A General Evaluation Criteria for BehavioralPower Amplifier Modeling

David Wisell1,2,3, Magnus Isaksson'<, Niclas Keskitalo':'

'University of Gavle, Dept. of Electronics, SE-80176 Gavle, Sweden.2Royal Institute of Technology, Signal Processing Lab, SE-IOO 44 Stockholm, Sweden.3Ericsson AB, SE-16480 Stockholm, Sweden. Phone: +46 730 477942, email: david.wisell@ericsson.com."Bricsson AB, SE-80006 Gavle, Sweden.

Abstract - In this paper a new goodness measure forbehavioral complex envelope power amplifier models is definedin the frequency domain. The measure can be calculated for anyinput signal using the same formulas, which makes it general andeasy to use. The results will however be dependent on the inputsignal.

The total model error, or normalized mean-square error, forpower amplifier models are normally dominated by the in-banderror, often mainly caused by the linear distortion.

The new measure is aimed at capturing the nonlinearmodeling performance of the amplifier model. This is of interestsince it is most often the nonlinear, rather than the linear,distortion that causes most harm in real-life power amplifierapplications.

I. INTRODUCTION

THIS paper treats an evaluation criteria for behavioralpower amplifier (PA) models, mainly with

telecommunication applications in mind. Behavioral PAmodels are most often evaluated with regard to the normalizedmean-square error (NMSE). This is a measure of obviousinterest since it tells us the total error of the model. A problemwith the NMSE, as was discussed in e.g. [I], is that it isnormally completely dominated by the in-band error. This is aproblem since most often it is the out-of-band error that is ofmost interest in real-life applications since it is the out-of-banderror that disturbs the communication in other channels.

The out-of-band performance of behavioral PA models hasoften been evaluated using the difference in the adjacentchannel leakage ratio (ACLR) as defined in [2] between theoutput signal of the model and the measured output signal ofthe PA. This will tell us something about the performance ofthe model in the adjacent channel, but is a rather crudemeasure.

In [3] the measure adjacent channel error power ratio(ACEPR) was proposed for measurements on WCDMAsignals. A similar measure for other standards could easily bederived. A clear drawback of the ACEPR, as defined in [3], isthat it is standard specific, which is unfortunate for the case of

0-7803-9763-01071$20.00 ©2007 IEEE

general purpose behavioral PA modeling.This drawback becomes even more pronounced as the

standardization of future telecommunication systems within3GPP proceeds toward the use of OFDM signals [4]. Due tothe multiple carrier structure of OFDM, the variablebandwidth of future systems as well as the fragmentation ofthe bandwidth, new measures will have to be defined in thestandardization . The same reasoning essentially holds forother proposed future wireless systems as well.

While standard specific measures certainly are needed andwill be defined and implemented, from a PA modelingperspective a more suitable and general measure is needed.The measure should preferably be general with respect to PAtechnology and telecommunication standard and of course, toother signals such as multiple tone signals as well. The PAtechnology is currently undergoing a rapid development, seee.g. [5], and numerous different technologies are possible forthe future, see. e.g. [6]. It is thus important that a generalmeasure, in order to be future proof, is independent of the PAtechnology. A high correlation between the measure andstandard specific measures such as the ACLR and ACEPR, asdefined in [3], as well as other measures such as two-tonethird-order intermodulation (1M3) would be desirable as well.As for the ACEPR it should focus on the out-of-banddistortion and be a complement to the NMSE, which, asalready stated, is a measure of obvious interest in anymodeling effort. The measure should also preferably be ableto capture nonlinear memory effects in the PA so that it can beused for guiding design efforts of PAs and digital predistorters(DPDs) for these PAs.

Some complementary measures have been proposed,especially for quantifying memory effects in the PAs, see e.g.[7], [8] and [9]. In [9] the measures MER (Memory EffectRatio) and MEMR (Memory Effect Modeling Ratio) wereproposed, in [7] some different measures were evaluated andin [8] a memory effect metric based on two-tonemeasurements were proposed. In [10] a frequency scaling ofthe ACEPR was also proposed which can be seen as a firstattempt to increase the generality of the ACEPR.

where J are the bins we use to calculate the signal power, Iare the bins we use to calculate the error power and W(k) is aweighting function . We could have used a weighting functionalso for Y(k), but we cannot currently see any meaning ofdoing so.

The simplest choice is probably to use W (k) = I for all k, J

= {all k for which U(k) > threshold} and 1= {all k for which

U(k) < threshold }. Here threshold is chosen to distinguish

the intended input signal from the noise and potentialspurioses. This essentially means that we compare the totalerror for all frequencies for which there is no input power tothe total power of the input signal.

Another choice is

In this paper we propose a generalization of the ACEPR tothe case of arbitrary input signals . As for the ACEPR we workwith complex envelope versions of the input and outputsignals of the PA, see [II]. Further we assume that thesampling frequency of the output signal is chosen sufficientlylarge to measure the entire spectrum, in the first spectral zone,of the output signal above the noise floor without aliasing .This requirement could probably be somewhat relaxed inpractice .

II. THE PROPOSED MEASURE

Suppose that we have the input signal u(n) and both the

measured output signal of the PA y(n) and the output signal

z(n) of some model that we are investigating on discrete-time

complex envelope form. The complex envelope c(t) of an RF

signal s(t) = r(t)cos(2nlet+rp(t)), where r( t), rp( t) are the

envelope and phase of the signal, respectively, and Ie is the

carrier frequency in Hz, is given by [12]

WESPR = 10 loglO

I: IW( i)E ( i)12

iEI (2)

c(t) = (s(t) + js(t))e- j 27rfJ = r(t)ei<p(t), (1) max (IE(k)l)W(m) = max (I E(k)l) + !U(m)1

(3)

15

-20Xal~

Qj -40 Z(k):s:

Y\>~d0a.Q)

-60.ziiiQ)~ v:

O r--~--~------,,;v-:;~-~--~----,

and J and I as above. In practice the choice J = 1= L could be

feasible . This choice essentially means that we do not make ahard , one or zero, decision on how to weight the error, but asoft weighting, which has some advantages that will bedescribed below.

The motivation for excluding (or putting a small weight for)bins for which there is a high input signal power is thatextremely small errors in the estimation of the linear gain vs.frequency will give errors large enough to mask the nonlineareffects, which after all are the main interest for our PAbehavioral modelling efforts in most cases.

:Adjacent: :Adjacent:.Channel : :Cha nel :

-10 -5 0 5 10Relative Frequency [MHz]

Fig. 1. Illustration of the channel and the adjacent channels as well as thespectra of the input (red), measured output (dark blue), simulated output (lightblue) and error (green) signals .

where s(t) is the Hilbert transform of s(t) .

Now assume that the parameters of the model has beenidentified using some method and our intention is now to

validate this model. Thus, u(n) and y(n) are validation signals,

which are different from the signals that have been used forthe model identification. All three signals are sampled withthe same sampling rate and this sampling rate fullfills theNyquist criteria for all of the signals. These three signals are

assumed to be time-aligned as in [II] . Further, let U(k), Y(k)

and Z(k) be the normalized discrete Fourier transforms (DFTs)

of size L of the signals respectively so that max ( U(k») = 1

and max(Y(k») ;::; max(Z(k»). Also, let E(k) be the DFT of

e(n) = y(n)-z(n) and normalized accordingly.

The ACEPR is illustrated in Fig. I in which the powerspectra of the input signal to the PA (red), the output signal ofthe PA (dark blue), the output signal of the model of the PA(light blue) and the difference (error) signal of the two latter(green) is shown. The ACEPR is the power of the error signalin the adjacent channel with the highest error power to thepower of the output signal inside the channel. The samplingfrequency was 40.8 MHz.

Rather than making a measure of the "goodness" of the PA

model based on E( k) dependent on a standard, we will make it

dependent on U(k). This could be done in several ways.

We first express what we will here call the weighted error­to-signal power ratio (WESPR) on the general form

Example 1 (Perfectly estimated nonlinear term and errorin the estimate of the linear term)

Suppose that our estimation algorithm gives us the PAmodel

Example 2: (Perfectly estimated linear term and error inthe estimate of the nonlinear term)

Suppose that our estimation algorithm gives us the PAmodel

where the small signal gain has been normalized to one. Anintroduction to complex envelope behavioral PA models canbe found in [II].

A. Two Simple Examples

III. EXAMPLES

Let u(n) = cos(am) , i.e. a two-tone signal in the complex

envelope notation. In the sequel, suppose that the true outputsignal of the PA is given by

centered around i: as illustrated in Fig. 2, where (0 = 21<f .If W(k) = 1 for all k we get a discountinuity in the WESPR

as a function of the threshold depending on if the threshold isset to include the two smaller tones at ±3(O or not. If thethreshold is set below the level of the two smaller tones theWESPR will be given by the the intermodulation productsgenerated between the larger and the smaller tones and thenoise floor, while if it is set above a quite substantial WESPRwill be measured as it will be dominated by the 1M3 from thetwo largest tones. A clear drawback is that the choice ofthreshold level must be explicitly done and influence theresults heavily.

Depending on the signal characteristics in the frequencydomain an infinite number of problematic cases as the oneabove could be found. However, by introducing weightingthis problem is overcome to a large extent. Here the weightingfunction that was given in (3) is used but other choices couldbe as good or even better. Obviosly it would be beneficial if acertain weighting function is always used since otherwise theWESPR could only be given and used together with thespecific weighting funcion used. The weighting function in (3)

by inspection gives W(k) ~ 1 if U(k) is small and W(k) ~

max(E(k)) if U(k) = max( U(k)) = 1, which should be a small

number, as we desire.

The WESPR for the case of W(k) = 1 for all k as a

function of the threshold (blue) and the WESPR using (3) forthe four-tone case given above is shown in Fig. 3 for the twoPA models estimated in Examples I. Example II gives asimilar result. Even if the result in Fig. 3 is somewhat of anextreme case it clearly supports the use of a weightingfunction rather than a fixed threshold. The smallerdiscontinuity is due to the DFT and windowing of the data andfurther discourage the use of a fixed threshold.

(4)

(5)

(6)z(n) = u(n) - 0.04Iu(n)12 u(n)

z(n) = 0.99u(n) - 0.05Iu(n)12u(n)

y(n) = u(n) - 0.05Iu(n)12 u(n) ,

and the error signal e(n) = O.Olu(n) = O.Olcos(am) .

This gives NMSE ~ -40 dB while the modelling of the 1M3will only be affected by the noise and will be close to the true

value. With W(k) = 1 for all k and threshold somewhere

between the level of the two tones and the noise floor, we getthe WESPR ~ noise floor.

and the error signal p

e(n) = 0.01Iu(n)12u(n) =

0.0075cos(am) + 0.0025cos(3am).(7)

Fig. 2. The four-tone input signal. The radio center frequency becomes zeroin the complex envelope notation.

This gives NMSE ~ -42 dB and the modelled 1M3 will be1.9 dB too low.

With W(k) = 1 for all k we get the WESPR = -52 dB »

noise floor in systems of the kind discussed in e.g. [11].Clearly, despite the better NMSE in Example 2, the out-of­

band performance in the form of 1M3 is considerably worsethan in Example 1. The WESPR is significantly lower inExample I which indicate its suitability as a measure of themodels ability to capture the nonlinear distortion.

B. Introducing Weighting

Consider the input signal u(n) = cos(am) + 0.0Icos(3am) .

In the complex envelope notation this is a four-tone signal

i-31 -1 1

a (=fc)

i31 1

-50 ,-------~---~--~---~-__,

-55

-60

co~-650:::0..

~ -70S

-75

-80

-85-80 -70

III

r---'I

-60 -50Threshold [dB]

-40

is derived explicitly for the WCDMA standard and a generalmeasure of the out-of-band modeling error cannot be expectedto be better than the standard specific measures for thatspecific standard.

The WESPR was further applied on the Doherty amplifiermeasurements previously reported in [13]. Thesemeasurements were carried out using a two carrier WCDMAsignal with the carriers spaced 15 MHz apart. This means thatthe problems with standard specific measurements start toemerge . Such a two carrier signal must conform to a numberACLR and spectral emission mask requirements, sometimesdifferent for different frequency bands and geographicalregions. While these requirements are necessary from aregulatory body perspective, they are unsuitable as targets foracademic PA behavioral modeling research.

5

--Order = 1--Order = 3--Order = 5--Order = 7--Order = 9--Order = 11--Order = 13

2 3 4Memory Length [Samples]

-65 '-----__-'----__...L.-__--L.-__---'-__---'

o

-60

-45iii'~

~ -50::2z

-55

-35 r--- --.- - - - - - - - - - - -,

~::::------------~

-40 r--~============1

Fig. 3. The WESPR as a function of a constant threshold level (blue). TheWESPR obtained using the proposed weighting function is shown forreference (red).

The WESPR could also be viewed upon as a filtering of theerror signal in the time domain prior to calculation of theNMSE . Due to our definition of W(k) , i.e. W(k) ~ I , this

means that WESPR s NMSE, where W(k) = 1 for all k gives

equality.The ACEPR can also be viewed as a special case of the

WESPR where J = {frequency bins inside the channel , see

Fig. I}, I = {frequency bins in the adjacent channel with

highest error power, see Fig. I} and W(i) = I, for i E I and

Wei) = 0, otherwise .

-35 ,-----.------~------,---.----------,

Fig. 4. NMSE. The NMSE does not increase significantly for higher ordersand memory lengths.

Fig. 5. ACEPR. The ACEPR continues to decrease as a function of bothmemory length and nonlinear order even when there is no significantimprovement in NMSE. SeeFig. 4 for legend.

51 234Memory Length [Samples]

-60

~-45

~ ~-------------~g: -50woc:( -55

-40

IV. EXPERIMENTAL

In this Chapter we apply the WESPR using the proposedweighting function to a complex envelope behavioral PAmodel and two different kinds of PAs.

First a comparison to the ACEPR and NMSE for WCDMAsignals is treated in order to see if the WESPR has the sameproperties as the ACEPR to judge the out-of-band modelerror. The comparison was done on data previously used in[II]. Thus, the reader is referenced to [11] for information onthe measurement set-up, the PA, measurement procedures, etcetera. The PA behavioral model used for the comparison wasthe widely used parallel Hammerstein (PH) model, again see[II]. The results are shown in Fig. 4 (NMSE) , Fig. 5(ACEPR) and Fig. 6 (WESPR). The same scale has onpurpose been used in the three figures in order to highlight thedifferences.

As has been noted in e.g. [1] the NMSE is unsuitable todetermine the necessary order and memory length of the PHmodel since the ACEPR decreases for higher orders andmemory lengths while the NMSE does not. The WESPR issomewhere in between . It does however have a similarperformance as the ACEPR for higher orders and memorylengths, as expected. It shall be remembered that the ACEPR

optimizing the parameters of the model towards a lowWESPR similar to what was proposed in [14] for the case ofACEPR by means of spectral weighting.

[I] M. Isaksson and D. Wisell , "Extension of the Hammerstein Model forPower Amplifier Applications," presented at ARFTG 63, Fort Worth ,2004.

[2] ETSI, "3GPP TS 25.141 latest rev."[3] M. Isaksson, D. Wisen , and D. Ronnow, "Nonlinear Behavioral

Modeling of Power Amplifiers Using Radial-Basis Function NeuralNetworks," presented at IEEE MTT-S , Los Angeles , 2005.

[4] 3GPP, "3GPP TS 25.212 v. 7.3.0 Multiplexing and Channel Coding(FDD)," 2006.

[5] B. Berglund, M. Englund , and J. Lundstedt, "Third design release ofEricsson 's WCDMA macro radio base stations ," Ericsson Review, pp.70-81, 2005.

[6] S. C. Cripps , Advanced Techniques in RF Power Amplifier Design.Boston : Artech House , 2002.

[7] M. Pirazzini , G. Fernandez, A. Alabadelah, G. Vannini, M. Barciela , E.Sanchez, and D. Schreurs , "A preliminary study of different metrics forthe validation of device and behavioral models," presented at 65 thARFTG Conf, Dig., Los Angele s, 2005.

[8] 1. P. Martins , P. M. Cabral , N. B. Carvalho , and J. C. Pedro, "A Metricfor the Quantific ation of Memory Effects in Power Amplifiers," IEEETrans. Microwave Theory and Tech. , vol. 54, pp. 4432-39, 2006.

[9] H. Ku and 1. S. Kenney , "Behavioral Modeling of Nonlinear RF PowerAmplifiers Considering Memory Effects ," IEEE Trans. MicrowaveTheory Tech., vol. 51, pp. 2495-2504,2003.

[10] M. Isaksson, D. Wisell , and D. Ronnow , "Wideband Dynamic Modelingof Power Amplifiers Using Radial-Basis Function Neural Networks,"IEEE Trans. Microwave Theory Tech., vol. 53, pp. 3422-28 , 2005.

[II] M. Isaksson , D. Wisell , and D. Ronnow , "A Comparative Analysi s ofBehavioral Models for RF Power Amplifier s," IEEE Trans. MicrowaveTheory Tech. , vol. 54, pp. 348-359 , 2006.

[12] J. Proakis , Digital Communications: McGraw-Hill, 1995.[131 D. Wisen, M. Isaksson , N. Keskitalo , and D. Ronnow , "Wideband

Characterization of a Doherty Amplifier Using Behavioral Modeling,"presented at ARFTG 67, San Fransisco, 2006 .

[14] M. Isaksson and D. Ronnow , "A Parameter-Reduced Volterra Model forDynamic RF Power Amplifier Modeling based on Orthonormal BasisFunction s," Int. J. RF and Microwave Computer-Aided Eng. In Press.

ACKNOWLEDGMENT

This work was supported by the Knowledge Foundation(KKS) , the University of Gavle (RiG), the Graduate School ofTelecommunications (GST), The Royal Institute ofTechnology (KTH), Ericsson AB, Freescale SemiconductorNordic AB, Infmeon Technologies Nordic AB, NOTE AB,Racomna AB, Rohde & Schwarz Sverige AB and SyntronicAB.

REFERENCES

5

30

234Memory Length [Samples]

-20 -10 0 10 20Relative Frequency [MHz]

-80-30

-60

-70

-65 '-------'------'------'-------'-----'o

Fig. 6. WESPR. The WESPR shows a similar behavior as the ACEPR . SeeFig. 4 for legend.

The power spectra of the input (blue), output (red),simulated output (green), error (cyan) and weighted error(black) signals are shown in Fig. 7. It is clear from Fig. 7 thatthe weighted error signal , compared to the error signal, issubstantially suppressed for the frequencies where the twoWCDMA signals have power. Thus, the WESPR is mostlydominated by the out-of-band distortion as intended.

-35 r-----.-------,---,..------.-------,

~-45eo~

g: -50CI)wS -55

Fig. 7. The two carrier WCDMA signal. The input (blue) , output (red),simulated (green) , error (cyan) and weighted error (black) signals are shown .

o

-10

-20

@-30;:Q.

CD -40:;:~ -50

-40 1:-- - - - - - - - - - - - - - - ---,l

-60

V. CONCLUSION & DISCUSSION

In this paper we have proposed a new measure, WESPR,for evaluation of behavioral PA models. The WESPR hassome clear advantages over existing measures. It can becalculated on any signal on any PA, regardless of standard orPA technology.

Further investigations are needed in order to correlate theWESPR to standard specific measures, which would be mostbeneficial. Also a discussion and standardization of theweighting function is needed .

Further investigations will focus on the effects of