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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 2, FEBRUARY 2014 485

Generalized Rational Functions forReduced-Complexity Behavioral

Modeling and Digital Predistortionof Broadband Wireless Transmitters

Meenakshi Rawat, Member, IEEE, Karun Rawat, Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE,Shubhrajit Bhattacharjee, and Henry Leung, Member, IEEE

Abstract— In this paper, we present and analyze rational-function-based digital predistortion (DPD) of transmitters forbroadband applications where system noise and prominent mem-ory effects contribute to the overall nonlinearity of the system.The performance is reported for simulation and measured resultsfor gallium nitride (GaN)-based class-AB and laterally diffusedMOS (LDMOS)-based Doherty power amplifiers (PAs) usingthree different wideband code division multiple access signalswith peak-to-average-power ratios of around 10 dB. The perfor-mance of the proposed model, in terms of normalized mean-square error, adjacent channel power ratio, matrix conditionnumber, and coefficient dispersion, is compared against thoseof a memory polynomial (MP) model and a previously proposedrational-function-based model. It is shown by simulation andmeasurement that the previously proposed absolute-term denomi-nator rational functions have limitations in the inverse modelingneeded for DPD. A new variation of the rational function isproposed to alleviate this limitation. Depending on the type ofPA and signals, a floating-point operation reduction of 8%–38%is reported as compared with a low-complexity MP model.

Index Terms— Adaptive filters, digital signal processing,modeling, nonlinear dynamical systems, predistortion, 3G mobilecommunication, transmitters.

I. INTRODUCTION

THE power amplifier (PA) is the most nonlinear compo-nent in a wireless transmitter, which is responsible for

most of the out-of-band distortion in the presence of envelope-varying third- and fourth-generation broadband signals [1].Accurate estimation of PA nonlinearity has been attemptedusing subsampled temporal data in order to avoid inher-ent modulator imperfections during modeling [2]. Moreover,

Manuscript received February 16, 2013; revised May 17, 2013; acceptedMay 22, 2013. Date of publication September 4, 2013; date of current versionJanuary 2, 2014. This work was supported in part by the iRadio Laboratoryteam, in part by the sponsors of the laboratory, in part by the Alberta InnovateTechnology Futures, in part by the Chair Program, in part by the CanadaResearch Chair Program, in part by the Canada Foundation of Innovation,and in part by the National Science and Engineering Council of Canada. TheAssociate Editor coordinating the review process was Dr. John Lataire.

M. Rawat, F. M. Ghannouchi, and H. Leung are with the Uni-versity of Calgary, AB T2N1N4, Canada (e-mail: [email protected];[email protected]; [email protected]).

K. Rawat is with the Indian Institute of Technology, Delhi 110 001, India(e-mail: [email protected]).

S. Bhattacharjee is with SNC Lavalin T&D, Calgary, AB T2B 3G4, Canada(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2013.2278598

frequency-domain approaches have also been reported bytaking the best linear approximations [3], [4].

A rather simple approach requires the measurement oftime-domain input/output voltage waveforms and behavioralmodeling in the digital domain [1], [5]. Digital-domain behav-ioral modeling has a more practical and popular applicationas indirect digital predistortion (DPD) to suppress spectralregrowth and signal distortion, where an inverse digital modelbased on input/output waveforms is synthesized and appliedin the baseband domain before the PA, to compensate for thePA nonlinearity.

At present, digital-domain compensation is favored inmany fields [6], [7] due to its compatibility with software-enabled transmitters and the simplicity of its implementationthrough the utilization of high-speed digital signal processorsand digital-to-analog converters [8]. Moreover, software-basedimplementation offers high reconfigurability for any change intransmitter hardware and configuration, and also supports theconcept of software-defined radio at the transmitter end foreffective communication [9].

In this paper, we propose a novel rational-function behav-ioral model for digital-domain baseband modeling of PAsalong with the DPD application. Section II provides a briefintroduction to PA behavioral modeling, the indirect DPDconcept, the need for inverse modeling, and the metricsused for PA behavioral modeling and DPD in this paper.Section III describes the experimental setup, signals, anddevices used for the measurements and indirect DPD imple-mentation. Section IV describes the rational-function-basedmodeling and DPD, and presents the proposed model. SectionV shows various modeling results for DPD. It is proven thatthe model offers certain advantages over memory polynomial(MP) models, such as reduced number of coefficients and lowcomplexity, while achieving comparable or better modelingperformance for practical PAs. Section VI reports the experi-mental results achieved for the proposed model compared witha state-of-the-art low-complexity MP model.

II. PA BEHAVIORAL MODELING AND INVERSE MODELING

FOR INDIRECT DIGITAL PREDISTORTION

A PA should provide a linearly amplified version of an inputsignal at a higher power. A typical application of a PA is

0018-9456 © 2013 IEEE

486 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 2, FEBRUARY 2014

Fig. 1. Block diagram of a transmitter, and data extraction for behavioral modeling in the digital domain.

found in a wireless communication transmitter, where the PAis the last active component in the transmitter chain before theantenna. Fig. 1 shows a schematic of a transmitter system [10].Data bits are encoded into digitally modulated waveforms.In the baseband, the input signal x(n) = I (n) + j Q(n) iscomprised of complex data, as shown in Fig. 1. Digital data areconverted to an analog signal and up-converted to the requiredcarrier frequency, which is then amplified using the PA andtransmitted to the receiver.

The PA contributes the most distortion to the transmitteroutput, due to its nonlinearity at high input power levels;therefore, modeling of the PA is an important technique,which is used in the simulation of the physical layer for acomplete communication system. Behavioral modeling of thePA captures the nonlinear relation between the complex inputdata x(n) and the complex output data y(n) available in thebaseband, as shown in Fig. 1. Any variation in the signalenvelope affects the amplitude and phase of the output signal,which are referred to as the amplitude modulation/amplitudemodulation (AM/AM) and the amplitude modulation/phasemodulation (AM/PM) characteristics of the PA.

An effective behavioral model should imitate the AM/AMand AM/PM characteristics accurately. The normalized mean-square error (NMSE) is the most popular metric for determin-ing the modeling performance of a PA, which is given by

NMSEdB = 10 log10

⎛⎜⎜⎜⎝

N∑k=0

|e(n)|2

N∑k=0

|ymeas.(n)|2

⎞⎟⎟⎟⎠ (1)

where N is the total number of samples, and

e(n) = ymeas.(n) − yest.(n) (2)

is the complex error between the measured output ymeas.(n)and the estimated model output yest.(n), for any sample n. TheNMSE is considered a measure of in-band performance [11].

The adjacent channel error power ratio (ACEPR) is used asa metric to assess the out-of-band modeling performance [11].

1G

Fig. 2. Principle of indirect learning architecture for DPD.

Its expression is given as follows [11]:

ACEPR= 1

2

⎛⎜⎜⎜⎜⎜⎜⎝

f1−�+ BW2∫

f1−�− BW2

|E( f )|2 d f +f2+�+ BW

2∫f2+�− BW

2

|E( f )|2 d f

f2∫f1

∣∣Ymeas.( f )2∣∣d f

⎞⎟⎟⎟⎟⎟⎟⎠

(3)

where E( f ) and Ymeas.( f ) are the discrete Fourier transformsof the error signal e(n) and ymeas.(n), respectively; f1 and f2define the limits of the output signal band in the frequencydomain; and BW defines the bandwidth of the adjacent channelat � frequency offset from the carrier frequency.

Digital behavioral models can also be used for DPD tocompensate for the distortion from the PA nonlinearity. Fig. 2shows the indirect learning architecture (ILA) for DPD. Theinput signal is passed through predistorter K with parame-ters α, creating the predistorted signal u(n). This predistortedsignal u(n) is passed through the PA to provide output y(n),which is passed through predistorter K to create u. ILA workson the principle that, when u(n) = u(n), x(n) = y(n)/G ory(n) = Gx(n), which represents the required linear poweramplification.

For the first iteration, u(n) = x(n), and DPD is a model thatmaps x(n) = K (y(n)/G), where K represents the predistorternonlinear function. The modeling of the complex input, with

RAWAT et al.: GENERALIZED RATIONAL FUNCTIONS FOR REDUCED-COMPLEXITY BEHAVIORAL MODELING 487

respect to the normalized PA output, is the inverse of the PAbehavioral model; therefore, DPD modeling is also knownas inverse modeling. This inverse modeling is the foremostrequirement for the successful application of a DPD model.Note that the AM/AM characteristics of a PA show compres-sion at high power levels, while the AM/AM characteristics ofan inverse PA model should have expansion characteristics athigh power levels; therefore, the requirements for PA modelingand PA inverse modeling are quite different.

Apart from PA nonlinearity, prominent memory effects dueto the use of wider band signals further contribute to signaldeterioration at the output of the transmitter [12]. Severalbehavioral models and inverse models for nonlinear PAs withmemory effects have been proposed in the literature, offering agood inverse model for indirect DPD in most cases [13]–[22].Among these models, the lookup table (LUT)-based modelsare generally assumed to be simple to implement; however,optimal spacing and proper bin selection are required forbest results [14], [15]. Moreover, when memory effects areconsidered, a large number of variables need to be stored forcascaded LUTs [16], [17].

Among the parametric models, the Wiener andHammerstein models assume that memory effects arelinear and, hence, can be separated from the memorylessnonlinear behavior of the PA [18]. However, it has beenestablished that this is not a valid assumption for all PAs[19]. Volterra models [20] and neural network models [21],[22], with theoretical support for their modeling properties,can be considered the most successful models, but their goodperformance is shadowed by their computational complexity.The Volterra series [9] is given as follows:

y(t) = H1[x(t)] + H2[x(t)] + · · · Hn[x(t)] + · · · (4)

Hn[x(t)] =∫ ∞

−∞· · ·

∫ ∞

−∞hn(τ1, · · · τn)x(t − τ1) . . .

× x(t − τn)dτ1 . . . dτn (5)

where integral Hn is called an nth-order Volterra operator.Although theoretically justified, the Volterra model, includ-

ing all nonlinear kernels, is too complex in practice to beidentified correctly. Some pruning approaches have, therefore,been proposed in order to keep only the required kernels,including the generalized MP [19] and pruned Volterra models[20], [23], where off-diagonal terms are pruned to eliminateineffective terms and keep only the effective terms. However,such pruning approaches have several practical shortcomings.For instance, the Volterra model given by (4) and (5) canextend to infinite values, and many selections may providesimilar solutions, and all combinations need to be consid-ered. Hence, a general direction for complexity reduction ishard to define. A pruned model can, therefore, create largeerrors, even with a small variation of the PA characteris-tics, due to the mutual dependencies of cross terms andthe numerical instabilities arising from the large size of thematrices.

These reductions of Volterra models may prove to be excel-lent, from a behavior modeling perspective, for a model thatonly needs to be identified once. However, DPD applications

Fig. 3. Measurement setup for data acquisition, preprocessing, and DPDevaluation.

may require frequent updating of the model coefficients andprocessing of the input signals when the PA characteristicschange, due to environmental and self-heating effects. There-fore, in addition to good modeling capability, simplicity ofthe model and a low number of coefficients are also desirablerequirements [24].

Due to these limitations, the simplified Volterra model withonly diagonal terms, also known as the MP model, seemsto be an attractive choice, due to its reasonable accuracy,compact parametric representation, simplicity, availability ofnoniterative least squares (LS) solutions, and the ability tomodel nonlinear and linear memory effects [25]. The MPmodel is given as [19]

y(n) =P∑

p=0

M∑m=0

ap,mx(n − m) |x(n − m)|p (6)

where P and M are the nonlinearity order and memory depth,respectively.

In this paper, we propose a rational-function-based modelthat aims to achieve a similar modeling performance as theMP model with fewer coefficients. This model is presentedin Section IV, along with previous rational-function-basedattempts for PA modeling.

III. DEVICE UNDER TEST AND EXPERIMENTAL SETUP

Fig. 3 shows the PA characterization setup used for dataextraction in forward and inverse PA modeling. A basebandsignal is uploaded into the vector signal generator (AgilentE4438C), where the signal modulation and frequency up-conversion to the required carrier frequency are carried out.The radio frequency (RF) modulated signal, which is tobe transmitted using an antenna, is amplified via the PA.For modeling and DPD, this PA output is down-converted,demodulated, and captured in the digital domain using a vectorsignal analyzer (Agilent 4440). The captured data are sent tothe digital processing unit (laptop/handheld computer), usinga general-purpose interface bus (GPIB). More details on eachcomponent of experimental setup can be found in [26].

The baseband output data are time aligned with the inputdata by using cross correlations between sampled input andthe output data. Normalized cross correlation ρxy between two


waveforms x and y is calculated as follows:

ρxy = 1

L

L∑n=0

(y(n) − μy)(x(n+τ ) − μx)

σyσx(7)

where μx and μy represent the mean values of the input (x)and output (y) waveforms, respectively; σx and σy representthe standard deviations of the x and y waveforms, respec-tively; L is the length of waveforms x and y and τ is thesample delay between the two waveforms. ρxy is observedwith the increase in τ values, and the τ value for whichρxy is the maximum represents the sample delay betweenx and y.

The ratio of this sample delay and sampling rate pro-vides time delay between two baseband waveforms. There-fore, the accuracy of the delay estimation is limited bythe sampling rate of the device. A detailed discussionon the fine-tuning of the delay alignment can be foundin [27]. This time-aligned input–output data can be usedfor PA modeling. Digital signal processing for time align-ment and model identification are carried out in MATLABsoftware.

In DPD, the output digital data are normalized with thesmall-signal gain of the PA. The normalized output data areused as the model input, and the original input data areused as the desired output to achieve an inverse model. Thecoefficients of each layer are copied to the predistorter, asshown in Fig. 3. Using this experimental setup, the modelingperformance of the proposed model was assessed for twodifferent PAs.

1) A commercial laterally diffused MOS (LDMOS)Doherty PA (Powerwave Technologies) with a saturationpower of 300 W at a carrier frequency of 2.14 GHz. ThePA had a gain compression of 1.7 dB [with an overallgain variation of 3.5 dB as shown in Fig. 4(a)] and aphase compression of 33° when driven by a signal ofpeak-to-power ratio (PAPR) of approximately 10.5 dB.The drain voltage for both the peaking and carrier ampli-fiers was 28 V and the total drain current was 1.7 A.

2) A 10-W gallium nitride (GaN) class-AB PA biasedat a drain voltage and current of 28 V and 780 mA,respectively, at a center frequency of 3.5 GHz. ThePA had a gain compression of 4.3 dB and a phasecompression of 30°.

The AM/AM and AM/PM characteristics of these amplifiersare shown in Fig. 4. Clearly, the Doherty PA had morenonlinearity with a slight expansion and then compression, interms of AM/AM; whereas the class-AB PA had a simpler yethighly compressed AM/AM, leading to high distortion. BothPAs had a phase compression of approximately 30°. Threedifferent wideband code division multiple access (WCDMA)signals with 10 and 15 MHz of bandwidth and a PAPR of10.5 dB were used in the experiments. All the WCDMAsignals had a chip rate of 3.84 MHz, and the chips/slot lengthwas 2560. Each chip had 24 samples. The sampling frequencywas 92.16 MHz, and the signal was captured for 2-ms timeframe for data acquisition.

(a) (b)

Fig. 4. AM/AM and AM/PM characteristics. (a) Doherty PA. (b) Class-AB PA.

IV. RATIONAL-FUNCTION-BASED APPROACHES

FOR PA MODELING

A rational function is defined as the ratio of two powerpolynomials given by

y(n) = a0 + a1x(n) + . . . + aJ x J (n)

b0 + b1x(n) + . . . + bK x K (n)=

J∑i=0

ai x i (n)

K∑j=0

b j x j (n)

(8)

where x(n) is the input and y(n) is the output of the rationalfunction at instance n.

In [28], the AM/AM and AM/PM were captured separatelyusing (8), where x(n) represented the input signal amplitudefor a traveling wave tube (TWT) amplifier and a solid-statePA (SSPA). The amplifiers were considered static nonlinearsystems; therefore, memory effects were not considered. Pre-distortion was reported only for simulations, approximating theSSPA with a cubic spline and the TWT amplifier with Saleh’smodel [29], [30]; however, the effects of measurement errorsand noise were neglected, which are necessary to validate amodel’s usefulness in a real practical scenario.

Saleh’s model [31] was used to represent the TWT amplifieritself, which is a form of a rational function; and, it is expectedthat the use of another rational function should easily be ableto model DPD. A modified Saleh’s model was also proposedfor SSPAs, but it did not compensate for memory effects and ittargeted the behavioral modeling of the PA [32]. Consideringthe importance of memory effects, a behavioral model basedon a rational function was proposed in [33], where memoryeffects were included as time delay taps in a complex-termnumerator and absolute-term denominator. The absolute-term-denominator rational function (ADRF) proposed in [33] isgiven by

y(n) =∑Kn

i=0

∑Mnmn=0 ai,mo x(n − m) |x(n − m)|2i

1 + ∑Kdj=0

∑Mdmd=0 b j,md |x(n − m)|2 j+1

(9)

where x(n) and y(n) are the input and output baseband signals,respectively; Kn and Kd are the nonlinearity orders for thenumerator and denominator, respectively; and Mn and Md arethe memory depths used in the numerator and denominator,respectively.

It was proposed in [33] that (9) converges to a finite valuefor large x(n) only when Kn = Kd = K and Mn = Md = M .Using a power series expansion, (9) can be rewritten as


(a) (b)

Fig. 5. NMSE performances of the MP and ADRF models of a Doherty PA. (a) PA modeling. (b) PA inverse modeling. X-axis represents nonlinearity orderP for MP model and K for ADRF model.

follows:

y(n) =⎛⎝1 −

K∑j=0

M∑md=0

b j, md |x(n − md)|2 j+1 + · · ·⎞⎠

×K∑

i=0

M∑mn=0

ai,mn x(n − mn) |x(n − mn)|2i . (10)

Thus, (10) contains odd- and even-order monomials, such asai,mn b j,md x(n − mn) |x(n − mn)|2i |x(n − md)|2 j+1, depend-ing on the values of i and j . Theoretically, the use ofonly odd-order terms should provide good baseband modelingperformance; however, it has been proven that the inclusion ofeven terms is essential for PA modeling and DPD, providinga richer basis function set and leading to much better in-bandand out-of-band performances with an overall reduced com-plexity [34], [35].

The performance of the ADRF model has been reportedfor PA behavioral modeling [33], which captures the AM/AMcompression characteristics; however, indirect DPD (explainedin Section II) for transmitters needs inverse modeling of thePA, which requires modeling of expansion characteristics. Tothe best of the authors’ knowledge, the use of a rationalfunction for inverse modeling (modeling of expansion char-acteristics) and DPD application has never been reported forpractical SSPAs containing measurement noise and memoryeffects.

Fig. 5 shows PA modeling and PA inverse modelingperformances, in terms of NMSE, for the ADRF modelproposed in [33] and the MP model. The x-axis of thegraph represents nonlinearity order P for the MP model

and Kn = Kd = K for the ADRF model. The PA undertest is the Doherty PA [AM/AM and AM/PM characteristicsshown in Fig. 4(a)] driven with a WCDMA11 signal (two-carrier WCDMA). We can perceive the following pointsfrom Fig. 5.

1) Convergence curves and the best performance achievedfor both models were different for the PA model and thePA’s inverse model.

2) Inclusion of delay taps M led to lower NMSE per-formances for both models, but as memory length(i.e., number of delay taps) increased, the performanceimprovement diminished. No performance improvementwas observed for M > 2.

3) The best NMSE performances of the ADRF and MPmodels for PA behavioral modeling were almost similar.

4) For PA inverse modeling, the ADRF model providedbetter NMSE performance for low nonlinearity orderthan that of the MP model. However, the best NMSEperformance of the ADRF model was ultimately notas good as the MP model for inverse modeling perfor-mance.

5) The MP model tended to converge to a constant valuewith the increase in the nonlinearity order; however,the curve for the ADRF model started diverging afterconverging to a best performance.

Indeed, the accuracy of the inverse model is of utmostimportance for a successful DPD operation. Moreover, it isalso desired that the number of coefficients be as low aspossible, in order to reduce the number of floating-pointoperations (FLOPs) per each input signal datum, leading tooverall savings on computation during transmission. In this


paper, we present a novel and modified rational-function modelfor application in DPD as a compromise between the ADRFmodel and the MP model, which provides better performancethan the ADRF model with lower number of coefficients thanthe MP model.

A. Rational Function With Memoryless Flexible-OrderDenominator Model

It can be perceived from (10) that, for any selected nonlin-earity order K , the ADRF model attempts approximation of apower series with a nonlinearity order of 2 × (2K+1), whichprovides the capability of modeling a highly nonlinear systemusing a low-power-term rational function. However, the ADRFmodel proposed in [33] has the following constraints.

1) It includes only odd-order power terms in the numeratorand denominator.

2) The numerator and denominator have the same nonlin-earity orders (Kn = Kd ).

3) The numerator and denominator terms have the samememory depths (Mn = Md ).

4) The model denominator is an absolute term that doesnot include signal phase information.

Due to the rigidity of the ADRF model, the best PA inversemodeling performance achieved with the ADRF model is notas good as that of the MP model. Therefore, we proposea dynamic rational function with memoryless flexible-orderdenominator (DRF-MFOD) model, which has memory effectsin the numerator polynomial, but has memoryless flexiblenonlinearity order in the denominator polynomial. Both thenumerator and denominator have even- and odd-order termpolynomials, resulting in a richer basis function domain tobe utilized in modeling. The proposed DRF-MFOD model isgiven as

y(n) =

Nn∑j=0

M∑m=0

a j,m x(n − m) |x(n − m)| j

1 +Nd∑

i=0bi x(n) |x(n)|i

(11)

where x(n) and y(n) denote the nth sample of the PA inputand output complex signals, respectively; M is the memorydepth; Nn and Nd are the nonlinearity orders in the numeratorand denominator polynomials, respectively; and a j,m and bi

denote complex coefficients for the numerator and denomina-tor, respectively.

Equation (11) represents the multiplications of static (mem-oryless) and dynamic (with memory) parts, after the powerseries expansion of the denominator can be rewritten as

y(n)=⎛⎝

Nn∑j=0

M∑m=0

a j,m x(n−m) |x(n−m)| j

⎞⎠

×⎛⎜⎝1−

Nd∑i=0

bi x(n) |x(n)|i +⎛⎝

Nd∑i=0

bi x(n) |x(n)|i⎞⎠

2

+. . .

⎞⎟⎠.

(12)

From (12), we find that the proposed model approximatesa case that has an Nn -order nonlinearity for memory-effectterms and an (Nd + Nn)-order nonlinearity term for the staticbranch of the MP. Using this representation, there is flexibilityto adjust the nonlinearity orders for the static and memory-tapbranches, which can be searched iteratively. The absence ofmemory terms in the denominator reduces the total number ofcoefficients to identify for the model.

An iterative search is part of the model selection for thedevice under test. This iterative search is carried out in adigital signal processor in an offline mode, which requires a3-D search for the denominator and numerator orders and thememory depth. This is achieved by the creation of a modelfor different values of Nn , Nd , and M and the comparison oftheir NMSE performances. An initial value of 0 is assumedfor each parameter, which is then incremented in steps of 1;and, performance improvement is observed. The dimensionvalues for which performance improvement becomes constantor deteriorates with increases in the dimensions are selected tobe appropriate model dimensions. Once the model dimensionsare selected, they are kept constant for any particular PA. Anyfluctuations in PA behavior due to environmental changes canbe adjusted by updating model coefficients without changingmodel dimensions.

One can argue that, unlike the ADRF model, the proposedDRF-MFOD model is not bounded for large x(n) values.However, the power series convergence criterion with aninfinite limit is not applicable for PA modeling and inversePA modeling, due to the following practical constraints.

1) Practical PAs have a drive limit, which is the inputpower saturation point (Pin,sat). After this limit, PAsare overdriven and may get damaged; therefore, inputsignals always have finite values bounded by the Pin,satvalue and can never tend to infinite/very large values fornonlinear nonclipped PA modeling applications.

2) For DPD applications, output power is normalized withthe gain of PA; therefore, the normalized output signal isa distorted version of the input signal within the powerranges of the actual input signal. Due to the PA inputpower drive limitation, as described in the first point ofthis list, the input for the inverse model can also onlyhave finite values.

3) It has been established that input values in either PAor inverse PA modeling are finite values; therefore, theradius of convergence is a finite value depending on thevalues of x(n), as well as on the complex coefficients.

B. Rational-Function Parameter Extraction

The DRF-MFOD model given by (11) can also be writtenin a recursive form

y(n) = −y(n)

Nd∑i=0

bi x(n) |x(n)|i

+Nn∑j=0

M∑m=0

a j, m x(n − m) |x(n − m)| j. (13)


U1(n) =

⎡⎢⎢⎢⎢⎢⎢⎣

−y(n)x(n) ... −y(n)x(n) |x(n)|Nd

−y(n − 1)x(n − 1) ... −y(n − 1)x(n − 1) |x(n − 1)|Nd

.

.

.

.

.

.

.

.

.

−y(n − L)x(n − L) ... −y(n − L)x(n − L) |x(n − L)|Nd

⎤⎥⎥⎥⎥⎥⎥⎦

(16)

U2(n) =

⎡⎢⎢⎣

x(n) ... x(n) |x(n)|Nn ... x(n − M) ... x(n − M) |x(n − M)|Nn

x(n − 1) ... x(n − 1) |x(n − 1)|Nn ... x(n − 1 − M) ... x(n − 1 − M) |x(n − 1 − M)|Nn

... ... ... ... ... ...

x(n − L) ... x(n − L) |x(n − L)|Nn ... x(n − L − M) ... x(n − L − M) |x(n − L − M)|Nn

⎤⎥⎥⎦ (17)

Moreover, parameter extraction is possible by rewriting(13) in a matrix form

y = AφDRF−MFOD (14)

where y is the output vector with the dimension of L × 1 andL is the length of the training data; and, φDRF−MFOD is thecoefficient vector, which is defined as

φDRF−MFOD

= [b0, . . . , bNd , a00, . . . , aNn 0, a01, . . . , aNn 1, . . . , aNn M ]T.

(15)

In (14), the observation matrix is denoted by A = [U1 U2],where U1 is a Vandermonde matrix containing recursive termsand is given as (16), shown at the top of the next page.

The Vandermonde matrix U2 is similar to the MP modeland is given by (17), shown at the top of the next page.

From (14)–(17), it is clear that the model identificationequation (16) is linear in its parameters and φDRF−MFODis obtained using the LS solution of φDRF−MFOD =(AT A)−1AT y, which is implemented according to the singularvalue decomposition method. Based on an individual PA’s data,the denominator complex coefficient automatically takes smallvalues to keep the denominator stable. This is possible becausethe PA model is created within the data range that falls in thePA power handling range. Once the coefficients are extracted,the output is calculated using (11).

It has been reported in [36] that recursive models are closerto the physical analogy of the PA and provide robust model-ing. In recursive models, the coefficients are extracted usingthe stored values of x(n) and y(n). However, for practicalapplications such as DPD, y(n) at the present instance isnot known beforehand, which is a limitation when applyingany recursive model. However, for a rational-function-basedmodel, the output is calculated using (11). Therefore, evenwith recursive model extraction, DPD application does notrequire the current output; and, the predistorted inputs canbe synthesized independently using only the input signals.

V. INVERSE MODELING REQUIREMENT FOR INDIRECT

LEARNING DIGITAL PREDISTORTION

A. In-Band and Out-of-Band Modeling Performances

Figs. 6 and 7 show the NMSE performances of the DRF-MFOD model in comparison to the MP and ADRF models

for the two PAs (characteristics shown in Fig. 4) and twosignals. Iteratively, the best memory length is found to be 3for all models. This is achieved by increasing the memorylength (delay taps) and observing the corresponding decreasein the NMSE performance until no performance improvementis observed for a fixed nonlinearity order.

For a memoryless case, the MP model is represented as afunction of the nonlinearity order (P), and the ADRF modelis represented as a function of K . However, the DRF-MFODmodel requires a two-dimensional search for the best valuesof Nn and Nd . Once the model dimensions are known forall models, we introduce delay taps for the memory effectsand observe the decrease in the NMSE with the increasein the number of the delay taps. The best memory-depthlength/number of delay taps is achieved when no furtherperformance improvement is observed.

Fig. 6 shows the NMSE performances of the MP, ADRF,and DRF-MFOD models for different Nn and Nd values forthe memory length of 3. The general trend of the NMSEfor both PAs shows that the ADRF converges for a verysmall nonlinearity order, followed by the DRF-MFOD and MPmodels. The DRF-MFOD model converges faster than the MPmodel, even for Nd = 0, due to the recursive nature of themodel.

For the class-AB PA and WCDMA101 signal, as shown inFig. 6(a), any increase in Nd in the lower Nn region leadsto a better performance. As the denominator does not containdelay taps, it leads to fewer coefficients. As an example, theMP model converges at N = 8 (8 × 4 = 32 coefficients),while a similar performance for the DRF-MFOD model can beachieved with Nn = 4 and Nd = 5 (4×4+5 = 21 coefficients).However, for the same PA when the WCDMA111 signal isused [Fig. 7(a)], the denominator terms in the DRF-MFODmodel have a negligible effect. It is also noted that, for theclass-AB PA with a WCDMA111 signal [Fig. 7(a)], the ADRFmodel also performs poorly.

One can observe from Figs. 6 and 7, that a maximumimprovement of 7 dB is achieved at Nn = 3 for the class-ABPA with the DRF-MFOD model compared with the MP model;and, a maximum improvement of 6 dB was achieved at Nn = 5for the Doherty PA for the DRF-MFOD model compared withthe MP model for both the WCDMA101 and WCDMA111signals. Clearly, due to the rational function properties, themaximum improvement is achieved for smaller nonlinearity


(a) (b)Fig. 6. NMSE performances for a WCDMA101 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K forADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

(a) (b)Fig. 7. NMSE performances for a WCDMA111 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K forADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

orders, as with the ADRF model; however, the transition ismuch smoother with the DRF-MFOD model.

Although the ADRF model also seems to providea comparable NMSE for the WCDMA101 signal, thisonly validates the in-band performance, as the NMSE is

more favorable to high-power data and is considered asan in-band performance figure of merit [11]. Therefore,as a metric to assess the out-of-band modeling perfor-mance, the ACEPR has much more significance for DPDapplications [11].


(a) (b)Fig. 8. ACEPR performance for a WCDMA101 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K forADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

(a) (b)Fig. 9. ACEPR performance for a WCDMA111 signal. (a) Class-AB PA. (b) Doherty PA. X-axis represents nonlinearity order P for MP model, K forADRF model, and Nn for DRF-MFOD model. Memory depth is 3 for all the curves.

Figs. 8 and 9 show the ACEPR performances for the twodifferent PAs and signals, and it can be observed that theDRF-MFOD and MP models have an improvement in the

ACEPR of 7 dB over that of the ADRF model for theclass-AB PA and a WCDMA101 signal. It is interesting tonote that, although each model converges to a different final


TABLE I

COMPLEXITY AND FLOPS PER DATUM

Fig. 10. Matrix condition number for the class-AB PA. X-axis representsnonlinearity order P for MP model, K for ADRF model, and Nn forDRF-MFOD model. Memory depth is 3 for all the curves.

performance, there are similar convergence trends for theNMSE and ACEPR.

B. Coefficient Dispersion and Matrix Conditioning

The MP and the proposed DRF-MFOD models requirean LS solution for model coefficient identification. Matrixconditioning is an important factor in the determination ofthe accuracy of the inversion operation in the LS algorithm,which is defined as [37]

Condition Number = λmax

λmin(18)

where λmax and λmin are the maximum and minimum singularvalues of the matrix to be inverted.

1

100

104

106

108

1010

0 2 4 6 8 10 12

M PAD RFDRF-M FO D (N

d=0)

DRF-M FO D (Nd=1)

DRF-M FO D (Nd=2)

DRF-M FO D (Nd=3)

DRF-M FO D (Nd=4)

DRF-M FO D (Nd=5)

Coe

ffic

ient

Dis

pers

ion

Nonlinearity Order

Fig. 11. Coefficient dispersion for the class-AB PA. X-axis representsnonlinearity order P for MP model, K for ADRF model, and Nn forDRF-MFOD model. Memory depth is 3 for all the curves.

A poorly conditioned matrix makes the pseudoinverse cal-culation very sensitive to slight disturbances.

It can also be shown that the condition number is an indi-cator of the transfer of error from the matrix to the solutions.As a rule of thumb, if a condition number is 10n , then onecan expect to lose at least n digits of precision in solving thesystem [38]. For example, the numerical precision is around10−7 for a single-precision floating-point calculation. Anycondition number greater than 102 leads to an approximateprecision of 10−5 and any condition number greater than 104

leads to an approximate precision of 10−3. Therefore, thecondition number should be as low as possible.

Another factor considered in this paper is the coeffi-cient dispersion, which is obtained as a ratio of maximumto minimum absolute coefficient value. If the coefficientsare much dispersed, the accuracy level of coefficientsin FPGA is compromised to cover the whole range ofcoefficients [39].


(a)

(b)

Fig. 12. Characteristics of the linearized class-AB PA. (a) AM/AM.(b) AM/PM.

Fig. 13. Predistorter performance with ACPR correction for the class-AB PA.(a) WCDMA11 signal with P = 8 for MP model and Nn = 4 and Nd = 6for DRF-MFOD model. (b) WCDMA111 signal with P = 8 for MP modeland Nn = 6 and Nd = 6 for DRF-MFOD model.

The Vandermonde-type matrices of (16) and (17) arenotoriously ill-conditioned for larger matrix sizes [40]. Ithas been claimed that the conditioning of a Vandermondematrix increases exponentially with its order [41]. It mayseem that the DRF-MFOD model matrix, which contains twoVandermonde-like matrices, may lead to a large conditionnumber; however, as the model converges for a smaller numer-ator order, the resultant matrix properties are similar to orbetter than that of the MP model. Fig. 10 shows the matrixcondition number for three models while Fig. 11 shows the

Fig. 14. Predistorter performance with ACPR correction for the DohertyPA. (a) WCDMA101 signal with P = 12 for MP model and Nn = 10 andNd = 0 for DRF-MFOD model. (b) WCDMA111 signal with P = 12 forMP model and Nn = 10, Nd = 0 for DRF-MFOD model.

coefficient dispersion with respect to the nonlinearity order(P , Nn , and K for MP, DRF-MFOD, and ADRF models,respectively) for the class-AB PA with a WCDMA101 signal.It can be observed that both the matrix condition numbers weresimilar, or slightly lower for the DRF-MFOD model (Nn = 4and Nd = 5) than for the MP model (P = 8), for the bestNMSE and ACEPR performance.

It is interesting to note that the ADRF model matrixcondition number trends are very similar to those of theMP model; however, the value of the condition number islarger by a factor of 100 compared with the MP model.The proposed DRF-MFOD model has a unique patternwhere the matrix condition number is almost constant forNn< (Nd+ 1).

C. Complexity Comparison

Table I shows the total number of FLOPs required for cal-culating the output of the ADRF, MP, and the proposed DRF-MFOD models. The equivalent numbers of FLOPs for eachoperation are based on [42]. The ADRF model has the lowestnumber of FLOPS however; the out-of-band performanceof the ADRF model is unacceptable for DPD applications.The proposed DRF-MFOD model needs fewer FLOPs forsimilar performance, and therefore, can be substituted forthe MP model for providing similar performance with lowercomplexity. It can be observed from Table I that the DRF-MFOD model requires 182 fewer FLOPs than the MP modelfor a class-AB PA. Similarly, in the case of a Doherty PA,the DRF-MFOD model requires 44 fewer FLOPs than theMP model. Note that these calculations are done only forsingle-input sample; however, for continuous transmission,each input needs to be processed through the predistorter, andthe saving in the number of FLOPs will be much significantcumulatively.

In the above case, we assumed that the system is undercontrolled environmental conditions and that the inverse modelcoefficients need to be extracted only once to identify thepredistorter digital model. However, where frequent processingis required due to changes in the environmental conditions,established linear adaptive algorithms such as the recursiveLS (RLS) and the least mean squares can be applied to (13)


TABLE II

MEASURED ACPR ACHIEVED WITH LINEARIZED PAS

instead of LS. The details of these algorithms can be foundin [27].

VI. APPLICATION IN DIGITAL PREDISTORTION

Fig. 12 shows the AM/AM and AM/PM characteristicsof the class-AB PA, the DPD (inverse PA characteristics)response using the proposed model, and the resulting lin-earized response. The excess scattering in the gain and phasecharacteristics is due to memory effects, which give multi-ple values of gain and phase for a single-input value. Ascan be seen from the figure, the DPD derived using theproposed model is an accurate inverse of the PA charac-teristics; and, the resulting DPD and PA characteristics arelinear.

The performance of the proposed DPD scheme is veri-fied for the above-mentioned PAs and signals, in terms ofadjacent channel power ratio (ACPR) performance at ± 5,± 10, and ± 15 MHz, according to industrial norms [43].The power spectral density curves are shown in Figs. 13and 14, which illustrate the performance of the proposedDRF-MFOD model for the class-AB and Doherty PAs withvarious WCDMA signals. The WCDMA signals used inthe experiment have a PAPR of approximately 10.5 dB,and the output power for each PA is maintained at analmost constant level. These figures also show curves withoutDPD for comparison. It can be seen that the static model(i.e., memoryless model) fails to compensate for memoryeffects in each case, whereas the MP and DRF-MFOD mod-els provide reasonable and comparable performances, withthe DRF-MFOD model performing slightly better than theMP model.

Table II shows the ACPR performance for the differentPAs and signals. The ACPR is a frequency-domain evaluationmetric used to assess the DPD performance. It is defined asthe ratio of the power of the output signal in an adjacentchannel to the power of one of the in-band carriers and isgiven as

ACPRdBc = 10 log10

(∫ ω2ω1

|YDPD( f )|2 d f∫ ω4ω3

|Yin( f )|2 d f

)(19)

where YPSD denotes the power spectrum density of the lin-earized output signal; w1 and w2 denote the lower and upper

frequency limits of the adjacent channel, respectively; and w3and w4 designate the lower and upper frequency limits for thein-band channel, respectively.

For comparison purposes, the ACPR values of the memo-ryless, MP, and the proposed models are reported in Table II.The ADRF model’s values are not included, as the modelfailed to provide acceptable predistortion performance. Thishas already been established for out-of-band performance interms of ACEPR in Figs. 8 and 9.

The output power back-off, with respect to the saturationpoint, is selected to achieve the desired linearization perfor-mance. This is achieved by adding the gain expansion ofthe predistorter to the PAPR of the signal, while ensuringthe linearized PA is not overdriven. It can be seen fromTable II that, in the case of the class-AB amplifier drivenby a WCDMA11 signal, an improvement of 5 dB in theACPR is achieved compared with that of the memorylessmodel at an offset of 5 MHz. It can also be noticed thatin all cases, the ACPR values at ± 5, ± 10, and ± 15 MHzremain well below the spectrum mask specified for WCDMAsignals [43].

VII. CONCLUSION

In this paper, we proposed a novel dynamic rational-function-based PA model, taking memory effects of the PAinto consideration. The modeling and predistortion perfor-mances were evaluated using class-AB and Doherty PAswith two- and three-carrier WCDMA signals as excitationsignals. A study of the proposed model and comparisonswith a previously proposed rational-function and memory-polynomial techniques showed that the proposed model couldachieve comparable performance in terms of NMSE andACEPR, but with a reduced number of coefficients in com-parison with the established MP model. The matrix condi-tioning and coefficient dispersion performance of the newmodel were also maintained, similar to that of the MPmodel.

ACKNOWLEDGMENT

The authors would like to thank the reviewers, whoenhanced the quality of this manuscript immensely with theirthoughtful detailed comments.


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Meenakshi Rawat (S’09–M’13) received theB.Tech. degree in electrical engineering from theGovind Ballabh Pant University of Agriculture andTechnology, Pantnagar, India, in 2006, and the Ph.D.degree from the Department of Electrical and Com-puter Engineering, Schulich School of Engineering,University of Calgary, Calgary, AB, Canada, in2012.

She is currently a Post-Doctoral Research Fellowwith the iRadio Laboratory, University of Calgary.She was with Telco Construction Equipment Co.

Ltd., Jamshedpur, India, from 2006 to 2007, and Hindustan Petroleum Corpo-ration Ltd., Mumbai, India, from 2007 to 2008. She is currently a Reviewer forseveral international transactions and journals. Her current research interestsinclude signal processing for software defined radios, communications, andmicrowave active and passive circuit modeling using nonlinear models andneural networks.

Karun Rawat (M’08–S’09–M’13) received the B.E.degree in electronics and communication engineer-ing from Meerut University, Meerut, India, in 2002,and the Ph.D. degree from the Department of Elec-trical and Computer Engineering, Schulich Schoolof Engineering, University of Calgary, Calgary, AB,Canada, in 2012.

He is currently an Assistant Professor with theIndian Institute of Technology, Delhi, India. Hewas a Scientist with the Indian Space ResearchOrganization from 2003 to 2007. He was a Post-

Doctoral Research Fellow with the iRadio Laboratory, Schulich School ofEngineering, University of Calgary, from April 2012 to April 2013. He is areviewer of several well known journals. His current research interests includemicrowave active and passive circuit design and advanced transmitter andreceiver architecture for software defined radio applications.

Dr. Rawat was the leader of a University of Calgary team that won firstprize and the Best Design Award in the Third Annual Smart Radio Challengein 2010, conducted by the Wireless Innovation Forum.

Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07)is currently a Professor and iCORE/CRC Chair withthe Department of Electrical and Computer Engi-neering, Schulich School of Engineering, Universityof Calgary, Calgary, AB, Canada, and the Directorof the Intelligent RF Radio Laboratory. He has heldnumerous invited positions at several academic andresearch institutions in Europe, North America, andJapan. He has provided consulting services to anumber of microwave and wireless communicationscompanies. His research activities have led to over

500 publications, two books, and ten U.S. patents (three pending). His currentresearch interests include microwave instrumentation and measurements, non-linear modeling of microwave devices and communications systems, design ofpower and spectrum efficient microwave amplification systems and design ofintelligent RF transceivers, and SDR Radio systems for wireless and satellitecommunications.

Shubhrajit Bhattacharjee received the M.Sc.degree in electrical and computer engineering fromthe University of Calgary, Calgary, AB, Canada, in2011, and the B.Eng. degree from the University ofPune, Pune, India.

He is currently a Project Engineer with SNCLavalin T&D, Calgary. His current research interestsinclude RF transceiver impairment mitigation, soft-ware defined radio, transmitter linearization, digitalsignal processing, and channel modeling.

Henry Leung (M’90) received the Ph.D. degree inelectrical and computer engineering from McMasterUniversity, Hamilton, ON, Canada.

He is a Professor with the Department of Electricaland Computer Engineering, University of Calgary,Calgary, AB, Canada. He was with the DefenceResearch Establishment Ottawa, Ottawa, Canada,where he was involved in the design of automatedsystems for air and maritime multisensor surveil-lance. His current research interests include chaos,computational intelligence, data mining, information

fusion, nonlinear signal processing, multimedia, sensor networks, and wirelesscommunications.

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