal makes that its second order nonlinear distortionchanges the operating point. This finally leads to third-ordernonlinear distortion. In order to simulate the changes in oper-ating point in an accurate way—by including the dynamics ofthe biasing networks—it is necessary to consider at least twocarriers (one around dc, one around). It shows that accuratedescription of the fundamental behavior of an amplifier needsmore that one carrier for a model generation.

In order to take into account out-of-band distortion withthe simulation efficiency of a complex low-pass representa-tion, we use this representation for signal’s frequency bandsaround a carrier and its harmonics. It is calledthe multicarrierlow-pass representation. This requires the translation of eachtransfer function to the several complex low-pass equivalentsin accordance with all possible bandpass signals in the place ofthe signal path of interest. This is illustrated in Fig. 10 for thecontribution of a 1-D nonlinearity to the third-order nonlinearbehavior of the complete circuit.

The meaning of descriptions used in Fig. 10(b) is the fol-lowing. The frequency in the left side of each transfer functionblock reflects the carrier frequency to which the particular com-plex low-pass equivalent model of the transfer function was de-rived. Also, frequency labels along signal paths reflects the car-rier frequency of a particular signal path.

If the resulting model is used in a digital dataflow simulator[17], [18], then, as a last step, the s-domain representations ofthe linear transfer functions are translated into digital filters.

V. EXAMPLE

The modeling approach has been implemented in a tool calledDISHARMONY. The first part, written in Perl, parses the orig-inal spice-like netlist of the circuit and generates modified netlistused for ac analysis to determine necessary transfer functions.The nonlinearity coefficients are computed by numerical differ-


Fig. 10. Contribution of the 1-D second-order coefficientK2 to thethird-order nonlinear behavior; (a) bandpass representation (b) complexlow-pass equivalent with multicarrier signal representation.

entiation on data from repetitive dc analyses or directly fromknown model equations of a device under inspection. Unfortu-nately, model equations usually describe a device behavior accu-rately up to the first-order (only guarantee continuous first-orderderivatives) and discontinuities in higher order characteristicscan occur on borders of operational regions. When an operatingpoint of a device lies in such a critical characteristic region, thenthe involved original model equations have to be replaced by ap-proximations that guarantee differentiability, not only for equa-tions themselves, but also for higher order derivatives up to thesecond-order.

Relevant transfer functions and nonlinearity coefficients, to-gether with a frequency limitation of an analysis inputted byuser, are processed by the second part of the tool that computescontributions of each nonlinear element to the complete VKTs.This part—implemented in C++—represents the most compu-tationally intensive task of the whole analysis and modeling pro-cedure.

Final postprocessing steps enable the users to specify accu-racy requirements for the approximated kernel by setting theerror criterion in the following way:

(16)

where is the maximal relative error at frequency, is the difference vector expressing the difference be-

tween the complete and the approximated VKT. It means

and are frequencies from the set of allfrequency components of an original multisine circuit’sexcitation. The condition (16) must hold for all possible

-combinations of , which satisfy the conditionand for all values of from frequency

interval in which the approximated model shouldbe valid. Based on this error specification, the approximatedhigh-level baseband model in the frequency domain can beconstructed.

A. Low-Noise Amplifier Design for a 5-GHz WLAN

The consecutive modeling steps are illustrated with the anal-ysis of a 5-GHz low-noise amplifier that is a part of a 5-GHzWLAN receiver front-end (see Fig. 11).

Fig. 11. 5-GHz WLAN receiverf-front-end described at the architecture level(the LNA block high-level specifications are indicated in more detail).

Fig. 12. 5-GHz low-noise amplifier. (a) The circuit scheme of the amplifier.(b) The scheme of the transistor subcircuitsX � � �.

The architecture description displayed in Fig. 11 represents ausual starting point for a mixed-signal design flow. The modelsfor this first high-level simulation are typically very rough. Forinstance, the LNA block is defined as a static nonlinearity, de-scribed by a polynomial of order three that relates output to theinput. The coefficients of this polynomial are related to the gainand intercept points [8].

In the process of the top-down path of the design flowexplained in the introduction section, we finally end up withanalog/RF circuit blocks circuit level descriptions expressed byfully specified circuit netlists. The corresponding result for theLNA block, is shown in Fig. 12.

The input of the low-noise amplifier is excited by digitallymodulated signal around the carrier frequency 5 GHz with abandwidth of 100 MHz. In the subcircuit , the intrinsicnpntransistor is the only active element as thenpntransistors ,

, the diode , and the passive elements model the parasiticstructures of the complete transistor subcircuits. In the LNAcircuit, the transistor of the subcircuit is the amplifyingtransistor, whereas the rest of the circuit provides the necessarybias.

In accordance with the goal of the design flow’s bottom-uppath (see Fig. 1), we can use the circuit descriptions of eachblock to derive a more accurate high-level model. These modelsshould cover the relevant signal degrading effects, while at thesame time they should not be too complex such that simula-tions are slowed down too much. This is especially a problem


Fig. 13. Magnitude of the second-order VKT and the important contributionsto it around dc and the second harmonic of the input carrier frequency.

for nonlinear behavior of analog blocks. The analysis and mod-eling approach described in this paper suggests the method forproducing behavioral bandpass or baseband models is usefulfor circuit or various high-level simulations. Resulting modelscover frequency-dependent weakly nonlinear behavior and, fur-thermore, yield insight in the linear and nonlinear operation ofthe circuits. The above claims are illustrated by the applicationof DISHARMONY with the initial circuit-level description ofthe LNA block shown in Fig. 12.

It is found by DISHARMONY that the largest contributionto the second-order nonlinear behavior originates from thesecond-order nonlinearity coefficients of the collector currentpower series expansion [see (1)].

The second-order VKT and the important contributions to itdisplayed in Fig. 13 are functions of two frequency variablesand , where each of them varies independently in the input ex-citation’s frequency band GHz GHz . The firstgraph corresponds to the intermodulation products expressed bythe combination of the two frequency arguments in the form( ), whereas the second graph corresponds to ( ).

Evaluation of the third-order VKT and its most importantcontributions leads again to the similar situation when only afew contributions are important. The graphic representation

TABLE IMOSTIMPORTANT CONTRIBUTIONS TO THE2ND- AND 3RD-ORDERVKT

of the third-order VKT is similar to the representation of thesecond-order one and includes again two different frequencyranges. The first of them—the band GHz GHzaround the first harmonic of the carrier frequency—reflectsthe intermodulation products corresponding to the frequencyarguments combination ( ) and the second—theband GHz GHz around the third harmonicof the carrier frequency—reflects the intermodulation productscorresponding to the combination ( ), wherefrequency variables , , and vary independently in theinput excitation’s frequency band GHz GHz .

Table I summarizes the list of the most important contribu-tions depending on the specified error criterion [see(16)].

Based on analysis results recorded in Table I, we can trade offbetween the accuracy and complexity of our high-level model.The column#MIC/#ACillustrates the number of necessary con-tributions for achieved accuracy in comparison with the overallnumber of contributions that participate on the nonlinear be-havior of the appropriate order.

Suppose we decided to prefer higher accuracy (bolded linesin Table I). To capture the second-order nonlinear behavior wehave to take into account only four contributions, all comingfrom nonlinear behavior of the intrinsic npn transistorof thetransistor macromodel . These four contributions—about 7%of the overall quantity of contributions—cover almost 98% ofthe second-order nonlinear behavior in the observed frequencyrange in terms of error criterion . For the third-order,again the most important contributions to the nonlinear behaviorcome from the transistor of the transistor macromodel .The eight most important contributions – about 6% of the overallquantity of contributions – cover about 96% of the nonlinear be-havior.

These results validate the previously stated assumption thatweakly nonlinear behavior is usually driven by only a few im-portant contributions. It facilitates to understand the sources ofnonlinear behavior of the circuit under analysis and makes it fea-sible to construct a compact and accurate high-level model. Thebandpass model of the LNA in the frequency domain is shownin Fig. 14.


Fig. 14. High-level model of the LNA; (a) the second-order nonlinear behaviorblock (b) the third-order nonlinear behavior block.

TABLE IIQUANTITATIVE SUMMARY OF DIFFERENTBLOCKS IN THE HIGH-LEVEL MODEL

Notice that the model for the third-order behavior comprisesalso blocks designated and that repre-sent the second-order VKT of the voltages over the appropriatecontrolling ports. It illustrates the way in which the second-order nonlinearity coefficients contribute to the third-order be-havior. The high-level model consists only of static nonlinearblocks, transfer function blocks, and scale factors. Structure ofthe model with respect to the number of different kinds of blocksis recapitulated in Table II.

Regardless of the used simulation algorithm, the simulationefficiency of the bandpass model is mainly affected by thenumber of static nonlinear blocks and the number of sinusoidalcomponents in the original excitation. Fig. 15 demonstratesdependency of the simulation time on the number of sinusoidalcomponents in the input excitation. Multitone harmonic bal-ance of APLAC circuit simulator [19] was used for both kindsimulations. With the circuit-level model, the simulation wasnot feasible for an input excitation frequency band coveredmore than 25 sinusoidal components, whereas the simulationwith the high-level model ran through in significantly short timeand had no limitation in the number of frequency components

Fig. 15. Comparison of the simulation efficiency with the circuit and thehigh-level model depending on the number of sinusoidal components coveringthe frequency band of the input excitation.

Fig. 16. Multicarrier complex low-pass equivalent model of the second-ordernonlinear behavior of the LNA.

covering an input excitation frequency band for practical cases.The accuracy comparison between the circuit-level simulationresults and simulation results with the approximated modelproves the accomplishment of the required error limits.

The multicarrier complex low-pass model of the LNAcould be derived from the high-level model by the techniquedescribed in the Section IV-C. Fig. 16 illustrates the resultingmodel for the second-order nonlinear behavior. It is easy to seethat in contrast to the classical low-pass equivalent approach,the out-of-band intermodulation products produced by non-linear operations on the basic nonlinear elements are preservedduring the multicarrier low-pass transformation.

For the third-order case, the block diagram is getting evenmore complicated and its explicit displaying probably does notbring any useful benefit. A better idea about the model com-plexity and its simulation performance provides Table III.

It summarizes the overall number of different blocks andtheir computational intensity during simulation in the time-do-main dataflow simulation environment. The detail informationabout computational intensity in Table III suppose that blocksof transfer functions were transformed into their digital equiv-


TABLE IIISTATISTIC FOR THEMULTI-CARRIER COMPLEX LOW-PASSMODEL OF THELNA

alents using four-tap FIR filters. This relatively small numberof taps used for digital filters representation was enabled bysmoothness of involved transfer functions in the frequencyband of interest.

The sum of all values at the last row of Table III shows thatthe processing of one sample requires 1296 flops. It representsabout 7.2 s of CPU time using an optimized dataflow simu-lator [17], which yields 180 Mflops on a Pentium III/500 MHzcomputer. While feasibility of a prediction of BER is assured byreplacing circuit-level models of all building blocks in a simu-lated RF chain by high-level models, the efficiency and accuracydepend on the complexity of used high-level models. The multi-carrier low-pass equivalent models guarantee enough flexibilityto perform comprehensible tradeoff between accuracy and effi-ciency.

VI. CONCLUSION

Modeling and understanding nonlinear behavior of analog in-tegrated communication circuits is not straightforward, since ina practical circuit many nonlinearities contribute to the overallnonlinear behavior. Moreover, these contributions depend onfrequency. This paper has described an approach that determinesthe most important contributions to second- and third-order non-linear behavior of the circuit including frequency dependency.A translation of these dominant contributions into a block di-agram yields high-level models that are useful for high-levelsimulations of front-end architectures. The models contain asmall number of linear transfer functions, static nonlinearities,and scale factors. In this way, the approach described in thispaper is useful to construct bottom-up models that yield efficienthigh-level simulations and provides insight in the nonlinear op-eration. A correct prediction of the BER of wireless transceiversis feasible since efficient cosimulation of analog and digitalparts can be obtained through the translation of the high-levelmodels from analog-domain to the digital domain. The gener-ated multicarrier low-pass equivalent model is suitable for usein dataflow simulation environment. It features higher simula-tion accuracy than classical complex low-pass models, sinceout-of-band distortion is taken into account. The simulation ef-

ficiency on the other hand is comparable to the use of classicalcomplex low-pass models.

REFERENCES

[1] V. Volterra, Theory of Functional and of Integral and Integro-Differen-tial Equations. New York: Dover, 1959.

[2] D. D. Weiner and J. F. Spina,Sinusoidal Analysis and Modeling ofWeakly Nonlinear Circuits. New York: Van Nostrand, 1980.

[3] P. Wambacq, G. Vandersteen, P. Dobrovolný, M. Goffioul, W. Eberle, M.Badaroglu, and S. Donnay, “High-level simulation and modeling toolsfor mixed-signal front-ends of wireless systems,” inProc. Advances inAnalog Circuit Design, Mar. 2002.

[4] D. Verkest, W. Eberle, P. Schaumont, B. Gyselinckx, and S. Vernalde,“C++ based system design of a 72 Mb/s OFDM transceiver forWLAN,” in Proc. IEEE Custom Integrated Circuits Conf., May 2001,pp. 433–439.

[5] G. Van der Plas, G. Debyser, F. Leyn, K. Lampaert, J. Vandenbussche, G.Gielen, W. Sansen, P. Veselinovic, and D. Leenaerts, “AMGIE : A syn-thesis environment for CMOS analog integrated circuits,”IEEE Trans.Computer-Aided Design, vol. 20, pp. 1037–1058, Sept. 2001.

[6] K. Lampaert, G. Gielen, and W. Sansen,Analog Layout Generationfor Performance and Manufacturability. Dordrecht, The Netherlands:Kluwer, 1999.

[7] M. Schetzen,The Volterra and Wiener Theories of Nonlinear Sys-tems. New York: Wiley, 1980.

[8] P. Wambacq and W. Sansen,Distortion Analysis of Analog IntegratedCircuits. Dordrecht, The Netherlands: Kluwer, 1998.

[9] I. Vassiliou and A. S. Vincentelli, “A frequency-domain, volterra series-based behavioral simulation tool for RF systems,” inProc. IEEE CustomIntegrated Circuits Conf., May 1999, pp. 21–24.

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[12] P. Wambacq, P. Dobrovolný, S. Donnay, M. Engels, and I. Bolsens,“Compact modeling of nonlinear distortion in analog communicationcircuits ,” in Proc. Design, Automation Test Eur., Mar. 2000, pp.350–354.

[13] P. Dobrovolný, P. Wambacq, G. Vandersteen, S. Donnay, M. Engels, andI. Bolsens, “Generation of multicarrier complex lowpass models of RFIC’s,” in Proc. IEEE MTT-S Int. Microwave Symp., vol. 1, May 2001,pp. 419–422.

[14] B. Razavi,RF Microelectronics. Englewood Cliffs, NJ: Prentice–Hall,1998.

[15] S. A. Maas,Nonlinear Microwave Circuits. Norwood, MA: ArtechHouse, 1988.

[16] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan,Simulation of Com-munication Systems. New York: Plenum, 1992.

[17] G. Vandersteen, P. Wambacq, Y. Rolain, P. Dobrovolný, S. Donnay, M.Engels, and I. Bolsens, “A methodology for efficient high-level dataflowsimulation of mixed-signal front-ends of digital telecom transceivers,”in Proc. 37th Design Automation Conf., June 2000, pp. 440–445.

[18] The Ptolemy Project–Modeling, Simulation, and Design of Con-current, Real-Time, Embedded Systems [Online]. Available:http://ptolemy.eecs.berkeley.edu/

[19] APLAC Solutions Corp. [Online]. Available: http://www.aplac.com/

Petr Dobrovolný (A’01–M’01) received the M.S.and Ph.D. degrees in microelectronics from theTechnical University Brno, Brno, Czech Republic,in 1987 and 1998, respectively.

From 1989 to 1998, he was a Research Assistant inthe Microelectronics Department, Technical Univer-sity Brno. Since 1999, he has been with the Interuni-versity MicroElectronics Center, Leuven, Belgium,working as a Senior Researcher in the Mixed-Signaland RF Applications Group (MIRA). His current re-search interests comprise modeling and simulation of

nonlinear behavior of RF front-ends and modeling and simulation of substratenoise coupling in mixed-signal ICs.


Gerd Vandersteen(A’00–M’01) received the elec-trical engineering and Ph.D. degrees from the FreeUniversity of Brussels (VUB), Brussels, Belgium, in1991 and 1997, respectively.

He is currently a Principle Scientist in theMixed-Signal Design Group, IMEC/DESICS,Leuven, Belgium. His main interest is in the field ofmodeling, measurement, and simulation of nonlinearmicrowave devices.

Piet Wambacq (S’89–M’91) received the M.S.degree in electrical and mechanical engineering andthe Ph.D. degree from the Katholieke UniversiteitLeuven, Leuven, Belgium, in 1986 and 1996,respectively.

From 1986 to 1996, he was a Research Assistantin the ESAT-MICAS Laboratory, Katholieke Univer-siteit Leuven. Since 1996, he has been with Interuni-versity MicroElectronics Center, Leuven, Belgium,working as a Principal Scientist on design method-ologies for mixed-signal and RF ICs. He is also a

Lecturer at the University of Brussels (Vrije Universiteit Brussel), Brussels, Bel-gium. His research interests are design and CAD of mixed-signal and RF ICs.He has authored or coauthored two books and more than 70 papers in editedbooks, international journals, and conference proceedings. He is the coinventorof two patents.

Dr. Wambacq regularly is a member of the program committees of inter-national conferences (e.g., DATE). He is an Associate Editor of the IEEETRANSACTIONS ONCIRCUITS AND SYSTEMS—PART 1: FUNDAMENTAL THEORY

AND APPLICATIONS.

Stéphane Donnay(S’88–M’90) received the M.S.and Ph.D. degrees in electrical engineering from theKatholieke Universiteit Leuven, Leuven, Belgium, in1990 and 1998, respectively.

He was a Research Assistant in the ESAT-MICASLaboratory of the Katholieke Universiteit Leuvenfrom 1990 to 1996, where he worked in the fieldof analog and RF modeling and design automation.In 1997, he joined Interuniversity MicroElectronicsCenter, Leuven, Belgium, where he is now re-sponsible for the mixed-signal and RF group. His

current research interests include integration of RF front-ends for digitaltelecommunication applications, in particular, 5 GHz-WLAN front-ends,chip-package codesign, modeling and simulation of substrate noise coupling inmixed-signal ICs, and modeling and simulation of RF front-ends.

Analysis and compact behavioral modeling of nonlinear distortion in analog communication circuits

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