schumacher 20rf power amplifier behavioral modeling based on takenaka–malmquist–volterra...

8/16/2019 Schumacher 20RF Power Amplifier Behavioral Modeling Based on Takenaka–Malmquist–Volterra Series15

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Circuits Syst Signal ProcessDOI 10.1007/s00034-015-0151-0

RF Power Amplifier Behavioral Modeling Based

on Takenaka–Malmquist–Volterra Series

Ricardo Schumacher1 · Eduardo G. Lima1 ·

Gustavo H. C. Oliveira1

Received: 5 April 2015 / Revised: 12 August 2015 / Accepted: 13 August 2015© Springer Science+Business Media New York 2015

Abstract In this paper, a Takenaka–Malmquist–Volterra (TMV) model structure isemployed to improve the approximations in the low-pass equivalent behavioral mod-eling of radio frequency (RF) power amplifiers (PAs). The Takenaka–Malmquist basisgeneralizes the orthonormal basis functions previouslyusedin this context. In addition,it allows each nonlinearity order in the expanded Volterra model to be parameterizedby multiple complex poles (dynamics). The state-space realizations for the TMV

models are introduced. The pole sets for the TMV model and also for the previousLaguerre–Volterra (LV) and Kautz–Volterra (KV) models are obtained using a con-strained nonlinear optimization approach. Based on experimental data measured on aGaNHEMTclassABRFPAexcitedbyaWCDMAsignal,itisobservedthattheTMVmodel reduces the normalized mean-square error and the adjacent channel error powerratio for the upper adjacent channel (upper ACEPR) by 1.6dB when it is compared tothe previous LV and KV models under the same computational complexity.

Keywords Low-pass equivalent behavioral modeling · Radio frequency power

amplifier · Orthonormal basis functions · Volterra series · Wireless communicationsystems

B Gustavo H. C. [email protected]

Ricardo Schumacher

[email protected]

Eduardo G. [email protected]

1 Department of Electrical Engineering, Federal University of Paraná (UFPR), Curitiba,PR 81531-980, Brazil

http://crossmark.crossref.org/dialog/?doi=10.1007/s00034-015-0151-0&domain=pdf


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Circuits Syst Signal Process

1 Introduction

Modern wireless systems must deal with the communication of a huge amount of information, at very high data rates, through a band-limited air interface and in a

power-efficient manner [23]. Spectral efficiency is achieved by modulating a radiofrequency (RF) carrier signal with a complex-valued envelope information having ahigh ratio between the peak and average amplitude levels. As a consequence, non-constant amplitude envelopes demand for linearity in the transmitter chain to avoidinterferences between adjacent channels. In this scenario, the RF power amplifier (PA)present at the transmitter chain plays a major role. Indeed, the traditional design of RF PAs, based on solid-state transistors operating in linear classes (A, B or AB), issubject to a trade-off between linearity and efficiency [4].

Toachieveahighlylinearandhighlyefficientamplification,acost-effectivesolution

is to include a digital baseband predistortion (DPD) to distort the baseband envelopesignal before the RF amplification [11,12,20]. The DPD is designed to have an inversetransfer characteristic regarding the RF PA. Therefore, the RF PA is able to operate innonlinear regimes to improve the power efficiency, and at the same time the overalltransmitter linearity is maintained within an acceptable value.

In DPD applications, the implementation of the RF PA inverse characteristic is thefinal goal, which may be accomplished using the direct or indirect learning architec-tures. In the indirect architecture, firstly a post-distorter (PoD) topology is a priorichosen and subsequently the PoD is copied as a DPD. In fact, both the PoD and the

DPD have the same PA inverse characteristic, but differ from each other because of their position in the cascade connection with the PA. The main motivation for the indi-rect learning is to simplify the identification of the inverse system, once the same set of input and output data can be used for the extraction of the PoD and PA model parame-ters if the roles of the input and output data are exchanged. In the direct architecture,the parameter identification of a DPD model is directly addressed without the use of a PoD. However, the DPD output signal is not available. In fact, only the input andoutput signals of the cascade connection of the DPD followed by the PA are available.As a consequence, the DPD parameter identification must minimize the error betweendesired and estimated PA output signals. Hence, the physical PA (or a PA model) mustbe included in the error expression. In this scenario, unless the inclusion of the physicalPA in a real-time optimization loop could be feasible, a model for the PA is manda-tory. Alternatively, the DPD can have a topology that is the inverse of a PA model, inwhich the DPD parameters are derived from the PA ones using analytical expressionsrelating them. Even though the nonlinear and dynamic behaviors observed at the for-ward and inverse PA characteristics can be different from each other [7,10,24], it is acommon practice in the literature [7,22] to apply the same topology for modeling boththe forward and inverse PA characteristics. In fact, it can be shown that forward andinverse models are approximately the same under certain conditions [7]. Therefore,the successful design of a DPD scheme is strongly conditioned by the availability of a high-accurate and low-complexity model for representing the inverse (in the caseof both the direct and indirect learning architectures) and also forward (only in thecase of the direct learning architecture) RF PA characteristics [1,15,19,21]. On theone hand, the model must be able to accurately represent the PA nonlinear behaviors


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at compressed gain regions, as well as to predict low-frequency and high-frequencydynamic effects, the former attributed to bias circuit and transistor self-heating andthe latter to matching networks. On the other hand, to reduce the computational com-plexity to an acceptable degree, low-pass equivalent behavioral models that relate just

the complex-valued envelopes at the RF PA input and output are mandatory.One class of dynamic nonlinear systems that is widely used for the low-pass equiv-

alent behavioral modeling of RF PAs is the Volterra series [24]. In fact, it was shown in[2] that only a subset of the discrete-time odd-order Volterra contributions is necessaryfor the low-pass equivalent PA modeling. Volterra series are linear in the parameters,and therefore, the parameter identification can be performed by standard linear tech-niques,liketheleast-squaresalgorithm.Inpractice,atruncatedseriesisused.However,the number of parameters needed to have an accurate Volterra model representationincreases very fast with the polynomial order and the memory length.

It is well known that fading memory systems, such as Volterra model kernels, can beexpanded by using an infinite series of orthonormal basis functions (OBFs), assumingthat the basis used in such representation is complete. OBFs present several interestingproperties and play an important role in many areas such as signal processing, control,and identification [6,14,17,25,26]. The expansion into an OBF is also a useful strategyto reduce the number of parameters required to have an accurate Volterra model rep-resentation. Concerning the low-pass equivalent PA behavioral modeling, such OBFVolterra models may have fewer parameters than classical Volterra (CV) models dueto the faster convergence of the corresponding expansion coefficients. However, the

reduction in the number of parameters in OBF models depends on the basis functionchoice and on the basis function pole choice, also referred to as dynamics.A well-known class of OBFs is the one based on rational functions parameterized

by one or more dynamics, such as the Laguerre basis, Kautz basis, or Takenaka–Malmquist basis. Takenaka–Malmquist functions are generalizations of the Laguerre(Kautz) functions. Indeed, Laguerre (Kautz) functions are built using only one real(complex) dynamic, while Takenaka–Malmquist functions are parameterized by pos-sibly multiple complex dynamics [16]. The CV model may be regarded as a specialcase concerning the OBF Volterra models. In particular, the CV model is obtainedwhen the poles of the OBF Volterra models are all set to zero. In this context, althoughLaguerre or Kautz basis can provide good approximations, more generalized bases(with possibly multiple complex poles) such as Takenaka–Malmquist basis can pro-vide better approximations using the same number of parameters.

Low-pass equivalent OBF Volterra models based on Laguerre expansion havealready been used for RF PA behavioral modeling in [28]. In [8], a variation of thestandard Kautz basis was also proposed, where the orthonormal function set for eachnonlinearity order of the Volterra model was parameterized by a single complex pole.However, to our knowledge the use of the most general case provided by Takenaka–Malmquist functions possibly with multiple complex dynamics for each nonlinearityorder, and a method to search for the function dynamics, has not been exploited in thecontext of the RF PA low-pass equivalent behavioral modeling.

Therefore, our paper proposes to apply the low-pass equivalent Volterra modelswith Takenaka–Malmquist expansion, namely Takenaka–Malmquist–Volterra (TMV)models, for the RF PA modeling. In addition, an approach based on state-space real-


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izations is used to describe such models. The set of poles which parameterizes themodel functions is chosen by using a constrained nonlinear optimization approach.

The paper is organized as follows. Some background information related to low-pass equivalent PA behavioral modeling is given in Sect. 2. In Sect. 3, the class of

low-pass equivalent OBF Volterra models is described. In particular, the use of modelsbased on Takenaka–Malmquist function sets and state-space realizations is motivatedin this section. Section 4 shows how TMV models can be extracted even withoutprior information about the basis function dynamics. In Sect. 5, the accuracy of theTMV model is fairly compared with the accuracies of the CV and the OBF Volterramodels previously reported in the literature for the low-pass equivalent PA behavioralmodeling. Finally, Sect. 6 addresses the conclusions of this work.

2 Low-pass Equivalent PA Behavioral Modeling

In order to amplify the power of an RF signal, a PA transfers energy from DC powersupplies to RF power at the output. In this process, some amount of power is dissipatedin the PA internal circuit. One strategy to improve the efficiency of the conversion fromDC to RF output power is to drive the PA at high power levels. However, in doingthat, significant nonlinear mechanisms are generated at the PA circuit, related to stronggain compression and saturation, which considerably deteriorates the quality of theamplifiedRFsignal.Infact,aspectralanalysisoftheRFoutputsignalclearlyillustratesthat the RF original signal is corrupted by the presence of significant spectral regrowththat increases the signal bandwidth. Moreover, non-ideal frequency responses of thePA bias and matching circuits also contribute to distortion of the amplified RF signal.Hence, accurate estimations of the amplified RF signal can only be achieved if thePA behavioral model is able to represent dynamic and nonlinear behaviors, as well astheir different kinds of interactions.

Let the PA input be excited by the real-valued RF signal:

x (t ) = e

˜ x (t )e j ωct

= R(t ) cos(ωct + θ (t )) (1)

where R(t ) and θ (t ) are the real-valued amplitude and angle components, respectively,of the complex-valued envelope signal ˜ x (t ), ωc is the RF carrier frequency, and t isthe continuous-time variable. Then, the PA output signal can be represented by:

y(t ) = e

˜ y(t )e j ωct

= S (t ) cos(ωct + ϕ(t ) + θ (t )) (2)

where S (t ) and ϕ(t ) + θ (t ) are the real-valued amplitude and angle components,respectively, of the complex-valued envelope signal ˜ y(t ).

In discrete-time PA behavioralmodels, the computational complexityvariesaccord-ing to the sampling interval. Shorter sampling intervals demand larger computationalcost, while longer sampling intervals require reduced complexity. Indeed, in a PAbehavioral model that relates the real-valued RF signals, x (t ) and y(t ), the samplinginterval is constrained to be very short, on the order of the reciprocal of the carrier


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frequency, to obey the Nyquist sampling criterion. In this way, the number of pre-vious samples required to an accurate prediction of the long-term memory effects isextremely high (on the order of a thousand past samples). For linearization purposes,where computational cost is a bottleneck, a low-pass equivalent description, work-

ing just with the complex-valued envelopes signals, ˜ x (t ) and ˜ y(t ), provides a moreappropriate approach. Actually, excellent predictions can be obtained by a low-passequivalent model requiring a small number of previous samples.

In wireless communication systems, a PA manipulates narrowband RF signals hav-ing bandwidths (on the MHz range) much lower than the carrier frequency (on theGHz range). To maximize the power transfer between the input and the output, band-pass passive filters with high quality factors, called matching networks, are designedin the PA circuit. In this way, a PA can only amplify in-band signals, e.g., signals hav-ing non-null energy only on the vicinity of the carrier frequency, and moreover, any

measurement taken on the PA output is constrained to be an in-band signal, once theout-of-band contributions (e.g., signals having non-null energy only on the vicinity of harmonic frequencies of ωc) are filtered by the output matching network. Exploitingsuch band-pass nature of PAs can significantly contribute to reduction in the compu-tational complexity of a low-pass equivalent behavioral model.

In low-pass equivalent models, the distinction between in-band and out-of-bandcontributions is not straightforward because spectral analysis can no longer be appliedto exactly determine the frequency location of a particular contribution. Indeed, thereduction in the sampling frequency in a low-pass equivalent model is only possible

because contributions physically located at very distinct regions of the frequencyspectrum (e.g., at baseband, fundamental, and harmonic bands) are allowed to sharethe same portion of the frequency spectrum. As a consequence, in low-pass equivalentmodels, extra effort must be spent on how to apply and manipulate complex-valuedenvelope signals in a way that the integer one that multiplies the carrier frequency isnot modified by the low-pass equivalent model. In fact, in doing that, it is guaranteedthat all the estimations generated by the model are in-band contributions, once thoseare the only contributions that can improve the accuracy of the model estimations.

3 OBF Volterra Series Generalization Based on Takenaka–MalmquistFunctions and State-Space Realizations

A Volterra series is the combination of a nonlinear system expressed in a Taylor seriesand a one-dimensional convolution integral representation for dynamic linear systems.The resulting multidimensional convolution integrals have been used in different areasof study for the description of time-invariant, causal, nonlinear dynamic systems withfading memory. The constitutive equation of a discrete-time Volterra series is givenby [13]:

y(n) =

P0 p=1

M 0τ 1=0

M 0τ 2=τ 1

· · ·

M 0τ p =τ p−1

h p(τ 1, . . . , τ p)

pi =1

x (n − τ i ) (3)


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where x (n) and y(n) are the input and output signals at the sample time n, respectively,P0 is the polynomial order truncation, and M 0 is the memory length. In (3), theassumption that the Volterra kernels, h p(τ 1, . . . , τ p), are symmetric functions of theirarguments was exploited. The kernel symmetry assumption does not imply any loss

of generality.Assuming a polynomial approximation at the RF system level, (3) relates the RF

output signal y of (2) to the RF input signal x of (1). To estimate the output signalat a single time instant, it is required the knowledge about the input signal at thatparticular time instant and also over a period of time in past. Indeed, due to the pres-ence of memory effects, each input applied at an arbitrary time instant is immediatelysensed by the output and continues to be sensed by the output at future time instants.In particular, the effects of past input samples over the instantaneous output samplebecome negligible only after the time interval between them is longer than the mem-

ory length of the slowest memory effect, which is usually called long-term memoryeffect, attributed to the bias circuit and on the order of the reciprocal of the envelopebandwidth.

Therefore, an accurate PA behavioral model at the RF level using (3) must formulatethe instantaneous sample of the output based on the instantaneous and previous (up tothe memory length M 0) samples of the input. Hence, the number of previous samplesrequiredto provide an accurateprediction of the long-term memory effects is extremelyhigh (on the order of a thousand past samples).

Following the procedure described in [2], the low-pass equivalent representation

of (3) that only generates physical in-band contributions, describing the relationshipbetween the complex-valued envelopes ˜ x (n) and ˜ y(n) at the PA input and output,respectively, is given by:

˜ yC V (n) =

P p=1

M τ 1=0

M τ 2=τ 1

· · ·

M τ p =τ p−1

M τ p+1=0

M τ p+2=τ p+1

· · ·

M τ 2 p−1=τ 2 p−2

×h̃2 p−1(τ 1, . . . , τ 2 p−1)

p

i =1˜ x (n − τ i )

2 p−1

i = p+1˜ x ∗(n − τ i ) (4)

where 2P − 1 = P0 is the polynomial order truncation, M is the memory lengthtruncation, h̃2 p−1(τ 1, . . . , τ 2 p−1) are the low-pass equivalent Volterra kernels, and(·)∗ denotes the complex conjugate. Kernel symmetry was assumed in (4). Observethat the integer one that multiplies the carrier frequency ωc is preserved in (4) becauseall the contributions are composed by the product of present (or past) samples of the complex-valued input envelope, in a way that the number of non-conjugate inputsamples is always equal to the number of conjugate input samples plus one. For similaraccuracies, the sampling frequency ( f s) required by the low-pass equivalent model of (4) is much lower than the sampling frequency demanded by (3). In fact, in (4) the f scan be set to several harmonics of the bandwidth of the envelope signal.

In this paper, a low-pass equivalent model obtained from (4) is referred to as aCV model. The terminology “Classical” is used to emphasize that the model does notpresent any basis function expansion of their kernels. As already pointed out, although


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low-pass equivalent models have lower computational complexities than models at theRFsystemlevel,CVmodelsstillleadtoahugenumberofparameters,sincethenumberof parameters increases very fast with P and M , and M is usually large to include allthe PA memory effects. In particular, [9] reports an analytical expression relating the

number of parameters in (4) as a function of P and M .It is well known that Volterra models with fewer parameters can be obtained if the

Volterra kernels are expanded into an infinite series of OBFs [18,27]. Concerning low-pass equivalent Volterra modeling, each kernel is approximated by a series truncatedin N 2 p−1 + 1 terms:

h̃2 p−1(τ 1, . . . , τ 2 p−1) ≈

N 2 p−1k 1=0

· · ·

N 2 p−1k 2 p−1=0

ck 1,...,k 2 p−1

2 p−1i =1

φ2 p−1,k i (τ i ). (5)

The CV model structure then becomes an OBF Volterra model structure described as

˜ yOBF(n) =

P p=1

N 2 p−1k 1=0

N 2 p−1k 2=k 1

· · ·

N 2 p−1k p =k p−1

N 2 p−1k p+1=0

N 2 p−1k p+2=k p+1

· · ·

N 2 p−1k 2 p−1=k 2 p−2

×ck 1,··· ,k 2 p−1

pi =1

l2 p−1,k i (n)

2 p−1i = p+1

l∗2 p−1,k i (n), (6)

where ck 1

,...,k 2 p−1

are the kernel expansion coefficients and l2 p−1,k (n) is the output of the function φ2 p−1,k (n) to the PA input ˜ x (n), which can be expressed as a function of the forward shift operator q , q ˜ x (n) = ˜ x (n + 1), as

l2 p−1,k i (n) =

M τ i =0

φ2 p−1,k i (τ i ) ˜ x (n − τ i ) = Φ2 p−1,k i (q) ˜ x (n). (7)

From now on, we define p = 2 p − 1 and the subscript associated with k in (5), (6),and (7) is omitted to simplify the notation.

For each nonlinearity order ( p) in (6), the set of orthonormal functions {Φ p,k (q)}

is completely parameterized by the set of poles {a p,k }. In general, the poles {a p,k }can assume real or complex values. Considering that low-pass equivalent modelsrelate complex-valued envelope signals, complex poles are not constrained to appearin conjugate pairs. Concerning OBF Volterra model approximations, the expansioncoefficients ck 1,...,k p may converge faster to zero if the model basis functions present avariety of poles and if these poles are selected in an optimal way. Therefore, when theOBF Volterra models are compared to the CV models under the same modeling accu-racy, low-pass equivalent models with a reduced number of parameters are obtainedby the OBF Volterra models.

3.1 Expansion Using Takenaka–Malmquist Functions

The Takenaka–Malmquist functions can be derived by imposing the orthonormalityproperty to the sequence of rational functions [6]


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Table 1 Volterra modelgeneralization

Model Takenaka–Malmquist function dynamics

CV a p,k = 0

LV a p,k = α p

KV a p,k = α p + jβ pTMV a p,k = α p,k + jβ p,k

Φ p,k (q) = qd q − a p,k

; d = 0 or 1 (8)

where a p,k can be either real or complex such that |a p,k | < 1. This procedure isessentiallya Gram–Schmidt orthonormalizationconstructionand leads to the so-calledTakenaka–Malmquist functions [6,16]

Φ p,k (q) = qd

1 − |a p,k |2

q − a p,k

k −1r =0

1 − a∗

p,r q

q − a p,r

; k = 0, . . . , N p (9)

It can be observed that when the functions in (9) are applied in (6), the choice of d determines if the resulting TMV model will be causal or strictly causal. The PAinstantaneous output ˜ y(n) is strongly dependent on the PA instantaneous input ˜ x (n),

and therefore, d = 1 represents the most appropriate choice for the basis functions.The generalization of the low-pass equivalent Volterra models based on the

Takenaka–Malmquist functions, for d = 1, is presented in Table 1. As shown in Table1, the simplest choice for the dynamics of {Φ p,k (q)} corresponds to {a p,k } = 0,which implies

Φ p,k (q) = q−k k = 0, · · · , N p (10)

and l p,k (n) = ˜ x (n − k ). In addition, if N p = M for p = 1, 3, . . . , P0 (or, equiv-

alently, for p = 1, . . . , P ), then the model structure of (6) is equivalent to the CVmodel structure in (4). Such CV models are widely applied in PA models even if prior information about the system dynamics cannot be introduced in the model basisfunctions.

In contrast, a priori knowledge about the system dynamics can be incorporatedinto a Laguerre–Volterra (LV) model. In LV models, the so-called Laguerre functionsgiven by

Φ p,k (q) =q

1 − α2 p

q − α p

1 − α p q

q − α p

k ; k = 0, . . . , N p (11)

are used. In this case, each set of orthonormal functions {Φ p,k (q)} in (9) is parame-terized by one dynamic which assumes a real value α p . In [28], the authors proposedsimplified LV models for the low-pass equivalent PA behavioral modeling, assumingthat all the model functions are parameterized by a single real dynamic {α p } = α .

In the same context, models with one complex pole parameterizing each set of Takenaka–Malmquist functions {Φ p,k (q)} were proposed in [8]. The authors referred


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to these models as Kautz–Volterra (KV) models. We will also use this terminology,although we acknowledge that (two-parameter) Kautz functions are differently definedin [6,16,26].

Table 1 also shows that the most general choice for the dynamics of {Φ p,k (q)} leads

to the TMV models, where the poles for each nonlinearity order of the OBF Volterramodel can assume different (real or complex) values. In these models, it is possible toincorporate a wide variety of system dynamics. For example, six different dynamicscan be incorporated into a third-order ( P0 = 3) TMV model with N 1 = N 3 = 2,whereas the LV and KV models with the same orders would have at most two differentdynamics introduced.

3.2 TMV Model Representation by State-Space Realizations

The TMV model output depends on the function output signals l p,k (n) =Φ p,k (q) ˜ x (n) for p

= 1, 3, . . . , P0. Therefore, the P0th order OBF Volterra modeloutput in (6) is rewritten as follows:

˜ yOBF(n) = H(l1(n), l3(n), · · · , lP0 (n)), (12)

where H(·) is a nonlinear operator acting on the state vectors l p (n):

l p (n) =

l p,0(n) · · · l p, N p (n)T

, (13)

and (·)T denotes the transpose. To calculate each state vector, firstly let us assume thatTakenaka–Malmquist functions in (9) are denoted by Φ p,k (q) for d = 0; then onecan define the pth state equation as

l p (n + 1) = A p l p (n) + B p ˜ x (n), (14)

where the pair of matrices A p ∈ C( N p +1× N p +1) and B p ∈ C

( N p +1×1) are com-pletely defined by the set of poles {a p,k } and constructed in such a way that the statesof (14) are given by

l p (n) =

Φ p,0(q) · · · Φ p, N p (q)T

˜ x (n)

=

l p,0(n) · · · l p, N p (n)

T

. (15)

This approach allows the shifted function output signals l p,k (n) = qd Φ p,k (q) ˜ x (n)

(where d = 1) to be obtained by the following state transformation in (14)

l p (n) = q l p (n) ⇒ l p (n) = q−1l p (n). (16)


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1 − a∗p ,0

z

z − ap ,0

1 − a∗p ,1

z

z − ap ,1

· · ·

1 − a∗p ,N

p

z

z − ap ,N p

X (z) =

Ú p ,0(z)

Ý p ,0(z) =

Ú p ,1(z)

Ý p ,1(z) =

Ú p ,2(z) Ý p ,N

p(z)

Fig. 1 Cascade connection of the N p + 1 first-order all-pass filters

Then, l p (n) can be written as

l p (n + 1) = A p l p (n) + B p ˜ x (n + 1). (17)

Observe that, since strictly causal Volterra models are more appropriate for the RF PA

behavioral modeling, we can use (17) rather than (14) to calculate the state vectorsl1(n), l3(n) , . . . , lP0 (n) throughout a set of input samples ˜ x (n), n = 0, . . . , K . Assuggested by [26], to get rid of the initial condition effects, it is possible to takel p (0) = 0 and consider each state vector l p (n) only for n ≥ n 0 (instead of n ≥ 0).

In (17), A p and B p for each nonlinearity order p can be obtained from a cascade

connection of N p + 1 first-order all-pass filters as presented in Fig. 1. Each filter of the cascade connection has a state-space realization defined in the Z -domain as

z L p,k ( z) = A p,k L p,k ( z) + B p,k Ú p,k ( z), (18)

Ý p,k ( z) = C p,k L p,k ( z) + D p,k Ú p,k ( z), (19)

where X ( z) = Ú p,0( z) and L p,k ( z) are the Z -transforms of ˜ x (n) and l p,k (n), respec-tively. In this approach, the matrices A p,k , B p,k , C p,k and D p,k are defined as

A p,k B p,k C p,k D p,k

=

a p,k

1 − |a p,k |2

1 − |a p,k |2 −a∗ p,k

, (20)

so that the transfer function from the input Ú p,k ( z) to the state L p,k ( z) is given by

L p,k ( z)

Ú p,k ( z)= ( z I − A p,k )

−1 B p,k =

1 − |a p,k |2

z − a p,k , (21)

and the input–output relation of each filter is satisfied, that is,

Ý p,k ( z)

Ú p,k ( z)=

1 − a∗ p,k

z

z − a p,k = C p,k ( z I − A p,k )

−1 B p,k + D p,k . (22)


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Hence, matrices A p and B p are obtained from the overall cascade constructionshown in Fig. 1, according to:

A p =

A p,0 0 · · · 0

B p,1C p,0 A p,1 · · · 0 B p,2 D p,1C p,0 B p,2C p,1 · · · 0 B p,3 D p,2 D p,1C p,0 B p,3 D p,2C p,1 · · · 0

..

....

. . ....

B p, N p D p, N p −1

· · · D p,1C p,0 B p, N p D p, N p −1

· · · D p,2C p,1 · · · A p, N p

B p =

B p,0 B p,1 D p,0

B p,2 D p,1 D p,0 B p,3 D p,2 D p,1 D p,0

..

.

B p, N p D p, N p −1

· · · D p,0

(23)

4 Model Extraction

Volterra models based on the OBF expansion are linear in the expansion coefficients.Hence, standard linear algorithms such as the least squares can be used to identifythe TMV model parameters throughout a set of data samples [ ˜ x (n), ˜ y(n)]. These data

samples can be obtained from either measurements or simulations.Firstly, let us assume that there is a priori knowledge about the pole sets {a p,k } for

each nonlinearity order of the low-pass equivalent PA model. One can then define theobjective function

J (θ ) =

K n=0

˜ y(n) − ϕ(n)Tθ 2 (24)where the vector ϕ(n) contains all products l p,k (n)l p,k (n) · · · l

∗ p,k

(n) from (6) andθ is a parameter vector containing all the expansion coefficients ck 1,··· ,k p so that

˜ yOBF(n) = ϕ (n)Tθ .Itcanbeobservedthattheproducttermsin ϕ(n) for n = 0, . . . , K can be easily evaluated using the state-space realizations proposed in Sect. 3.

The estimate θ̂ which minimizes (24) with respect to θ , i.e.,

θ̂ = argminθ

J (θ ) (25)

is obtained by the standard least-squares solution

θ̂ = XHX−1 XHỹ, (26)where (·)H denotes the Hermitian transpose (conjugate transpose) and

X =

ϕ(0) ϕ(1) · · · ϕ(K )T

, (27)


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Table 2 Pole structures of the models

Nonlinearity order 1st 3r d 5th

CV model 0 0 0

0 0 0LV model −0.133 0.442 0.382

−0.133 0.442 0.382

KV model −0.105 + j 0.109 0.443 + j0.009 0.376 + j 0.012

−0.105 + j 0.109 0.443 + j0.009 0.376 + j 0.012

TMV model −0.166 − j 0.181 0.499 + j0.161 0.678 − j 0.013

+0.699 + j 0.147 0.581 − j0.138 0.473 − j 0.037

power ratios (PAPRs) of the input signals used for the modeling identification andvalidation are equal to 5.0 and 4.7dB, respectively.

The modeling accuracy of the proposed TMV, CV, LV, and KV models are carefullycompared under the same computational complexity. For a fair comparison among thedifferent models, the polynomial order truncation and the number of basis functionsfor each nonlinearity order were set to the same values for all the studied models.In particular, all the models are fifth-order Volterra models (P0 = 5) with two basisfunctions for each nonlinearity order ( N 1 = N 3 = N 5 = 1). There is no theoretical

justification for these choices. These specific values were empirically chosen becausethey provide an excellent trade-off between modeling error and number of parame-ters. Table 2 shows the pole sets for the first, third, and fifth nonlinearity orders usedby each model. These sets are obtained using the constrained nonlinear optimiza-tion approach described in Sect. 4, with initial pole sets taken at the origin, that is,{a p0,k

} = 0, ∀ p0, k , since it is considered that there is no prior information about thesystem dynamics. Here, we have used the interior point algorithm [3] included in theMATLAB® toolbox.

The modeling accuracy is evaluated based on error signals containing the differencebetween the desired and estimated outputs. Specifically, two metrics are computed:the normalized mean-square error (NMSE) and the adjacent channel error power ratio(ACEPR), according to their definitions reported in [9].

The performances of the models in terms of NMSE and ACEPR are reported inTables 3 and 4, respectively. The ACEPR computations have employed bandwidths of 3.84 MHz for the main and adjacent channels, as well as a 5-MHz separation betweenthe main and adjacent channels. It is clear that great improvements in NMSE andACEPRs (for the lower and upper adjacent channels) occur when the OBF Volterramodels are used. While the accuracies of the LV and KV models are very close toeach other (in fact, their pole sets shown in Table 2 are also very close), a clear andsignificant reduction in modeling error is provided by the TMV model. For instance,reductionsofupto1.6dBinNMSEandACEPRvaluesareobservediftheTMVmodelis used instead of the LV or KV models. In fact, by considering two different complexpoles for each nonlinearity order, considerable improvements in NMSE and ACEPRmetrics are achieved by the TMV model in comparison with previous approaches


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Table 3 NMSE performancesof the models

Model NMSE (dB)

CV −27.7

LV −36.2

KV −36.2TMV −37.8

Table 4 ACEPR performancesof the models

Model Upper ACEPR (dB) Lower ACEPR (dB)

CV −36.2 −35.4

LV −45.2 −43.7

KV −45.2 −43.8

TMV −46.8 −45.1

885 890 895 900 905 910 915−110

−100

−90

−80

−70

−60

−50

−40

−30

Frequency (MHz)

P o w e r S p e c t r a l D e n s i t y ( d B m / H z ) measured out

error for CV

error for LV

error for KV

error for TMV

Fig. 2 PSDs of the measured output signal and the error signals obtained from the different models

having a unique pole (either real or complex) for each nonlinearity. An illustration of the superior modeling accuracy of the TMV model is provided by Fig. 2, which showsthe power spectral densities (PSDs) of the error signals for the different models. ThePSD of the measured (and normalized) output signal is also included in Fig. 2 by tworeasons: to emphasize the bandwidth and the center frequency of the main channel,and to illustrate how the error power levels are small in comparison with the outputpower levels. Observe that the PSD of the TMV error signal is clearly lower than thePSDs of the other error signals, especially at the main and adjacent channels.

Up to this point, attention was paid to confirm, in this case study, the better per-formance of the TMV model with respect to the CV, LV, and KV models. From nowon, focus is turned on investigating how close the TMV estimations are from the mea-surements. Therefore, in Figs. 3, 4, and 5, only measured and TMV-estimated dataare included. Figure 3 shows the instantaneous (normalized) amplitude of the output


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0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

Normalized |Vin| (V)

N o r m a l i z e d | V

o u

t | ( V )

measured

TMV model

Fig. 3 Instantaneous AM-to-AM conversion: measured and estimated by TMV

0.2 0.4 0.6 0.8 1 1.2 1.4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Normalized |Vin

| (V)

P h a s e d i f f e r e n c e ( r a d )

measured

TMV model

Fig. 4 Instantaneous AM-to-PM conversion: measured and estimated by TMV

signal as a function of the instantaneous (normalized) amplitude of the input signal,e.g., the instantaneous amplitude modulation-to-amplitude modulation (AM-to-AM)conversion. Figure 4 shows the difference between the phase (or polar angle) compo-nents of the instantaneous output and input signals as a function of the instantaneous(normalized) amplitude of the input signal, e.g., the amplitude modulation-to-phasemodulation (AM-to-PM) conversion. The TMV estimations are in close agreementwith the measurements shown in Figs. 3 and 4. Figure 5 shows the real and imag-inary components of the measured and estimated outputs as a function of time. Asexpected, Fig. 5 also confirms that the TMV estimations are in great accordance withthe measurements.

At this point, the TMV, LV, KV, and CV models are applied to the modeling of thePA inverse transfer characteristic. To that purpose, the roles of the input and output


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0 0.5 1 1.5 2 2.5−1

0

1

time (us)

R e a l p a r t

0 0.5 1 1.5 2 2.5

−1

0

1

time (us)

I m a g i n a r y p a r t

measured

TMV model

Fig. 5 Top) real and bottom) imaginary components of the output envelope as a function of time: measuredand estimated by TMV

Table 5 NMSE performances of the different models when applied to the modeling of the PA inversecharacteristic

Model NMSE (dB)for P0 = 3

NMSE (dB)for P0 = 5

NMSE (dB)for P0 = 7

NMSE (dB)for P0 = 9

CV −25.1 −26.3 −26.8 −26.9

LV −27.2 −31.0 −32.5 −33.7

KV −27.9 −31.5 −32.9 −34.7

TMV −29.4 −32.4 −35.1 −36.1

Table 6 ACEPR performances of the different models when applied to the modeling of the PA inversecharacteristic

Model Worst ACEPR(dB) for P0 = 3

Worst ACEPR(dB) for P0 = 5



CV −32.5 −34.6 −34.7 −35.2

LV −32.7 −38.2 −39.4 −41.4

KV −33.4 −38.6 −40.3 −41.9

TMV −36.3 −40.8 −43.6 −44.1

signals are exchanged. Therefore, the complex-valued envelope measured at the PAoutput is applied as input for the different models, and the complex-valued envelopemeasured at the PA input is treated as desired output for the different models. Particularinstances of the different models, differentiated among them by the polynomial ordertruncation P0, are first identified and then validated. In all cases, each nonlinearityorder has exactly two basis functions. The nonlinear optimization approach described


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885 890 895 900 905 910 915−100

−90

−80

−70

−60

−50

−40

−30

Frequency (MHz)

P o w e r S p e c t r a l D e n s i t y ( d B m / H z ) measured out

error for CV

error for LV

error for KV

error for TMV

Fig. 6 PSDs of the measured output signal and the error signals obtained from the different models, whenapplied to the modeling of the PA inverse characteristic

in Sect. 4 is employed to obtain the pole sets. The initial poles for the LV model are allsetto0.15,andtheinitialpolesfortheKVandTMVmodelsareallsetto0 .15+ j 0.15.The modeling accuracy of the TMV, CV, LV, and KV models are compared under thesame computational complexity. Tables 5 and 6 report the NMSE and ACEPR results,

respectively. In Table 6, only the worst ACEPR result (between the ACEPR valuesfor the lower and upper adjacent channels) is reported. Again, the TMV model showsa higher accuracy than the LV, KV, and CV under the same number of parameters.In particular, for P0 = 7, the TMV improves the NMSE and ACEPR results by 2.2and 3.3dB, respectively, in comparison with the KV model. Furthermore, the superiormodeling accuracy of the TMV model is observed by the PSDs of the error signalsshown in Fig. 6.

6 Conclusion

This paper presents how the RF PA low-pass equivalent behavioral modeling can beperformed using Volterra models based on the Takenaka–Malmquist function expan-sion(TMV models). This new approach generalizes classical and OBF Volterra modelspreviously used in this context since Takenaka–Malmquist function sets enable eachpole to have its unique value (either real-valued, complex-valued or even zero). Theuse of state-space realizations for the TMV model representation is highlighted. Anew approach for selecting the model basis function poles using a constrained non-linear optimization is also proposed. Based on measurements taken on a GaN HEMTclass AB RF PA, it is possible to observe that TMV models with multiple complexdynamics can provide better approximations than CV, LV, and KV models under thesame computational complexity, quantified by improvements of 1.6dB in NMSE andof 1.6 and 1.3dB in upper and lower ACEPRs, respectively. Moreover, when appliedto the modeling of the PA inverse transfer characteristic, the proposed TMV proved


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superior performance than the KV model, quantified by improvements in the NMSEand ACEPR metrics by up to 2.2 and 3.3dB, respectively.

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