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MANUSCRIPT FOR IEEE MICROWAVE MAGAZINE — FINAL VERSION PUBLISHED IN VOLUME 19, ISSUE 1, PAGES 69-77, JAN.-FEB. 2018. 1

Competitive Linearity for Envelope Tracking:Dual-Band Crest Factor Reduction and

2D-Vector-Switched Digital PredistortionHarald Enzinger, Karl Freiberger, and Christian Vogel

Abstract—This is the preprint version of the paper describingthe winning solution of the 2017 IMS student design competition"PA linearization through DPD", published in IEEE MicrowaveMagazine, volume 19, issue 1, pages 69-77, Jan.-Feb. 2018.

As wireless communication standards evolve to supportever-higher data rates, the required linearity and bandwidthmust increase, which leads to higher energy consumption [1].The problem of high energy consumption has become more se-rious due to the wide-spread adoption of orthogonal frequencydivision multiplexing (OFDM), which is used in LTE andWiFi applications [2]. The low energy efficiency of OFDM-based systems results from the statistical distribution of OFDMsignals, combined with the efficiency characteristics of radiofrequency (RF) power amplifiers (PAs). OFDM signals spendmost of their time at low magnitudes, where the energyefficiency of PAs is low; however, they also exhibit peaksthat must be linearly amplified as well. A detailed analysisof the joint linearity-efficiency behavior of RF PAs, excitedby single-carrier and OFDM signals can be found in [3].

To reduce the energy consumption of wireless transmitters,several highly efficient PA architectures have been develo-ped [4]. For base stations, the Doherty PA dominates [5],while, for handset devices, envelope tracking is now widelyused [6]. The idea behind envelope tracking is to use a time-varying supply that tracks the instantaneous envelope of thecommunication signal. An important advantage of envelopetracking is that the supply modulator can be designed inde-pendently of the carrier frequency [6]. This makes envelopetracking very attractive for multi-band applications, wheresignals at different frequencies are sent through the same PA.

A common characteristic of many high-efficiency PAs istheir reduced linearity with respect to conventional PAs. Con-sequently, linearization is often required, with digital predis-tortion (DPD) being one of the most used techniques. Theimportance of DPD as a microwave technique is illustratedby the fourth edition of the student design competition "PALinearization through DPD", sponsored by IEEE MicrowaveTheory and Techniques Society (MTT-S) Technical Coordi-nating Committees 9 and 11, which took place at the 2017IEEE MTT-S International Microwave Symposium (IMS) inHonolulu, Hawaii, last June.

Harald Enzinger ([email protected]) and Karl Freiberger ([email protected]) are with the Signal Processing and Speech Communica-tion Laboratory at Graz University of Technology, Austria. Christian Vogel([email protected]) is with FH JOANNEUM, University of Applied Sciences,Austria and the Signal Processing and Speech Communication Laboratory atGraz University of Technology, Austria.

TABLE I: An overview of the score computation.

Target Score Conditions

ACPR −45 dB 1 point/dB (∗) —

NMSE −33 dB 0.5 point/dB (∗) (1)

Power 20 dBm 10 points/dBm (∗) (1) (2) (3)

Efficiency — 1 point/percent (1) (2) (3)

(∗) Relative to target value.(1) ACPR target is achieved.(2) NMSE target is achieved.(3) Power difference between bands is within ±0.5 dBm.

COMPETITION DETAILS

The aim of the 2017 DPD competition was to linearizean envelope tracking PA, driven by a concurrent dual-bandOFDM signal. The measurement setup, illustrated in Fig. 1,consisted of a vector signal generator (VSG), a signal andspectrum analyzer (SSA), and a DC power supply from Ro-hde & Schwarz. The device under test was an evaluation boardfrom Texas Instruments, equipped with the envelope trackingsupply modulator chips LM3290 and LM3291, and the multi-band, multi-mode power amplifier module SKY77621 fromSkyworks. Several months before the competition, the mea-surement setup was accessible over the internet, so studentsfrom all over the world could experiment with it and preparetheir DPD algorithms in MATLAB. Two weeks before thecompetition, the participants had the chance to compare theirinitial solutions in the first edition of the "Rohde & SchwarzDPD Cup" online competition. With seven teams taking part inthe online competition and eight teams competing in Honolulu,the number of participants was the highest since the firstpresentation of this event in 2014.

SIGNAL AND SCORING

The MATLAB code for signal generation, signal analysis,and score computation was provided by the organizers. Thecharacteristics of the input signal in the frequency and timedomain are shown in Fig. 2. Without DPD, the spectrum of theoutput signal in Fig. 2 (a) shows severe spectral regrowth. Thesignal waveform in Fig. 2 (b) shows rapid envelope variationscaused by the frequency separation of the two transmissionbands. The sampling rate was 122.88 MHz which is an integermultiple of the LTE subcarrier spacing of 15 kHz. Eachmeasurement contained 122880 complex baseband samples.


ATTENUATORS+

DIR. COUPLER

SSA (R&S FSW8)1950 MHz RF

wwwremoteUPCLab

SERVER

INTERNATIONALPARTICIPANTS

ENV.TRACKER

PASKY77621

DC PWR (R&S HMP4040)

VSG (R&S SMW200A) Texas Instruments

LM3290-91-EVM10 MHz REF

Fig. 1: The measurement setup of the 2017 DPD competition. See also www.dpdcompetition.com.

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Frequency (MHz)

-180

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-140

-120

-100

-80

-60

PS

D (

dB

m/H

z)

Input

Output

10 MHz 5 MHz

(a)

0 0.2 0.4 0.6 0.8 1

Time (microseconds)

0

0.2

0.4

0.6

0.8

1

Ma

gn

itu

de

(fu

ll-sca

le)

Input

Slow Envelope

Supply

(b)

Fig. 2: The dual-band OFDM signal: (a) power spectral density (b) time domain waveforms.

Scoring was based on four criteria: adjacent channel powerratio (ACPR), normalized mean square error (NMSE), outputpower, and drain efficiency. As TABLE I shows, output powerhad a strong influence on the score. The goal was clearlyto maximize the output power, while achieving the targetsfor ACPR and NMSE. Failing in either ACPR or NMSEimmediately leads to a score approaching zero.

SYSTEM ARCHITECTURE

Since the task was to linearize the PA in concurrent dual-band operation, we used a concurrent dual-band linearizationarchitecture [7]. In such an architecture, the dual-band signalis split into two single-band signals, which are processedjointly at a lower sampling rate. This renders the processingindependent of the center frequencies of the two bands.

To divide the dual-band signal into two single-band signals,we transformed the dual-band signal to the frequency domain,applied frequency translation, filtering and decimation, andconverted the individual bands back to the time domain. Torecombine the single-band signals, we used a similar method.The sampling rate of the single-band signals was 80 MHz.

CFR

s1DPD PAs2

Supply

u1u2 x2

x1y2

y1

z

Fig. 3: A block diagram of our linearization approach.

An overview of our system architecture is shown in Fig. 3.We follow the common practice of separating the signalprocessing before the PA into crest factor reduction (CFR)and DPD. The purpose of CFR is to limit the peaks of thesource signal, such that the operating range of the DPD isclearly defined. Within this range, the DPD ideally implementsthe inverse PA transfer characteristic, such that the PA outputsignal is a scaled replica of the CFR signal.

In our system architecture, the supply signal is derived fromthe CFR signal. We chose the CFR signal as the basis forthe supply signal because the supply signal should track theinstantaneous envelope of the PA output signal, which, afterthe DPD is operating, is very similar to the CFR signal.


ClipGain

s1

s2

Filter

Filter

Delay

Dual Band Clipper 1 Error Shaping Filter

Delays1

s2

ClipGain

Dual Band Clipper 2

u1

u2

g

Fig. 4: The dual-band CFR architecture.

Initially, we also experimented with a three-dimensional(3D) DPD architecture, where the supply signal is used asa third input to the DPD [8]. However, in the present setup,the 3D-DPD did not improve the linearization performance.

SUPPLY SIGNAL GENERATION

The supply signal generation was specified by the com-petition organizers and involved a two-step process. First, aslow envelope of the dual-band signal is computed from themagnitudes of the single-band signals by

uenv[n] =∣∣u1[n]

∣∣+∣∣u2[n]

∣∣ . (1)

Fig. 2 (b) shows that this slow envelope follows the peaks ofthe dual-band envelope. Second, the supply signal is derivedfrom the slow envelope by the memoryless shaping function

z[n] =(z6min + u6env,norm[n]

)1/6, (2)

where zmin is the desired minimum value of the supply signaland uenv,norm[n] is the normalized slow envelope, obtained bydividing uenv[n] by its maximum. As Fig. 2 (b) shows, thesupply signal tracks the slow envelope, but remains abovezmin = 0.2 at all times. Other methods for generating the dual-band envelope tracking supply signal are summarized in [9].

An important detail concerning the supply signal is thatit reaches the PA over a different path than the RF inputsignal. Any delay mismatch between these paths deterioratesthe linearity. To equalize the delay mismatch, we time-shiftedthe supply signal such that the ACPR was optimized.

CREST FACTOR REDUCTION

The many available CFR methods [10] can be divided intotwo types: 1) methods that add a certain amount of tolerabledistortion and 2) methods that do not add distortion, butmodify the signal generation. For the DPD competition, thesignal generation was fixed, so we had to use a CFR method ofthe first type. Most CFR methods of this type use some sortof clipping and filtering to trade-off more inband distortionfor less out-of-band distortion. In the variant proposed in[11], which was also used by previous winners of the DPDcompetition [12], [13], the filtering is applied on the errorsignal rather than on the clipped signal. In this way the desiredsignal is not affected by the filtering, and, consequently, wehad more flexibility for designing the error-shaping filter.

For the DPD competition we adapted the method from [11]to the dual-band case [14], which lead us to the CFR architec-ture in Fig. 4. We apply clipping on the single-band signals,but with the objective of reducing the crest factor of the dual-band signal. For this purpose, we estimate the peak envelopeof the dual-band signal from the single-band signals by

senv[n] =∣∣s1[n]

∣∣+∣∣s2[n]

∣∣ . (3)

If the estimated peak envelope senv[n] is larger than the desiredmaximum umax, the magnitude of the single-band signals mustbe reduced. This magnitude reduction is implemented by themultipliers in Fig. 4, which apply the time-varying gain

g[n] =

umax

senv[n]senv[n] > umax ,

1 otherwise .(4)

The resulting clipping distortion is very broadband and af-fects the inband and out-of-band performance nearly equally.However, practical targets for the out-of-band performance,measured by ACPR, are typically much stricter than thosefor the in-band performance, measured by NMSE, error vec-tor magnitude [15], noise power ratio [16], or error powerratio [17]. To account for this asymmetry, we use an errorshaping filter to trade-off more inband distortion for less out-of-band distortion. In the error shaping filter, the clipping erroris computed by subtracting the clipped signal from the originalsignal. Afterwards, the clipping error is filtered by a lowpassfilter to reduce its effect on the ACPR. Finally, the outputsignals are obtained by subtracting the filtered error signalsfrom delayed versions of the original signals.

Due to the error shaping filter, a certain amount of peak-regrowth occurs. To obtain a dual-band signal with a clearlydefined maximum magnitude, we added another dual-bandclipper after the error shaping filter. With the clipping thres-hold of the second clipper being equal to the first one, thedistortion produced by the second clipper could be toleratedwithout filtering.

To set the clipping threshold, we simulated CFR with100 realizations of the input signal and different clippingthresholds. These simulations showed that the ACPR andNMSE targets of TABLE I can be achieved with sufficientmargin by using a clipping threshold that produces a crestfactor of 8.6 dB.


abs

x1

x2

u1

u2

Model for

Band 1

absModel

forBand 2

(a)

Linear

IntraBand

Σ CrossBand

MixedBand

u1 x1

u1

u2

(b)

Fig. 5: The dual-band DPD structure: (a) top level, (b) model for band one.

STRUCTURE OF THE DIGITAL PREDISTORTER

Linearity with respect to the coefficients is an importantproperty of many DPD structures. This property allows theuse of efficient training algorithms, which are based on solvingsystems of linear equations. For concurrent dual-band systems,a model that is linear in the coefficients is given by

x1[n] =

I1∑i=1

c1,i Φ1,i

{u1[n], u2[n]

}, (5)

x2[n] =

I2∑i=1

c2,i Φ2,i

{u1[n], u2[n]

}, (6)

where c1,i and c2,i are coefficients, Φ1,i{·} and Φ2,i{·} arebasis functionals, i enumerates the basis functionals, and I1and I2 are the numbers of coefficients and basis functionalsper frequency band. For the DPD competition, we used basisfunctionals of the form

Φb,i

{u1[n], u2[n]

}=

ub[n−m0]∣∣u1[n−m1]

∣∣p1∣∣u2[n−m2]

∣∣p2, (7)

which are inspired by the generalized memory polynomial(GMP) [18]. With this type of basis functionals, the top-levelDPD structure can be drawn as in Fig. 5 (a), where "abs"represents the absolute value operation.

To further specify our model, we divided the basis functi-onals into four groups: linear basis functionals where both p1and p2 are zero, intra-band and cross-band basis functionalswhere either p1 or p2 is zero, and mixed-band basis functionalswhere both p1 and p2 are larger than zero. Using this classifi-cation, the model for band one can be drawn like in Fig. 5 (b).In the following discussion, we specify the behavior of eachof the blocks in Fig. 5 (b) in terms of equations. The modelfor band two has the same structure as the model for band oneand is obtained by swapping the band indices of all variables.

The linear block in Fig. 5 (b) contains a linear finite impulseresponse filter given by

x(linear)1 [n] =

M∑m=0

cm u1[n−m] . (8)

m1

m0 Static Nonlinearity

Dynamic Main Diagonal

Dynamic Lagging Diagonals

Dynamic Leading Diagonals

M

M

Fig. 6: The structure of the modified GMP.

The intra-band and cross-band blocks in Fig. 5 (b) containmodified versions of the GMP [18]. The model for the intra-band block is given by

x(intra)1 [n] =

M1∑m=0

P1[m]∑p=1

c(1)m,p u1[n−m]∣∣u1[n−m]

∣∣p+

K2∑k=1

M2[k]∑m=0

P2[k,m]∑p=1

c(2)k,m,p u1[n−m]

∣∣u1[n−m− k]∣∣p

+

K3∑k=1

M3[k]∑m=0

P3[k,m]∑p=1

c(3)k,m,p u1[n−m− k]

∣∣u1[n−m]∣∣p .

(9)

The model for the cross-band block is obtained from (9)by using u2 instead of u1 inside the magnitude operators.Compared to the original formulation of the GMP [18], weapplied the following modifications:

• We excluded the linear part, since it is modeled by aseparate block.

• We reformulated the third line of (9) such that only causalterms are included.

• We rearranged the sums and used summation limits thatdepend on the indices of previous sums.

The third modification allows us to use different nonlinearorders for different memory depths such that we could hea-vily prune the model without sacrificing too much modelingaccuracy. An illustration of the sample lags used by (9) withrespect to m0 and m1 from (7) is shown in Fig. 6.

The mixed-band block in Fig. 5 (b) contains a modified


RegionalModel 1

Regional Model K

ModelSelector

u1u2 x2

x1

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6

Magnitude Lower Band (full-scale)

0

0.1

0.2

0.3

0.4

0.5

0.6

Mag

nitu

de U

pper

Ban

d (f

ull-s

cale

)

Centroids of theVector Quantizer

(b)

Fig. 7: The 2D-vector-switched model: (a) block diagram, (b) modeling regions.

two-dimensional (2-D) memory polynomial [19], given by

x(mixed)1 [n] =

M∑m=0

P1[m]∑p1=1

P2[m,p1]∑p2=1

cm,p1,p2

× u1[n−m]∣∣u1[n−m]

∣∣p1∣∣u2[n−m]

∣∣p2. (10)

Again, we use summation limits that depend on the indices ofprevious sums to heavily prune the model.

In all models, we use both odd-order and even-order terms.Many authors exclude even-order terms from baseband mo-dels [20] because conventional even-order terms in passbandmodels produce no output near the carrier frequency [21].However, in [22], [23] we have analytically shown that even-order terms in baseband models correspond to modified, odd-symmetric, even-order terms in passband models that doproduce output near the carrier frequency. Including even-order terms in baseband models helps to keep the nonlinearorder low, which improves the numerical properties of the mo-del [23]. Results obtained by automatic pruning also indicatethat odd-order and even-order terms contribute approximatelyequally to the modeling accuracy [24].

To further increase the modeling accuracy, several authorshave proposed piecewise models [25]–[29]. These models cir-cumvent the problems associated with high-order basis functi-onals by using low-order basis functionals with higher locality.Most of them are based on splines, which enforce a certainlevel of continuity between the segments. An exception is thevector-switched model [26] that does not enforce continuity.Dropping the continuity constraint has two advantages: 1) itsimplifies the model formulation since any model can be usedfor the individual segments, and 2) it improves the numericalproperties since the support of each basis functional is confinedto the modeling region of the respective segment. Potentialproblems due to discontinuities between the segments havenot been observed when the model is trained by least squaresfitting with long signal vectors [26].

For the DPD competition, we adapted the vector-switchedtechnique to the dual-band case, which lead us to the 2D-vector-switched model in Fig. 7. Each of the regional modelsin Fig. 7 (a) contains the dual-band DPD structure discussed

above. The switching between the regional models is based onthe magnitude of the current sample in each of the frequencybands. For mapping sample magnitudes to modeling regions,we used a vector quantizer with manually selected centroidsas shown in Fig. 7 (b). In the overall model, we includedeight identical regional models, each having 72 coefficientsper frequency band. By using identical regional models, thevector-switched model can be implemented efficiently by asingle regional model with switched coefficients. The identifi-cation complexity grows linearly with the number of regionalmodels. Like in any other polynomial or Volterra-based model,Horner’s method can be used to efficiently compute the basissignals required for filtering and training [30].

TRAINING OF THE DIGITAL PREDISTORTER

Two architectures are commonly used to train digital pre-distorters: indirect learning and direct learning [31]. Bothoriginate from earlier work on adaptive control [32]. To give anoverview of these architectures, we introduce the signal vectorsu, x, and y, containing the CFR signal, the PA input signal,and the delay-, phase-, and gain-corrected PA output signal ofone frequency band. Similarly, we introduce the matrices U ,X , and Y , which contain columns of basis signals obtained bysending the respective signals of both frequency bands throughthe basis functionals of one frequency band model.

The training is applied for each frequency band and foreach modeling region of the 2D-vector-switched model. Fortraining the regional models, only the samples belonging tothe respective region are used. In the following, we discussthe training for one frequency band and one modeling region.

A model of the PA is given by the system of linear equations

y = X cPA , (11)

where cPA contains the PA coefficients. An postinverse modelof the PA is given by

x = Y cDPD , (12)

where cDPD contains the postinverse PA coefficients.


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Frequency (MHz)

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PS

D (

dB

m/H

z)

CFR

without DPD

DPD, 1st iter.

DPD, 10th iter.

(a)

0 0.1 0.2 0.3 0.4 0.5

Input Magnitude (full-scale)

-40

-20

0

20

40

Pha

se S

hift

(deg

ree)

0

0.1

0.2

0.3

0.4

Out

put M

agni

tude

(V

olt)

without DPD

DPD, 10th iter.

(b)

Fig. 8: The linearization performance: (a) power spectral density, (b) AM-AM and AM-PM.

In the indirect learning architecture, (12) is solved for cDPDand the result is used in the DPD, which implements

xDPD = U cDPD . (13)

A serious drawback of indirect learning is that solving (12)finds the optimal postdistorter, which is in general differentfrom the optimal predistorter. The goal of direct learning is tofind a better solution by directly optimizing the predistorter.This can be done by inverting the PA model in (11) usingiterative methods [33], or by closed-loop adaptation based onrepeated measurements [34]. Direct learning is in general moredifficult than indirect learning since the cascade of the DPDand the PA is not linear in the DPD coefficients. The objectivefor direct learning can be either to minimize the error signal

e = u− y (14)

or to optimize standard-relevant performance metrics [35].Adaptive algorithms for minimizing (14) either include anonlinear model of the PA [36], or they approximate the PAby a complex gain [37]. The latter leads to a rather simplealgorithm which practically converges to better solutions thanindirect learning [38]. This algorithm can be formulated asfollows. Estimate the error in the DPD coefficients by solving

e = U cerror (15)

for cerror and update the DPD coefficients by

c(i+1)DPD = c

(i)DPD + µ c(i)error , (16)

where i is the iteration index and µ is the step size.For the DPD competition, we initialized the predistorter

with indirect learning, followed by several iterations of thesimple direct learning algorithm described above. For thedelay and gain correction, we used the methods describedin [39]. By using the root-mean-square gain estimate fromthe measurement without the DPD in all iterations, the outputpower with the DPD converged to the same value as in thefirst measurement without the DPD. The phase correction wasbased on a least squares estimate of the complex gain.

PERFORMANCE EVALUATION

At the competition, each team had 20 minutes for training.After this period, a new realization of the input signal wasused to evaluate the score. Our solution achieved a scoreof 71.8 points, corresponding to an ACPR of -49.2 dB, anNMSE of -35.7 dB, an output power of 24.4 dBm, and adrain efficiency of 22.3 %. Higher drain efficiency, like in [8],can be reached with a dynamic bias, as discussed in [40].

Plots of the linearization performance are shown in Fig. 8.From Fig. 8 (a), we see that after ten iterations of training, theDPD reduces the out-of-band distortion to levels close to theCFR-induced distortion. The scatter plot in Fig. 8 (b) showsthe amplitude modulation to amplitude modulation (AM-AM)and amplitude modulation to phase modulation (AM-PM) forthe lower frequency band, before and after linearization.

CONCLUSIONS

The DPD competition is an excellent opportunity for youngresearchers to apply their digital signal processing skills on arealistic measurement setup. The direct comparison of soluti-ons also facilitates a benchmarking of research results, whichis otherwise very hard to achieve. The main ingredients forwinning the 2017 DPD competition were:

• a dual-band CFR based on clipping and filtering• a dual-band DPD based on a modified GMP• the 2D-vector-switched technique• direct learning with several iterations.

These ingredients combine several state-of-the-art methodsfor CFR and DPD with new extensions we have developedspecifically for the 2017 DPD competition.

ACKNOWLEDGMENTS

We thank Prof. Pere L. Gilabert, Prof. Gabriel Montoroand Mr. David López-Bueno from Universitat Politècnica deCatalunya (UPC), and Hermann Boss from Rohde & Schwarzfor the excellent organization of the 2017 DPD competition.The research leading to these results has received fundingfrom the Austrian Research Promotion Agency FFG under theproject numbers 835187 and 4718971.


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