Amplification of signals with fluctuating envelopes inevitably leads to distortion because of nonlinear behavior of the power amplifier (PA). Digital predistortion can counteract these nonlinear effects. In this paper, a digital predistortion architecture is presented which is based on the calculation of correlation functions using coarsely quantized signals. The crosscorrelation functions are transformed to the frequency domain and the spectra are used to calculate the coefficients of the predistorter memory polynomial. This method has reduced complexity and slightly improved average performance in comparison with existing schemes

Kokkeler, Andre B.J.

English

Amplification of signals with fluctuating envelopes inevitably leads to distortion because of non-linear behavior of the Power Amplifier (PA). Digital Predistortion can counteract these non-linear effects. In this paper, a novel Digital Predistortion architecture is presented which is based on the calculation of crosscorrelation functions using coarsely quantized signals. The crosscorrelation functions are transformed to the frequency domain and the spectra are used to calculate the coefficients of the predistorter memory polynomial. This method has reduced complexity and slightly improved average performance in comparison with existing schemes

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A Crosscorrelation Predistorter using Memory Polynomials

Kokkeler, André B.J.

University of Twente Research Information

A CROSSCORRELATION PREDISTORTER USING MEMORY POLYNOMIALSAndré B.J. KokkelerUniversity of TwenteP.O. Box 217, 7500 AE Enschede, The Netherlands.Phone: +31 53 4894291, Fax: +31 53 4894571,kokkeler@cs.utwente.nlABSTRACTAmplification of signals with fluctuating envelopes inevitablyleads to distortion because of non-linear behavior of thePower Amplifier (PA). Digital Predistortion can counter-act these non-linear effects. In this paper, a novel Digi-tal Predistortion architecture is presented which is based onthe calculation of crosscorrelation functions using coarselyquantized signals. The crosscorrelation functions are trans-formed to the frequency domain and the spectra are used tocalculate the coefficients of the predistorter memory poly-nomial. This method has reduced complexity and slightlyimproved average performance in comparison with existingschemes.1. INTRODUCTIONOne of the characteristics of modern, spectrally efficienttransmission formats (CDMA, OFDM) is their relativelyhigh peak-to-average ratio. Non-linear Power Amplifiers(PA) fed by signals with high peak-to-average ratio intro-duce severe distortion. Distortion within the band of trans-mission degrades signal detection. Distortion outside theband of transmission (spectral regrowth) interferes with thesignals transmitted by other users at adjacent channels andis therefore called Adjacent Channel Interference (ACI).Power Amplifiers are costly and their operation is power-inefficient because of the measures to limit distortion. Tocounteract the non-linearity of a PA, several techniques havebeen investigated: feedforward, feedback and predistortion.In this document we will concentrate on predistortion. Thegeneral principle is to apply the inverse input-output relationof the PA to the signal at the input of the PA. Predistortionfollowed by the PA (and its inherent distortion) should resultin linear amplification. Predistortion is applied in the analogdomain (see [1]) and in the digital domain (see [2]). In dig-ital predistortion, the baseband signal is predistorted beforeit is converted to the analog domain, frequency translated toRF and amplified. (see figure 1).Because the input-output relation of the PA changes intime due to temperature changes and aging of componentsxpControlPredistortionyRF to DigitalxDigital to RF PAPredistortionFig. 1. Digital Predistortionin the analog part, a control mechanism constantly adaptsthe predistortion. A small fraction of the PA output is fedback and converted from RF to baseband. An adaptation al-gorithm compares this signal with the output of the predis-torter. Two different approaches exist: Direct- and IndirectLearning. When using Direct Learning, the input-output re-lation of the PA is determined. This relation is inverted andused for predistortion. For an example, see [3]. When us-ing Indirect Learning, the input-output relation of the PA isdetermined indirectly, the inverse relation directly. This in-verse relation is used as predistorter. Recent Indirect Learn-ing schemes (see [4]) involve multiplications of relativelylarge matrices. If the number of samples involved in calcu-lating a new setting of the Predistorter equals T , the num-ber of multiplications is O(T ). PA’s suffer from memoryeffects. The output signal of the PA at any time-instant de-pends on the input at that instant and on the input signal atprevious time-instants. Recent Indirect Learning schemesare suitable for counteracting the memory effects. The pur-pose of this paper is to introduce a novel predistortion archi-tecture with reduced complexity. In Section 2, the architec-ture will be presented. It is based on the Indirect Learningscheme. In Section 3, implementation issues are consideredand in Section 4, simulation results are given.III - 3450-7803-8251-X/04/$17.00 ©2004 IEEE ISCAS 2004➠ ➡2. THE CROSSCORRELATION PREDISTORTERThe predistorted signal xp and the PA output (convertedback to baseband) y are used to determine the output-to-input relation of the PA. This relation can be described bymeans of a memory polynomial (see [3]). The polynomialused in our paper is similar to the polynomial used in [4]:xp(t) =K∑k=1τmax−1∑τ=0akτy(t − τ) | y(t − τ) |k−1 (1)The memory length of the predistorter equals τmax andthe polynomial order equals K . To find the values akτ , pow-ers of y and of delayed versions of y are generated:ykτ = y(t − τ) | y(t − τ) |k−1 (2)for t ∈ Z. Via linear combination of these signals, an esti-mate of signal xp can be constructed using the least squarescriterion:xp = [xp(1), ..., xp(T )]T (3)ykτ = [ykτ (1), ..., ykτ (T )] (4)Y = [y10, ...,yK0, ...,y1τmax−1, ...,yKτmax−1](5)â = (YHY)−1YHxp (6)where H indicates the complex conjugate transpose. T is thenumber of consecutive samples, available to the predistorter.In the remainder of this paper the predistorter based on thissolution will be called: LS-predistorter.In the crosscorrelation predistorter, the signals ykτ arenot combined directly. First ykτ and xp are crosscorrelatedwith a reference signal. We chose to crosscorrelate with asingle-bit representation xQ of xp, because the quantizedsignal xQ has significant power in the adjacent channelswhen the predistorter is operational. Single-bit quantizationis defined as:xQ(xp) = sign(Re(xp)) + jsign(Im(xp)) (7)where sign() is defined as:sign(x) ={−1 x < 01 x ≥ 0 (8)Re(x) indicates the real part of a complex value x and Im(x)the imaginary part. Using a single-bit quantized value asone of the operands of a crosscorrelation drastically reducesthe complexity. The multiplications involved become verysimple to implement: when the first operand equals 1 or-1, the second operand of the multiplication is either leftunchanged or its sign is reversed respectively. The numberof points (or lags) of the crosscorrelation is even and equalsN . Using vector notations, the single-bit quantized signalequals:xQ =[xQ(xp(12N + 1)), ..., xQ(xp(T − 12N + 1))]T(9)We define the matrices Ykτ for k = 1, ..., K and τ =0, ..., τmax − 1, and Xp as:Ykτ =[[ykτ (1), ..., ykτ (T − N + 1)]T , ...,[ykτ (N), ..., ykτ (T )]T]TXp =[[xp(1), ..., xp(T − N + 1)]T , ...,[xp(N), ..., xp(T )]T]T(10)The crosscorrelations are defined as:rkτ = Ykτx∗Q (11)rp = Xpx∗Q (12)The crosscorrelation vectors rkτ and rp are tapered with aHanning taper:h(j) =12(1 − cos(2π(j − 1)/(N − 1))) (13)H = diag(h(1), ..., h(N)) (14)where ’diag()’ indicates a diagonal N × N matrix. A Dis-crete Fourier Transform is applied to the tapered crosscor-relation vectors:fkτ = WHrkτ (15)fp = WHrp (16)where W equals the DFT kernel. The elements wpq of thekernel are defined as:wpq = exp−i2πp−1N (q−1) (17)The vectors (spectra) fkτ are concatenated into a matrix F:F = [f10, ..., fK0, ..., f1τmax−1, ..., fKτmax−1] (18)The vector a is defined as:a = [a10, ..., aK0, ..., a1τmax−1, ..., aKτmax−1] (19)The memory polynomial predistorter is then described inthe frequency domain as:fp = Fa (20)III - 346➡ ➡The least squares solution minimizes the absolute er-ror over the frequency domain. Because the signal has arelatively low power spectral density in the adjacent chan-nels, the relative errors in the adjacent channels can be (too)large. For that reason, we minimize the relative error. Thisis realized by normalizing the spectra with fp:gkτ (n) =fkτ (n)fp(n)(21)gkτ = [gkτ (1), ..., gkτ (N)]T (22)G = [g10, ...,gK0, ...,g1τmax−1, ...,gKτmax−1] (23)gp(n) = 1, n = 1, ..., N (24)The normalized version of equation (20) becomes:gp = Ga (25)The least-squares solution â equals:â = (GHG)−1GHgp (26)To reduce the effects of noise on the correlation functionsgkτ , we select only those vector elements which representthe power in the main channel and the first adjacent channelleading to vectors gp and gkτ with limited length.3. IMPLEMENTATION ASPECTSThe complexity of current polynomial predistorters is at leastO(T ); the number of complex multiplications scales lin-early with the number of samples used to update the pre-distorter polynomial coefficients â. In the crosscorrelationpredistorter, the size of the vectors fp and fkτ is reducedto N (instead of T ). N equals the number of points (orlags) of the crosscorrelation functions and in general N ismuch smaller than T . If an FFT is used to transform thevectors from the time domain to the frequency domain, thecomplexity is O(N log2 N). The reduction of the length ofthe vectors is due to the crosscorrelation. The crosscorrela-tion is easy to implement. Because of the single-bit quanti-zation of xp, no full-precision complex multiplications arerequired, so the overall complexity of the crosscorrelationpredistorter is determined by the FFT.4. SIMULATION RESULTSA simulator, based on the structure presented in the previ-ous sections has been implemented. As a signal source weused two independent data generators: one for the real partof the signal and one for the imaginary part. A data gener-ator is based on the signal source used in [5] and generates416 sinewaves with different consecutive frequencies, equalamplitudes and randomly selected phase. The oversampling−1.5 −1 −0.5 0 0.5 1 1.5−100−80−60−40−200Distorted signal Corrected signal Original signal Adjacent channel Primary channel Adjacent channel Fig. 2. Spectrafactor is 10. The PA model used is a Wiener-Hammersteinmodel as presented in [4]. We chose this model because itincludes PA memory effects. The model consists of an IIRfilter H(z) followed by a memoryless polynomial which onits turn is followed by a second IIR filter G(z). The specifi-cations are:H(z) =1 + 0.5z−21 − 0.2z−1 , G(z) =1 − 0.1z−21 − 0.4z−1The memoryless polynomial equals:w(n) =K∑k=1bk v(n) | v(n) |k−1where v indicates the output of H and w the input of G. Thecoefficients are: b1 = 1.0108 + 0.0858j, b3 = 0.0879 −0.1583j, b5 = −1.0992 − 0.8891j. The standard deviationof the complex input signal is 4.24 10−2. In our simulations,we used blocks of 8K samples (T = 8192) and 64-pointcrosscorrelation functions (N = 64). For the predistorter,we used only odd polynomials up to the fifth order (k =1, 3, 5) and the memory length is 2 (τ = 0, 1).In figure 2, a typical double sided power spectral den-sity (PSD) of the original signal is given, together with thebaseband equivalent output of the PA with and without pre-distortion. The PSD’s are normalized with the power of theoriginal signal.We see that the Adjacent Channel Interference (ACI)levels, have been reduced significantly. Outside the Ad-jacent Channels, the power has increased slightly but re-mains below -80 dB. The reduction depends on the inputIII - 347➡ ➡0 10 20 30 40 50 60 70 80 90 100−75−70−65−60−55−50−45cycle numberACI levels (dB)LS−predistorterCrosscorrelation predistorter No predistortion Fig. 3. ACI levels with and without predistortion for PAmodel 1signal. New settings of the predistorter are based on datathat already has been transmitted. New data might containsignal-excursions which have not been accounted for. Thiscan lead to limited suppression of ACI levels and in severecases to settings from which the predistorter cannot recover.For that reason, we simulated 100 cycles where every cycleconsisted of the following stages: generation of an 8K sam-ples dataset for cycle i, predistortion of this dataset using thepredistorter settings found in cycle i−1, distortion, determi-nation of the memory polynomial and determination of thenew predistorter settings by averaging the memory polyno-mial found (weight = 0.25) with the old predistorter settings(weight = 1). The crest-factors (ratio of the peak amplitudeand standard deviation) of the generated data range from2.5 to 3.4 dB (8 to 11 dB). The ACI levels with and withoutpredistortion for both the LS-predistorter and crosscorrela-tion predistorter are given in figure 3. The bandwidth of thetwo adjacent channels equals the bandwidth of the primarychannel. The ACI-levels reflect the maximum of the totalpower of the two adjacent channels related to the power inthe primary channel.We see that after 10 cycles, the predistorter system sta-bilizes. For the crosscorrelation predistorter, the average re-duction of ACI-levels is around 20 dB. For the LS-predistorter,the average reduction is slightly less and varies significantlyfrom cycle to cycle. At cycle 48, the LS-predistorter evenincreases the ACI-levels.5. CONCLUSIONSIn this paper a novel digital predistortion architecture, thecrosscorrelation predistorter, is presented. It is based on theIndirect Learning scheme.Simulations using this crosscorrelation predistorter haveshown significant reductions of ACI levels. Compared withthe LS-predistorter, the crosscorrelation predistorter has re-duced complexity and shows a slightly increased reductionof ACI-levels. The complexity is reduced because the num-ber of complex multiplications depends on the number ofpoints (lags) of the crosscorrelation functions (N ) and equalsO(N log2 N). This complexity is independent of the sizeof the data set (T ), used to update the predistorter. In ex-isting schemes, the number of complex multiplications isO(T ). In our simulations, N = 64 and T = 8192. Thecrosscorrelation predistorter is based on the calculation ofcrosscorrelation functions. Because one of the inputs of thecrosscorrelation is a single-bit signal, no complex multipli-cations are involved in that part of the predistorter.6. REFERENCES[1] C.G. Rey, “Predistorter linearizes CDMA power am-plifiers,” Microwaves RF, Oct. 1998, pp. 114-123.[2] F. Zavosh, M. Thomas, C. Thron, T. Hall, D. Ar-tusi, D. Anderson, D. Ngo, and D. Runton, “Digi-tal predistortion techniques for RF power amplifierswith CDMA applications,” Microwave Journal, Oct.1999. linearization using predistortion-Experimentalresults”, IEEE Trans. Veh. Technol., vol. 43, no. 2,May 1994, pp. 323-332. performance of an adaptivedigital linearized power amplifier”, IEEE Trans. Veh.Technol., vol. 41, no.4, Dec. 1992, pp. 395-400.[3] J. Kim and K. Konstantinou, “Digital predistortionof wideband signals based on power amplifier modelwith memory,” Electronics Letters, vol. 37, no. 23,Nov. 2001, pp. 1417-1418. memory-less nonlinearityproceded by a dynamic linear system”, Proc. GLOBE-COM 1995, pp. 152-156.[4] L. Ding, G.T.Zhou, Z. Ma, D.R. Morgan, J.S. Kenney,J. Kim, and C.R. Giardina, “A robust digital basebandpredistorter constructed using memory polynomials,”IEEE Trans. on Communications, March 2002 (sub-mitted).[5] J. K. Cavers, “Amplifier linearization using a digitalpredistorter with fast adaptation and low memory re-quirements”, IEEE Trans. Veh. Technol., vol. 39, no. 4,Nov. 1990, pp. 374-383. predistortion linearization de-sign based on the indirect learning architecture,” Proc.ICASSP 2002, May 13-17, 2002.III - 348➡ ➠

A crosscorrelation predistorter using memory polynomials

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