Measurement 46 (2013) 2735–2745

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Measurement

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Test bed for power amplifier behavioral characterizationand modeling

0263-2241/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.measurement.2013.04.013

⇑ Corresponding author. Tel.: +65 6790 6587; fax: +65 6791 7320.E-mail addresses: fu0001ai@ntu.edu.sg (K. Fu), ECLLAW@ntu.edu.sg

(C.L. Law), TTThein@ntu.edu.sg (T.T. Thein).

Kai Fu ⇑, Choi Look Law, Than Tun TheinSchool of EEE, Nanyang Technological University, Singapore 637553, Singapore

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 September 2012Received in revised form 19 January 2013Accepted 5 April 2013Available online 20 April 2013

Keywords:Power amplifierBehavioral characterization and modelingTest bedImpairments compensationEnvelope recovery

Traditional test bed of power amplifier (PA) behavioral characterization and modelingbased on vector signal analyzer (VSA) and often equipped with specialized software is bothexpensive and inflexible to modify to suit different scenarios. In this paper, a new test bedbased on an oscilloscope or other general purpose data acquisition systems, which worksas analog to digital converter (ADC) with a proper (radio frequency) RF bandwidth andmaximum sampling rate, is proposed. The common impairments, e.g. transmitter IQ imbal-ances, channel delay, frequency offset, and carrier phase offset, are all well compensated.The accurately recovered envelopes of the PA’s input and output signals are used for aPA behavioral characterization and modeling. Furthermore, Relative envelope error (REE)parameter is proposed to evaluate the accuracy of envelope recovery. The experimentshows a very accurate RF signal envelope recovery, and a good performance of PA behav-ioral modeling.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

An accurate, efficient, and flexible test bed is critical toPA behavioral characterization and modeling. Since PA AM/AM and AM/PM characterization and behavioral modelingare only concerned with the envelopes of the correspond-ing radio frequency (RF) signals, the test bed is requiredto accurately recover the envelopes of the PA’s input andoutput signals in the form of digital samples. The testbed requires almost all the functions of a physical receiver,e.g., down conversion, ADC, demodulation, and basebandsignal processing. Moreover, it should perfectly compen-sate all kinds of impairments occurring in the channelswhile leaving the PA nonlinear memory influence un-touched, e.g., channel delay, frequency offset, or IQ imbal-ances, etc. Meanwhile, the test bed also needs a flexiblesignal generator which can output enough linear powerto drive the PA and generate all kinds of signal forms, such

as two-tone, OFDM, and white Gaussian signals, to facili-tate the PA behavioral characterization and modeling.

Currently a typical test bed is based on a specializedinstrument called vector signal analyzer (VSA) [1–3], oftenequipped with a specialized digital signal analysis soft-ware, e.g., Agilent 89600 VSA software. Meanwhile, real-time spectrum analyzer (RSA) works in the similar wayas the VSA [4], and can serve as the receiver in a testbed. However, the major drawback is their high price.Moreover, all the functions are packed into the instrumentand embedded analysis software. Little modification can bemade, rendering it inflexible or unable to work for scenar-ios which require different test methods to compensate forthe various kinds of impairments present in the test bed.Large signal network analyzer (LSNA) is used in [5] to mod-el the PA behavior over a wide range of frequency span.However, Fager et al. [6] argues that LSNA requires themodulation to be periodic, which limits the type of testingsignals. Moreover, whether it can measure PA’s instanta-neous behavior or not is still unknown.

Instead of using specific instruments or software, theoscilloscope or other general purpose data acquisition

3dBcoupler

Oscilloscope

E4438C(Signal Gen)

3dBcoupler

RF

FIFI

DownConverter

(A)

Down Converter

(C)

Attenuator

DUT

Attenuator

Power Amplifier

LO

A C

Channel A Channel C

CgiSAgiSPersonal Computer

Fig. 1. Overall structure of test bed for PA behavioral modeling.

2736 K. Fu et al. / Measurement 46 (2013) 2735–2745

systems, which works as analog to digital converter (ADC)with a proper RF bandwidth and maximum sampling rate,is a good alternative for a PA behavior test bed. If the RFbandwidth of the oscilloscope is higher than the signal’scarrier frequency, the signal can be acquired directly;otherwise, an external down-converter is required. It turnsout to be low cost, more flexible to implement differentsignal recovery methods, and with no limitation on thetype of input signals. There are several PA behavior testbeds based on oscilloscope [7,6]. However, none of themgive details on how the test bed is implemented, nor com-pensated for different kinds of impairments of the channel.The authors in [8] show a detailed WiMAX system test bedbased on oscilloscope and the steps to implemented andcompensate channel impairments. However, the test bedis only for WiMAX system quality measurement, and notdemonstrated for PA behavioral characterization andmodeling.

In this paper, a PA behavior test bed based on generalpurpose oscilloscope is proposed, and detailed steps onhow to implement it is shown.The major impairments suchas IQ imbalances, channel delay, frequency offset andphase offset, are considered, analyzed and compensated.After compensating all the impairments, the envelopes ofthe PA’s input and output signals are accurately recoveredin the form of digital samples for behavioral modeling.Moreover, to quantify the nonlinearity and memory effectsof the PA – memory effects refer to the phenomenon thatthe PA output is determined not only by the simultaneousinput signal but also by the past ones – both adjacent chan-nel power ratio (ACPR) and error vector magnitude (EVM)are measured. Welch method [9] is employed to numeri-cally estimate the spectrum and compute the ACPR perfor-mances with arbitrary resolution.

The structure of the paper is organized as follows: InSection 2, the overall test bed structure is presented, fol-lowed by detailed steps and explanations of software sig-nal processing implementation to compensate the majorchannel impairments. In Section 3, performance metricsare designed, and the performance of the test bed is as-sessed by experiments, and the results are presented. Fi-nally, Section 4 gives the conclusion.

2. Proposed test bed and signal processing

2.1. Test bed overall structure

Fig. 1 shows the overall structure of the Test Bed. Herethe RF bandwidth of the oscilloscope is assumed to be low-er than the carrier frequency of the RF signals, so twoexternal down converter channels are used to convert thesignals to intermediate frequency (IF). If the RF bandwidthof the oscilloscope is high enough, the external down con-version can be omitted. Throughout the paper, it is as-sumed that external down converters are required.

Agilent E4438C ESG is a vector signal generator that cangenerate arbitrary waveform and up-convert it to RF sig-nal. In cases where the RF frequency of the signal generatoris lower than the required carrier frequency, an up-con-verter may be used. In order to characterize its nonlinear

behavior, the RF signal power level should be high enoughto drive the device under test (DUT: high power amplifier)to work in its nonlinear zone. The input and output signalsof the DUT pass through separate down conversion chan-nels – Channel A and Channel C. In each channel, the signalpasses through a variable attenuator before down conver-sion. This ensures that the input signal level presented tothe down-converter is within its linear zone.

Finally, the PA input and output signals in IF frequen-cies, denoted as Sig A and Sig C, are acquired by two chan-nels of the oscilloscope. The signals from the oscilloscopeare digitized and are passed to a personal computer (PC)for digital demodulation, impairments compensation, andother digital signal processing. The accurately recoveredDUT input and output signals in baseband forms are usedto characterize and model the DUT behavior.

2.2. Software overall structure

The overall structure for digital signal processing whichare implemented in software running on the PC is shown inFig. 2. At the transmitter, the baseband waveform of thetest signal is generated, followed by IQ imbalances linearpre-distortion. At the receiver, channel delay, frequencyoffset, and phase offset are compensated.

2.3. TX IQ imbalances compensation

In our test bed, IQ imbalances in the demodulator areavoided by the use of heterodyne structure and digitaldemodulation. However, IQ imbalances are present in themodulator (Agilent E4438C ESG VSG) due to its analog di-rect up conversion structure. Although they do not influ-ence the relationship between the envelopes of the PA’sinput and output signals, where they suffer from the sameIQ imbalances, IQ imbalances in the modulator degradethe performance of PA behavioral characterization and

Fig. 2. Block diagram of digital signal processing.

Fig. 3. Linear predistorter to compensate IQ imbalances in modulator.

K. Fu et al. / Measurement 46 (2013) 2735–2745 2737

modeling in two ways. Firstly, IQ imbalances severely dis-tort the training sequences, and in turn degrade the perfor-mance of time alignment, frequency offset compensationand phase offset compensation. As a result, the accuracyof the recovered envelopes is significantly decreased. Sec-ondly, IQ imbalances degrade the evaluation metrics suchas adjacent channel power ratio (ACPR) and error vectormagnitude (EVM). In this case, the metrics cannot preciselyevaluate the performance of PA behavioral modeling andlinearization, since part of the degradation is due to IQimbalances in the modulator instead of the PA nonlinearand memory effects. So the frequency-dependent IQ imbal-ances in the modulator should be eliminated.

There are compensation methods based on either signalprocessing [10] or hardware architecture [11]. In this pa-per, a dual-input linear pre-distorter is proposed by Dinget al. [10] to compensate IQ imbalances of a ESG in a sim-ilar way that traditionally a nonlinear pre-distorter is usedto linearize a nonlinear PA. Contrary to other methodsusing compensation at the receiver side that require train-ing sequences or complex mathematical computation, the

linear pre-distorter only requires to inversely model theIQ imbalances using linear FIR filters. The method is shownin Fig. 3.

The IQ imbalances including crosstalk are modeled by 4FIR filters, and the linear predistorter is also modeled byanother 4 FIR filters correspondingly, as shown in Fig. 4.

As shown in Fig. 4, the complex signal u(n) is the origi-nal signal, x(n) is the predistorted signal that goes into theDAC, and y(n) is the lowpass equivalent of the RF signalafter up conversion, where:

uðnÞ ¼ uiðnÞ þ j � uqðnÞxðnÞ ¼ xiðnÞ þ j � xqðnÞyðnÞ ¼ yiðnÞ þ j � yqðnÞ

8><>: ð1Þ

Assuming 4 FIR filters in I/Q linear predistorter have thesame length M, and using vector expression, a sequence ofdata x(n) and u(n) can be written as:

x ¼ Uipi þ Uqpq ¼ Ui;Uq½ �pi

pq

" #ð2Þ

where

x ¼ ½xðM � 1Þ; . . . ; xðN � 1Þ�T

pi ¼ ½pið0Þ; . . . ;piðM � 1Þ�T

pq ¼ ½pqð0Þ; . . . ; pqðM � 1Þ�T

8>><>>: ð3Þ

with

piðnÞ ¼ p11ðnÞ þ j � p21ðnÞ;0 6 n 6 M � 1pqðnÞ ¼ p12ðnÞ þ j � p22ðnÞ;0 6 n 6 M � 1

(ð4Þ

and Ui = Re{U}, Uq = Im{U} for

U ¼

uðM � 1Þ uðM � 2Þ � � � uð0ÞuðMÞ uðM � 1Þ � � � uð1Þ

..

. ... . .

. ...

uðN � 1Þ uðN � 2Þ � � � uðN �MÞ

266664

377775 ð5Þ

Here pi and pq can be identified without even knowing theIQ imbalances model. Since I/Q linear predistorter is the in-verse function of the IQ imbalances model, in perfect con-ditions, u(n) and y(n) are the same except for some delay.So replace u(n) with y(n), Eq. (2) turns out to be

x ¼ Yipi þ Yqpq ¼ Yi;Yq½ �pi

pq

" #ð6Þ

Fig. 4. IQ imbalances model and linear predistorter.

Fig. 5. Digital demodulation.

Fig. 6. General block diagram of down sampler.

2738 K. Fu et al. / Measurement 46 (2013) 2735–2745

Least square method is used to identify the coefficients[10]

pi

pq

" #¼

YTi Yi YT

i Yq

YTqYi YT

qYq

" #�1YT

i x

YTqx

" #ð7Þ

To use Eq. (7), a sequence of data x(n) and y(n) need tobe acquired. Here x(n) is ready as it is the digital signal thatgoes into the DAC. y(n) can be acquired using the test bedshown in Fig. 1 with DUT removed. Either Channel A orChannel C can be used, and the acquired IF signal is furtherdigitally processed following the procedure in Fig. 2 to re-cover y(n), i.e., the envelope of the RF signal that suffersfrom IQ imbalances. The identification of the IQ imbalancespredistorter is done off-line.

2.4. Digital demodulation

The advantage of digital demodulation [12] is that thereare no IQ imbalances impairments in demodulation. Thedrawback is that it requires very high sampling rate – nor-mally four times of IF carrier frequency or higher. Luckily, amodern oscilloscope usually has a RF bandwidth of severalGHz, and a maximum sampling rate faster than 10 Gsps.The high sampling rate makes it easy to acquire the signal;moreover, it increases the time resolution of signal timealignment to counter the channel delay effects.

The received IF digitalized signal can be denoted as:

~rðnTsÞ ¼ I0ðnTsÞ cosð2pðf0 þ Df ÞnTs þ /Þ � Q 0ðnTsÞ� sinð2pðf0 þ Df ÞnTs þ /Þ þ NðnTsÞ ð8Þ

where fs is the sampling frequency, Ts = 1/fs is the sampleinterval, f0 is the nominal intermediate frequency, Df isthe frequency offset from the nominal intermediate fre-quency due to the local oscillator (LO) frequency differencein the down converters, / is a constant phase, and N(nTs) isthe channel noise. The digital demodulation is shown inFig. 5. It uses incoherent demodulation, so there is a phaseoffset h in LO. The higher order components are filtered outby digital low pass filters (LPFs), which are also down sam-plers that reduce signals to baseband sampling rate. HenceLPFs can be seen as the boundary line that divides overalldigital signal processing into high sampling rate domainand baseband sampling rate domain. Let fbb denote the

baseband sampling rate, and Tbb = 1/fbb is baseband sam-pling interval. With perfect time alignment and down sam-pling operation, the baseband form signal is

rðmTbbÞ ¼ ~rlðmTbbÞejð2pDfmTbbþhÞ

¼ ½I0ðmTbbÞ þ j � Q 0ðmTbbÞ�ejð2pDfmTbbþhÞ ð9Þ

where ~rlðnTsÞ is the low pass equivalent of the received IFsignal ~rðnTsÞ. Here it is shown that there are both fre-quency offset Df and phase offset h impairments in the re-ceived signals.

2.5. Efficient implementation of down sampling and LPF

As shown in Fig. 5, the LPF accomplishes three goals:

(1) It filters out the high frequency components.(2) It reduces the sampling rate to baseband sampling

rate, based on which the PA behavior is character-ized and modeled. The down sample ratio is D = f/fs.

(3) It completes the first step of time alignment by find-ing the optimal down sample position out of every Dhigh rate samples.

The first two goals and the corresponding designs areintroduced and analyzed in this subsection, while the thirdone is analyzed in the next subsection.

Fig. 6 shows the general block diagram of the downsampler. The input signal x(n) is convoluted with an anti-aliasing filter h(n), and then down sampled by a factor D.The down sampled output y(m) is given by:

yðmÞ ¼ zðmDÞ ¼XN

k¼0

hðkÞxðmD� kÞ ð10Þ

However, the above structure is very inefficient, becausethe filter is operating in high sampling rate, and onlyone out of every D filtered samples is a valid output. Inother words, most of the filtering computation is wasted.

Fig. 7. Polyphase filter structure.

−20 −15 −10 −5 0 5

−10

−5

0

5

10

15

20

25

30

35AM/AM relation

PA o

utpu

t pow

er (d

Bm)

PA input power (dBm)

not alignedaligned

Fig. 8. The time misalignment influence on AM/AM scattering.

10 short training sequences

2 long training sequences

Data payload characterizing PA behavior

Fig. 9. Signal structure.

K. Fu et al. / Measurement 46 (2013) 2735–2745 2739

To improve the efficiency, the down sampling operationcan be embedded into the filter. In direct-form FIR filterstructure [13], all the multiplications and additions are cal-culated in baseband sampling rate, which reduces thecomputational complexity by a factor D.

The computation efficiency can be further improved byreducing the length of the FIR, from a M-tap filter into a setof smaller filters of length K = M/D, where M is selected tobe a multiple of D. The structure is shown in Fig. 7. It iscalled Polyphase filter [13], which can be seen as a set ofsubfilters. Each time, only one subfilter of length K = M/Dis calculated, where the efficiency is further increased byanother factor D. The subfilters are defined as

Q lðnÞ ¼ hðnD� lÞ; l ¼ 0;1; . . . ;D� 1 ð11Þ

2.6. Time alignment

2.6.1. Channel delay differenceAs shown in Fig. 1, the PA’s input and output signals

passes through Channel A and Channel C respectively andtheir baseband signals are used to characterize the PAbehavior. Since time delays through Channel A and Chan-nel C are different, Sig A and Sig C are misaligned in time.This misalignment shows up as widely scattered points inthe AM/AM plot as shown in Fig. 8. The results are forchannel delay difference between Sig A and Sig C of19.1 ns.

2.6.2. Training sequencesTo counter channel delay, frequency offset and phase

offset impairments, training sequences are transmitted.Here 10 short training sequences and two long training se-quences, both defined in IEEE802.11a [14], are transmitted,followed by data payload that is used to characterize PAbehavior. The signal structure is shown in Fig. 9.

2.6.3. Time alignment methodInstead of finding out the channel delay difference be-

tween Channel A and Channel C and then compensatingthe difference, both received signals, Sig A and Sig C, arealigned with the receiver timer. Based on the signal struc-ture shown in Fig. 9, the first sample in the short trainingsequences marks the start of the signals. If the start sam-ples of both Sig A and Sig C can be precisely found, the startsamples can be used as the time origins, from which the

receiver starts to receive and process the two channel sig-nals. In this way, Sig A and Sig C are aligned in time withreceiver timer, and hence are aligned to each other.

Now the task is to precisely find the start sample ofeach signal. This is often called Time Synchronization inOFDM systems. There is a lot of research work on how tosolve Time Synchronization problem [15–17]. A classicalmethod is to do cross correlation between received shorttraining sequences and ideal ones – distortion-free versionof the short training sequences stored in receiver [18].Depending on the number of ideal training sequences used,the magnitude of cross correlation results has one or morepeaks, the first of which marks the start time of the re-ceived signal. Let v(n) denote the ideal short training se-quences, r(n) denote the received signal, p denote thefirst sample of the signal or the first peak position, andRxx denote the normalized cross correlation.

RxxðmÞ ¼PL

n¼0v�ðnÞrðnþmÞPLn¼0jv�ðnÞj

2 ð12Þ

where L is length of ideal short training sequences, and

p ¼ arg maxmjRxxðmÞj ¼ arg max

m

PLn¼0v�ðnÞrðnþmÞPL

n¼0jv�ðnÞj2

���������� ð13Þ

Here a modification is made on the classical method – in-stead of computed in baseband sampling rate after thedown sampler, here cross correlation is done in highsampling rate before down sampling operation, in orderto increase the resolution. An up sampling operation byinterpolation may even be needed to increase timesynchronization resolution, which in turn increases theaccuracy of signal recovery and PA behavioral character-

2740 K. Fu et al. / Measurement 46 (2013) 2735–2745

ization and modeling. The cost is to use more computationresources.

2.6.4. Two steps of time alignmentTo implement time alignment scheme, two steps are

designed:

(i) The first step is to find the optimal down samplingposition – which sample out of every D samples tobe chosen to output in the down sampler. It is donein high sampling rate, and determines the resolutionof time alignment.

(ii) The second step is to find the start sample in base-band sampling rate. This is the same as the classicalTime Synchronization method.

In each step, a cross correlation between the receivedshort training sequences and the ideal ones is calculated,where the first is in high sampling rate, while the secondis in baseband sampling rate. For the first step, the first cor-relation peak determines the optimal down sampling posi-tion, or which sample out of every D samples to output inthe down sampler. Let p denote the first correlation peakposition, and d denote the optimal down sampling posi-tion, then

d ¼ p mod D ð14Þ

For implementation simplicity, the down sampler rateis fixed, while a buffer of adjustable length is inserted be-fore the down sampler to facilitate changing down sam-pling position. The final diagram of demodulation withLPFs, down samplers, and the first step of time alignmentis shown in Fig. 10.

2.7. Frequency offset compensation

Due to the inaccuracy of LOs in the up converters anddown converters, frequency offset is a typical channelimpairment which can severely distort signals and degradethe accuracy of envelope recovery for the PA behavioralcharacterization and modeling. The algorithm proposedin [19] is used to estimate the frequency offset Df, and thencompensate it. For 2H repetitive training sequences, each

Fig. 10. Final digital demodulation with LPFs, down samplers, and thefirst step of time alignment – finding optimal down sampling position.

having a length of P samples, the normalized frequency off-set is estimated as

De ¼ HpXH

l¼1

xðlÞ argfRðlÞR�ðl� 1Þg ð15Þ

where the weights are

xðlÞ ¼ 3 � ð2H � lÞð2H � lþ 1Þ � H2

Hð4H2 � 1Þð16Þ

and the lP-lag autocorrelations are

RðlÞ ¼XN�lP

k¼1

rðkþ lPÞr�ðkÞ ð17Þ

In the test bed, two steps are designed to estimate thefrequency offset.

(1) In the first step, 10 short training sequences areused, where H = 5. By averaging over five weightedestimates, the noise influence is greatly reduced.However, due to their short length, short trainingsequences cannot estimate small frequency offset,or the estimate error is relatively large. So this stepis called coarse frequency offset estimation.

(2) In the second step, two long training sequences areused, where H = 1. Since their length is five timeslonger than that of the short training sequences, theestimation resolution is also increased by a factor 5.Hence it is called fine frequency offset estimation.

2.8. Phase offset compensation

After compensating the frequency offset, the receivedsignals still suffer from phase offset impairments. Phaseoffset is a constant phase shift added to the recoveredenvelopes due to inherent demodulation. Moreover, sincephase offsets are different in different experiments due tothe unknown carrier phase, the PA AM/PM characteriza-tion and modeling suffer from random phase shift or phaseuncertainty over different experiments. Hence phase offsetshould be compensated before the envelopes of the PA’s in-put and output signals can be accurately recovered.

Similar to the time alignment, a classical method toestimate phase offset is based on the cross correlation be-tween received long training sequences and ideal ones –distortion-free version of the long training sequencesstored in the receiver [20]. Let u(n) denote the ideal longtraining sequences, w(n) denote the received long trainingsequences suffering from phase offset h. Without consider-ing noise, and assume perfect time alignment and fre-quency offset compensation

wðnÞ ¼ uðnÞejh ð18Þ

Then the normalized cross correlation Rxx is

Rxx ¼PK

n¼0u�ðnÞwðnÞPKn¼0ju�ðnÞj

2 ¼ ejh ð19Þ

where K is the length of the long training sequences. Hencethe phase offset estimation is the angular of the cross cor-relation results

K. Fu et al. / Measurement 46 (2013) 2735–2745 2741

h ¼ argfRxxg ¼ argPK

n¼0u�ðnÞwðnÞPKn¼0ju�ðnÞj

2

( )ð20Þ

3. Performance assessment

3.1. Measurement setup

The structure shown in Fig. 1. Agilent E4438C ESG vec-tor signal generator with output frequency ranging from250KHz to 3 GHz is used to generate arbitrary waveformsignals, where modulation bandwidth is 100 MHz. A 4-channel oscilloscope model Infiniium 54832D with 1 GHzanalog bandwidth, 4GS/s maximum sample rate and 8-bitvertical resolution is used. The DUT is a Class A Mini-Cir-cuits ZVE-8G high power amplifier (HPA) with carrier fre-quency of 2.4 GHz.

To assess the test bed, signals with training sequences asshown in Fig. 9 are generated with output frequency at2.4 GHz, and then passed through the DUT. The input andoutput signals of the DUT are down converted separately toIF of 50 MHz, and received into the oscilloscope with a sam-pling rate of 2GS/s. The signal generator and oscilloscope areboth connected to a PC through GPIB. Advanced Design Sys-tem (ADS) and Matlab are cooperatively programmed toimplement digital signal processing, PA behavioral modelingand metrics calculation. For instance, IEEE802.11a basebandwaveform is firstly generated in ADS, and then downloadedinto E4438C through GPIB to generate the standardIEEE802.11a signal. The received signals in oscilloscope aretransferred to PC through GPIB for further processing.

The resources cost, both spatially and temporally, are asfollows: The analog memory size of the oscilloscope Infin-iium 54832D is set to be 4,194,304 points, or 2,097,152points per channel (If more samples are needed, just re-fresh the memory, and then read new samples into thememory. The progress can be repeated continuously andautomatically, until the number of samples satisfies therequirement). And under the analog sampling rate of2 Gsps, it means that the oscilloscope can read into thememory about 1 ms length of signals of both Channel Aand Channel C at once. It takes around 12 s to transferthe sampled signal of each channel from oscilloscopememory to PC, which has a two-core Pentium 4 CPU run-ning at 3.00 GHz. After that, the digital signal processingblocks for each channel, which are implemented in ADS,takes around 34 s to find the optimal down sampling posi-tion, 19 s for digital demodulation and down sampling, andfinally 59 s for all the others to recover the envelope. Forthe computer memory resources, it requires about 180 Mbytes of memory to find the optimal down sampling posi-tion alone, and then about 40 M bytes for all the other pro-cesses put together. If longer than 1 ms length of signalsare needed, the whole process above is repeated with pro-portionally increased processing time. Meanwhile, thereare some other metrics used in VSA [21] – Vector (IQ) dia-gram shows the transitions between adjacent constella-tions, eye diagram shows IQ versus time behavior – bothcan help evaluate the distortion of the whole test bed,and will be added in the future.

3.2. Recovery accuracy metric

The difficulty in evaluating the accuracy of signal recov-ery is that the envelopes of the PA’s input and output sig-nals cannot be directly acquired to be compared with therecovered ones. Here a new way to evaluate the recoveryaccuracy is introduced to overcome this problem. The basicidea is to use the transmitter baseband digital waveform toapproximate the complex envelope of the RF signal whenthe DUT is removed.

Assuming that there is no distortion or impairments inthe signal generator (Agilent E4438C ESG), in anotherword, assume the DAC and up-conversion is absolutely lin-ear. Then the transmitter baseband digital waveform justbefore the DAC can be regarded as the lowpass equivalentor complex envelope of the RF signal at point A (or C) point.The envelope differences between the transmitter base-band digital waveform and one of the recovered envelopes(Env. A or Env. C) is a competent metric to evaluate therecovery accuracy. It is defined as the relative envelope er-ror (REE) given by:

REEðdBÞ ¼ 20log10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNn¼1jsðnÞ � rðnÞj2PN

n¼1jsðnÞj

2vuut ð21Þ

where s(n) and r(n) are the transmitter baseband digitalwaveform and recovered envelope respectively. If the val-ues of REE of both Channel A and Channel C are very small,the test bed is regarded as accurately recovering the enve-lopes of the PA’s input and output signals.

3.3. Measurement results on envelope recovery

3.3.1. EVM performanceThe accuracy of the recovered envelope is totally depen-

dent on the linearity of the channels. After removing theDUT in the test bed, EVM measurement can be used to as-sess the linearity of the channels and how well all kinds ofimpairments are compensated. The constellation of the re-ceived signal through Channel A is shown in Fig. 11 andEVM is measured to be �42.3 dB.

The EVM measurements for Channel A and Channel C isshown in Table 1. Both EVM values are below �41 dB,which justifies that the channels are quite linear and allthe impairments can be compensated very well, and thetest bed is able to accurately recover the envelopes ofPA’s input and output signals.

3.3.2. REE performanceREE define in Eq. (21) is measured to directly assess the

accuracy of envelope recovery. Tables 2 and 3 show theREE measurement results under different baseband sam-pling rate. A range of baseband sampling rate is requiredto satisfy PAs of different memory effects and signals of dif-ferent bandwidth. From the measured REE values, thebaseband sampling rate does not influence the accuracyof the envelope recovery very much. All the relative enve-lope errors between transmitter baseband digital wave-form and the recovered envelopes are below �25 dB or5.5%, representing a good envelope recovery.

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Constellation with compensating IQ imbalances

Fig. 11. Constellation: 16 QAM + BPSK, with compensated IQ imbalancesin modulator, EVM is �42.3 dB.

Table 1EVM measurement results for Channel A and Channel C.

Channel A Channel C

EVM value �42.3 dB �41.3 dB

Table 2Channel A: REE values versus baseband sampling rate.

Baseband sampling rate (MS/s) REE value (dB)

20 �25.640 �26.180 �26.4

100 �25.7200 �26.2

Table 3Channel C: REE values versus baseband sam-pling rate.

Baseband sampling rate(MS/s)

REE value(dB)

20 �25.440 �26.880 �27.1

100s �25.3200 �26.4

Table 4Channel A: REE value versus time alignment resolution.

Time resolution (ns) Interpolation ratio REE value (dB)

0.5 1 �26.20.25 2 �26.00.1667 3 �26.50.125 4 �26.60.1 5 �26.00.05 10 �28.1

9200 9300 9400 9500 9600 9700

−1

−0.5

0

0.5

1

1.5

2

time (ns)

real

par

t of w

avef

orm

s (v

olta

ge)

instantaneous waveform comparison: real partTX baseband wfmCh A: compensated envCh A: non−compensated envCh C: compensated envCh C: non−compensated env

Fig. 12. Time-domain waveforms comparison: with and without trans-mitter nonlinearity and IQ imbalance compensation.

2742 K. Fu et al. / Measurement 46 (2013) 2735–2745

Table 4 shows the REE measurement results under dif-ferent resolution of time alignment by interpolation. Aninterpolation can be inserted before time alignment anddown sampler as in Fig. 10 to increase the resolution andtime alignment. From the results, with 10 times interpola-tion the REE performance improved by around 2 dB, other-wise there is no significant difference. It suggests that alarge interpolation ratio is required to significantly im-prove the time alignment accuracy and envelope recoveryaccuracy.

Fig. 12 compares recovered envelopes with and withoutcompensation when the DUT is removed, showing the non-ideal influence of the two channels, and the ability of thecompensation algorithm to eliminate it. The basebandsampling rate at the receiver is 100 MS/s, and the resolu-tion of time alignment is 0.5 ns. The waveform with in-verted triangular symbol is the real part of transmittedbaseband waveform, which can be seen as the ideal one.If the test bed is perfectly linear and with no IQ imbalances,all the recovered envelopes should be exactly the same asthe transmitted waveform when DUT is removed. How-ever, due to the non-ideality of the channels, the recoveredwaveforms from different channels are quite different fromeach other and from the transmitted one. After using thecompensation methods proposed in the previous sections,the recovered envelopes from the two channels areapproximately coincided with the transmitted basebandwaveform, and coincided with each other, justifying thecompetency of the test bed to accurately recover the enve-lopes of the PA’s input and output signals.

Finally, in Figs. 13 and 14, the behavior of the DUT ischaracterized from the recovered envelopes of the PA’s in-put and output signals – the AM/AM and AM/PM figures.Due to the memory effects and a wide 16.6 MHz band-width signal, the AM/AM and AM/PM are not curves butscattered bunches of points. The 1-dB gain compression in-put power point is around 0dBm, after which, the outputpower quickly saturates. As illustrated in Figs. 13 and 14,

−60 −50 −40 −30 −20 −10 0 10−30

−20

−10

0

10

20

30

40

input power (dBm)

outp

ut p

ower

(dBm

)

power amplifier AM/AM relation

compensatednon−compensated

Fig. 13. Recovered AM/AM behavior of ZVE-8G HPA: Twith and withouttransmitter nonlinearity and IQ imbalance compensation.

−60 −50 −40 −30 −20 −10 0 10−200

−150

−100

−50

0

50

100

150

200

input power (dBm)

phas

e sh

ift in

deg

ree

power amplifier AM/PM relation

compensatednon−compensated

Fig. 14. Recovered AM/PM behavior of ZVE-8G HPA: with and withouttransmitter nonlinearity and IQ imbalance compensation.

Table 5Identified coefficients of PA memory polynomial model.

a00 55.72 + 3.83j a10 �223.95�j26.15j a20 323.96 + 37.58ja01 �0.80�6.01j a11 �6.07 + 31.66j a21 5.47�6.38ja02 �0.27 + 1.94j a12 14.59�6.35j a22 �11.78�28.36j

0 100 200 300 400 500 600 700 80010−4

10−3

10−2

10−1

100

101Learning Curve (averaged over 72 runs)

Number of iterations, k

MSE

[dB]

RLSNLMS

Fig. 15. learning curve comparison: RLS versus normalized LMS.

0 100 200 300 400 500 600 700 800 9000

20

40

60

80Evolution of the 1st coefficient (amplitude)

Number of iterations, k

Coe

ffici

ent a

mpl

itude

RLSNLMSLS

0 100 200 300 400 500 600 700 800 900−6

−4

−2

0

2

4Evolution of the 1st coefficient (angle in degree)

Number of iterations, k

Coe

ffici

ent p

hase

in d

egre

e

RLSNLMSLS

Fig. 16. Evolution of the 1st coefficient: NLMS, and RLS. The one by LS isalso shown as a constant as a reference.

K. Fu et al. / Measurement 46 (2013) 2735–2745 2743

without compensation, the non-ideality leads to a muchlarger diffusion in the AM/AM and AM/PM relations, plusa significant phase shift in AM/PM relation. With the com-pensation methods, the diffusion becomes much moreconcentrated, and the phase shift is eliminated.

3.4. Measurement results on PA model assessment

The PA behavioral is characterized and modeled using awidely used PA model – memory polynomial model [22]defined as

yðnÞ ¼XN�1

k¼0

XM�1

m¼0

akmxðn�mÞjxðn�mÞj2k ð22Þ

where x(n) and y(n) are the sampled envelopes of the PA’sinput and output signals, and only odd order terms areconsidered. A Class A Mini-Circuits ZVE-8G HPA is modeledwith maximum nonlinear order 2N � 1 = 5 or N = 3 and

memory depth M = 3. The recovered envelopes of the PA’sinput and output signals by the proposed test bed are usedto identify the coefficients shown in Table 5. The identifica-tion is done by solving the least squares (LS) problem:

Ax ¼ b ð23Þ

where the matrix A contains all the linear and nonlinearforms of the input, vector x contains the coefficients, and

−50 0 50−80

−75

−70

−65

−60

−55

−50

−45

−40spectrum comparison: received vs calculated

Frequency (MHz)

Spec

trum

Mag

nitu

de (d

Bm)

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

diffe

renc

e (d

B)

received(real PA)calculated(PA model)

difference

Fig. 17. Spectrum comparison: The right y-axis is the difference betweenthe two spectrums.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5Constellation of real PA output signal

Fig. 18. Constellation of received real PA output signal: The signal is inthe form of BPSK + 16QAM.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5Constellation of PA model output

Fig. 19. Constellation of PA model output: The PA model is Memorypolynomial model, and the signal is in the form of BPSK + 16QAM.

Table 6EVM of real PA output signal received byoscilloscope, and EVM of the output signal ofPA model.

EVM of real PA output �14.1 dBEVM of PA model output �15.5 dB

2744 K. Fu et al. / Measurement 46 (2013) 2735–2745

vector b contains the output. It is solved by PseudoinverseEquation [23]:

x ¼ ðATAÞ�1

ATb ð24Þ

Moreover, adaptive identification methods, namely the,normalized LMS (NLMS) and RLS [24], are also used toidentify the coefficients for comparisons. Fig. 15 showsthe learning curve of NLMS and RLS, where the step-sizeparameter l in NLMS is 0.05, and the forgetting factor kin RLS is 0.97. RLS shows a better performance by achievinga lower error floor. Fig. 16 shows the comparison of theevolution of the 1st coefficients by all three identificationmethods.

The spectrums of the output of the identified PA modeland the real PA output are estimated by Welch method [9].The two spectrums are compared in Fig. 17, where they areapproximately coincided with each other, demonstratingthe relatively high accuracy of the PA model. Moreover,

the difference between those two is also illustrated, whichis shown by the dotted line with scale given by the right y-axis in the figure (mostly less than 1 dB).

The output constellation of the real PA output versusthat of the identified PA model output are shown in Figs. 18and 19 respectively. Results show that they are approxi-mately the same. The EVM values of both signals are calcu-lated and shown in Table 6, where the difference is lessthan 1.5 dB. These results show that the behavioral modelis relatively accurate.

4. Conclusion

A new test bed is proposed for PA behavioral modelingand characterization. It generally accomplishes two goals:The first goal is to accurately recover the envelopes ofthe PA’s input and output signals to facilitate PA behavioralcharacterization and modeling. The second goal is to pro-vide necessary metrics to evaluate the accuracy of enve-lope recovery, and PA modeling and linearization.

The proposed test bed is based on general purpose labequipments with software running on a PC. This opensup flexibility for changes to the test bed hardware and soft-ware changes to facilitate different test scenarios or for en-hanced performances. The proposed test bed can outputenough linear power to drive the PA under test and gener-ate signals of arbitrary waveform. Software signal process-ing methods are implemented to compensate the majorimpairments, such as transmitter IQ imbalances, channeldelay, frequency offset, and carrier phase offset.

K. Fu et al. / Measurement 46 (2013) 2735–2745 2745

The test bed also provides the facilities and metrics toevaluate the accuracy of envelope recovery, and PA behav-ioral modeling and linearization schemes. Relative enve-lope error (REE) is for the first time proposed as a metricto evaluate the accuracy of envelope recovery. Signal spec-trum, Constellation and EVM value are also used to evalu-ate the linearity of the channel, the accuracy of a PA model,and the wellness of a PA linearization scheme.

The competence of the test bed is demonstrated by anumber of experiment results. It achieves high accuracyfor envelope recovery under different baseband samplingrates or different interpolation ratios in time alignmentscheme. The AM/AM and AM/PM behavior is characterizedbased on the recovered envelopes. Using memory polyno-mial model as an example, metrics such as spectrums,ACPRs, constellations and EVMs, are calculated and com-pared between the received real PA output, and the identi-fied PA model output.

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