Abstract – This paper presents a multiple LUT digital adaptivepredistorter based on a Hammerstein model that uses the returnchannel to feed back information from the receiver, concretelythe bit error rate (BER), in order to train and later adapt thespecific LUT gains that permit always operating at the bestback-off level. This new predistorter architecture is aimed atcoping with modern communication standards that use adap-tive modulation (such as IEEE 802.11 or IEEE 802.16) andtherefore continuously searching the best linear amplificationto maximize power efficiency at the time that a certain qualityof service (BER) in reception is guaranteed. Simulations pro-vided will show the advantages of this multi-LUT configura-tion, where in front of different channel conditions, linear andefficient amplification (minimum back-off) is achieved at thetime that a certain level of BER at reception is ensured.Index terms – Lookup tables (LUT), digital adaptive predis-tortion, Hammerstein models, adaptive modulation systems,peak to average power ratio (PAPR), bit error rate (BER).

I. Introduction

Power Amplifier (PA) linearization is a well knownproblem extendedly presented in literature, but alwaysopen since it has to continuously cope to new communi-cation scenarios where linear amplification is a must andpower efficiency is an important figure of merit. Sincethe former medium and short wave radio transmitters,many linearization techniques have been proposed [1]and significant advances have been reported [2]. Thecurrent allowance of high speed digital signal processors(DSP’s) not only have revived classical analogue solu-tions (such as the Kahn’s envelope elimination andrestoration [EE&R] or the Chireix ones [3]) but alsohave facilitated new approaches to the linearizationproblem. Few years ago, the use of digital processors forlinearizing PA’s was considered a good but dispropor-tionate solution because of the costs and the DSP ener-gy consumption. Nowadays, most of the communicationequipment already incorporates some digital processorfor mandatory issues within the wireless standards (i.e,coding, interleaving, OFDM…), which can be also usedto implement digital-based linearisers avoiding the needfor a specific digital device devoted to this function.Among all linearizers, some digital-based linearizersproposed in literature are: Linear amplification usingNonlinear Components (LINC, which is actually arevived Chireix structure) [4], and its variation CAL-LUM, the EE&R and RF, BB or IF Digital Predistortion.Most of the digital predistorters proposed in the recentpast years have been modeled by using memoryless

techniques. This PA memoryless model might be anacceptable approximation for narrowband signals (e.g.nearly-constant envelope modulations). However,memoryless predistortion has shown insufficient can-cellation performance since new multilevel modulationformats claim for power handling capability and high-er baseband bandwidth.Therefore, for current wideband multilevel modulationformats it is necessary to consider PA memory models.The proposed models used as memory predistortiondevices are usually based on Volterra series (or prunedVolterra) [5, 6], memory polynomials [7], Wienner-Hammerstein models [8], or neural networks [9].Moreover, current communication standards such asIEEE 802.11g, IEEE 802.16, ETSI HiperLAN-2,UMTS or ETSI DVB, look for high spectral efficiencyover moderate channel bandwidths (over 20 MHz), byusing multilevel modulations, multicarrier or combin-ing both (e.g. M-QAM, π/4 DQPSK, WCDMA, SingleCarrier or OFDM). These modulation schemes haveinformation in both amplitude and phase, thus becom-ing very sensitive to PA nonlinearities. Taking intoaccount that they present high peak to average powerratios (PAPR), significant back-off levels are requiredfor linear amplification, thus penalizing PA’s powerefficiency. In order to guarantee maximum power effi-ciency in power amplification, this paper presents amultiple LUT digital adaptive predistorter based on aHammerstein model (considering memory effects).The advantages of using multiple lookup tables fordigital predistortion regarding short-term variations inPA characteristics have been advanced in [10].The predistorter architecture presented in this paperuses the bit error rate (BER) information comingfrom the receiver (through the return channel speci-fied in most of modern wireless standards) to designtwo specific LUT’s for each modulation, providing anormal and a safe mode operation. Thus, this archi-tecture permits setting the best back-off level of oper-ation at the time that a certain quality of service(QoS) in reception is achieved.

Proceedings of the European Microwave Association Vol. 1; December 2005; 304-311

Received: July 28, 2005. Revised: September 12, 2005

Department of Signal Theory and Communications,Polytechnic University of Catalonia (UPC). Av. CanalOlímpic s/n, 08860 Castelldefels, Barcelona, Spain. Phone:+34 934137000; Fax: +34 934137032; E-mail: [plgilabert1,bertran2, montoro3]@tsc.upc.edu

Multiple lookup table predistortion for adaptive modulationPere Ll. Gilabert1, Eduard Bertran2, Gabriel Montoro3

EuROPEAN MICROWAVE ASSOCIATION - INVITED PAPER

PERE L.I. GILABERT ET AL.

Proceedings of the European Microwave Association 305

In contrast to previous digital predistorters proposals[5-11], this multi-LUT based predistorter solves theproblem of coping with different modulation formats,by adaptively calculating the optimum PA +Predistorter gain for operating as close to saturation aspossible in order to maximize efficiency.

II. Problem statement

Since some of modern communication standards per-mit adaptive modulation, that is, changing the modula-tion format depending on the channel conditions, dif-ferent PAPR’s will be considered for linear amplifica-tion. The input back-off can be defined as in [11]:

(1)

where denotes the input power that corresponds toan output saturated power, denotes the input aver-age power level, and PBO stands for the peak back-off,and it is defined as:

(2)

Being S the fraction (value ranging between 0 and 1) ofthe saturation power that is considered in order todefine the maximum input power fed at the PA input toobtain linear amplification:

(3)

So then, considering a communications transmitter thatuses different modulation formats presenting differentPAPR’s it seems reasonable to adapt the input back-offin the PA taking into account the specific modulationused at any time.In order to adjust the optimum back-off, it is possible to:– vary the input mean power level for each modulation

format. Then, some kind of additional power controlwill be required;

– alternatively, see eq. (1), maintaining fixed the meaninput power, to vary the overall chain gain, com-posed by the Predistorter plus the PA, thus varyingthe saturation power level.

Therefore, as it is shown in Figure 1, it is possible toset the input back-off level by adjusting the overallPredistorter + PA gain. Moreover, a multilevel modu-lated carrier only reaches peak envelope power (PEP)occasionally, as it can be observed in Figure 1. Most ofthe time the modulated carrier is at an average powerlevel significantly lower than the PEP. In order to pro-vide maximum efficient amplification, a certain levelof signal clipping will be sometimes desirable and tol-erable. So then, the objective of this paper is to proposean adaptive predistorter capable to find the best back-off level in order to obtain power efficient amplifica-tion and preserve a specific BER at reception.

III. Predistortion model

Although Volterra series is a general nonlinear modelwith memory, its predistortion is complex and its real-time implementation difficult. Apart from more complexmodels such Neural Networks, there are two simple buteffective possibilities for describing the predistortionmodel taking into account memory effects. Those arememory polynomials or Hammerstein (Wiener) models.But, memory polynomial can be seen as a particular con-figuration of a more general Hammerstein model.

A) Hammerstein modelHammerstein models are composed by a memorylessnonlinearity followed by a linear time-invariant system,as it is shown in Figure 2. While the Wiener model con-sists of the same subsystems but connected in the reverseorder. The predistorter based on a Hammerstein modelcan compensate the PA nonlinearities (by means of thememoryless nonlinearity block) as well as the PA fre-quency-dependent characteristics (by means of the lineartime-invariant system). The use of a FIR filter in the lin-ear time invariant block will correspond to a memorypolynomial model, as it is described in eq. (4):

( ), ,SAT AVGin inIBO dB P P PBO PAPR= − = +

,SAT AVGin inI P −

, AVGi inI P

10log( )

PBO S= −

( ), ,

0 1MAX S ATin inP P S S= ⋅ < <

Fig. 1. AM-AM curves of a PA considering different amplifica-tion gains. Fig. 2. Hammerstein model.

H(z)z Z

z zN

N

D

D= + +⋅ ⋅ +

+ + ⋅ ⋅ +

−−

− −

− −

β β βα α

0 1

1

1

1 1

1

11

( )x(k) y u(k)+ y u (k)+

+y u (k)+ y u (k)P-

P- P-

P

= ⋅ ⋅⋅ ⋅ ⋅ ⋅

0 1

2

2

1 1

LTI (linear time-invariant)Memoryless nonlinearity

MULTIPLE LOOKUP TABLE PREDISTORTION FOR ADAPTIVE MODULATION

306 Proceedings of the European Microwave Association

(4)

Where and are the input and output signals ofthe predistorter model, while arecomplex gains, N and P are the number of delays andthe order of the nonlinearity considered in the modelrespectively.In this paper we have considered to use an IIR filter inthe linear time invariant block, since IIR filters presentbetter resolution than FIR filters at the same filterorder, so then, reducing computational load in the DSP

B) Indirect learning of the predistorterTo identify the Hammerstein model parameters, themost common solution is the use of the indirect learn-ing structure (also called translation method [12]),depicted in Figure 3. In a first step, the post-distorterparameters are identified by using the PA output signaland the PA input signal multiplied by a particular lin-ear gain ( , see the PAPR of Fig. 1) whose valuewill be related to the PAPR of the input signal and thus,to its modulation scheme. Eq. (5) shows the resultinglinear amplification derived from the cascade connec-tion of the PA and the post-distorter:

(5)

Once the post-distorter coefficients are estimated, anexact copy of the post-distorter is translated in cascadewith the PA, so obtaining the predistorter.By observing the block diagram in Figure 3, the post-distorter (predistorter) learning process can be definedby the following equations:

(6)

(7)

Being y(k) the PA output and v(k) the output of thememoryless non-linear block of the Hammersteinmodel. P is the order of the memoryless polynomial,while N and D are the general zero/pole order respec-tively.By combining eq. (6) and (7) we obtain a closedexpression for the predistorter input-output:

(8)

Rewriting eq. (8) in a more compact matrix notation, itresults:

(9)

where

and

where vectors are noted in bold and matrices are notedin bold and with a hat. The superindex H denotesHermitian Transpose.

IV. Identification algorithms

In order to estimate the post-distorter coefficients, thecost function to be minimized (J(k)), is defined as themean square error between the output samples of thepost-distorter and the input samples (previously multi-plied by the desired linear gain) of the PA, as is shownin eq. (10)

(10) -

(11)

To minimize this cost function two algorithms are pro-posed: the Least Mean Square (LMS) algorithm and

x k b y k n y k nnpn

N

p

P P

( ) = ⋅ −( ) ⋅ −( )∑∑== 00

( )y n x n( ) 1.. ; 1...npb n N p P= =

ilinearG

( )( )postilinearF G G xx = ⋅

Fig. 3. Indirect learning architecture.

v k y k y kp

p

p

P

( ) = ⋅ ( ) ⋅ ( )∑=

γ0

˜ ˜( ) ( )x k x k d v k ndd

D

nn

N

( ) = ⋅ −∑ + ⋅ −∑= =α β

1 0

˜ ˜

( )

x k x k d

y k y k

dd

D

n p

p

p

P

n

N

( ) = ⋅ −( )∑ +

+ ⋅ ⋅ ⋅ ( )∑

∑

=

==

α

β γ

1

00

x k c qk

H

k( ) = ⋅

cd ,d ,..,d ,d ,d ,..,d ..

...,d ,d ,..,d ,.., ,..,k

P P

N N NP D

T

=

01 02 0 11 12 1

1 1 1α αdnp = ⋅α βn p

qk =

( ) ( ) ( ) −( ) −( ) −( )−( ) −( ) −( )

−( ) −( ) ( )

−

y k y k y k y k y k y k

y k N y k N y k N

x k x k

P P

P

T

x k D

,.. , ,.., ,

.., ,...., ,

˜ , ˜ ,..., ˜

1 1 1

1 2

e k x k x k x k( ) ˜= ( ) − ( ) = ( ) ⋅

J k e k x k x k( ) ˜= ( ) = ( ) ( )2 2

-

,

qk⋅

c qk

H

k⋅

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Proceedings of the European Microwave Association 307

the Fast-Kalman Filter. Other algorithms such LeastSquares or Recursive Least Squares are suitable fortraining the predistorter, but result more computationalcomplex for the adaptation process in front unexpectedchanges in the PA characteristics.

A) Least Mean Square (LMS)The well-known Least Mean Square algorithm isdescribed in eq. (12).

(12)

being m the error step (trade-off between speed of con-vergence and accuracy), and its bounds are:

(13)

where Tr[·] denotes the trace of the data (signal)matrix.

B) Fast-Kalman FilterFast-Kalman algorithms [13] use the optimum Kalmanfiltering technique to adaptively estimate the predis-torter coefficients without the need of knowing a pri-ori any transition matrix (unlike the conventionalKalman filter). The Fast-Kalman equations describingthe algorithm are [13]:

(14)

being,

(15)

where QM and are related to the estimation andmeasure error variance [13]. The matrix is recur-sively updated like:

(16)

C) Look-up table (LUT) coefficientsSince real-time adaptation algorithms would require ahigh speed DSP capable to calculate complex algorithmswithin a symbol period, the use of pre-trained look-uptables simplifies the DSP computational load and relaxesthe adaptation time constants. Therefore, one possibleimplementation of a Hammerstein based predistorter willconsist of the use of LUT’s to compensate the memory-less nonlinearities of the PA, followed by an IIR filter(see Fig. 4), to compensate possible memory effectsderived from electrical of thermal effects.But, for dividing the predistorter into a LUT plus a IIRfilter, it is necessary to firstly have the Hammerstein

parameters separately. Having a look at eq. (12) and/oreq. (14), it is possible to notice how by using the LMSor Fast-Kalman algorithms we obtain the ck coefficientvector, where . To identify gammas andbetas separately, it is possible to apply the two stagesidentification algorithm proposed by Bai in [14], con-sisting in a singular value decomposition of the matrix described in eq. (17).

(17)

where the matrix is defined as:

(18)

being the matrices andwhere are N-P-dimen-sional orthonormal vectors respectively, and

Finally, the gammas and betas are estimated asdescribed in eq. (19) and eq. (20).

(19)

(20)

where su denotes the sign of the first nonzero elementof µ1.Alternatively, by using the Narendra-Gallman (NG)algorithm [15], based on the estimation of the predis-torter coefficients by means of the Least Squares tech-nique, it is possible to separately obtain the alphas,betas and gammas coefficients. The NG algorithm canbe perfectly used to synthesize multiple LUT’s in thetraining process, but for later adaptation the LMS orthe Fast Kalman algorithms are better fitted since theyare simpler and faster.

V. Multiple lookup tables

The block scheme of the multi-LUT based predistorteris shown in Figure 4. It consists of a set of 2 × Q LUT’sand their associated IIR filters, being Q the number ofmodulation schemes considered. Each modulation for-mat has assigned two LUTs with different gains, thenormal gain and the safe mode gain, as well as theircorresponding filters. This bank of LUT’s aims at com-pensating the PA nonlinear behavior, and IIR filters atcompensating the PA memory effects.

c c q e kk 1 k k

*

+ = + ⋅ ⋅ ( )µ

02< <⋅[ ]µ

Tr q qk k

H

c c h e kk k k

*

+ = + ⋅ ( )1

hkk k

k

H

k k M

q

q q Q=

+−

−

λλ

1

1

QP

λk

λ λ λk k 1 k k

H

k 1 ph q Q= − +− −

d np n p= ⋅β α

H

H ii

N,P

i i

H H= ∑ ⋅ ⋅ ==

µ σ ν1

min( )

U VΘ

H

H =

=

d d d

d d

d d d

P

N N NP

P

N N N P

00 01 0

10 11

0 1

0 0 0 1 0

1 0 1 1

0 1

L

M

M O M

L

L

M

M O M

L

β γ β γ β γβ γ β γ

β γ β γ β γ

U V= ( ) = ( )µ µ µ ν ν ν1 2 1 2, , , , ,L LN N

µ νi ii N i P=( ) =( )1 2 1 2, , , , , , ,K K

σ σ σ1 2 0≥ ≥ ≥L min( , ) .N P

β µ= ⋅sµ 1

γ σ ν= ⋅ ⋅sµ 1 1

The normal gain LUT operates with the minimum nec-essary back-off that accomplishes BER specificationsat reception, so then, aiming at always operating in themost efficient dynamic range. While the safe modegain LUT provides a significant back-off operation in

order to compensate any degradation in the PA charac-teristics or the channel conditions.The “LUT Gain Decision” block in Figure 4, performsthe LUT gain decision algorithm described in Figure 5.In a first step, taking into account the modulation for-mat of the input signal, a normal gain LUT (with itscorresponding filter) is assigned. If the minimum BERrequired at reception is not guaranteed (BER > 10–γ),the LUT gain decision switches into the safe mode gainLUT. Then, if the minimum BER is accomplished,starts in parallel (in a different time scale) the adapta-tion process of the normal gain LUT in order to read-just any possible variation suffered in the PA character-istics. But, if operating with the safe mode gain LUTthe desired BER at reception is not accomplished, thatmeans that PA nonlinear behavior does not contributethat much in BER degradation as channel conditionsdo. Thus the transmitter has to choose a more robustmodulation format.

VI. Experimental results

A) Power amplifier model characterizationFor the experimental results, we have considered a PAHammerstein model estimated from modulated inputand output data of a class-A PA designed with theAgilent ATF-54143 PHEMT transistor. The obtainedcoefficients are listed in Table 1, and its static AM-AMcurve is shown in Figure 6.

B) Training Process: calculating Multi-LUT GainsIn the training process, four different types of modula-tion schemes (all of them supported by the IEEE802.16 standard) filtered by a root raised cosine filterwith a roll-off factor of α = 0.35 have been considered:QPSK, 16-QAM, 64-QAM and 256-QAM (all Grayconstellation ordered). The communications channel isan Additive White Gaussian Noise (AWGN) channelwith a signal to noise ratio (SNR) in reception of 10

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308 Proceedings of the European Microwave Association

Fig. 4. Structure of an adaptive Multi-LUT based predistorter.

Fig. 5. LUT gain decision algorithm.

Tab. 1. PA Hammerstein model coefficients.

Gammas Alf=as Betas

0 –4.42 – 9.16i –0.098 + 0.00705i –0.843 – 0.0378i1 49.0 + 44.43i 0.25444 – 0.00903i 0.21 + 0.00089989i2 –472.76 – 363.65i –0.1187 + 0.0129i –0.0986 + 0.00705i3 2226.2 + 1513.3i -- --4 –5610.1 – 3463.6i -- --5 7704.4 + 4383.5i -- --6 –5400.7 – 2844.6i -- --7 1509.5 + 734.93i -- --

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Proceedings of the European Microwave Association 309

dB, nor multipath nor Doppler have been considered.Assuming 50 Ω adaptation for the overall system, the1 dB compression point (see Fig. 6), will approximate-ly correspond to an input amplitude of 0.75 Vrms, thatis, an input power of 7.5 dBm, while the output satura-tion power is 23 dBm approximately (corresponding to4.5 Vrms).The PAPR of the considered modulations can be calcu-lated using eq. (21):

(21) PAPR(dB) = 10*log(CF)2

being the crest factor (CF) the ratio of the peak to r.m.samplitude of a signal, as it is described eq. (22):

(22)

For our particular power amplifier, with an input satu-rated power of 7.5 dBm, and taking into account a frac-tion of the saturation power of S = 0.95 (see eq. (1), (2)

and (3)), what corresponds to a peak back-off of PBO= 0.22 dB, the maximum input power is set at 7.13dBm. So then, considering the most restrictive PAPR(belonging to the 256 QAM modulation, see Table 2)and by using eq. (1), the minimum recommendableIBO for linear amplification will be IBO ≥ 7.5 db. But,by using Multi-LUT predistortion, the mean inputpower is fixed at 3 dBm, so then considering a generalIBO of IBO = 4.5 db, independently of the modulationused.In order to accomplish BER specifications at recep-tion, all normal and safe mode gain LUT’s with theircorresponding filters have to be calculated inadvanced. Figure 7 shows the Bit Error Rate for differ-ent LUT gains and for different modulations schemes,considering an input mean power of 3 dBm, an AWGNchannel with SNR = 10 dB and LUT’s with 128entries.Considering a BER restriction of BER < 10–3 (prior tocode gain inclusion) and observing Figure 8 (a zoom of

Fig. 6. Power amplifier AM-AM curve (7th order).

CFx

xpeak=

Fig. 7. BER versus LUT Gains (Pin = 3 dBm, LUT’s of 128bins).

Fig. 8. BER versus LUT Gains (Zoom on the y-axis of Fig. 7).

Fig. 9. BER and modulation schemes for different SNR channelconditions.

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310 Proceedings of the European Microwave Association

Fig. 7), the LUT-Gains for each modulation are listedin Table 2.

C) Normal processFigure 9 shows the BER and the different modulationformats for different SNR of the AWGN channel.Modulations 8 and 7 correspond to the 256 QAM nor-mal and safe mode gain LUT’s; modulations 6 and 5correspond to 64 QAM; 4 and 3 to 16 QAM; finally, 2and 1 to QPSK normal and safe mode gain LUT’srespectively.Looking at Figure 9 it is possible to identify the LUTdecision algorithm described in Figure 5. Until effi-cient amplification is possible the normal gain LUT ispreferred, but when channel conditions get worse, ifthe safe mode gain LUT is not enough, then the mod-ulation format is changed. Note that after adaptationthe algorithm retries with a more effective amplifica-tion (or modulation format if applies).

VII. Conclusion

In this paper a multi-LUT digital adaptive predistortercapable of supporting different modulation formats atthe time that assures high efficient linear amplificationby feedbacking the BER information at the receiverhas been presented. The design process and basic prin-ciples have been reported, as well as simulation resultsthat show its good performance. Among different pos-sibilities for model parameters identification, thoseshowing less computational effort have been here con-sidered, aiming to set up the predistorter algorithminside the already existent DSP infrastructure in mod-ern communication systems, thus minimizing pertur-bation to other DSP functionalities.

The main advantage of this multi-LUT predistorter isthat even when the dynamic range is adjusted at thebest back-off level, there is no penalization in theamplification gain, since it always operates as close tosaturation as the specified BER at reception permits.

Acknowledgment

This work was partially supported by SpanishGovernment (MCYT) under project TIC2002-04084-C03-01, by the EU network TARGET “Top AmplifierResearch Group in a European Team” (IST-1-507893-NOE) and by the EU STREP CAPANINA“Communications from Aerial Platform Networks deliv-ering ‘Broadband for All’” (FP6-IST-2003-506745).

References

[1] Kenington, P.B.: High-linearity RF amplifier design.Boston, Artech House Publishers, 2000.

[2] O’Droma, M.S.; Portilla, J.; Bertran, E.; Donati, S.; Brazil,T.J.; Rupp, M.; Quay, R.: Linearisation issues inmicrowave amplifiers. GAAS Symp. Amsterdam, October04, 199-202.

[3] Cripps, S.C.: RF power 2005. Microwave Journal 48(2005), 22-36.

[4] García, P.; Ortega, A.; de Mingo, J.; Valdovinos, A.:Nonlinear distortion cancellation using LINC transmittersin OFDM systems. IEEE Trans. On Broadcasting 51(2005), 84-93.

[5] Euns, C.; Powers, E.J.: A new Volterra predistorter basedon the indirect learning architecture. IEEE Tran. on SignalProcessing 45 (1997), 223-227.

[6] Zhu, A.; Brazil, T.J.: An adaptive Volterra predistorter forthe linearization of RF high power amplifiers. MTT-IMS2002 1 (2002), 461-464.

Tab. 2. Modulation requirements and LUT-Gains results.

Modulation QPSK 16-QAM 64-QAM 256-QAMscheme

PAPR 4.15 dB 6.43 dB 6.97 dB 7.2 dB(roll-off = 0.35)

IBO 4.37 dB 6.65 d 7.19 Db 7.44 dB

Theoretical 3,13 dBm 0.85 dBm 0.31 dBm 0.08 dBmrequired meaninput power

LUT Gain atNormal and Safe N: 9.4 N: 7.5 N: 6 N: 4.8ModeP?? = 3 dBm SM: 9 SM: 7 SM: 5.4 SM: 3.2

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Proceedings of the European Microwave Association 311

[7] Kim, J.; Konstantinou, K.: Digital predistortion of wide-band signals based on power amplifier model with memo-ry. Electronics Letters 37 (2001), 1417-1418.

[8] Ding, L.; Raich, R.; Zhou, G.T.: A Hammerstein predistor-tion linearization design based on the indirect learning.IEEE Int. Conf. on Acoustics, Speech, Signal Proc. 3(2002), 2689-2692.

[9] Benvenuto, N.; Piazza, F.; Uncini, A.: A neural networkapproach to data predistortion with memory in digital radiosystems. IEEE Int. Conf. Comm. ICC 93 1 (1993), 232-236.

[10] Jung, W.J.; Kim, W.R.; Kim, K.M.; Lee, K.B.: Digital pre-distorter using multiple lookup tables. IEEE ElectronicsLetters 39 (2003), 1386-1388.

[11] Baudoin, G.; Jardin, P.: Adaptive polynomial pre-distor-tion for linearization of power amplifiers in wireless com-

munications and WLAN. Int. Conf. on Trends in Comm.EUROCON 2001 1, 157-160.

[12] Marsalek, R.; Jardin, P.; Baudoin, G.: From post-distortionto pre-distortion for power amplifiers linearization. IEEEElectronics Letters 7 (2003), 308-310.

[13] Haykin, S.: Adaptive filter theory. New York: PrenticeHall, 1991.

[14] Bai, E.W.: An optimal two stage identification algorithmfor Hammerstein-Wiener nonlinear systems. Proc.American Contr. Conf., Philadelphia, Pennsylvania, June1998, 2756-2760.

[15] Narendra K.S.; Gallman P.G.: An iterative method for theidentification of nonlinear systems using a Hammersteinmodel. IEEE Trans.Automat. Contr. AC-11 (1966),546–550.

Pere Ll. Gilabert Pinal received the M. Sc.degree in Telecommunication Engineeringfrom the Universitat Politècnica de Catalunya(UPC), Barcelona, in 2002, and did hisMaster thesis at the University of Rome “LaSapienza”. He is currently a Ph.D. candidateand his research is focused in the study ofpower amplifier linearization techniques inthe department of Signal Theory and

Communications (TSC) at the UPC. His research interestsinclude control, communication electronics, signal processingand circuit theory. He is currently lecturing at the EscolaPolitècnica de Castelldefels (EPSC) as assistant professor.

Eduard Bertran received the Engineer(option Communication Electronics) andDoctor Engineer degrees in telecommunica-tion engineering, both from the UniversitatPolitècnica de Catalunya (UPC), Barcelona,Spain, in 1979 and 1985, respectively. Hejoined the department of Signal Theory andCommunications (TSC) in 1987, where he iscurrently a full professor. He has been the

head of studies of the TSC department and an associate dean indifferent telecommunications schools. His research interestsinclude control, communication electronics, signal processingand circuit theory, topics where he has produced different booksand peer-reviewed journal and conference papers. Member ofIFAC and senior member of IEEE. He has collaborated in dif-ferent national and European research projects and teams.

Gabriel Montoro López received the M.Sc.degree in Telecommunication Engineeringfrom the Universitat Politècnica de Catalunya(UPC) in 1989 and did his Master thesis in thearea of Adaptive Control. He joined thedepartment of Signal Theory andCommunications (TSC) in 1991, where he iscurrently an associate professor. He receivedthe Ph.D. degree of Telecommunications in

1996, being Robust Control the central core of his Ph.D. thesis.He is a lecturer on pregraduate courses at the Escola Politècnicade Castelldefels (EPSC), where he has been an associate deanfor laboratories. He also teaches postgraduate courses onAdaptive and Non-Linear Systems in the TSC department. Hisresearch interests include control and consumer electronics. Heserved as referee for some journals and conferences andreceived a grant from the Spanish Ministry of Foreign Affairs.