Volterra Series Based Time-domain Macro-modeling of Nonlinear Circuits
Xiaoyan Y. Z. Xiong, Li Jun Jiang, Jose E. Schutt-Aine, Weng Cho Chew
aa r X i v : . [ m a t h - ph ] M a y Volterra Series Based Time-domainMacro-modeling of Nonlinear Circuits
Xiaoyan Y. Z. Xiong,
Member, IEEE,
Li Jun Jiang,
Senior Member, IEEE,
Jos´e E. Schutt-Ain´e,
Fellow, IEEE, andWeng Cho Chew
Fellow, IEEE,
Abstract —Volterra series representation is a powerful math-ematical model for nonlinear circuits. However, the difficultiesin determining higher-order Volterra kernels limited its broaderapplications. In this work, a systematic approach that enables aconvenient extraction of Volterra kernels from X-parameters ispresented. A concise and general representation of the outputresponse due to arbitrary number of input tones is given.The relationship between Volterra kernels and X-parameters isexplicitly formulated. An efficient frequency sweep scheme andan output frequency indexing scheme are provide. The leastsquare linear regression method is employed to separate differentorders of Volterra kernels at the same frequency, which leads tothe obtained Volterra kernels complete. The proposed Volterraseries representation based on X-parameters is further validatedfor time domain verification. The proposed method is systematicand general-purpose. It paves the way for time domain simulationwith X-parameters and constitutes a powerful supplement toexisting blackbox macro-modeling methods for nonlinear circuits.
Index Terms —Volterra series, X-parameters, nonlinear cir-cuits, blackbox macro-modeling.
I. I
NTRODUCTION T HE macro-modeling of nonlinear devices and systems isa topic of growing interest due to the dramatic increasesin the complexity and size of modern systems [1]–[6]. Thebasic idea of macro-modeling of a circuit system is to replacethe original system by an approximating system which requiresmuch less design time and fewer resources. Building macro-models is key for enabling complete system verification andhigh-level design exploration. Recently developed nonlinearmacro-modeling methods can be largely categorized into twomajor classes: nonlinear model order reduction (MOR) andnonlinear blackbox macro-model generation. Both approachesare substantially harder than their linear counterparts. The firsttype works directly on the SPICE level schematics, i.e., stateequation derived from modified nodal analysis (MNA) method.For nonlinear MOR, a traditional method is to first linearizethe system then extend projection-based MOR techniques forlinear systems to accommodate nonlinear systems [7]–[10]. Bydoing this, the nonlinear MOR task is re-cast as the reductionof a series of linear systems. However, this method suffers
X. Y.Z. Xiong and L. J. Jiang are with the Department of Electrical andElectronic Engineering, University of Hong Kong, Pokfulam Road, HongKong, China (e-mail: [email protected]; [email protected]).J. Schutt-Ain´e and W. C. Chew are with the Department of Electrical andComputer Engineering, University of Illinois at Urbana-Champaign, Urbana,IL 61801 USA (e-mail: [email protected]; [email protected]). W. C.Chew was with the Faculty of Engineering, University of Hong Kong, HongKong. from the exponentially increased dimension. Its relevant sys-tem transfer function is prohibitively too difficult to expandand to generate moments, essentially limiting the approachto weakly nonlinear systems. The trajectory-based piecewise-linear (TPWL) approximation approach models a nonlinearsystem using a collection of weighted linear models basedupon a state trajectory generated by a training input [11]. Theneach linear model is reduced using linear MOR techniques.This approach has the potential capability to handle largenonlinearities, but is limited by the training input dependency.The above-mentioned nonlinear MOR methods are basedon SPICE level modeling. However, in some situations, it isdifficult to obtain the SPICE model due to intellectual propertyrestrictions and limited information. Some systems cannotbe described by SPICE level models because of couplingeffects, distributed elements, and higher-order modes (excitedby via, connector), etc. [12]. Consequently, blackbox macro-modeling becomes a viable alternative. The goal of blackboxmacro-modeling is to find a mathematical relation that canreproduce the electrical behavior at the ports without anyassumption about the device’s internal structure. However, it isdifficult to find accurate and efficient models to characterizethe nonlinear behavior of devices under arbitrary loads andinput signals. S-parameters are the network parameters thathave been used widely as a blackbox macromodel in the signalintegrity and RF/microwave frequency domains [13]–[16]. Butthe applicability of S-Parameters has been limited to smallsignals and linear behaviors.The recently developed X-parameters from the poly-harmonic distortion (PHD) model [17] are a superset of S-parameters. They describe the relationships between incidentand scattered waves by using not only port-to-port but alsoharmonic-to-harmonic interactions under certain large signaloperating points (LSOPs). They have been successfully usedto describe various nonlinear devices [18]. In the descriptivefunction concept, input signals are restricted to fundamentalcomponents consisting of LSOPs superposed with small har-monics. Consequently, X-parameters, essentially a frequency-domain tool, cannot support time domain simulations and havedifficulty handling input signals with high peak-to-average ra-tios that will excite the device over broad linear and nonlinearoperating ranges.Dynamic X-parameters [19]–[21] is a fundamental exten-sion of X-parameters to modulation-induced baseband mem-ory effects. It can be used to model hard nonlinear behaviorand long term memory effects and is valid for all possiblemodulation formats, for all possible peak-to-average ratios and for a wide range of modulation bandwidths. The model canbe implemented in a commercial complex envelope simulator.The core of dynamic X-parameters is the memory kernelfunction derived from hidden variables concept. The memorykernel can be regarded as the nonlinear impulse response of thesystem and can be uniquely identified from the set of complexenvelope time domain measurement from initial states to finalstates. However, the current dynamic X-parameters model isdefined in the envelope domain under several basic assump-tions. The incident waves and scattered waves are restricted tobe complex envelope representations of modulated carriers.The Volterra series representation is another popular black-box macro-modeling approach for describing nonlinear de-vices with memory [22]–[24]. It can support time domainsimulation with arbitrary input and is valid for signals thatcan excite both linear and nonlinear responses. Without know-ing the state equation, the difficulty in determining higher-order Volterra kernels has restricted its application. We pre-viously proposed a method to get Volterra kernels from X-parameters [25]. However, in [25], the Volterra kernel modelis used as a frequency domain model. The input is restrictedto harmonic input. It provides no more information than X-parameters and is unable to conduct time domain simulation.In this paper, we have extended the Volterra kernel modelto time domain simulation. To make the paper more easilyreadable and self-contained, some formulas are rewritten andadditional formulas are provided with detailed explanation.The accuracy and capability of Volterra kernel model for timedomain simulation have been verified. A concise and generalrepresentation of the output responses due to arbitrary numberof input tones is given. The generalized relationship betweenVolterra kernels and X-parameters is explicitly formulated. Inaddition, the requirements of the input signal are discussedin detail. An efficient frequency sweep scheme and an outputfrequency indexing scheme are provide. With these schemesand the symmetry property of Volterra kernels, there is no needto determine the irreducible frequency sweeping region, whichmakes the kernel determination procedure more convenientfor computer programming. The Volterra kernels are extractedonly once, which can be repeatedly used for different types ofinput. The proposed method is systematic and general-purpose.It is very useful for the blackbox macro-modeling of nonlinearcircuits.The organization of the paper is as follows. Section IIprovides a brief description of Volterra series theory. SectionIII gives the X-parameters formalism with incommensuratemulti-tone input. Section IV presents the detailed technicaldescription of the Volterra kernel extraction process. In SectionV, numerical examples of extracting Volterra series from theX-parameters are provided. Time domain outputs are presentedto validate the proposed method. Finally, a conclusion is givenin Section VI. II. V
OLTERRA S ERIES
Volterra series have been widely used to characterize non-linear systems with memory [26]. For a system with input u ( t ) , the output y ( t ) can be expressed using the expansion y ( t ) = ∞ X n =1 y n ( t ) (1)with y n ( t ) = 1 n ! Z + ∞−∞ . . . Z + ∞−∞ h n ( τ , . . . , τ n ) · (2) u ( t − τ ) . . . u ( t − τ n ) dτ . . . dτ n where h n ( τ , . . . , τ n ) is the n th-order time domain Volterrakernel or impulse response. In particular, y ( t ) is the usualfirst-order convolution having its frequency domain represen-tation Y ( ω ) = H ( ω ) U ( ω ) (3)where H ( ω ) = R + ∞−∞ h ( τ ) e − jωτ dτ is the linear transferfunction or the first-order Volterra kernel. U ( ω ) is the Fouriertransform of u ( t ) . However, the nonlinear higher-order outputcannot be written in a form similar to (3). By replacing thesingle time axis by multiple time axes, (2) becomes y n ( t , . . . , t n ) = 1 n ! Z + ∞−∞ . . . Z + ∞−∞ h n ( τ , . . . , τ n ) · (4) u ( t − τ ) . . . u ( t n − τ n ) dτ . . . dτ n The frequency domain representation of (4) can be conve-niently written in a form similar to (3) Y n ( ω , . . . , ω n ) = H n ( ω , . . . , ω n ) U ( ω ) . . . U ( ω n ) (5)with the nonlinear transfer function H n defined as H n ( ω , . . . , ω n ) = Z + ∞−∞ . . . Z + ∞−∞ h n ( τ , . . . , τ n ) · (6) e − jω τ . . . e − jω n τ n dτ . . . dτ n To restore y n ( t ) , one then evaluates along the diagonal line inthe multi-time hyperplane y n ( t ) = y n ( t , . . . , t n ) | t = t = ... = t n = t (7)where y n ( t , . . . , t n ) is the multidimensional inverse Fouriertransform of Y n ( ω , . . . , ω n ) .III. X- PARAMETERS F ORMALISM
X-parameters, derived from the polyharmonic distortion(PHD) model [17], [27], are a superset of S-parameters andcan be used to describe the behavior of nonlinear devices in thefrequency domain. For better clarification and without losinggenerality, we take the incommensurate 3-tone excitation caseto illustrate the formalism of X-parameters [28]. Suppose theincident signal A q ( t ) at port q has three large incommensuratefundamental tones with frequencies ω , ω and ω . They areincommensurate if the ratio of their frequencies is irrational: k ω + k ω + k ω = 0 ⇒ k = k = k = 0 (8)for k , k , k ∈ Z The scattered signal at port p contains numerous frequencycomponents. They are the combinations of the input tones ω = k ω + k ω + k ω and can be indexed as B p, [ k ,k ,k ] . Then, X-parameters are used to link the scattered signal B withincident signal A under linearization around specific LSOPs(amplitudes and phases of input tones, bias, loads, etc.). B p, [ k ,k ,k ] = X ( F ) p, [ k ,k ,k ] P k [1 , , P k [0 , , P k [0 , , (9) + X q,k ′ ,k ′ ,k ′ h X ( S ) p, [ k ,k ,k ]; q, [ k ′ ,k ′ ,k ′ ] P k − k ′ [1 , , P k − k ′ [0 , , P k − k ′ [0 , , A q, [ k ′ ,k ′ ,k ′ ] i + X q,k ′ ,k ′ ,k ′ h X ( T ) p, [ k ,k ,k ]; q, [ k ′ ,k ′ ,k ′ ] P k + k ′ [1 , , P k + k ′ [0 , , P k + k ′ [0 , , A ∗ q, [ k ′ ,k ′ ,k ′ ] i The X ( F ) term includes the information of the three largefundamental tones A q, [1 , , , A q, [0 , , and A q, [0 , , . P [1 , , , P [0 , , and P [0 , , are the initial phases of input tones. X ( S ) and X ( T ) are scattering parameters describing the small signalinteractions under spectral linearization around the LSOPs. X ( S ) p, [ k ,k ,k ]; q, [ k ′ ,k ′ ,k ′ ] is a scattering parameter of type S that accounts for the contribution to the frequency indexed as [ k , k , k ] of the scattered wave at port p from the [ k ′ , k ′ , k ′ ] -th harmonic of the incident wave at port q . X ( T ) is a scatteringparameter of type T . The definitions of the subscripts of X ( T ) are the same as those of the X ( S ) term except that they accountfor the contribution from harmonics of the conjugate of theincident wave. The existence of a scattering parameter of type T is due to the nonanalyticity of the spectral mapping fromthe time domain to the frequency domain [27]. The sum runsover all q and all integers k ′ , k ′ , k ′ .IV. V OLTERRA K ERNEL E XTRACTION FROM X- PARAMETERS
A. Determination of Volterra Kernels
Volterra kernels are transfer functions of a nonlinear sys-tem and are widely used to characterize a nonlinear systemwith memory. Previously, Volterra kernels are calculated bythe harmonic input method [26], [29], [30]. However, thedetermination process is very tedious and time-consuming. Itrequires the generation of input tones, waiting for steady-state,sampling the output, computation of Fourier transfer to get theoutput response at distinct frequencies. When frequency sweepis needed, the determination task becomes extremely chal-lenging especially for high-order kernels. On the other hand,X-parameters can be conveniently obtained from harmonic-balance (HB) simulation or measured by modern nonlinearvector network analyzers (NVNA). Inspired by the formalismof the incommensurate multi-tone X-parameters, we propose asystematic method to obtain the Volterra series representationof X-parameters based on the concept of harmonic inputmethod. In this work, we take the negative frequencies intoaccount by adding the complex conjugation terms and givethe general representation of the output responses due toharmonic inputs. In general, to get the complete descriptionof the M th-order Volterra kernel, an M -tone excitation isrequired. Suppose the input signal is the superposition of M incommensurate tones u ( t ) = M X m =1 V m e jω m t + c.c. = M X m = − Mm =0 V m e jω m t (10)where “c.c.” denotes the complex conjugate terms, and ω − m = − ω m , and V − m = V ∗ m , m is an integer. Substituting the inputrepresentation (10) into (2), the n th-order output is y n ( t ) = 1 n ! Z + ∞−∞ . . . Z + ∞−∞ h n ( τ , . . . , τ n ) · (11) n Y i =1 (cid:20) V e jω ( t − τ i ) + . . . + V M e jω M ( t − τ i ) + c.c. (cid:21) d ¯ τ = 1 n ! M X m = − M . . . M X m n = − M " n Y i =1 V m i · H symn ( ω m , . . . , ω m n ) · exp j n X i =1 ω m i t ! where m i is an integer and m i = 0 , d ¯ τ = dτ . . . dτ n . H symn ( · ) in (11) is defined as the n th symmetric frequencydomain Volterra kernel or transfer function. The symmetrickernel satisfies H symn ( ω , ω , . . . , ω n ) = H symn ( ω m , ω m , . . . , ω m n ) (12)where the subscript of the argument m i denotes any permuta-tion of the integers , . . . , n . It can be obtained by setting [30] H symn ( · ) = 1 n ! X all permutations of { ω m ,...,ω mn } H asymn ( ω m , . . . , ω m n ) (13)In this paper, we use symmetric Volterra kernel. Thesuperscript “ sym ” is omitted for simplicity. Some of thevalues ω m , . . . , ω m n may be repeated. Thus, many termsin (11) contain identical exponents. Taking a three-toneexcitation ( M = 3 ) as an example, the third-order output y ( t ) in (11) contains third-order Volterra kernels. Thearguments ω m i of these kernels H ( ω m , ω m , ω m ) can be ± ω , ± ω and ± ω . The Volterra kernels included in y ( t ) are H ( − ω , ω , ω ) , H ( ω , − ω , ω ) , H ( ω , ω , − ω ) , H ( ω , ω , − ω ) , H ( ω , − ω , ω ) , H ( ω , ω , − ω ) , H ( − ω , ω , ω ) , H ( ω , − ω , ω ) , H ( − ω , ω , ω ) ,etc. Due to the permutation symmetry, the first threekernels are the same and corresponding to the outputfrequency ω = ω m + ω m + ω m = 2 ω − ω = ω .The next six kernels are the same and also correspondingto the output frequency ω = ω + ω − ω = ω .These symmetric Volterra kernels coinciding at the samefrequency can be collected together. We introduce a concisekernel G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( ω , . . . , ω M ) to denote allsymmetric kernels at frequency k ω + k ω + · · · + k M ω M ( k m is an integer and ≤ | k m | ≤ n ). G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( ω , . . . , ω M ) is H n (cid:0) ω m , . . . , ω m n (cid:1) with k + 2 r + . . . + k M + 2 r M = n (14)first k + r of ω m i = + ω next r of ω m i = − ω ...next k M + r M of ω m i = + ω M next r M of ω m i = − ω M The number of arguments of G ( · ) is equal to the numberof excitation tones. The number of arguments of H ( · ) is k + 2 r + . . . + k M + 2 r M . The m th subscript of G ( · ) kernel [ k m + r m , r m ] corresponds to the arguments ω m of H ( · ) (with k m + r m positive ω m and r m negative ω m arguments). Hence, in the three-tone excitation example, thefirst three kernels, e.g., H ( ω , ω , − ω ) , can be rewrittenas G [1+1 , , [0+0 , , [0+0 , ( ω , ω , ω ) (with k = r = 1 , k = r = k = r = 0 in the general representationof G ( · ) ). That is, the first k + r = 2 arguments of H are ω ; the next r = 1 argument is − ω . Similarly,the next six kernels, e.g., H ( ω , ω , − ω ) , can be denotedas G [1+0 , , [0+1 , , [0+0 , ( ω , ω , ω ) (with k = r = 1 , r = k = k = r = 0 in the general representation of G ( · ) ). That is, the first k + r = 1 argument is ω ; the next r = 0 argument is − ω ; the next k + r = 1 argumentis ω ; and the next r = 1 argument is − ω . By using thenew kernel representation and collecting all terms at frequency k ω + k ω + · · · + k M ω M , we get a simplified representationof (11) y n ( t ) = X r . . . X r M " (cid:18) V (cid:19) k + r (cid:18) V ∗ (cid:19) r . . . (cid:18) V M (cid:19) k M + r M (cid:18) V ∗ M (cid:19) r M · " ( k + r )! r ! . . . ( k M + r M )! r M ! − · G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( ω , . . . , ω M ) · exp [ j ( k ω + . . . + k M ω M ) t ] (15)where r m are nonnegative integer indices that satisfy k + 2 r + k + 2 r + . . . + k M + 2 r M = n since G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( · ) is the n th-orderkernel. The number of symmetric kernels denoted by G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( · ) is n !( k + r )! r ! ... ( k M + r M )! r M ! .When k m < , the signs of ω m are reversed, e.g., if k < ,the first | k | + r arguments of H n are − ω and the next r arguments are + ω . For example, H ( − ω , ω , ω ) corresponds to the output frequency ω = − ω + 2 ω . Itis denoted by G [ − , , [2+0 , , [0+0 , with k = − < .Then we reverse the sign of ω . Now the first | k | + r = 1 argument becomes − ω .The general representation of the total output y ( t ) due toan M -tone excitation is similar to (15) except that all r m go from zero to infinity. y ( t ) = ∞ X r =0 . . . ∞ X r M =0 " (cid:18) V (cid:19) k + r (cid:18) V ∗ (cid:19) r . . . (cid:18) V M (cid:19) k M + r M (cid:18) V ∗ M (cid:19) r M · " ( k + r )! r ! . . . ( k M + r M )! r M ! − · G [ k + r ,r ] ,..., [ k M + r M ,r M ] ( ω , . . . , ω M ) · exp [ j ( k ω + . . . + k M ω M ) t ] (16)The Volterra kernels can be determined once the magnitudesand phases of the corresponding frequency components ofthe output signal Y ( k ω + . . . + k M ω M ) are known. Theinformation is provided by X-parameters. Once we knowthe mapping relationship between the output response due toan m -tone excitation and the m -tone X-parameters, we cancalculate the m th-order Volterra kernels. B. Relationship between Volterra Kernels and X-parameters
As discussed above, the determination of the M -th Volterrakernel requires an M -tone input signal. With multi-tone input,mixing products will occur. The maximum mixing order M is defined as the maximum order of the intermodulation termsincluded in the output frequency list. For example, assumethere are two fundamental tones ω and ω . If M = 0 or , no mixing products are included in the output frequencylist; if M = 2 , the ±| ω ± ω | intermodulation terms areincluded; if M = 3 , additional ±| ω ± ω | and ±| ω ± ω | terms are included. For better presentation, the M = 3 caseis used to illustrate the relationship between Volterra kernelsand X-parameters. The maximum order of each tone is M and the maximum mixing order M is also set to M = M .In addition, thanks to the time invariant property [27], theinitial phase of each tone is set to zero ( V m = V ∗ m ). Based on(9) and (14), we can link Volterra kernels with X-parametersby equating the same frequency component of the outputsignal, i.e., by setting Y ( k ω + k ω + k ω ) = B p, [ k ,k ,k ] .Table I gives the mapping between Volterra kernels and X-parameters of the first four frequency components. As shownin Table I, each frequency component contains the contributionfrom numerous Volterra kernels. Take the output frequencywith index [ k , k , k ] = [0 , , as an example. The outputfrequency is ω = k ω + k ω + k ω = ω . According to(16), the phasor of this frequency component can be writtenas: Y ( ω ) = ∞ X r =0 ∞ X r =0 ∞ X r =0 (cid:20) ( V / r ( V / r ( V / r ( r !) ( r !) (1 + r )! r ! (cid:21) · G [0+ r ,r ] , [0+ r ,r ] , [1+ r ,r ] ( ω , ω , ω )= B p, [0 , , (17) TABLE IT HE M APPING BETWEEN V OLTERRA K ERNELS AND
X-P
ARAMETERS OF A T WO P ORTS N ETWORK
Frequencyindex [ k , k , k ] Phasor Y ( k ω + k ω + k ω ) Volterra Kernels X-parameters [0 , , Y ( ω ) V H ( ω ) + V H ( ω , ω , − ω ) + V V H ( ω , ω , − ω ) + V V H ( ω , ω , − ω ) + V H ( ω , ω , ω , − ω , − ω ) + . . . B , [0 , , [0 , , Y (2 ω ) V H ( ω , ω ) + V H ( ω , ω , ω , − ω ) + V V H ( ω , ω , ω , − ω ) + . . . B , [0 , , [0 , , Y (3 ω ) V H ( ω , ω , ω ) + V H ( ω , ω , ω , ω , − ω ) + V V H ( ω , ω , ω , ω , − ω ) + . . . B , [0 , , [0 , , − Y ( ω − ω ) V V H ( ω , − ω , − ω ) + V V H ( ω , − ω , − ω , − ω , ω ) + . . . B , [0 , , − when r , r and r run from zero to infinity, we can list theVolterra kernels included in (17) according to (14) r = r = r = 0 : H ( ω ) (18) r = r = 0 and r = 1 : H ( ω , ω , − ω ) r = r = 0 and r = 1 : H ( ω , − ω , ω ) r = r = 0 and r = 1 : H ( ω , − ω , ω ) r = r = 0 and r = 2 : H ( ω , ω , ω , − ω , − ω ) ... ...Consequently, (17) can be expanded and rewritten as Y ( ω ) = V H ( ω ) + V H ( ω , ω , − ω ) + (19) V V H ( ω , − ω , ω ) + V V H ( ω , − ω , ω ) + V H ( ω , ω , ω , − ω , − ω ) + . . . = B p, [0 , , As presented in (19), the output at frequency ω containsdifferent orders of Volterra kernels: the linear term H ( ω ) , thecompression term H ( ω , ω , − ω ) , H c , the desensitizationterms H ( ω , − ω , ω ) , H d , H ( ω , − ω , ω ) , H d and other higher-order terms with n > . We need to separatethese kernels. C. Separation of Volterra Kernels
In general, the magnitude of high-order output decreasesdrastically as the magnitude of the input signal decreaseseven though the magnitude of higher-order kernel may belarge. For instance, supposing the input signal is denoted as αu ( t ) (with αu ( t ) < ), then the n th-order output y n ( t ) isproportional to α n u n ( t ) according to (2). Thus, if the inputis reduced by dB, y ( t ) falls by dB, y ( t ) falls by dBand so on. The separation of Volterra kernels makes useof this property. Provided that the magnitude of the inputsignal u ( t ) is smaller than some upper bound, the high order terms above M are negligible. By setting an input signal withsuitable magnitude, the infinite summation for each frequencycomponent is truncated to a finite one by ignoring the higher-order terms. Then the kernels can be separated based on theleast square linear regression method [31] in the frequencydomain. The basic idea consists of arranging the magnitudesof the input tones so that a matrix can be constructed for thekernels. Take (19) as an example and ignore the higher-orderswith n > by changing V , a matrix equation is constructed: V (1)3 (cid:16) V (1)3 (cid:17) V (1)3 V V (1)3 V V (2)3 (cid:16) V (2)3 (cid:17) V (2)3 V V (2)3 V V (3)3 (cid:16) V (3)3 (cid:17) V (3)3 V V (3)3 V V (4)3 (cid:16) V (4)3 (cid:17) V (4)3 V V (4)3 V H H c H d H d (20) = (cid:2) Y ( ω ) (1) Y ( ω ) (2) Y ( ω ) (3) Y ( ω ) (4) (cid:3) T where V ( i )3 , i = 1 , . . . , , are some properly chosen magni-tudes of the third tone and Y ( ω ) ( i ) are the correspondingphasors of the frequency component ω . Different Volterrakernels are the solutions of (20). It is essential that themagnitudes of the input tones are properly chosen. Theyshould be smaller than some upper bound so that lower-order Volterra kernels will not be skewed by high order termsbut not so small that the higher-order terms will be buriedin the noise. One improvement is to add additional inputmagnitudes and use the least square solutions of the resultingoverdetermined equation as the estimate of the kernels. Themagnitudes of the other two tones V and V are also changedto form the overdetermined matrix equation. However, withthe increasing number of higher order Volterra kernels beingincluded, the kernel separation becomes much harder evenwith the least square method. The least square method maygive poor estimates of the kernels. One remedy is that wecan first get good estimates of low order kernels by properlychoosing small magnitudes of input tones; then conduct anadditional least square estimate for the higher order kernelsby setting low order kernels as knowns to reduce the errorpropagation. D. Design of Input Signal
To obtain a complete description of Volterra kenels H ( ω ) , H ( ω , ω ) and H ( ω , ω , ω ) , frequency sweep along axes ω , ω and ω in the interested region is required. In ad-dition, for convenient kernel separation, output frequenciesare required to be distinct from each other. Hence, carefulattention must be paid in choosing the frequency componentsincluded in the input signal. With incommensurate input tones,this condition is satisfied automatically. However, it is difficultto ensure the incommensurate condition for all combinationsof sweeping frequencies. In practice, the incommensuraterequirement can be relaxed as follows: suppose the numberof frequency sweeping points along the ω α axis is N α with α = 1 , , ( M = 3 case); by carefully choosing the inputfrequencies ω ,i , ω ,j , and ω ,k , the output frequencies willnot overlap with each other. Here, i, j and k are integers with ≤ i ≤ N , ≤ j ≤ N , and ≤ k ≤ N . Hence, we needto choose the input frequencies such that k ω ,i + k ω ,j + k ω ,k = k ′ ω ,i + k ′ ω ,j + k ′ ω ,k (21)iff k = k ′ , k = k ′ and k = k ′ Equation (21) must be satisfied for all k , k , k , k ′ , k ′ , k ′ ∈ { , ± , ± , . . . , ± M } . E. Notation of Output Frequency
For systems with real input and output signals in thetime domain, the spectra in the frequency domain are doublesided with conjugate symmetry, e.g., H ( ω , ω , − ω ) = H ∗ ( − ω , − ω , ω ) . Hence, the spectra contain redundantinformation by a factor of two. The output frequency isdetermined by indices k , k , and k , and by values of inputfrequencies. For some set of [ k , k , k ] , the output frequencycan be negative. The information about these negative fre-quencies can be obtained from the corresponding positivefrequencies with indices [ − k , − k , − k ] . Here, we presentan indexing scheme that includes all output frequencies andmeanwhile removes the redundancy. Supposing the highest-order of Volterra kernels is M = 3 and the maximum mixingorder M = M , then the indices of the output frequency [ k , k , k ] are arranged according to the following scheme:1) the summation of absolute values of all indices is lessthan or equal to the maximum mixing order (e.g., | k | + | k | + | k | ≤ M );2) the first index is always nonnegative (e.g., k ≥ );3) the first nonzero index begins with positive number (e.g.,if k = k = 0 , k > );4) the second nonzero index begins with the availableminimum integer (e.g., if k = 0 , k begins with − ( M −| k | ) ) and so does the next nonzero index (e.g.,if k = 0 and k = 0 , k begins with − ( M −| k |−| k | ) );These conditions are listed in descending order accordingto priority. By using the above indexing scheme, instead ofrecording all combinations of [ k , k , k ] , only 31 frequenciesneed to be recorded for the M = M = 3 case (seeAppendix A for more details). When the index [ k , k , k ] results in a negative frequency, the complex conjugate of thecorresponding phasor is considered and its contribution isattributed to the corresponding positive frequency.In addition, Volterra kernels also have the permutationsymmetry property, e.g., H ( ω , ω , ω ) = H ( ω , ω , ω ) = H ( ω , ω , ω ) = H ( ω , ω , ω ) = H ( ω , ω , ω ) = H ( ω , ω , ω ) . For each triplet ( ω ,i , ω ,j , ω ,k ) , as shownin Fig. 1, it will determine points in the H ( ω , ω , ω ) space. Meanwhile, and points will be determined inthe H ( ω , ω ) and H ( ω ) space, respectively. Due to thepermutation and conjugate symmetry properties of kernels,only points need to be recorded for the third-order ker-nel, and and points for the second-order and first-orderkernels, respectively. This leads to an efficient Volterra kerneldetermination process. Fig. 1. Distribution of points in the H ( ω , ω , ω ) space with one triplet (cid:0) ω ,i , ω ,j , ω ,k (cid:1) . V. N
UMERICAL E XAMPLES
A. A Benchmark Case1) Description of the Nonlinear System:
The first numericalexample is meanwhile to benchmark a nonlinear system withknown Volterra kernels. Fig. 2 shows the system diagram. H a , H b and H c are frequency domain transfer functions of linear-time-invariant systems. The symbol Π denotes the time domainmultiplication even though the subsystems are represented inthe frequency domain. The V-I relationship at port of themultiplier is i ( t ) = [ v ( t ) v ( t ) − v ( t )] /Z f (22)where v j and i j are the voltage and current at port j respec-tively, j ∈ { , , } . Z f = R = 50 Ω .Because there are two multipliers, the whole system hasup to the third-order nonlinearity. In the frequency domain,Volterra kernels of the whole system have the following idealanalytical expression: H ( ω ) = c H a ( ω ) (23) H ( ω , ω ) = c [ H a ( ω ) H b ( ω )] sym H ( ω , ω , ω ) = c [ H a ( ω ) H b ( ω ) H c ( ω )] sym Fig. 2. System diagram of a third-order nonlinear circuit. where [ · ] sym indicates a symmetrized Volterra kernel accordingto (13); c i is a constant determined by circuit parameters(e.g., Z f , R ). However, due to the mismatch between differentblocks, reflections will occur and slightly modify the idealkernel representations. We set H a = H b = H c as the transferfunction of a low pass filter (LPF) system. Fig. 3 shows themagnitude of the transfer function H a which is equivalent toS-parameters S . | H a | LC C
Fig. 3. Magnitude of the transfer function H a of the low pass filter; the insetshows the circuit schematic diagram with L = 42 . nH and C = 8 . pF.
2) Frequency Domain Volterra Kernels:
Volterra kernels ofthe nonlinear system shown in Fig. 2 are extracted from X-parameters. The 3-tone X-parameters of the nonlinear systemare generated by applying the ADS X-parameters genera-tor [32]. The frequency sweep scheme of each axis is givenin Table II. Although the frequency step of each axis is
MHz, the equivalent frequency step is MHz thanksto the symmetry properties of kernels. To separate different
TABLE IIF
REQUENCY S WEEP S CHEME FOR THE TONE X- PARAMETERS G ENERATION
Frequency Start Step Stop ω MHz
MHz . GHz ω MHz
MHz . GHz ω MHz
MHz . GHz order of Volterra kernels, the power of each input tone isset to P in = { , } dBm. The proposed method describedin Sec. IV is used to extract Volterra kernels H , H and H . Fig. 4 shows the magnitude and phase of the first-orderVolterra kernel. They agree well with the S-parameters of thesystem with small signal input. Fig. 5 and Fig. 6 present themagnitudes and phases of the second-order Volterra kernel and a slice of the third-order Volterra kernel, respectively, with ω fixed at . GHz. They have reasonable distributions comparedto the ideal analytical expression in (19). In addition, they alsoagree with the permutation and conjugate symmetric proper-ties. For example, the magnitude of H has two symmetryplanes: ω = ± ω ; the phase of H has a symmetry plane ω = ω and an anti-symmetry plane ω = − ω ; while thethird-order kernel loses the conjugate symmetry property since ω is fixed. It only has one symmetry plane ω = ω . Fig. 4. Magnitude and phase of the linear transfer function H ( ω ) . (a)magnitude; (b) phase.Fig. 5. Magnitude and phase of the second-order Volterra kernel H ( ω , ω ) ;(a) magnitude; (b) phase.Fig. 6. Magnitude and phase of the third-order Volterra kernel H ( ω , ω ,ω = ω c ) with ω c fixed to . GHz; (a) magnitude; (b) phase.
3) Time Domain Output:
To validate the accuracy of theextracted Volterra kernels and to demonstrate the capability ofVolterra series for time domain simulation with arbitrary input,the time domain outputs are presented. Without losing gener-ality, the input is chosen to be a rectangular pulse as shown inthe inset of Fig. 7(b). The magnitude is V = 1 V. The inputhas rich frequency components as shown in Fig. 7(a). The X-parameter macro-model requires numerous fundamental tonesto represent the input [33]. In addition, once the input signalchanges, one needs to regenerate the X-parameter macro-model for the same system since X-parameter is defined under certain LSOPs (fixed magnitude and phase of eachfundamental tone). In contrast, the same Volterra series modelcan be used for arbitrary input. The linear, second-order andthird-order time domain responses y ( t ) , y ( t ) and y ( t ) aregiven in Fig. 7(b), (c) and (d), respectively. They are calculatedaccording to (5) and (7) with extracted Volterra kernels. Thetotal response is the summation of both linear and nonlinearresponses as indicated by (1). Fig. 8 presents the total responsecalculated by the Volterra series representation. It agrees verywell with that obtained by ADS transient simulator with theoriginal circuit model. However, as shown in Fig. 8, thelinear response calculated by S-parameter does not agree withthe response of the original circuit model. The higher-orderVolterra kernels are crucial to capturing the nonlinearities ofthe systems. The linear response ignores the contributions ofhigher order Volterra kernels and hence it is incorrect. Fig. 7. Input and output signal of the benchmark nonlinear system as shownin Fig. 2. (a) Spectrum of the rectangular pulse input. Time domain output:(b) linear response y ( t ) ; (c) second-order response y ( t ) ; (d) third-orderresponse y ( t ) . The inset in (b) shows the shape of the rectangular pulsewith raise and fall time t r = t f = 1 ns, and width t w = 5 ns. y ( t ) ( V ) Volterra Series (X−paramter)ADS Transient (Circuit Model)Linear Response (S−paramter)
Fig. 8. Comparison of the time domain output of the benchmark nonlinearsystem as shown in Fig. 2 calculated by the Volterra series representation andADS transient simulator.
B. Low Noise Amplifier1) Frequency Domain Volterra Kernel:
The second numer-ical example is a low noise amplifier (LNA). The amplifierschematic used in this work is taken from
X-parameter gener-ation tutorial of the example directory of ADS [32]. The insetin Fig. 9(a) shows the macro-model of the LNA. The saturationlimit of input is mV. The 3-tone X-parameters are generatedwith the same frequency sweep scheme shown in Tab. II. Thepower of each tone is set to P in = {− , − } dBm. Theproposed method is used to extract Volterra kernels from X-parameters. Fig. 9(a) and (b) show the magnitude and phaseof the first-order Volterra kernel H . They agree well withthe small signal S-parameters. This is reasonable since X-parameters becomes S-parameters in the small signal limit.Fig. 9(c) and (d) present the magnitudes of the second-orderVolterra kernel and a slice of the third-order Volterra kernelwith ω = 0 . GHz, respectively. Again, H has two symmetryplanes due to permutation and conjugate symmetry while H only has one symmetry plane due to the lost of the conjugatesymmetry. Fig. 9. Volterra kernels of the low noise amplifier (LNA). (a) and (b) are themagnitude and phase of the linear transfer function (first-order Volterra Kernel H ( ω ) ); (c) magnitude of the second-order Volterra kernel H ( ω , ω ) ; (d)a slice of the magnitude of the third-order Volterra kernel H ( ω , ω , ω = ω c ) with ω c = 0 . GHz. The inset in (a) is the macro-model of the LNA.
2) Time Domain Output:
After obtaining the Volterra ker-nels, different orders of time domain responses are calculatedaccording to (5) and (7). The rectangular pulse shown in theinset of Fig. 7(b) is used with V = 0 . V. Fig. 10 displaysthe total output y ( t ) calculated based on the Volterra seriesrepresentation. It captures the distortion due to nonlinearitiesand agrees well with the result simulated by the ADS transientsimulator with the original circuit model.VI. C ONCLUSION
This paper presents a systematic method for extractingVolterra series representation from X-parameters. By com-pletely separating different order of Volterra kernels based onthe least square linear regression method, the complete de-scription of Volterra kernels can be determined very efficiently. y ( t ) ( V ) Volterra Series (X−paramter)ADS Transient (Circuit Model)Linear Response (S−paramter)
Fig. 10. Comparison of the time domain output of the low noise amplifiercalculated by the Volterra series representation and ADS transient simulator.
Time domain output can be obtained conveniently based on thedetermined Volterra kernels with arbitrary input. Numericalresults show the capability of Volterra series representationfor describing nonlinear devices in a broad input powerregion. The procedure for extracting Volterra kernels with thetruncation order M = 3 is illustrated in detail. The papergives the general relationship between Volterra series and X-parameters and the method can be applied to the extraction ofhigher-order Volterra kernels.A PPENDIX AO UTPUT F REQUENCIES AND V OLTERRA K ERNELS FOR THE T HREE -T ONE I NPUT C ASE
For the three-tone input case, the output frequencies are themixing of the input frequencies ω = k ω + k ω + k ω and can be indexed as [ k , k , k ] . Suppose the maximummixing order is M = 3 , the first-order output contains output frequencies and the corresponding Volterra kernelsas listed in Tab. III.
TABLE IIIF
REQUENCIES AND V OLTERRA K ERNELS I NCLUDED IN THE F IRST -O RDER O UTPUT
Frequency Notation Volterra Kernel ω [1,0,0] H ( ω ) ω [0,1,0] H ( ω ) ω [0,0,1] H ( ω ) The second-order output contains output frequencies andthe corresponding Volterra kernels as shown in Tab. IV.Table V shows the output frequencies and the corre-sponding Volterra kernels included in the third-order output.It should be noticed that both the first-order and the third-order outputs contain the output frequencies ω , ω , ω .Hence, the total number of the output frequencies is .A CKNOWLEDGMENT
This work was supported in part by NSFC 61271158, USAOARD 124082 contracted through UTAR, Hong Kong UGCAoE/P-04/08, and by the US National Science Foundation
TABLE IVF
REQUENCIES AND V OLTERRA K ERNELS I NCLUDED IN THE S ECOND -O RDER O UTPUT
Frequency Notation Volterra Kernel ω [2,0,0] H ( ω , ω )2 ω [0,2,0] H ( ω , ω )2 ω [0,0,2] H ( ω , ω ) | ω , ± ω | [1 , ± , H ( ω , ± ω ) | ω , ± ω | [0 , , ± H ( ω , ± ω ) | ω , ± ω | [1 , , ± H ( ω , ± ω ) TABLE VF
REQUENCIES AND V OLTERRA K ERNELS I NCLUDED IN THE T HIRD -O RDER O UTPUT
Frequency Notation Volterra Kernel ω [1,0,0] H ( ω , ω , − ω ) , H ( ω , ω , − ω ) , H ( ω , ω , − ω ) ω [0,1,0] H ( ω , ω , − ω ) , H ( ω , ω , − ω ) , H ( ω , ω , − ω ) ω [0,0,1] H ( ω , ω , − ω ) , H ( ω , ω , − ω ) , H ( ω , ω , − ω )3 ω [3,0,0] H ( ω , ω , ω )3 ω [0,3,0] H ( ω , ω , ω )3 ω [0,0,3] H ( ω , ω , ω ) | ω , ± ω | [1 , ± , H ( ω , ω , ω ) , H ( ω , − ω , − ω ) | ω , ± ω | [2 , ± , H ( ω , ω , ω ) , H ( ω , ω , − ω ) | ω , ± ω | [0 , , ± H ( ω , ω , ω ) , H ( ω , − ω , − ω ) | ω , ± ω | [0 , , ± H ( ω , ω , ω ) , H ( ω , ω , − ω ) | ω , ± ω | [1 , , ± H ( ω , ω , ω ) , H ( ω , − ω , − ω ) | ω , ± ω | [2 , , ± H ( ω , ω , ω ) , H ( ω , ω , − ω ) | ω , ± ω ± ω | [1 , ± , ± H ( ω , ω , ω ) , H ( ω , ω , − ω ) , H ( ω , − ω , ω ) , H ( ω , − ω , − ω ) Award 1218552. The authors thank Dr. Ngai Wong for theconstructive suggestions and Keysight Technologies Inc., forsupporting this work and providing the ADS X-parametergeneration platform; especially, we thank David Root, LorenBetts, Steve Fulwider and Bill Wallace of Keysight for fruitfuldiscussions, insightful comments and helpful suggestions. X-parameters is a registered trademark of Agilent Technologies.R
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