A Hybrid Approach to Nonlinear Macromodel Generation for Time-Varying Analog Circuits†

Peng Li, Xin Li, Yang Xu and Lawrence T. Pileggi

ECE Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA

ABSTRACTModeling frequency-dependent nonlinear characteristics of com-

plex analog blocks and subsystems is critical for enabling efficientverification of mixed-signal system designs. Recent progress has

reduction of large linear IC interconnects in the past decade (e.g.[4]-[7]). The recent research effort in this direction was to extendprojection-based model order reduction techniques for LTI (lineartime invariant) systems to accommodate nonlinear system reduc-

been made for constructing such macromodels, however, their accu-racy and/or efficiency can break down for certain problems, partic-ularly those with high-Q filtering. In this paper we explore a novelhybrid approach for generating accurate analog macromodels fortime-varying weakly nonlinear circuits. The combined benefits ofnonlinear Padé approximations and pruning by exploitation of thesystem’s internal structure allows us to construct nonlinear circuitmodels that are accurate for wide input frequency ranges, and there-by capable of modeling systems with sharp frequency selectivity.Such components are widely encountered in analog signal process-ing and RF applications. The efficacy of the proposed approach isdemonstrated by the modeling of large time-varying nonlinear cir-cuits that are commonly found in these application areas.

1. INTRODUCTIONBuilding compact analog circuit macromodels is a key for

enabling complete system verification and high-level design explo-ration for mixed-signal system design [3]. Various analog macro-models can be categorized according to their intended application (agood taxonomy can be found in [17]). Performance models such asthose in [17] and [18] are mainly employed in synthesis for designspace exploration; While other research work has focused on mod-els with a primary application of simulation and system verification[8]-[10][12]-[16]. It should be noted, however, that the latter cate-gory is not excluded from use in design exploration when appropri-ate.

This paper addresses the modeling problems that largely fall intothe second category. To this end, two requirements exist for anysuch circuit model: efficiency and accuracy. The latter is often mea-sured by certain performance metrics such as gain, bandwidth andlinearity, etc. It is worth emphasizing that modeling system distor-tion is crucial for most analog applications, since circuit linearity isone of the most important design specifications. This casts the mod-eling task into a nonlinear model generation or nonlinear modelreduction problem, both of which are substantially more challengingthan their linear time-invariant model counterparts[8]-[10][12]-[16].

Several nonlinear model generation approaches have beenrecently reported, and can be largely categorized into two majorclasses: pruning-based and transformation-based. The first type ofapproaches work directly on the original circuit structure, and applypruning-based simplification for model generation. In particularly,symbolic modeling of weakly nonlinear circuits has been used tobuild system-level models [14]-[16]. The underlying mathematicaldescription employed is Volterra series [1]-[2], and the resultingmodel is in a block-diagram form suitable for signal-flow type sim-ulation using tools such as Matlab Simulink [19].

The second class of methods, categorized as transformation-based, have been stimulated by the progress made for model order†

This work was funded by the Semiconductor Research Corporation under con-tract 2000-TJ-779, and by the MARCO Center for Circuits, Systems and Soft-ware, under contract 2001-CT-888 and DARPA grant MDA972-02-1-0004.

45

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tion. The piecewise-linear approximation approach in [12] modelsa nonlinear system using a collection of weighted linear modelsbased upon a state trajectory generated by a training input. Theneach of these linear models are reduced using projection-basedtechniques for LTI systems [5]-[7]. This approach has the potentialcapability of handling large nonlinearities, but limited by the train-ing-input dependency. This piecewise-linear approximation wasrecently extended to a piecewise-polynomial approach to bettermodel nonlinear distortion [24]. Another approach that does notcompletely fall into either of the above two categories uses modeltemplates for building system-level simulation models [23].

For a broad class of weakly nonlinear systems in analog signalprocessing and RF communication ICs, the low distortion level, aswell as the required modeling accuracy, seemingly makes Volterraseries a more suitable choice for system description. The work of[8]-[10] extends the projection-based LTI techniques to weaklynonlinear systems that are described by Volterra series, therebyfacilitating a direct transformation-based macromodeling.Recently, several improvements to this nonlinear model reductionframework were proposed in [13]. Based on a careful momentanalysis under a general nonlinear system context, the nonlinearmodel order reduction algorithm NORM [13] uses a minimum setof Krylov subspaces for projection, leading to a significant reduc-tion of model size.

As we move towards modeling larger analog circuit buildingblocks and subsystems, it becomes increasingly beneficial toexploit the model order reduction techniques that were so success-ful for LTI systems. This is particularly the case when various fil-tering blocks, along with the parasitics, are included as part of thesystem model. However, extending LTI based techniques for ana-log model generation is nontrivial, as the nonlinear aspects of theproblem greatly complicates the situation. For the reduction frame-work of [8]-[10][13], one difficulty is that the reduced high-ordermodel matrices are extremely dense, and increasingly more expen-sive to form as the model size grows. This difficulty makes controlof the model size growth a critical issue for an effective modelreduction [13]. Thus, a full projection-based approach is moreappropriate when a low-order approximation is adequate.

In this paper, we propose a hybrid approach for generating non-linear analog macromodels that can efficiently capture weaklynonlinear effects up to third order. The low-order system nonlinearresponses are captured efficiently using recently developed projec-tion-based NORM algorithm of [13], while high-order responsesare approximated via direct matrix pruning, in conjunction withthe projection-based reduction of a linear adjoint network. Thepruning technique is similar to what was used in symbolic model-ing [14]-[16], with the difference being that the pruning is applieddirectly to the high-order system matrices for our approach, andthe model order reduction is embedded with the pruning to accel-erate this relatively slow procedure. The overall model generationcost is bounded by a time complexity of ,where is the number of the nonzeros in the second and third

O sE E( ) kN3+log( )E

4

order system matrices, is the problem dimension, is numberof matrix factorizations for building projection-based reducedmodels, and is total number of frequency samples evaluated dur-ing the pruning.

By exploiting projection-based model order reduction and theinternal circuit structure simultaneously, this hybrid approachallows us to model system nonlinear distortion over a wide rangeof input frequency of interest, thereby extending the applicabilityof nonlinear model generation. We demonstrate the efficacy of ourapproach on several large time-varying circuit examples, wherefrequency-domain nonlinear characteristics vary dramatically dueto active or high Q filtering.

2. BACKGROUND

2.1 Nonlinear System Modeling Via Volterra SeriesVolterra series has been widely used to characterize weakly non-

linear systems [1]-[2]. It is usually applicable to cases where the in-put amplitudes to the system are small and nonlinearities are excitedin a weakly nonlinear fashion. For a circuit with input , the re-sponse (circuit unknowns) can be expressed using the expan-sion

, (1)

where is the nth order response, which is related to the inputvia convolution in time domain:

. (2)

In the above equation, is the nth order Volterra kernel, andcan be thought as an extension to the impulse response function ofa linear system. (2) can also be cast into the frequency domain,where the Laplace transform of the nth order Volterra kernel

, or the nth order nonlinear transfer function, isused.

Before using Volterra series for analysis, consider the followingordinary differential-algebraic equations for a nonlinear circuit

, (3)

where and represent resistive and dynamic nonlineari-ties. To apply Volterra series, first expand circuit nonlinearitiesabout a dc bias (this can be extended to a varying operating pointfor time-varying systems) into Taylor series. Correspondingly, wedefine

. (4)

It follows that the first through third order nonlinear small-signal re-sponses, based upon , are computed recursively as

, (5)

where is the small-signal input to the circuit.

2.2 Nonlinear Model Order Reduction Equation (5) can be immediately interpreted as a set of linear

systems with different inputs. Following the work in [8][10], thisnonlinear system model can be reduced using an extension to pro-

jection-based LTI reduction techniques (e.g. [5]-[7]). The nonlin-ear model reduction was achieved by successively reducing theselinear systems with increasing number of inputs, for approximatingthe first order response , second order response and thirdorder response , respectively. However, the efficacy of modelorder reduction eventually suffers from the rapid increase in thenumber of inputs to the successive linear networks. In [9], a wellstructured projection-based moment matching scheme was devel-oped based on a theoretically interesting bilinear form of weaklynonlinear systems. However, a bilinear form of a nonlinear systemhas significantly more state variables than the original state-equa-tion representation. Therefore, the use of a bilinear form leads to areduction problem with a much larger size.

In [13], a nonlinear model order reduction algorithm NORM wasproposed to improve the reduced model compactness under the pro-jection-based framework by carefully examining the problem undera general nonlinear context. For the expanded MNA formulation of(5), this was accomplished by first deriving a general matrix-formnonlinear transfer function and the associated mo-ments, and then applying a minimum set of Krylov-subspace pro-jection vectors to match the moments of nonlinear transferfunctions. A complete description of the NORM algorithm is out-side of the scope of this paper. To provide sufficient background tounderstand the hybrid approach proposed in this paper, we brieflydiscuss the algorithm and limit it to single-input multi-output (SI-MO) time-invariant systems.

The first order linear transfer function of (5) is given by . (6)

Due to the recursive nature of (5), the symmetrized second ordernonlinear transfer function is determined by

, (7)

where ,. It can be shown that the symmetrized third order trans-

fer function can be computed using

(8)

where , is the arith-metic average of terms over all possible permutations of frequencyvariables in the Kronecker product, and

(9)

To perform a projection-based nonlinear Padé approximation,moments of the corresponding nonlinear transfer functions are re-quired. For this purpose, (6)-(8) are expanded into a power seriesw.r.t frequency variables at the origin

, (10)

where , and are a kth order moment of , and , respectively. We define a reduced or-

der model as a kth order model in , or if and only if the respective moments are preserved in

the reduced model up to the kth order. In NORM, to match a specificnumber of moments, a set of Krylov subspace vectors with a mini-mum total dimension is used in projection-based reduction for im-proving the model compactness. For instance, if we define

N k

s

u t( )x t( )

x t( ) xn t( )n 1=

∞

∑=

xn t( )

xn t( ) … hn τ1 … τn, ,( )u t τ1–( )…u t τn–( ) τ1d … τnd∞–

∞

∫∞–

∞

∫=

hn ·( )

Hn s1 s2 … sn, , ,( )

f x t( )( )td

d q x t )( )(+ bu t( )=

f ·( ) q ·( )

x0

Gi1i!---

xi

i

∂∂ f

x x0=Rn n

i×∈ Ci1i!---

xi

i

∂∂ q

x x0=

Rn ni×∈=,=

x0

tdd C1x1( ) G1x1+ bus t( )=

tdd C1x2( ) G1x2+

tdd– C2 x1 x1⊗( )( ) G2 x1 x1⊗( )–=

tdd C1x3( ) G1x3+ G2 x1 x2 x2 x1⊗+⊗( ) G3– x1 x1 x3⊗ ⊗( )–=

tdd C2 x1 x2 x2 x1⊗+⊗( ) C3 x1 x1 x3⊗ ⊗( )+( )–

us

x1 x2x3

Hn s1 s2 …sn, ,( )

G1 sC1+( )H1 s( ) b=

G1 sC1+[ ] H2 s1 s2,( ) G2 sC2+[ ] H1 s1( ) H1 s2( )⊗⋅–=

H1 s1( ) H1 s2( )⊗ 12--- H1 s1( ) H1 s2( ) H1 s2( ) H1 s1( )⊗+⊗( )=

s s1 s2+=

G1 sC1+[ ] H3 s1 s2 s3, ,( ) G3 sC3+[ ] ⋅–=

H1 s1( ) H1 s2( ) H1 s3( )⊗⊗ 2 G2 sC2+[ ] H1 s1( ) H2 s2 s3,( )⊗⋅ ⋅–

s s1 s2 s3+ += H1 s1( ) H1 s2( ) H1 s3( )⊗⊗

H1 s1( ) H⊗ 2 s2 s3,( ) 16--- H1 s1( ) H⊗ 2 s2 s3,( ) H1 s2( ) H⊗ 2 s1 s3,( )+(=

H+ 1 s3( ) H⊗ 2 s1 s2,( ) H2 s1 s2,( ) H1 s3( )⊗+

H+ 2 s1 s3,( ) H1 s2( ) H2 s2 s3,( ) H1 s1( )⊗+⊗ )

H1 s( ) skM1 k, H2 s1 s2,( ) s1l s2

k l– M2 k l, ,

l 0=

k

∑k 0=

∞

∑=,k 0=

∞

∑=

H3 s1 s2 s3, ,( ) s1l s2

m

m 0=

k l–

∑ s3k l– m– M3 k l m, , ,

l 0=

k

∑k 0=

∞

∑=

M1 k, M2 k l, , M3 k l m, , , H1 s( )H2 s1 s2,( ) H3 s1 s2 s3, ,( )

H1 s( ) H2 s1 s2,( )H3 s1 s2 s3, ,( )

455

, ,it can be shown that the following Krylov subspaces are the re-quired minimum set containing up to the kth order moment direc-tions of :

, (11)

where

(12)

A minimum set of Krylov subspace vectors can also be found formoment matching of the third order transfer function . If wecombine Krylov subspaces for moment matching of the firstthrough the third order nonlinear transfer functions and compute aorthonormal basis V for the combined subspaces, the reduced ordermodel can be computed via a projection of the form:

(13)

As seen from (13), the dense third order reduced system matricesare of dimension , where is the number of columns for pro-jection V, which represents the size of the reduced order model.Thus, controlling model size by a proper selection of projectionvectors is critical, as addressed in NORM:

• The use of a minimum set of Krylov subspace vectors avoidsunnecessary inflation of the model size, thereby leading to a sig-nificant improvement in model compactness. Furthermore, it isshown that the order of moment matching for a transfer functionat a higher order cannot exceed that of a transfer function of alower order. In NORM, the order of moment matching for thenonlinear transfer function at each order can be chosen individ-ually, provided that the above condition is satisfied, and the sizeof overall reduced order model is minimized for each of thesechoices.

• Unlike the counterpart of LTI system reduction, adopting amulti-point method can improve the model compactness undernonlinear context as recognized in NORM. This stems from thefact that the total dimension of optimal Krylov subspaces usedin a single-point method actually exceeds the number ofmoments matched in the resulting reduced order model. How-ever, the use of a multi-point method, in particular, a zerothorder multi-point NORM, where high order nonlinear transferfunction values (zeroth order moments) are matched simulta-neously at several expansion points, can produce a reducedorder model with a size equal to the number of momentsmatched. That means multi-point methods can further increasethe model compactness that is essential for nonlinear systemreduction.

To compare different methods, we define the size of a reduced or-der model as the number of state variables that it contains. Theworst-case model sizes for the system in (5) are compared inTable 1, where reduced order models match the moment of or

and up to certain order. NORM-SP is the single-point ver-sion of NORM, and NORM-MP is the “equivalent” zeroth ordermulti-point NORM in which the same number of transfer functionmoments are matched. It is evident that the use of NORM, especial-

ly multi-point NORM, dramatically reduces the model size.

2.3 Modeling of Time-Varying Weakly NonlinearSystems

A large class of networks, such as clocked switching networksand RF current switching mixers, can be characterized as time-varying weakly nonlinear systems with respect to the small inputsignal being processed. This would require us to treat the large ex-citations to the system, such as clock and LO, as part of the systemunder modeling, and analyze the small-signal input upon the time-varying operating point caused by the large excitations. One specialcase of practical interest is that the system has a periodically time-varying operating point introduced by a periodic clock or LO.

Assume that the large-signal periodic excitation to the system hasa period , then the weakly nonlinear system re-sponse to the small-signal input can be characterized using period-ically time-varying Volterra series. Now the nonlinear transferfunction becomes a periodic function of time withthe same period , and can be expanded into Fourier series

. (14)

Using (14), the nth order response is in the form

(15)

It can be seen from the above equation that the kth harmonic of specifies the nth order nonlinear effect that is frequencytranslated by due to the time variation of the system.

By extending the formulation for linear periodically time-vary-ing systems in [20][11][8] via either frequency-domain collocationor time-domain backward difference, the periodically time-varyingnonlinear transfer functions can be formulated in a matrix form sim-ilar to that of (6)-(8). In this paper, a time-domain backward Euleris employed to discretize over sample points

within a period of. Analogous to the use of in (6)-(8), a

set of matrices can be formulated basedon discretization of the system nonlinear characteristics over a pe-riod of for time-varying systems. Based on these matrices, (6)-(8) can be reformulated in terms of . Each of these equa-tions now has unknowns, where is the number of physicalcircuit unknowns of the system. To accurately capture the time-varying operating point due to the large-signal excitation, manysample points are often required, leading to a set of large time-vary-ing system equations. These equations must be reduced to be usedefficiently in simulation. In the remainder of this paper we focus onthe time-varying systems as the results for time-invariant systems

A G– 11– C1 r1, G 1– b= = Km A p,( ) colspan p Ap … Am 1– p, ,, =

H2

K2 k( ) KmiA vi,( )

i∪=

vi

G11– G2 Amr1( ) Anr1( ) Anr1( ) Amr1( )⊗+⊗( )m∀ 0≥ n∀ 0 m n k≤+,≥, m n with mi k m– n– 1+=,≥,

G11– C2 Amr1( ) Anr1( ) Anr1( ) Amr1( )⊗+⊗( )m∀ 0≥ n∀ 0 m n k 1–≤+,≥, m n with mi k m– n–=,≥,

=

H3

G1˜ VTG1V C1

˜ VTC1V G2˜ VTG2 V V⊗( ) C2

˜ VTC2 V V⊗( )=,=,=,=

G3˜ VTG3 V V V⊗ ⊗( ) C3

˜ VTC3 V V V⊗ ⊗( ) b VTb=,=,=

q q3× q

H2H2 H3

Model Order 3 4 5 6 7 8[8][10]:H2 116 230 402 644 968 1386

[8][10]:H2-H3 3,700 11,480 28,914 63,070 123,848 224,460NORM-SP:H2 24 40 62 91 128 174

NORM-SP:H2-H3 66 118 194 301 446 636NORM-MP:H2 16 23 30 39 48 59

NORM-MP:H2-H3 37 56 78 108 141 182

Table 1. Worst-case reduced order model size

T0 2π( ) ω0⁄=

Hn t s1 …sn, ,( )T0

Hn t ω1 …ωn, ,( ) Hn k, ω1 …ωn,( ) ejkω0t

⋅k ∞–=

∞

∑=

xn t( ) 12π------ n

… Hn t ω, 1 …ωn,( ) U ω1( ) … U ωn( )⋅ ⋅⋅∞–

∞

∫∞–

∞

∫=

e⋅j ω1 … ωn+ +( )t

ω1…d ωnd

12π------ n

ejkω0 t

… Hn k, ω1 …ωn,( )∞–

∞

∫∞–

∞

∫⋅k ∞–=

∞

∑⋅=

U ω1( ) … U ωn( ) ej ω1 … ωn+ +( )t

ω1…d ωnd⋅ ⋅⋅ ⋅⋅Hn k,

Hn t ·,( )kω0

Hn t ·,( ) MHn Hn t1 ·,( )T Hn t2 ·,( )T … Hn tM ·,( )T, , ,[ ] T=T0 G1 C1 b1 G2 C2 G3 C3, , , , , ,

G1 C1 b1 G2 C2 G3 C3, , , , , ,

T0H1 H2 H3, ,

M N× N

456

can be easily deduced from this more general description.

3. HYBRID APPROACH

3.1 An Illustrative ExampleTo motivate the need for a hybrid approach, consider the simple

nonlinear circuit shown in Fig. 1(a). This circuit has only two non-linearities, resistors and . Each of them is modeled as a thirdorder polynomial .The small-signal currentsource drives a high Q resonator (tuned at 1GHz), which is followedby 500 RC segments. A cross-section of the third order nonlineartransfer function at the output is plotted in Fig. 1(b). It is evidentfrom Fig. 1(b) that when coupled with high Q resonator (part of lin-ear network), the two nonlinearities create a sharp transition for thethird order nonlinear transfer function within a neighborhood of theresonant frequency. To capture accurately the third-order nonlineareffects around the resonant frequency using a projection-based im-plicit moment-matching, many projection vectors may be required.In the reduced model, although the number of states can be signifi-cantly reduced, a large and dense reduced third order matrix may result( nonzeros, is the number of states in the reducedmodel). This is in contrast to the original third order matrix thatis of much higher dimension but only has 2 nonzeros.

Projection-based nonlinear model reduction is very effective if alow-order approximation is sufficiently accurate. In such cases, thereduced order model can also be constructed rather efficiently (thedominate cost is no more than a few matrix factorizations). Howev-er, for the cases where either a good accuracy is required over awide range of frequency or the nonlinear frequency-domain charac-teristics varies dramatically within a band of interest, a completeprojection-based approach might be difficult to apply. For practicalexamples, the primary reason for this difficulty is that projectinghigh order system matrices (e.g. ) becomes less and less efficientas the dimension of the Krylov subspaces increases.

3.2 Hybrid ApproachAs clearly seen from Table 1, if we use multi-point NORM to

only approximate the first and second order nonlinear transferfunctions, the size of the reduced order model is much less con-strained. More specifically, as the third order nonlinear transferfunctions are not considered in the projection-based reduction, thereduced model size now grows much slower as the order ofmoment-matching increases. For the same reason, forming thereduced third order system matrices is not an issue any more, nei-

ther is it a limiting factor for the model size. Moreover, the adop-tion of the projection-based approach allows us to efficientlymodel the first and second order system properties, especially forlarge circuits with detailed parasitics. From a pruning-based mod-eling perspective, although numerous nonlinearities can exist in acircuit, there is often a natural tendency for only a few of them tobe dominant due to the specific circuit structure. Thus, identifyingthese dominant nonlinearities under the original circuit structurecan be a very useful component of model generation. In the pro-posed hybrid approach, the low-order (first and second) responsesare matched using projection while high-order (third) response areapproximated by exploiting both the circuit internal structure andprojection-based reduction.

3.2.1 Approximation of low-order responsesWe begin with application of multi-point NORM to approximate

the first and second nonlinear transfer functions. We denote the cor-responding projection matrix as , where is the num-ber of physical circuit unknowns, is number of time-domain samples to discretize the time-varying nonlinear transferfunctions (assume an odd number of samples for simplicity), and is the size of reduced order model. We further denote the reducedfirst and second order matrices produced by NORM as

. (16)

Substituting (16) into (6)-(7), the first and second order transferfunctions and of the reduced model can be computed. Thetime-sampled first and second order transfer functions of the origi-nal system can be approximated as

. (17)

As the reduced model of (16)-(17) represents a time-invariant sys-tem, the approximation of the first and second order responses ofthe original time-varying system involves the use of discrete Fouri-er transform. It can be shown that the following th order systemis a time-domain realization of the corresponding reduced model

, (18)

where , is the DFT matrix converting time sam-ples to the corresponding Fourier coefficients,

, is the

identity matrix.The first and second order responses of the original

system can be approximated in time-domain by .

3.2.2 Approximation of the third order responseTo approximate the third order transfer function based on the re-

duced model of (16)-(17) or (18), substitute (16) into (8)

, (19)

where

(20)

As seen from the above equations, to compute the third order trans-

R1 R2i a1v a2v2 a3v3+ +=

G3˜

q4 qG3

G3

... ...

i(t)R2R1

+

-Vout

High Q Resonator

(a)

(b)Fig. 1. (a) A simple nonlinear circuit, and (b) its third order

nonlinear transfer function

0.90.95

11.05

1.1

x 109

0.90.95

11.05

1.1

x 109

−80

−60

−40

−20

0

20

Frequency (Hz)Frequency (Hz)

H3

(1/V

2 dB

)

V RNM q×∈ NM 2K 1–=

q

G1˜ VTG1V Rq q×∈ C1

˜ VTC1V Rq q×∈ b VTb Rq∈=,=,=

G2˜ VTG2 V V⊗( ) Rq q

2×∈ C2˜ VTC2 V V⊗( ) Rq q

2×∈=,=

H1˜ H2

˜

H1ˆ VH1

˜ H2ˆ VH2

˜=,=

q

tdd C1

˜ x1˜( ) G1

˜ x1˜+ bus t( )=

tdd C1

˜ x2˜( ) G1

˜ x2˜+

tdd– C2

˜ x1˜ x1

˜⊗( )( ) G2˜ x1

˜ x1˜⊗( )–=

y t( ) y1˜

y2˜

J 00 J

x1˜

x2˜

= =

J D Γ V⋅ ⋅= Γ M

D ejKω0t–

IN N× … IN N× … ejKω0 t

IN N×, , ,,[ ]= IN N× N N×

x1 y1 x2, y2= =

G1 sC1+[ ] H3ˆ s1 s2 s3, ,( ) g=

g G3 sC3+[ ] u1 G2 sC2+[ ] u2⋅+⋅=

u1 VH1˜ s1( )( ) VH1

˜ s2( )( ) VH1˜ s3( )( )⊗ ⊗–=

u2 2 VH1˜ s1( )( ) VH2

˜ s2 s3,( )( )⊗–=

457

fer function, one needs to solve a linear system in terms of and, and form the input to the linear system in terms of that is a

function of high-order system matrices and low-or-der transfer functions . As are approximated using(17), to further reduce (19) requires reduction of the dimension ofthe linear problem in (19) as well as the high-order system matrices

.This goal is accomplished by a combination of pro-jection-based reduction of an adjoint network and direct matrixpruning.

In analog signal processing and RF applications, a circuit blockusually has only one or at most a few output nodes. For time-vary-ing systems, typically only a small number of sidebands (w.r.t.clock or LO) for the outputs are of interest. Thus, (19) can beviewed as a system with potentially many inputs but few outputs,and can be reduced as an adjoint network. Without loss of general-ity, let us assume only the ith sideband of the voltage response atnode p is of interest (e.g, ). Define as a vector that has a 1 at the p-th location and zeros at allother locations, and

. (21)

The ith sideband of the third order transfer function at node p, ,(corresponding to the ith harmonic of the third order transfer func-tion for node p) can be obtained from the adjoint network

. (22)

We can apply Krylov subspace projection (such as [7]) to reduce thelinear adjoint network of (22)

, (23)

where is a orthonormal basis of the Krylov subspace with .

As the DFT vector is absorbed into , to perform a real projec-tion, can be split into real and imaginary parts before reduction[21]. Based on (22)-(23), we can approximate as

(24)

To speed up the computation of the third order transfer functionor response, high order system matrices need to bereduced. To avoid forming dense reduced matrices (particularly forthe third order ones) in a projection-based approach, here we ex-ploit the internal structure of the problem by applying a direct ma-trix pruning technique, where nonzero elements of are pruned according to their contributions to the third order trans-fer function at the output node of interest. Although a nonlinear cir-cuit may contain many nonlinearities, a few of them tend to bedominant. Therefore, retaining the original circuit structural infor-mation by avoiding using projection and pruning these high-ordermatrices in the original coordinates can be very effective. This sameidea has been used in symbolic modeling of nonlinear circuits,where dominant circuit nonlinearities are identified and kept forbuilding system-level models [14]-[16].

The pruning process proceeds as follows: A set of sampled fre-quency points are first selected within the frequency band of in-terest. Then for each of these frequency samples, the third ordertransfer function are computed, as well as the individual contribu-tions of nonzeros in . These contributions are sortedby magnitude, and non-dominant ones are discarded provided thata user-specified error tolerance on is not exceeded. The endproducts of the process are a set of pruned matrices

that satisfy the error tolerance for all the sam-pled frequency points in . As the computation of and individ-

ual contributions may take place on many sample points, matrixpruning can be a slow process. However, to speed up this process,we embed the projection-based model reduction step directly intothe pruning procedure.

To compute at a frequency point, and need to be com-puted at related points. As in (20), to form vector we can use thereduced order model in (16)-(17) to compute approximate first andsecond order transfer functions. Then, instead of solving a poten-tially large linear problem in (19), the reduced adjoint network of(24) can be used to find an approximate vector . Computation ofeach individual contribution term is straightforward, based on (20)and (24). For instance, the contribution of the nonzero at lo-cation of can be simply computed as

, (25)

where is the value of the nonzero, is the nth elementof , and is the mth element of . Since reduced order modelsare used, factorizing the original large system matrices at manysampled points are avoided. The cost for constructing projection-based reduced order models of (16) and (24) is dominated by thecost of a few matrix factorizations, therefore it is bounded by

, where is the number of matrix factorizations, isthe number of sampled points used to discretize the time-varyingtransfer functions, and is the number of physical circuit un-knowns. The cost of the pruning process is dominated by evaluationand sorting of the different contributions, assuming that the use ofreduced order models makes other cost much less. The overall costfor model generation is, therefore, bounded by

, where is the number frequency samplesevaluated in the pruning, and is the number of nonzeros in thehigh order matrices being pruned.

Using pruned matrices , (20) can be further ap-proximated as

(26)

To derive a time-domain model for generating the desired third or-der response, we substitute (26) into (24) and transpose both sidesof the equation

. (27)

Defining

, (28)

it is not hard to show that the corresponding time-domain model is

, (29)

where are given in (18), is the desired time-domainthird order response. Note that since only the ith harmonics of thethird order transfer functions are considered at the output, in the above equation is complex. To recover the corresponding realsignal, we can simply add the conjugate component that corre-sponds to the -ith harmonic. The first and second order responses atnode p, and , can be obtained from (18) by selectingthe proper entries from . Also note that when more than one out-put or set of side bands are of interest, multiple reduced adjoint net-works can be incorporated into the model in a straightforward way.

3.2.3 Flow of Hybrid Model GenerationWe summarize the model generation flow in Fig. 2. It should be

G1C1 g

G2 C2 G3 C3, , ,H1 H2, H1 H2,

G2 C2 G3 C3, , ,

i 0 1±,= l 0 0 … 1… 0T

=N 1×

L diag lT lT … lT, , ,( ) RM NM×∈=

d 1 e j2πi– M⁄ … e j2π M 1–( )i– M⁄ TM⁄ ba, LTd= =

ypi

G1T

sC1T

+[ ] z s1 s2 s3, ,( ) ba=

ypigTz=

Gaˆ Va

TG1TVa Ca

ˆ VaTC1

TVa ba

ˆ,=, VaTba= =

Vacolspan ra Aara Aa

2ra…,, Aa G1T–C1

Tra,– G1

T–ba= =

d baba

ypi

ypi

ˆ gTz z, Va Gaˆ sCa

ˆ+[ ]1–baˆ= =

G2 C2 G3 C3, , ,

G2 C2 G3 C3, , ,

Ω

G2 C2 G3 C3, , ,

H3

G2p C2p G3p C3p, , ,Ω H3

H3 H1 H2g

z

m n,( )G3

δg3 m n, , G3 m n u1 n, zm⋅ ⋅, ,=

G3 m n, , u1 n,u1 zm z

O kM3N3

( ) k M

N

O sE E( ) kM3N3

+log( ) sE

G2p C2p G3p C3p, , ,

g G3p sC3p+[ ]– VH1˜ s1( )( ) VH1

˜ s2( )( ) VH1˜ s3( )( )⊗ ⊗⋅=

2 G2p sC2p+[ ] VH1˜ s1( )( ) VH2

˜ s2 s3,( )( )⊗⋅ ⋅–

ypi

ˆ baˆ T

Gaˆ T

sCaˆ T

+[ ]1–Va

Tg=

x111 Vx1˜ Vx1

˜ Vx1˜ x12

12--- Vx1

˜ Vx2˜ Vx2

˜ Vx1˜⊗+⊗( )=,⊗ ⊗=

tdd Ca

ˆ Txa( ) Ga

ˆ Txa+ =

V– aT G3p x111 td

d C3p x111⋅( ) 2G2p x12 2td

d C2p x12⋅( )+⋅+ +⋅

yp 3,ˆ t( ) e

jiω0 tbaˆ T

xa=

x1˜ x2

˜, yp 3,ˆ t( )

yp 3,ˆ t( )

yp 1,ˆ t( ) yp 2,

ˆ t( )y

458

noted that the hybrid model can be either realized in the time-do-main based on (18) and (29), or the reduced, pruned matrices andassociated projections , , , , , , , , , ,

, , and can be directly used in a frequency-domainVolterra-based analysis. The corresponding model structure is de-picted in Fig. 3.

4. RESULTS4.1 A switch-capacitor filter

Switch-capacitor filters are often found in RF receivers as chan-nel-select filters [22]. If the input signal is small, then these circuitscan be characterized by time-varying Volterra series. A Butter-worth lowpass switch-capacitor biquad is shown in Fig. 4. Thetwo-phase clock is at 20MHz, and the 3-dB frequency of the filteris about 700KHz. Each circuit nonlinearity in the filter is modeledas a third order polynomial about the periodically varying operat-ing point generated by the clock. The resulting full model has 8142time-sampled circuit unknowns. To view the third order nonlineareffects within and close to the signal band, we plot the dc compo-nent of the time-varying third order nonlinear transfer function

, where ,, in Fig. 5(a). It is clearly observable that the third order

nonlinear characteristics vary dramatically between passband andstopband. To capture not only the nonlinear distortions due to the

1. Apply multi-point NORM to compute a projection for approximating the first and second order transfer functions:

.

2. Form the adjoint network as in (22) and reduce it using a projection-based reduction:

.

3. For each frequency point (pre-selected):

3.1. Compute , , , ,

, using (6)-(7) based on

reduced matrices .

3.2. Approximate the first and second order transfer functions of the original system using (17).

3.3. Compute the desired sideband of at

the output node, and individual contributions of

nonzeros in based on reduced order

models using (20), (24) and (25).

3.4. Prune by sorting all the per-nonzero

contributions and retaining dominant ones to satisfy an error tolerance .

4. Form the time-domain reduced models using (18) and (29) if needed.

V

G1˜ VTG1V C1

˜ VTC1V b VTb=,=,=

G2˜ VTG2 V V⊗( ) C2

˜ VTC2 V V⊗( )=,=

Gaˆ Va

TG1TVa Ca

ˆ VaTC1

TVa ba

ˆ,=, VaTba= =

s1 s2 s3, ,( ) Ω∈

H1˜ s1( ) H1

˜ s2( ) H1˜ s3( ) H2

˜ s1 s, 2( )

H2˜ s1 s, 3( ) H2

˜ s2 s, 3( )

G1˜ C1

˜ b G2˜ C2

˜, , , ,

H3 t s1 s2 s3, , ,( )

G2 C2 G3 C3, , ,

G2 C2 G3 C3, , ,

ε

Fig. 2. Nonlinear hybrid macromodel generation

G1˜ C1

˜ b G2˜ C2

˜ V Gaˆ Ca

ˆ baˆ Va

G2p C2p G3p C3p

Linear Networku(t)

Linear Network

b~

22

~,

~CG

)~

,~

( 11 CG

)~

,~

( 11 CG

Linear Network

pp CG 22 , pp CG 33 ,

)ˆ,ˆ( aa CGLinear Network

pp CG 22 , pp CG 33 ,

)ˆ,ˆ( aa CG

(Reduced by NORM)

(Pruning followed by projection)

)(ˆ 3, ty p

1~x

2~x )(ˆ 1, ty p

)(ˆ 2, ty p )(ˆ ty p

V

Fig. 3. Hybrid model structure

+- +

-

Vin+

Vin-

Φ1

Φ1 Φ2

Φ2 Φ1

Φ1

Φ2

Φ2

+- +

-Φ1

Φ1 Φ2

Φ2 Φ1

Φ1

Φ2

Φ2

Φ1

Φ1 Φ1

Φ1

Φ1 Φ2

Φ2

Vout+

Vout-

Fig. 4. A switch-capacitor biquad low-pass filter

Fig. 5. (a) Original H3 of the biquad, and(b) relative error of H3

(a)

(b)

100

105

100 10

2 104 10

6

−60

−50

−40

−30

−20

−10

0

Frequency (Hz)Frequency (Hz)

Nor

mal

ized

H3

(dB

)

100

105

100 10

2 104 10

6

0

0.02

0.04

0.06

0.08

Frequency (Hz)Frequency (Hz)

H3

Rel

ativ

e E

rror

H3 t j2πf1 j2πf2 j2πf3,,,( ) 1Hz f1 f2 10MHz≤,≤f3 1– Hz=

459

signals within the passband, but also the third order mixing of thelarge out of channel interferers into the passband, the nonlinear fre-quency-domain characteristics of the filter must be modeled accu-rately over a frequency range of 7 decades. As such, a completeprojection-based approach becomes inefficient, since a sufficientnumber of moments have to be matched for accuracy, while theresulting model size leads to expensive dense projected third ordermatrices.

To apply our hybrid approach, we first used the multi-pointNORM algorithm to accurately capture the first and second trans-fer functions using a reduced order model with 27 states (SVD wasused to deflate the Krylov subspaces), where 6 moments of and24 moments of were matched. The adjoint network describingthe propagation of third order nonlinear effects from various non-linearities to the output was reduced to an 11th order model. Basedon these reduced models, were significantly prunedin the original coordinates on a set of sampled frequencies points.The final hybrid model has a maximum relative modeling error of6% (or about 0.5dB) for , as shown in Fig. 5(b).

This hybrid model was also validated in a frequency-domainVolterra-like analysis, where 6 sinusoidals with various phaseswere selected from passband, transition band and stopband as inputsignals. The simulation result was compared against that of the fullmodel, as shown in Fig. 6. As can be seen from the figure, thehybrid model captures the frequency-domain nonlinear character-istics of the filter over a wide range of input band very accurately.In additional to excellent accuracy, for this example, a runtimespeed up of 64x over the full model was achieved by using thehybrid model in simulation.

4.2 A 900MHz Receiver Front-End

Next consider the 900MHz receiver frond-end depicted inFig. 7. It is composed of a LNA, a mixer and an IF amp as well ashigh-Q bandpass and lowpass filters. To model the weakly nonlin-ear behavior of the receiver with respect to the RF input, weextracted a time-varying third order polynomial nonlinear model

of the system upon the varying operating point due to the 880MHzLO. The extracted full model has 8496 unknowns. To see the thirdorder intermodulation effects of the system, we plot the first har-monic of the time-varying third order nonlinear transfer function

of the output about the RF center fre-quency in Fig. 8(a). This harmonic specifies the amount of down-converted (by one LO frequency) third order intermodulation(IM3) at the output. In the figure, ,

. As high-Q filterings are included as part of the system, due to

their high frequency selectivity, varies dramatically about thecenter frequency. To account fully for the receiver distortion per-formance under the presence of large out-of-band interferers (e.g.large blocking signals), the receiver third order intermodulationmust be characterized accurately not only in the channel, but alsoat frequencies tens or hundreds of MHz away from the center fre-quency. This requirement makes it very difficult to directly apply aprojection-based reduction to the complete receiver, and maynecessitate modeling each nonlinear block individually in order toisolate the high-Q filtering blocks.

To apply the proposed hybrid approach to the complete receiver,multi-point NORM was first applied to approximate and using a reduced order system with 52 states, where 16 moments of

and 44 moments of are matched (the Krylov subspaces

were compacted using SVD). The adjoint network for propagatingthird-order nonlinear effects to the first harmonic of time-varying

at the output was reduced to a 40th order system. The second

and the third order system matrices were again pruned on a set of

H1H2

G2 C2 G3 C3, , ,

H3

0 0.5 1 1.5 2 2.5 3

x 106

−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

Frequency (Hz)

Norm

aliz

ed O

utp

ut (d

B)

reducedfull

Fig. 6. Biquad output due to a six-tone sinusoidal input

LNA IF AMPVRF

LO

Output

Mixer

Fig. 7. A 900MHz receiver front-end

H3 t j2πf1 j2πf2 j2πf3,,,( )

850Hz f1 f2 1050MHz≤,≤f3 950MH– z=

8.59

9.510

10.5

x 108

8.5

9

9.5

10

10.5

x 108

0

0.02

0.04

0.06

0.08

Frequency (Hz)Frequency (Hz)

H3 R

ela

tive E

rror

Fig. 8. (a) Original H3 of the front-end, and (b) relative error of H3

(b)

(a)8.5

99.5

1010.5

x 108

8.59

9.510

10.5

x 108

−250

−200

−150

−100

−50

0

Frequency (Hz)Frequency (Hz)

No

rma

lize

d H

3 (

dB

)

H3

H1 H2

H1 H2

H3

460

sampled frequency points for meeting the accuracy requirementfor based on these reduced models. The corresponding relative

modeling error on of the hybrid model is less than 8% (0.7dB)

as plotted in Fig. 8(b). Since varies as much as 230dB in the

frequency band shown in Fig. 8, the relative modeling error was

computed as , where is a small positive num-

ber that is added to avoid overestimation of the error for an

extremely small value of .

To further verify the accuracy of the receiver model, we per-formed a frequency-domain Volterra-based analysis where the RFinput consisted of 12 sinusoidals over a frequency span of100MHz. To test the receiver performance under the interferenceof large blocking signals, the largest amplitude difference betweenthe in-band and out-of-band signals was as high as 60dB. Thecomplete output spectrum consisted of 147 frequencies, and thefrequency components close to the IF band are plotted in Fig. 9. Asclearly seen from the plot, the hybrid model accurately captures thereceiver linearity performance over a wide range of input fre-quency. In addition, the use of the hybrid model brought more than20x simulation efficiency improvement over the full model. Weshall also note that model accuracy can be easily traded off withsimulation efficiency for specific cases. For instance, as the accu-racy requirement permits, simpler hybrid models can be con-structed by adopting lower order projection-based approximations,as well as less stringent error tolerance for pruning.

5. CONCLUSIONSWe have shown that generating fully-encompassing nonlinear

models for complex analog circuit blocks or subsystems presentschallenging modeling problems. For these cases, the requirementson model accuracy and applicability can often render existingmethods ineffective. To address these challenges, a combination ofdifferent techniques are required. Our proposed hybrid model gen-eration approach combines the benefits of both pruning-based andtransformation-based methodologies into a unifying model. Theutility of this approach is demonstrated for modeling circuits inanalog signal processing and RF applications.

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weakly nonlinear circuits, Van Nostrand Reinhold, 1980.[2] S. A. Maas, Nonlinear Microwave Circuits, Artech House,

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[5] P. Feldmann and R. Freund, “Efficient linear circuit analysisby Padé approximation via the Lanczos process,” IEEETrans. CAD, vol. 14, pp. 639-649, May 1995.

[6] L. Silveira, M. Kamon and J. White, “Efficient reduced-ordermodeling of frequency-dependent coupling inductances asso-ciated with 3-d interconnect structures,” Proc. of ACM/IEEEDAC, 1995.

[7] A. Odabasioglu, M. Celik and L. Pileggi, “PRIMA: passivereduced-order interconnect macromodeling algorithm,” IEEETrans. CAD, vol. 17, no. 8, pp. 645-653, August 1998.

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[15] P. Wambacq, P. Dobrovolny, S. Donnay, M. Engels and I.Bolsens, “Compact modeling of nonlinear distortion in ana-log communication circuits,” Proc. of ACM/IEEE DATE,2000.

[16] P. Dobrovolny, G. Vandersteen, P. Wambacq, S. Donnay,“Analysis and white-box modeling of weakly nonlinear time-varying circuits,” Proc. of ACM/IEEE DATE, 2003.

[17] H. Liu, A. Singhee, R. Rutenbar and L. Carley, “Remeber-ance of circuits past: macromodeling by data mining in largeanalog design spaces,” Proc. of ACM/IEEE DAC, 2002.

[18] W. Daemes, G. Gielen and W. Sansen, “An efficient optimi-zation-based technique to generate posynomial performancemodels for analog integrated circuits,” Proc. of ACM/IEEEDAC, 2002.

[19] Matlab Simulink Reference, http://www.mathworks.com/.[20] M. Okumura, T. Sugawara and H. Tanimoto, “An efficient

small-signal frequency analysis method of nonlinear circuitswith two frequency excitations,” IEEE Trans. CAD, vol. 9,no. 3, pp. 225-234, March 1990.

[21] A. Ruhe, “The rational Krylov algorithm for nonsymmetriceigenvalue problems III: complex shifts for real matri-ces,”BIT, vol. 34, pp. 165-176, 1994.

[22] P. Chang, A Rofougaran and A. Abidi, “A CMOS channel-select filter for a direct-conversion wireless receiver,” IEEEJ. of Solid-state Circuits, vol. 32. no. 5, pp. 722-729, May1997.

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H3

H3

H3

δH3

ori H3app·–

H3ori α+

------------------------------= α

H3ori

0 0.5 1 1.5 2

x 108

−160

−140

−120

−100

−80

−60

−40

−20

0

Frequency (Hz)

Norm

aliz

ed O

utp

ut (d

B)

reducedfull

Fig. 9. Front-end receiver output due to a 12-tone sinusoidal input

461