DATA PREDISTORTION FOR NONLINEARSATELLITE CHANNELS

Jakob Harmon, Stephen G. WilsonCharles L. Brown Dept. of Electrical and Computer Engineering

University of VirginiaCharlottesville, VA 22904{jbh5x, sgw}@virginia.edu

Abstract—Power amplifiers, an integral component tomost wireless communication systems, are inherently non-linear devices that introduce out-of-band spectral regrowthand constellation degradation to the amplifier output. Onesolution is to operate in a highly backed-off state to achievequasi-linear performance. Operating in this region requiresa higher-saturated power rating for a given output powerand reduces DC-to-RF efficiency. A more effective solutionis to apply digital predistortion which pre-compensates forthe harmful nonlinear effects. Compensation can be done atthe waveform level or operate at the symbol rate, known asdata predistortion. In this work we study data predistortionfor commonly adopted higher-order modulations, and studyperformance as a function of amplifier drive level relative tosaturation, and as a function of the predistorter’s memoryspan. A math model for a nonlinear satellite with transpon-der filtering is employed.

I. INTRODUCTION

POWER amplifiers (PAs) operated in their most effi-cient regime, near saturation, are nonlinear, produc-

ing amplitude-to-amplitude modulation (AM-AM) con-version as well as amplitude-to-phase modulation (AM-PM) conversion of a narrowband signal. These effectshave been known for many years in the context oftraveling-wave tube (TWT) amplifiers, a common typeof amplifier used in satellite communication. The non-linearity induces several harmful effects, including in-termodulation in multicarrier environments, and spectralregrowth in single-carrier settings. Out-of-band spectralpower can then increase to the point of not meetingregulatory specifications. Moreover, the in-band distortionproduces degradation in bit error rates (BER) relative tothat of linear amplifiers.

Newer satellite transmission standards that usehigher-order modulation schemes, e.g. DVB-S2, are in-creasingly vulnerable to these nonlinearities because of

This research was supported by the Virginia Space Grant Consortium(VSGC) and iDirect.

their high peak-to-average ratios. Complicating matters,PAs exhibit memory effects where the output not onlydepends on the current input, but also on the magnitude ofprevious values of the input, producing additional distor-tion. To meet frequency mask requirements of regulatoryagencies and bit-error rate (BER) performance require-ments of customers, many satellite system operators arebeing driven to confront the effects of nonlinear poweramplifiers.

One solution is to backoff the amplifier, i.e. reducethe drive level so that operation is more confined to thequasi-linear range of amplification. The disadvantages ofthis approach are two-fold: 1) a higher saturated powerrating is needed to achieve a given desired linear poweroutput, and 2) the DC-to-RF power efficiency suffersgreatly.

The technique of waveform predistortion employsan approximate inverse nonlinearity ahead of the poweramplifier such that the cascade operation is close to ideal,i.e. the output amplitude is just some constant G timesthe input amplitude, and phase at the output is the sameas that of the input. The earliest predistorters were analogdevices with RF feedback, which were difficult to align,prone to instability, and seldom performed well. Morerecently, digital signal processing (DSP) technology hasbeen applied, along with adaptive training of the predis-torter, to achieve substantial improvement in the lineariza-tion task. These systems act on a baseband I/Q discrete-time version of the input signal (hence the commonname digital predistorter), producing a precompensatedI/Q waveform that is converted to continuous-time, up-converted, and amplified by the PA.

An alternative is to apply predistortion at the symbolrate, prior to pulse shaping, to pre-warp the complex sym-bol sequence so that the resulting receiver constellationsuffers less distortion due to nonlinearity. The simplestdata predistorters simply warp the constellation in am-plitude and phase to make gross compensation for AM-AM and AM-PM effects. However, the satellite channel

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has memory due to both pulse shaping and matchedfiltering, as well as transponder filtering surroundingthe PA itself. Thus, the data predistorter benefits frommemory, and can be viewed as a nonlinear pre-equalizerfor the discrete-time channel. Our focus in this work is tostudy data predistortion, and its performance for typicalconstellations, as a function of amplifier drive level andpredistorter memory.

A. Applicability to NASA mission

The Tracking and Data Relay Satellite System(TDRSS), developed by NASA, relays mission and track-ing data through nine operational satellites (with twocurrently under development) to/from orbiting user satel-lites including the Space Shuttle, the Hubble Telescope,and the International Space Station. Figure 1 shows theTracking and Data Relay Satellite (TDRS) spacecraftsrelaying user satellite data to a ground-station. Each relaysatellite incorporates multiple inherently nonlinear travel-ing wave tube (TWT) power amplifiers that require severeinput power backoff, which limits output power andamplifier efficiency. A high-order modulation scheme,such as 16-point quadrature amplitude modulation (16-QAM) recently proposed for TDRSS in [1], is especiallyvulnerable to the PA nonlinearities.

Fig. 1. TDRSS system with TDRS spacecrafts relaying user satellitedata to a ground-station [1]

A robust digital predistortion technique along witha field-programmable gate array (FPGA) implementationcan linearize the power amplifiers and significantly en-hance TDRSS communication. Aside from application toTDRSS, other NASA communication systems such asthe Deep Space Network, which connects with currentscience missions on Mars and those spacecraft passingother planets, would significantly benefit from usinglinearized power amplifiers.

B. System modeling

The simplest model for a nonlinear PA is a memory-less model, typified by models of Saleh [2] for example.

It is now known, however, that memory effects need to beincorporated into PA models (and predistorters), at leastfor wideband operation, in order to achieve high perfor-mance. Memory can be incorporated in simple mannerby Wiener models (linear filter followed by memorylessnonlinearity), by Hammerstein models (memoryless non-linearity followed by linear filter), or both. More generalVolterra models, as seen in equation (1), have been well-studied for this application, which relate the complexbaseband output of the amplifier (in discrete-time) to thecomplex baseband input according to Schetzen [3].

yn = H [xn]

=∑i

Hi [xn]

=Q−1∑q1=0

a1(q1)xn−q1

+Q−1∑q1=0

Q−1∑q2=0

Q−1∑q3=0

a3(q1, q2, q3)xn−q1xn−q2x∗n−q3

+ ...(1)

The order (number of terms retained) and memoryQ are key measures of model complexity. Some predistor-tion work has attempted to define a precise Volterra repre-sentation for an amplifier, then analytically solve for whatis known as the pth-order Volterra inverse. A pth-orderinverse for a nonlinear system H is another nonlinearsystem that, when cascaded with H, leaves an end-to-end Volterra model whose kernels vanish through orderp, except for the identity kernel. Schetzen shows that thepre- and post-inverses of order p are identical. Such apredistortion technique, however, requires knowledge ofthe Volterra PA model, and moreover does not directlyminimize end-to-end distortion.

The alternative is to formulate a data predistorter,either in the form of a lookup table or an algorithm(perhaps a memory polynomial [4]), and train it to haveoptimal characteristics, given some constraints. This isakin to training an adaptive receiver equalizer for ISIchannels.

II. PREDISTORTER STRUCTURE AND TRAINING

The adopted structure for the data predistorter,shown in Figure 2, utilizes a lookup table (LUT) thatis addressed by a span of 2L + 1 contiguous modulatorsymbols. The lookup table outputs a complex value vnwhich can be viewed as the output of a nonlinear discrete-time filter, or equalizer, operating on a sequence ofmodulator symbols. Its function is to try to invert thenonlinear discrete-time channel with memory, operating

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PredistorterLUT

Rotateindices

Un-rotate vnto properquadrant

× hT (n) Satellitechannel hR(n)

PredistorterLUT

batch update

+Rotate cn

to firstquadrant

Complexsymbolmapper

Rotate znto first

quadrant×

~d = (dn−L, ...dn, ...dn−(L+1))

dn un

Complexsymbols

vn w(t) y(t)

Complexsymbols

zn

en+ −

α

β

Fig. 2. Data predistortion block diagram. Four-quadrant symmetry is assumed

at the modulator symbol rate. Advantages of the lookuptable approach include simplicity of hardware and highspeed operation, but the number of table entries is M2L+1

for M -ary signaling, and this can become infeasible forlarger M and L. For example, with 8-PSK transmissionand L = 2 (span = 5), the table size is a modest 32Kwords, but for 16-APSK with the same memory, theLUT address space grows to 1 Meg. In such cases, itmay be necessary to adopt an algorithmic form for thepredistorter.

The training method is a member of the so-calleddirect learning family of adaptive algorithms, whereinwe sequentially update the table entries to minimize theerror between the ideal constellation and the output of theeventual receiver. With reference to Figure 2, the targetvalue for a given transmitted data symbol is denotedun, whereas the received complex value correspondingto this transmitted data symbol is denoted zn, the resultof the nonlinear system with memory. The ‘satellitechannel’ is composed of an input demultiplexing filterhi(t), nonlinear PA model g(·), follwed by an outputmultiplexing filter ho(t) as seen in Figure 3. To maintaina desired input backoff (IBO) drive level, e.g. 3 dB, tothe satellite channel during training, the output of thepredistorter LUT is scaled by α. This scale factor isadjusted following each iteration based on the averagepower of the LUT and the desired drive level.

hi(t) g(·) ho(t)w(t) y(t)

Fig. 3. Satellite channel block diagram

The training update is completed in batches of sizeN symbols, typically at least several thousand in size. Wefirst scale the mean-square value of a batch of collectedoutput samples so that the mean-square value is 1, thesame as our reference constellation. The correspondingscale factor is denoted β. Subsequently we define theerror for the nth symbol as

en = un − βzn . (2)

This error value will depend not only on the currentsymbol dn, but also on past and future data symbols dueto the memory of the system. Such memory is manifestin the plot of receiver output when no predistortion isapplied–a large cluster of points is seen correspondingto a given constellation point, due to this nonlinear ISIeffect. (The cluster size is not due to additive noise, whichis assumed to be negligible here.) We note that in eachtraining batch, we desire several occurrences of a givendata pattern of span 2L+1 so that 1) additive noise can beaveraged out, when present; and 2) the effect of systemmemory that exceeds our adopted memory can also beaveraged out. This suggests a typical batch size N on theorder of K×M2L+1, where K is an integer on the rangeof say 1 to 10.

At each batch update, we adjust each LUT entry,according to the observed errors, in such a direction thatshould make the squared-error smaller for the next batch,conditioned on this specific data pattern. To do so, we firstgather all measurements zn that correspond to a certaindata vector ~d = (dn−L, dn−L+1, ...dn, ...dn−(L+1)). Wefind the centroid of this cluster of points, denoted ej , scaleby β as above, and form the error signal for the table entryj that corresponds to this data pattern. The updating isan LMS-style update, currently separately updating themagnitude and phase of the table entry at location jaccording to

A(i)j = A

(i−1)j + µA|ej | (3)

θ(i)j = θ

(i−1)j + µθ 6 {ej} (4)

where i is an iteration index. This training algorithm haspreviously been studied by the European Space Agency[5], although restricted to the case of memory span3, apparently for complexity reasons. Presently, we usevalue 1 for both updating scale factors, choices that seem

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a good balance between convergence speed and tableadjustment ‘noise.’

It is beneficial to exploit constellation symmetryin training the table. All constellations studied here andthat are typically utilized in satellite communication havefour-quadrant symmetry. This means that the number ofunique patterns that must be studied is M2L+1/4. Intraining we group data clusters that correspond to a ro-tation of a first-quadrant constellation point for purposesof defining ej . This allows a smaller batch size (by 4)and thus faster convergence time. It also provides a bettertraining quality when additive noise is present, since casescorresponding to a quadrant rotation should really all havethe same table entry. When we actually apply the lookuptable in transmission however, we have to ‘un-rotate’ fromthe first quadrant to the proper quadrant for symbol un.If we wish to implement this rotation on the fly, the LUTsize can also be reduced by a factor of 4.

III. SIMULATIONS AND RESULTS

In this section, we present simulation results of thetechniques outlined. As previously mentioned, a Wiener-Hammerstein structure is adopted for the satellite chan-nel. Input backoff (IBO) and output backoff (OBO) aredefined relative to the amplifier saturation power levels.The PA model is based on the nonlinear AM-AM andAM-PM characteristics of the DVB-S2 TWT amplifierin [5]. Since the polynomial fit was completed for thesecharacteristics in dB scale, a conversion to and back fromdB is incorporated. The PA polynomial model is

|x(n)|dB = 20 log10 (|x(n)|) (5)

a(n) =K∑k=0

bk |x(n)|kdB (AM-AM) (6)

θ(n) =K∑k=0

ck |x(n)|kdB (AM-PM) (7)

p(n) = 10a(n)/20ej(θ(n)+ 6 x(n)) (8)

with complex input x(n) and output p(n), and uses thecoefficients

b0 = −1.150× 10−2

b1 = 5.454× 10−2

b2 = −3.974× 10−2

b3 = −5.128× 10−4

c0 = 7.360× 10−1

c1 = 6.661× 10−2

c2 = 3.032× 10−4

c3 = −1.285× 10−4

c4 = −4.295× 10−6

c5 = −4.118× 10−8 .

This PA model alone, like those of [2], does notincorporate so-called memory effects, which are an in-creasingly recognized distortion effect associated withPAs, particularly for wideband signals. If the real PAhad memory effects not modeled here, it is believedthat the direct learning LUT approach is well-suited toprecompensating for these effects as well, perhaps withminor losses in performance. The primary source of mem-ory effects however in our model is transponder filteringat the PA input demultiplexer and output multiplexer.These filters have been chosen to be 4-pole Chebyshevcharacteristics with bandwidth-symbol duration productB3T = 1.02.

Predistorter LUT batch updates continue until eitherof two stopping criteria are met. The first limits the num-ber of training batches to 20 and the second relates to theconstellation error. If the constellation error of the currentbatch is less than 1% better than the previous batch,then the LUT has essentially converged and additionaliterations are not completed.

Figure 4 shows the set of points produced by theLUT as a result of training with 16-APSK modulation andmemory span 3, where the full LUT address space is amodest 1 K. The amplifier drive level has input backoff 3dB. The training batch size was 2 K symbols meaning themean number of occurrences of each data pattern is 2 perbatch. The figure shows the four clusters (out of 16) thatare stored in the predistorter LUT, highlighted in black.The other three quadrants, a complex rotation of the firstquadrant points, are plotted solely for reference but arenot stored in the LUT. Figure 5 shows the ‘learning curve’for the algorithm, which is a plot of the absolute errorvalues versus time. There is a batch nature to this curve asexpected since updates are performed every N samples.Clearly only a few batches are needed to gain convergenceto an equilibrium condition. We note however that whenmemory span is increased, the convergence time is longer;both batch size and then number of batches need toincrease to extract the benefits of more equalizer memory.

Next, Figure 6 presents the received constellation,following training, with 16-APSK and memory span 3,labeled ‘precompensated output.’ Again, the IBO is 3dB. For comparison, we show the output data when nocompensation is applied; the uncompensated points areclearly more scattered, due to both linear and nonlinearISI, and exhibit gain compression of outer points aswell as phase rotation effects due to AM-PM in thenonlinear amplifier. Also, Figure 7 shows the receivedconstellation when a predistorter with memory span 5 isapplied. A larger batch size of 256 K was used in training

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Fig. 4. Predistorter LUT contents (black) rotated to the other quadrants(grey) for reference for a memory 3 predistorter trained at 3 dB IBOfor the 16-APSK constellation

Fig. 5. Absolute value of the constellation error decreasing versustraining samples and converging to an equilibrium minimum error level

and the number of samples required for convergence isapproximately 1000 times more than for the memory 3predistorter due to the 256x increase in the LUT size.

As these figures show, predistortion is effectiveat removing these gross shifts of clusters, as well astightening the receiver output clusters around a givenconstellation point. We measure the quality of the re-ceived constellation by the measured called error vectormagnitude (EVM).

Fig. 6. Comparison of constellation improvement with a memory 3predistorter driven at IBO = 3 dB for the 16-APSK constellation

Fig. 7. Comparison of constellation improvement with a memory 5predistorter driven at IBO = 3 dB for the 16-APSK constellation

Table I lists the EVM measure in decibels as a func-tion of IBO, for cases of no compensation, a memoryless(span 1) predistorter, as well as performance with memoryspan 3 and 5. The larger memory span offers significantgain in EVM, or reduction in cluster scatter, due to itsability to better equalize both linear and nonlinear ISI, butprimarily the latter. Again, Figure 7 presents the receivedconstellation at IBO=3 dB, visually demonstrating theperformance gain.

The discussion here has assumed noiseless operation

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EVM (dB)IBO (dB) PA Predistorter with memory

only span 1 span 3 span 5

0.5 -15.6 -15.4 -18.9 -24.2

1.0 -15.6 -15.7 -19.6 -25.3

2.0 -15.8 -16.1 -20.6 -27.6

3.0 -16.0 -16.6 -21.4 -29.8

4.0 -16.3 -17.0 -22.0 -31.8

5.0 -16.7 -17.4 -22.6 -30.0∗

TABLE ICOMPARISON OF ERROR VECTOR MAGNITUDE REDUCTION

BETWEEN NO COMPENSATION AND SPAN 1 (MEMORYLESS), 3, AND5 MEMORY PREDISTORTERS FOR 16-APSK DATA. ∗THIS TEST

REACHED THE STOPPING CRITERION ON CONSTELLATION ERRORWHEREAS THE OTHERS WITH SPAN 5 REACHED THE MAX NUMBER

OF ALLOWED BATCHES, CAUSING THIS VALUE TO DIP SLIGHTLY

to study the predistorter’s performance. In a satellite link,downlink noise will also contribute to constellation scatterin two-dimensional Gaussian form, and this variance willbe additive to the variance resulting from nonlinear andlinear satellite distortion. If we model the distortion ef-fects as two-dimensional Gaussian, then we can computea measure called total degradation that expresses the lossin SINR relative to the what an ideal amplifier having thesame average output energy Esat,r would produce. Thisloss is measured at a symbol error probability Ps of 0.1,corresponding to a typical operating point for a strongly-coded transmission. The total degradation expression for16-QAM is

Dtot(OBO) =1

OBO ·[1− 5 [Q−1(Ps/4)]2 · EVM(OBO)

](9)

and its value is plotted in Figure 8 as a function ofdrive level for different predistorter memory. We notea memoryless predistorter gains only a little relative tono predistortion, but that span-3 and span-5 predistortionbuys an effective gain in the line of 2 and 2.5 dB at thebest operation point, namely around 2 dB IBO. Also notethat the coefficient ‘5’ would be slightly different for the16-APSK constellation being studied.

IV. CONCLUSIONS

Data predistortion, with memory span of 3 or 5symbols, represents an effective means of compensatingfor satellite nonlinear distortion, and can improve linkperformance by 2-3 dB over no compensation. Trainingof the predistorter’s LUT has been described which

Fig. 8. Comparison of total constellation degradation due to noisecorresponding to Ps = 10−1 and satellite channel distortion vs. IBO(dB) between no compensation and span 1 (memoryless), 3, and 5memory predistorters

employs loop-back reception to progressively improve theconstellation quality.

Predistortion with memory span 3 can be easilyimplemented using LUT architectures, even for con-stellations such as 32-APSK, and has been developedin a Stratix II FPGA implementation. Implementationof memory 5 predistortion however, will likely requireadoption of a functional form expressing the desired I/Obehavior of the predistorter, e.g. memory polynomial orVolterra forms, with predistorter output computed in real-time in response to the data sequence. This remains anitem for further study.

REFERENCES

[1] B. Gioannini, Y. Wong, J. Wesdock, and C. Patel, “Bandwidthefficient modulation and coding techniques for NASA’s existingKu/Ka-band 225 MHz wide service,” Aerospace Conference, 2005IEEE, Mar. 2005.

[2] A. M. Saleh, “Frequency independent and frequency dependentnonlinear models of TWT amplifiers,” IEEE Trans. on Commu-nications, Nov. 1981.

[3] M. Schetzen, The Volterra and Wiener Theories of NonlinearSystems. Wiley, 1980.

[4] L. Ding, G. Zhou, D. R. Morgan, Z. Ma, J. S. Kennedy, J. Kim, andC. R. Giardina, “A robust digital baseband predistorter constructedusing memory polynomials,” IEEE Trans. on Communications, Jan.2004.

[5] E. Casini, “DVB-S2 modem algorithms design and performanceover typical satellite channels,” European Space Agency, Tech.Rep., Oct. 2004.

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