Quantifying Distortion of RF Power Amplifiers for Estimation ofPredistorter Performance
Paul J. Draxler1,\ Anding Zhu\ Jonmei J. Yan2
, Pavel Kolinko2, Donald F. Kimball\ and Peter M. Asbeck2
1Qualcomm Inc., 5775 Morehouse Dr., San Diego, CA, USA2 University of California, San Diego, La Jolla, CA, USA
3 University College Dublin, Dublin 4, Ireland
Abstract - This paper demonstrates a method to quantify theaccuracy of memory models and the effectiveness of digitalpredistortion of power amplifiers with memory. By usingassumptions of periodic stationarity, coherent ensembleaveraging and a stable measurement system, we are able todecompose the waveform distortion into memoryless,deterministic memory, and random memory contributions. Wedemonstrate how this can be used to evaluate the performance ofa power amplifier and project its optimal performance withpredistortion. We also show experim~ntall~ ~hat the ~yna~c
deviation reduction-based Volterra serIes dIgItal predlstortIontechnique has convergent behavior, moving towards t~ese
quantified targets when applied to a class AB power amplIfierimplemented with GaN FETs.
Index Terms - Behavioral modeling, linearization, poweramplifier, predistorter, Volterra series.
compare to the predictions, quantifying the perfonnancelevels.
1 Digital RF Power 1:'-X Predistortion xp. :y -- Gox••: (OPO) Amplifier
I
i tL k::1
1 -----------------------------
Fig. 1. A predistortion system
where the function G(n) is the complex describing function ofthe amplifier's nonlinear gain, and &..n) specifies the phaseresponse. In memoryless cases, IG(n)1 and &..n) are functionsof the current value of xp(n) only; however, the more generalcase results in a gain that is dependent on additionalparameters or context. This context dependence is referred toas memory and has two components: deterministic and non-
II. DISTORTION ESTIMATION
A technique has been previously presented in [4] thatdemonstrates the elimination of deterministic memory from awaveform by using mUltiple measurements, ensembleaverages and successive approximations. In this section wepresent an extension of this technique which can be used topredict the optimal linearization targets for an amplifierthroughout the DPD process. This is achieved by extracting amemoryless model, estimating the random components andusing known waveforms to generate estimates of the totalsystem performance (where the system includes both thepredistortor and amplifier as shown in Fig. 1).
In the discrete complex envelope domain when a signal,x(n), is modified by the predistortion block to generate xp(n),which is applied to an amplifier, the output, y(n), is describedas:
I. INTRODUCTION
Many behavioral models and predistortion basedlinearization techniques for RF power amplifiers (PAs) havebeen introduced in recent years [1-3]. A significant problemfor the practicing engineers is how to determine which modelshould be used for a given amplifier and how muchimprovement can be achieved with each type of digitalpredistorter (DPD). Specific questions include: willmemoryless DPD be sufficient to achieve the specification orwill memory effects need to be compensated? If so, what levelof complexity does the memory compensation need to have toachieve the target specification?
In this paper, we present a measurement oriented distortionestimation technique to enable answering these questions. Wealso demonstrate a DPD algorithm with progressively bettermemory compensation which is applied experimentally to apower amplifier.
The rest of the paper is organized as follows. In section IT,we present the details of the methodology that quantifiesmemoryless distortion, deterministic and nondeterministicmemory in the PA. In section III, we briefly introduce theopen loop DPD derived from dynamic deviation reductionbased Volterra series. In section IV, we test a class AB poweramplifier that is driven with a WCDMA signal and present theresults. This includes a series of DPD experiments which we
where f) (n) is a unifonn random variable between 0 and 21f.As we take more samples, we can a obtain more accurateestimate of the deterministic output signal and output randomeffects.
associated with other random discontinuous events or chaoticbehavior. By perfonning a coherent averaging on a number ofthese outputs, one can generate a noise reduced estimate of theoutput waveform:
deterministic. Following equation (1), we decompose the lefthand side into three terms: a memoryless term, Ymln), adeterministic memory term, Ymem(n) , and a non-deterministicterm, Yn(n) (which includes noise, non-reproducible memory,transient events, and other random events):
yen) = Yml(n) +Ymem(n) + yn(n) (2)
The memoryless tenn can be determined with onemeasurement of a complex input sequence (such as aWCDMA signal) covering a wide range of input power levels,since this is the expected gain characteristic, or memorylessmodel, of the amplifier across the range of input envelopevalues. To fonn appropriate averages, we group values ofinput power into a set of bins centered around xi (for the ith
bin), and then define:
Yml(n)=G(xp)·xp(n) (3) N=(Ym/(n) +Ymem(n)) (6)
where: lim(N-too )
(4)1 N
Yn(n) ~ y(n)-- Ly;(n)N ;=1
(7)
where the constant amplification of the compensated system Qis Go' the system gain term. For another example, withmemory predistortion, the system, Qmem, has compensatedmemory effects. If all memoryless distortion and deterministicmemory have been removed, the output of the systembecomes:
By quantifying and decomposing the output into these threeterms, one can quantify the level of model completeness aswell as perform estimates of different predistortionconfigurations. For example, if one would like the optimalmemoryless performance for the output of the system, Qml, wecan use equations (2) and (3) to get:
The remaining effect is the non-deterministic term which isn'tcompensated in the system.
Once we have these optimal performance targets for theamplifier, we can properly evaluate memory effect models andthe associated predistortion algorithms relative to theperformance specifications of the system.
(9)
(8)
III. DYNAMIC DEVIATION REDUCTION DPD
To validate the distortion evaluation methodology describedin section II, we employ an open loop DPD technique [5] asour test. This DPD approach was derived from the dynamicdeviation reduction-based Volterra series [4], that allowscompensation for nonlinear distortion and memory effectsinduced by RF power amplifiers in a very efficient andeffective way. In this approach, the Volterra model is pruned
where xp are the input bins. The actual gain G(n), (1), andexpected gain (4) define gain residue which can be used toquantify the memory effect and build the memory model.
In order to separate out the memory terms into thedeterministic and non-deterministic components, we need toperform repeated measurements of the system operating on thesame data. By looking at the output as a composite of arepeated memoryless component and deterministic memorycomponent with a random component that is different fromone waveform to the next, we can obtain an estimate of thenon-detenninistic memory term.
The deterministic memory term can be extracted byassuming that the sequence has periodic stationarity with therepeat length of the measurement sequence. This is onlypossible if the measurement is repeatable. This requires thatthe local oscillators of the upconverter and down converterhave low phase noise, are phase and frequency locked to oneanother and the clocks on the DAC and ADC's are carefullycontrolled. The output of a periodically stationary system hasan identical "context" each time a specific sample point isexperienced. By taking the measurement repeatedly andperforming proper time alignment on the output sequences wecan obtain a sample by sample difference to quantify the levelof waveform randomness. If for 2 repeated measurements withthe same input data, xp(n), we obtain outputs YJ(n), and Yin),then an initial estimate of the non-detenninistic componentcan be obtained:
YIn (n) - Y2n (n) =Yl (n) - Y2 (n) (5)
where the non-deterministic term, Yin), is a random variable(which, for a properly adjusted measurement system, weexpect to be independent, identically distributed - iid). Ideally,it is a white noise signal with random phase, but can be
by controlling the order of dynamics involved in the system,which dramatically simplifies the system implementation. Forexample, if only the first-order dynamics are taken intoaccount, the DPD function in the discrete complex envelopedomain can be written as
Fig. 4. Original PSD of the measurement and the estimated nondeterministic waveform PSD.
The optimal system performance estimates (8,9) arecalculated and the power spectral densities calculated for theseare presented in Fig. 3. Fig. 4 shows the PSD of the outputwaveform and the non-deterministic waveform obtained fromequation (5). The low value of the non-deterministicwaveform (of order 0.9%) demonstrates the high accuracy ofthe test signal generator and output captured. Comparablevalues are measured with "through" connections from input tooutput. The power in the error signal can be quantified by theNRMSE.
One thing to note is that one of the local oscillators isresponsible for the spurious tone in Fig. 3-4 at a -16MHzoffset (i.e. LO leakage). We believe this is a 3ni orderintermodulation product from the LO.
Experiment 1, on Table I, predicts a target of a NRMSE of6.03% for the memoryless DPD floor, even though the currentmeasurement has a NRMSE of 15.2% when no predistortion isapplied. Experiment 2 and 3 have increasing predistorterpolynomial order, and yield similar memoryless DPDperformance estimates. Experiment 3 with P= 11 perfonnswithin 0.05% of the memoryless DPD goal. Measurements 3-7have the same polynomial order (P=II), but a progression in
(10)
P-l
2 M
lien) =LLgZk+l,l(i)li(n)IZk
x(n-i)k=O ;=0
P-l
2 M _ • _ Z(k-l) _Z _* .+LLgzk+l,Z(I)lx(n)1 x (n)x (n-I)
k=1 ;=1
PC withMATLAB
& Labview
IV. EXPERIMENTAL RESULTS
A Nitronex GaN HEMT based PA operated in Class ABwas used for experimental measurements. The device wasbiased with Vds=28V, Vgs=-1.49V, and an idle drain currentof 0.8A, at 1.95 GHz. The system includes a pattern generator,DAC, an upconverter, and the output was captured with alogic analyzer after going through a down-converter and anADC as shown in Fig.2. Seven predistortion configurationsare evaluated, where the system (predistorter + amplifier) wasexcited with a WCDMA wavefonn (decrested to a peak toaverage ratio (PAR) of 7.5 dB). A summary of these results ispresented in Table I. The sampling rate is 15.36 MHz andthere are 9360 samples.
For each test case, the input waveform was modified by thespecified predistorter (P is the polynomial order and M is thememory depth), then applied to the PA. Then, four outputmeasurements were aquired. From the PA input and outputwaveforms, an estimate of the memoryless waveform wasmade (8) and the nonnalized root mean squared error(NRMSE [7]) to the target wavefonn was computed andreported (Est. ML in Table I). From each pair of outputmeasurements, the NRMSE of the deterministic memory limit(N=2 est. Non-Det in Table I) is computed (5).
where x(n) and u(n) are the complex envelopes of the input
and the output, respectively, and g2k+l,j (-) is the complex
Volterra kernel of the system. (-)* represents t e complex
conj ate operat on an '·1 ret rns t e rna n tease on
t e pt -or er post- nverse t eor [6], t e parameters 0 t sDPD can be recti est llRte rom t e meas re np t ano tp tot e PA w t a s mple 0 -I ne c aracter zat onprocess
TABLE I. - Experimental Results and Perfonnance EstimatesAMI." (K*vob) "aw.d Iblu. dots) Ideal Ired) Curveftt I...n)
Fig. 5c. DPD + Amplifier performance when DPD has 11 th
order, 4 memory taps DDR DPD (Experiment #7).
Pin PoNRMSE(%)
Exp P M(dBm) (dBm)
Est. N=2 est.Meas
ML Non-Det
(1) 0 0 27.77 40.98 15.2 6.03 0.7-08
(2) 5 0 25.28 38.86 6.45 6.08 0.6-0.9
(3) 11 0 25.2 38.78 6.13 6.08 0.6-1.0
(4) 11 1 25.17 38.75 4.42 6.20 0.5-1.0
(5) 11 2 25.13 38.74 1.91 6.36 0.6-1.0
(6) 11 3 25.25 38.8 1.80 6.36 0.7-1.0
(7) 11 4 25.26 38.76 1.78 6.36 0.6-1.0
ACKNOWLEDGEMENT
V. CONCLUSION
REFERENCES
[1] M. Isaksson, D. WiseH, D. Ronnow, '''A comparative analysis ofbehavioral models for RF power amplifiers," IEEE Trans.Microw. Theory Tech., vol. 54, no. 1, pp. 348-359, Jan. 2006.
[2] J.e. Pedro, S.A. Maas, ,')' IEEE Trans. Microw. Theory Tech.,vol. 53') no. 4') pp. 1150-1163, April 2005.
[3] J. Wood and D.E. Root, Eds., Fundamentals of NonlinearBehavioral Modeling for RF and Microwave Design. Norwood,MA: Artech House, 2005.
[4] A. Zhu, 1. C. Pedro, and T. J. Brazil, "Dynamic deviationreduction-based Volterra behavioral modeling of RF poweramplifiers," IEEE Trans. Microw. Theory Tech., vol. 54, no. 12,pp. 4323-4332, Dec. 2006.
[5] A. Zhu, P. Draxler, 1. Van, TJ. Brazil, D.F. Kimball, and P.M.Asbeck, "Open loop digital predistorter for RF power amplifiersusing dynamic deviation reduction-based Volterra series'')submitted to IEEE Trans. Microw. Theory Tech.
[6] M. Schetzen, The Volterra and Wiener Theories of NonlinearSystems, reprint ed. Malabar, FL: Krieger, 2006.
[7] P. Draxler, J. Deng, D. Kimball') I. Langmore') P.M. Asbeck,"Memory Effect Evaluation and Predistortion of PowerAmplifiers,'') 2005 IEEE MTT-S Digest., TH2B, 2005
An experimental methodology has been demonstrated toquantify the performance floor of memoryless and memorymodeling and predistortion. With this approach, one canquantify the nonlinear distortion and memory effects in poweramplifiers and evaluate the extent to which these effects canbe compensated by specific linearization algorithms. With onemeasurement of the system with a complex modulatedwaveform, one can estimate the expected gain characteristicsof the amplifier and the limits of memoryless DPD. Withadditional measurements, one can determine the repeatabilityof the measurement system and estimate the deterministicmemory effects that could be compensated.
The authors wish to acknowledge the assistance and supportof the Center for Wireless Communications, Qualcomm Inc.,Science Foundation Ireland, Agilent Technologies, andNitronex.
0.'
u
0,4 e.'Normalized YIn
0'" '.1Nonn"lzed Yin
0.2
'.2
O.t
0.2
O.S
0.'
0.'
0.1
10.•
10.5
JOA
Fig. 5b. Amplifier performance with 11 th order memorylessDDR DPD (Experiment #3).
the memory tap depth (M=O, 1,2,3,4). The benefit of additionalterms is diminishing by experiment 7, although the estimatesreveal that a portion of the memory effect remains to becompensated. The overall DPD AMAM performanceprogression through these experiments is presented in Fig.5.
