A novel measurement method for the behavioral modeling of power amplifiers is presented.The proposed method is based on the use of a specific configuration of a vector network analyzer.The principle consists in measuring under two excitation,a parametric and a conversion gain of an amplifiers versus the magnitude of the large input signal carrier

Barataud, D.

Buret, H.

Le Gallou, N.

Nebus, J.M.

Ngoya, E.

English

AMS Acta

A NOVEL MEASUREMENT METHOD FOR THE EXTRACTIONOF DYNAMIC VOLTERRA KERNELS OF MICROWAVE POWER AMPLIFIERSN. Le Gallou*, D. Barataud**, H. Buret*, J.M. Nebus **, E. Ngoya**,* ALCATEL SPACE INDUSTRIES, 26 Av. J.F. Champollion, 31037 Toulouse cedex (France)** IRCOM, , UMR CNRS 6615, University of Limoges, 123 av Albert Thomas , 87060 Limoges cedex (France)e-mail : Nicolas.Le-Gallou@space.alcatel.frABSTRACTA novel measurement method for the behavioral modeling of power amplifiers is presented. The proposed method isbased on the use of a specific configuration of a vector network analyzer. The principle consists in measuring under twoexcitation, a parametric and a conversion gain of an amplifiers versus the magnitude of the large input signal carrier.INTRODUCTIONWith increasing  complexity of signals travelling through RF and microwave analog devices, accurate measurementbased system level (behavioral) models are rewarded. The difficulty for modeling nonlinear systems or subsystems likepower amplifiers [5], linearizers or mixers, is to take into account the nonlinear memory exhibited in such equipments.Classical behavioral memoryless models made with AMAM and AMPM characteristics stay very poor, namely forwide band applications with complex modulated signals.  Recently a new rigorous modified Volterra model [1] hasbeen presented for accurate modeling of nonlinear distributed memory in subsystems like power amplifiers. This modelis based on both single tone and two tones measurements. In this paper we present the required characteristics forbuilding the model and the measurement setup used to practically extract the model. A particular highlight is made onthe solution found to measure the two tones nonlinear complex frequency conversion gain which is one of the basis ofthe model. The solution proposed, based on the use of a vector network analyzer (VNA) doesn't require specific testequipment and is easy to introduce in an industrial environment.THEORICAL CONSIDERATIONS ON DYNAMIC VOLTERRA APPROACHAt system level low pass equivalent signals are used andreal signals are replaced by associated input/outputcomplex envelopes )(tX and )(ˆ tY  (Fig. 1.) .As formerly described in [1] the output of a nonlinearsystem with memory could be expressed with sufficientaccuracy as the first order development as following :ΩΩΩ−π+ΩΩΩπ+=−Ω−Φ−ΩΦ de)(Xe),)t(X(H21  de)(X),)t(X(H21  e))t(X(Y)t(Y2/BW2/BWtj*222/BW2/BWtj1jdc)t(X)t(X  (1)where :   BW is the modulation bandwidth,  )(ΩX thespectrum of input signal.)())(( tXjdc etXYΦ  represents the purely static response ofthe system : this is the response calculated withMemoryless Model.  ),)((1 ΩtXH and ),)((2 Ω−tXHare the Dynamic Volterra Kernels accounting  fornonlinear memory effects, expressing the distance fromthe real response to the purely static one (withoutmemory).Measurement of static part of the model ))(( tXYdc iseasily done with classical AMAM and AMPMmeasurements performed on a nonlinear vector networkanalyzer at the center frequency ω0 of amplifier, with aninput sweept power. The typical shape of such characteristics is as shown in Fig.2 .[ ]tjetXetx 0)()( ωℜ= [ ]tjetYety 0)()( ωℜ=Fig.1: Nonlinear system with memory510152025303540-28 -23 -18 -13 -8 -3 2 7Input Power  ( dBm )Outpout Power  ( dBm )130140150160170180190200210220230Input / Output phase shift ( ° )AM AM Characteristic AM PM Characteristic Fig.2 : Power Amplifier AMAM and AMPM CharacteristicsNonlinear transfer functions 1H and 2H are howeverextracted from specific characterization as shown in Fig 3.The signal at the input of the device is constituted of twotones at different frequencies ω0 and ω0 +Ω havingdifferent amplitudes.The first tone, depicted as a pump, is fixed at the mid bandfrequency ω0(carrier frequency). The pump power 0Xˆ  isswept from small signal region of amplifier to stronglynonlinear region. The second tone, depicted as a probingsignal at frequency ω0 +Ω, has a constant amplitude fixedin linear region of amplifier. The modulation frequency Ωis swept for covering the entire bandwidth of the amplifier.The resulting signal at the output of the device is composed of three tones. We are especially interested in the two tones+ΩYˆδ  and −ΩYˆδ . Indeed building nonlinear transfer functions 1H and 2H consists of calculating the following ratios :( )0ˆ)0()0(01ˆˆ,ˆXPOWERINPUTFORXYXYXH−=Ω=Ω+=ΩΩ+Ωδδδδ      ( )0ˆ)0()0(02ˆˆ,ˆXPOWERINPUTFORXYXYXH−=Ω−=Ω−=ΩΩ−Ωδδδδ (2)In facts Ω+ΩXYδδ ˆis a small signal parametric gain Gp at the frequency ω0 +Ω where the parameter is the input power ofpump 0Xˆ .  In the same time Ω−ΩXYδδ ˆ is a frequency conversion gain Gc from the input frequency ω0+Ω to the outputfrequency ω0-Ω also depending of input power of pump  0Xˆ . Hence equation (2) becomes :( ) ( ) ( )000,ˆ 01 XpXp GGXH −Ω=Ω      ( ) ( ) ( ) 00 0,ˆ 02 XcXc GGXH −Ω−=Ω−  (3)Measuring complex parametric gain Gp could be done without any problem with a vector analyzer, as it is done forAMAM and AMPM characteristics, because input and output signals are at the same frequency.However directly measuring Gc is difficult with a VNA, because input and output signal are not at the same frequency.The extraction of such a characteristic necessitates an intermediate reference signal as it will be shown in the following.A close look to eq. (3) could provoke some interrogation. Of course it is impossible into a practical measurement todetermine Gp(0) and Gc(0), because in the case where Ω=0, the pump signal and the probing signal are at the samefrequency at the input of the device and result on a unique single tone signal.In facts this is not a problem. Indeed Gp(0) and Gc(0) are the nonlinear parametric gain and the conversion gain of theamplifier considered as memoryless. For this reason these two nonlinear transfer function could be easily obtained insimulation by applying the setup depicted in Fig.3 to the measurement based memoryless static part of the modelobtained at center frequency ω0 with AMAM and AMPM characteristics (Fig.2).MEASUREMENT OF NONLINEAR FREQUENCY CONVERSION GAINThe problem faced for measurement of complex frequency conversion gain is that tones to be characterized are not atthe same frequencies. This problem has found some solutions presented in [3] for signals constituted of harmonic tonesand in [2] for any kind of signal with  low frequency conversion and a time domain measurements. However solutionsproposed require specific equipment or rather complicated calibration procedures.Here the measurement procedure proposed for model extraction is not a fully and absolute determination of all phaserelations between tones at input or output of the device, but a more simple characterization allowing recording of phaseevolution as well as amplitude evolution of the different tones at input  and output ports of the device with swept inputpower. Distance between tones is not limited to any size. The few unknown relations not directly obtained bymeasurements are resolved both by general knowledge of phase relations in two tones fed nonlinear devices as well asoptimization.The proposed principle is based on the use of a modified vector network analyzer, used in receiver mode [4] , calibratedwith a classical SOLT procedure and power calibration. In this operation mode, the analyzer is capable of measuringpower of any tone constituting input or output device spectrum by using an internal power reference, as it could be doneby a spectrum analyzer. For this reason determination of input output power relation between tones for proceduredepicted in Fig. 3  is not a problem. In other words Gp(Ω) and Gc(-Ω) are directly determined by vector analyzerwithout any difficulty.0)( XcG Ω−+ΩYδ0ω0Y0ω0XΩXδΩ−ΩYδ0)( XpG ΩFig. 3 : Signal setup for model extractionFurthermore determination of phase Φ(Gp(Ω)) is  easy todo because it is a simple S21 measurement, tones being atthe same frequency at the input and output of the device.The difficulty takes place in the extraction of the phaseΦ(Gc(Ω)) because input and output tones are not at thesame frequency.  The principle used for this measurementis to use a reference fixed signal constituted of allnecessary spectral frequencies. This signal is placed on areference port of network analyzer, and input or outputtones of devices are measured relatively to the referencesignal, as shown in Fig. 4 . The key for this measurementis to force the reference to stay constant in amplitude andphase during extraction. In this case we have eq. (4) .The ideal reference generator would be a three tonesgenerator with perfectly known amplitudes and phaserelationships and having the capability of selecting anyfrequency spacing : for example from 100 Hz to 1 GHz.Hence determination of Gc(-Ω) would be  complete andabsolute. However such a standard signal is difficult toobtain. We decided to use as reference signal a more simple to obtain signal constituted of intermodulation of the twotones at ω0 and ω0+Ω passed through a nonlinear amplifier.Therefore exact phase relation between ( )Ω+0ωRef  and ( )Ω−0ωRef  is not known, but reference staying constant,measurement of relative drift of Φ(Gc(Ω)) versus input power 0Xˆ  is known using both measurements M1 and M2with an undetermined constant K :      ( ) ( )( ) KGMMXcX+Ω−Φ=Φ0021*   (5)On the other hand we are interested in evolution of Φ(Gc(-Ω)) with input power 0X , and K is independent of 0X . Sothe evolution of measurement (5) with 0X  gives us the evolution of ( )( )0XcG Ω−Φ  with 0X . In fact the constant Kcorresponds to the unknown value of  Φ(Gc(-Ω)) for very small input power.Determination of K is done afterwards by a numerical treatment. Indeed, given the model (1), we know that this modelis a very good representation of amplifier response at least for the quasi linear region of amplifier. Thus, a model isbuild for a given modulation frequency Ω, with an arbitrary small signal phase K on Gc. Then an optimization isperformed for minimizing model error compared to single tone power measurement at frequency Ω in quasi linearregion of the amplifier. The phase corresponding to this minimum is found to be always unique and gives us the valueof K.MEASUREMENT SETUPThe bench corresponding to the extraction methodologydescribed above is finally relatively simple, as shown in Fig 5.It is important to note that in this bench, reference signal,pump and probing signals are coming from same sources. Thismethod avoids any problem of phase coherence, or jitterbetween reference signal and input signals used to feed theDUT.The method described in the previous paragraph is applied tocharacterize the amplifier under test  with f1 placed at thecenter bandwidth of amplifier and f2 covering step by step theentire bandwidth of amplifier. For each step of f2, variableattenuator allows to sweep input power of the device fromlinear to strongly nonlinear region of DUT.APPLICATIONThe behavioral modeling of a L band (1.5 GHz) Hybrid 4 Watt power amplifier for space applications has beenachieved with this method. Static part of the model has been extracted with AMAM AMPM measurements at centerfrequency of amplifier bandwidth (Fig. 2). Dynamic kernels are obtained by measurements of nonlinear parametric gainand frequency conversion gain by the described technique. The next figures show the obtained characteristics for Gp+ΩYδ0ω0Y0ω0XΩXδ−ΩYδω0-Ω      ω0      ω0+Ω     ω0+2ΩReferenceFig. 4 : Input Output relations using reference signal50 Ωf2MicrowaveSource 2CouplerVector Network Analyzer W360BRB IN RA IN TA IN TBINAmpli..f110 MHzAmpli10 MHz10 MHzVar. Atten.Couplerf1 +f2f1  f2 f1 f2DUTf1 f2f1 f2Fixed frequency, Variable amplitudeMicrowave Source 1Reference signalN.L Amp.Fig. 5 : Measurement setup      ( ) ( )( )Ω−ΩΩ−Ω Ω+Ω−==Ω−XRefRefYXYGXc δωωδδδ 00*ˆˆ0   (4)2 Ratios measured by VNA:  M1         M2and Gc for a swept input power. For each curve the probing signal has been swept in frequency from -10 MHz to +10MHz around the carrier pump signal at 1.53 GHz.After the previously described numerical optimization onthe model build with parametric gain and conversion gain,the small signal phases K for conversion gain are recoveredfor each modulation frequency as shown in the next graph .With this information Φ(Gc(-Ω)) is now completely known,and behavioral completely determined.CONCLUSIONA simple and easy to carry out measurement principle hasbeen exposed for nonlinear behavioral model extraction.The particularity of the method proposed is to allownonlinear frequency conversion gain measurement both inphase and amplitude. Application to dynamicalcharacterization of a power amplifier has shown that method could give rather accurate results. Based on a nonlinearvector network analyzer this measurement setup doesn't necessitate complicated calibration procedure or specificequipment. Hence it could easily be introduced in an industrial environment.REFERENCES[1] E Ngoya, N. Le Gallou & Al , "Accurate RF and microwave system level modeling of wide band nonlinearcircuits",  IEEE MTT-S International Micowave Symposium  2000 Digest, Boston, Vol 1 pp 79-82[2] C.J. Clark , AA Moulthrop, M.S. Muha and C.P. Silva, "Transmission response measurements of frequencytranslating devices using a vector network analyzer", IEEE Trans. Microw. Theory Tech., vol 44,pp 2724-2737, dec 96[3] J. Verspecht, P. Debie, A. Barel and L. Martens,  "Accurate on-wafer measurement of phase and amplitude of thespectral components of incident and scattered voltage waves at the signal ports of a non-linear microwave device",IEEE MTT-S IMS Digest Orlando May 1995, TH1C-1, pp. 1029-1032.[4] D. Barataud, & Al, "Measurements of time domain voltage/current waveforms at R.F. and microwave frequencies, based onthe use of a Vector Network Analyzer, for the characterization of nonlinear devices. Application to high efficiency power amplifiersand frequency multipliers optimization". IEEE Trans. on Instrum. and Measurement, vol. 47, n°5,Oct. 1998, pp.1259-1264.[5] M. Zoyo, N. Cartier  & Al.,"X-Band 22W  SSPA for Earth observation satellites", GAAS 1999 Digest pp 190-1932025303540455055-20 -15 -10 -5 0 5Input power (dBm) Parametric Gain (Linear Amplitude )Fig. 6 : Amplitude of Parametric Gain formodulation frequency of +/- 10 MHz around pump130140150160170180190200210220-20 -15 -10 -5 0 5Input power (dBm) Parametric Gain (Phase in Degree )Fig. 7 : Phase of Parametric Gain for modulationfrequency of +/- 10 MHz around pump carrier02468101214161820-20 -15 -10 -5 0 5Input power (dBm) Frequency Conversion Gain (Linear Amplitude )Fig. 8 : Amplitude of conversion Gain for modulationfrequency of +/- 10 MHz around pump carrier-120-100-80-60-40-2002040-20 -15 -10 -5 0 5Input power (dBm) Parametric Gain (Phase in Degree )Unknown Phase starting point K for each frequencyFig. 9 : Phase of Conversion Gain for modulationfrequency of +/- 10 MHz around pump carrierFig. 10 : Distribution of Small signal values (K)  for phasesof Frequency conversion Gain at each modulation frequency556065707580859095100-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12Modulation frequency  ( MHz )Smal Signal Phase of frequency Conversion Gain  ( Degree)

A novel measurement method for the extraction of dynamic Volterra Kernels of microwave power amplifiers

AMS Acta - Alm@DL - Università di Bologna

A NOVEL MEASUREMENT METHOD FOR THE EXTRACTION
OF DYNAMIC VOLTERRA KERNELS OF MICROWAVE POWER AMPLIFIERS
N. Le Gallou*, D. Barataud**, H. Buret*, J.M. Nebus **, E. Ngoya**,
* ALCATEL SPACE INDUSTRIES, 26 Av. J.F. Champollion, 31037 Toulouse cedex (France)
** IRCOM, , UMR CNRS 6615, University of Limoges, 123 av Albert Thomas , 87060 Limoges cedex (France)
e-mail : Nicolas.Le-Gallou@space.alcatel.fr
ABSTRACT
A novel measurement method for the behavioral modeling of power amplifiers is presented. The proposed method is
based on the use of a specific configuration of a vector network analyzer. The principle consists in measuring under two
excitation, a parametric and a conversion gain of an amplifiers versus the magnitude of the large input signal carrier.
INTRODUCTION
With increasing  complexity of signals travelling through RF and microwave analog devices, accurate measurement
based system level (behavioral) models are rewarded. The difficulty for modeling nonlinear systems or subsystems like
power amplifiers [5], linearizers or mixers, is to take into account the nonlinear memory exhibited in such equipments.
Classical behavioral memoryless models made with AMAM and AMPM characteristics stay very poor, namely for
wide band applications with complex modulated signals.  Recently a new rigorous modified Volterra model [1] has
been presented for accurate modeling of nonlinear distributed memory in subsystems like power amplifiers. This model
is based on both single tone and two tones measurements. In this paper we present the required characteristics for
building the model and the measurement setup used to practically extract the model. A particular highlight is made on
the solution found to measure the two tones nonlinear complex frequency conversion gain which is one of the basis of
the model. The solution proposed, based on the use of a vector network analyzer (VNA) doesn't require specific test
equipment and is easy to introduce in an industrial environment.
THEORICAL CONSIDERATIONS ON DYNAMIC VOLTERRA APPROACH
At system level low pass equivalent signals are used and
real signals are replaced by associated input/output
complex envelopes )(tX

 and )(ˆ tY  (Fig. 1.) .
As formerly described in [1] the output of a nonlinear
system with memory could be expressed with sufficient
accuracy as the first order development as following :
ΩΩΩ−
π
+ΩΩΩ
π
+=
−
Ω−Φ
−
ΩΦ de)(Xe),)t(X(H
2
1
  de)(X),)t(X(H
2
1
  e))t(X(Y)t(Y
2/BW
2/BW
tj*2
2
2/BW
2/BW
tj
1
j
dc
)t(X)t(X

  (1)
where :   BW is the modulation bandwidth,  )(ΩX

 the
spectrum of input signal.
)())(( tX
j
dc etXY
Φ  represents the purely static response of
the system : this is the response calculated with
Memoryless Model.  ),)((1 ΩtXH

 and ),)((2 Ω−tXH

are the Dynamic Volterra Kernels accounting  for
nonlinear memory effects, expressing the distance from
the real response to the purely static one (without
memory).
Measurement of static part of the model ))(( tXYdc

 is
easily done with classical AMAM and AMPM
measurements performed on a nonlinear vector network
analyzer at the center frequency ω0 of amplifier, with an
input sweept power. The typical shape of such characteristics is as shown in Fig.2 .
[ ]tjetXetx 0)()( ωℜ= [ ]tjetYety 0)()( ωℜ=
Fig.1: Nonlinear system with memory
5
10
15
20
25
30
35
40
-28 -23 -18 -13 -8 -3 2 7
Input Power  ( dBm )
O
u
tp
o
u
t 
P
o
w
e
r 
 (
 d
B
m
 )
130
140
150
160
170
180
190
200
210
220
230
In
p
u
t 
/ 
O
u
tp
u
t 
p
h
a
s
e
 s
h
if
t 
( 
° 
)
AM AM Characteristic 
AM PM Characteristic 
Fig.2 : Power Amplifier AMAM and AMPM Characteristics
Nonlinear transfer functions 
1H

 and 
2H

 are however
extracted from specific characterization as shown in Fig 3.
The signal at the input of the device is constituted of two
tones at different frequencies ω0 and ω0 +Ω having
different amplitudes.
The first tone, depicted as a pump, is fixed at the mid band
frequency ω0(carrier frequency). The pump power 0Xˆ  is
swept from small signal region of amplifier to strongly
nonlinear region. The second tone, depicted as a probing
signal at frequency ω0 +Ω, has a constant amplitude fixed
in linear region of amplifier. The modulation frequency Ω
is swept for covering the entire bandwidth of the amplifier.
The resulting signal at the output of the device is composed of three tones. We are especially interested in the two tones
+
ΩYˆδ  and −ΩYˆδ . Indeed building nonlinear transfer functions 1H

 and 
2H

 consists of calculating the following ratios :
( )
0
ˆ)0(
)0(
01
ˆˆ
,ˆ
XPOWERINPUTFOR
X
Y
X
Y
XH





−=Ω
=Ω
+
=Ω
Ω
+
Ω
δ
δ
δ
δ      ( )
0
ˆ)0(
)0(
02
ˆˆ
,ˆ
XPOWERINPUTFOR
X
Y
X
Y
XH





−=Ω−
=Ω
−
=Ω
Ω
−
Ω
δ
δ
δ
δ (2)
In facts 
Ω
+
Ω
X
Y
δ
δ ˆ
is a small signal parametric gain Gp at the frequency ω0 +Ω where the parameter is the input power of
pump 
0Xˆ .  In the same time 
Ω
−
Ω
X
Y
δ
δ ˆ
 is a frequency conversion gain Gc from the input frequency ω0+Ω to the output
frequency ω0-Ω also depending of input power of pump  0Xˆ . Hence equation (2) becomes :
( ) ( ) ( )
00
0,ˆ 01 XpXp GGXH −Ω=Ω      ( ) ( ) ( ) 00 0,ˆ 02 XcXc GGXH −Ω−=Ω−  (3)
Measuring complex parametric gain Gp could be done without any problem with a vector analyzer, as it is done for
AMAM and AMPM characteristics, because input and output signals are at the same frequency.
However directly measuring Gc is difficult with a VNA, because input and output signal are not at the same frequency.
The extraction of such a characteristic necessitates an intermediate reference signal as it will be shown in the following.
A close look to eq. (3) could provoke some interrogation. Of course it is impossible into a practical measurement to
determine Gp(0) and Gc(0), because in the case where Ω=0, the pump signal and the probing signal are at the same
frequency at the input of the device and result on a unique single tone signal.
In facts this is not a problem. Indeed Gp(0) and Gc(0) are the nonlinear parametric gain and the conversion gain of the
amplifier considered as memoryless. For this reason these two nonlinear transfer function could be easily obtained in
simulation by applying the setup depicted in Fig.3 to the measurement based memoryless static part of the model
obtained at center frequency ω0 with AMAM and AMPM characteristics (Fig.2).
MEASUREMENT OF NONLINEAR FREQUENCY CONVERSION GAIN
The problem faced for measurement of complex frequency conversion gain is that tones to be characterized are not at
the same frequencies. This problem has found some solutions presented in [3] for signals constituted of harmonic tones
and in [2] for any kind of signal with  low frequency conversion and a time domain measurements. However solutions
proposed require specific equipment or rather complicated calibration procedures.
Here the measurement procedure proposed for model extraction is not a fully and absolute determination of all phase
relations between tones at input or output of the device, but a more simple characterization allowing recording of phase
evolution as well as amplitude evolution of the different tones at input  and output ports of the device with swept input
power. Distance between tones is not limited to any size. The few unknown relations not directly obtained by
measurements are resolved both by general knowledge of phase relations in two tones fed nonlinear devices as well as
optimization.
The proposed principle is based on the use of a modified vector network analyzer, used in receiver mode [4] , calibrated
with a classical SOLT procedure and power calibration. In this operation mode, the analyzer is capable of measuring
power of any tone constituting input or output device spectrum by using an internal power reference, as it could be done
by a spectrum analyzer. For this reason determination of input output power relation between tones for procedure
depicted in Fig. 3  is not a problem. In other words Gp(Ω) and Gc(-Ω) are directly determined by vector analyzer
without any difficulty.
0
)( XcG Ω−
+
ΩY
δ
0ω
0Y

0ω
0X

ΩX
δ
Ω
−
ΩY
δ
0
)( XpG Ω
Fig. 3 : Signal setup for model extraction
Furthermore determination of phase Φ(Gp(Ω)) is  easy to
do because it is a simple S21 measurement, tones being at
the same frequency at the input and output of the device.
The difficulty takes place in the extraction of the phase
Φ(Gc(Ω)) because input and output tones are not at the
same frequency.  The principle used for this measurement
is to use a reference fixed signal constituted of all
necessary spectral frequencies. This signal is placed on a
reference port of network analyzer, and input or output
tones of devices are measured relatively to the reference
signal, as shown in Fig. 4 . The key for this measurement
is to force the reference to stay constant in amplitude and
phase during extraction. In this case we have eq. (4) .
The ideal reference generator would be a three tones
generator with perfectly known amplitudes and phase
relationships and having the capability of selecting any
frequency spacing : for example from 100 Hz to 1 GHz.
Hence determination of Gc(-Ω) would be  complete and
absolute. However such a standard signal is difficult to
obtain. We decided to use as reference signal a more simple to obtain signal constituted of intermodulation of the two
tones at ω0 and ω0+Ω passed through a nonlinear amplifier.
Therefore exact phase relation between ( )Ω+0ωRef  and ( )Ω−0ωRef  is not known, but reference staying constant,
measurement of relative drift of Φ(Gc(Ω)) versus input power 0Xˆ  is known using both measurements M1 and M2
with an undetermined constant K :      ( ) ( )( ) KGMM
XcX
+Ω−Φ=Φ
0021
*   (5)
On the other hand we are interested in evolution of Φ(Gc(-Ω)) with input power 
0X , and K is independent of 0X . So
the evolution of measurement (5) with 
0X  gives us the evolution of ( )( )
0Xc
G Ω−Φ  with 0X . In fact the constant K
corresponds to the unknown value of  Φ(Gc(-Ω)) for very small input power.
Determination of K is done afterwards by a numerical treatment. Indeed, given the model (1), we know that this model
is a very good representation of amplifier response at least for the quasi linear region of amplifier. Thus, a model is
build for a given modulation frequency Ω, with an arbitrary small signal phase K on Gc. Then an optimization is
performed for minimizing model error compared to single tone power measurement at frequency Ω in quasi linear
region of the amplifier. The phase corresponding to this minimum is found to be always unique and gives us the value
of K.
MEASUREMENT SETUP
The bench corresponding to the extraction methodology
described above is finally relatively simple, as shown in Fig 5.
It is important to note that in this bench, reference signal,
pump and probing signals are coming from same sources. This
method avoids any problem of phase coherence, or jitter
between reference signal and input signals used to feed the
DUT.
The method described in the previous paragraph is applied to
characterize the amplifier under test  with f1 placed at the
center bandwidth of amplifier and f2 covering step by step the
entire bandwidth of amplifier. For each step of f2, variable
attenuator allows to sweep input power of the device from
linear to strongly nonlinear region of DUT.
APPLICATION
The behavioral modeling of a L band (1.5 GHz) Hybrid 4 Watt power amplifier for space applications has been
achieved with this method. Static part of the model has been extracted with AMAM AMPM measurements at center
frequency of amplifier bandwidth (Fig. 2). Dynamic kernels are obtained by measurements of nonlinear parametric gain
and frequency conversion gain by the described technique. The next figures show the obtained characteristics for Gp
+
ΩY
δ
0ω
0Y

0ω
0X

ΩX
δ
−
ΩY
δ
ω0-Ω      ω0      ω0+Ω     ω0+2Ω
Reference
Fig. 4 : Input Output relations using reference signal
50 Ω
f2
Microwave
Source 2
Coupler
Vector Network Analyzer W360B
RB IN RA IN TA IN TB
IN
Ampli.
.
f1
10 MHz
Ampli
10 MHz1
0
 M
H
z
Var. Atten.
Coupler
f1 +f2
f1  f2 f1 f2
DUT
f1 f2f1 f2
Fixed frequency, 
Variable amplitude
Microwave 
Source 1
Reference signal
N.L Amp.
Fig. 5 : Measurement setup
      ( ) ( )
( )
Ω
−
Ω
Ω
−
Ω Ω+
Ω−
==Ω−
X
Ref
Ref
Y
X
Y
G
Xc δ
ω
ω
δ
δ
δ 0
0
*
ˆˆ
0
   (4)
2 Ratios measured by VNA:  M1         M2
and Gc for a swept input power. For each curve the probing signal has been swept in frequency from -10 MHz to +10
MHz around the carrier pump signal at 1.53 GHz.
After the previously described numerical optimization on
the model build with parametric gain and conversion gain,
the small signal phases K for conversion gain are recovered
for each modulation frequency as shown in the next graph .
With this information Φ(Gc(-Ω)) is now completely known,
and behavioral completely determined.
CONCLUSION
A simple and easy to carry out measurement principle has
been exposed for nonlinear behavioral model extraction.
The particularity of the method proposed is to allow
nonlinear frequency conversion gain measurement both in
phase and amplitude. Application to dynamical
characterization of a power amplifier has shown that method could give rather accurate results. Based on a nonlinear
vector network analyzer this measurement setup doesn't necessitate complicated calibration procedure or specific
equipment. Hence it could easily be introduced in an industrial environment.
REFERENCES
[1] E Ngoya, N. Le Gallou & Al , "Accurate RF and microwave system level modeling of wide band nonlinear
circuits",  IEEE MTT-S International Micowave Symposium  2000 Digest, Boston, Vol 1 pp 79-82
[2] C.J. Clark , AA Moulthrop, M.S. Muha and C.P. Silva, "Transmission response measurements of frequency
translating devices using a vector network analyzer", IEEE Trans. Microw. Theory Tech., vol 44,pp 2724-2737, dec 96
[3] J. Verspecht, P. Debie, A. Barel and L. Martens,  "Accurate on-wafer measurement of phase and amplitude of the
spectral components of incident and scattered voltage waves at the signal ports of a non-linear microwave device",
IEEE MTT-S IMS Digest Orlando May 1995, TH1C-1, pp. 1029-1032.
[4] D. Barataud, & Al, "Measurements of time domain voltage/current waveforms at R.F. and microwave frequencies, based on
the use of a Vector Network Analyzer, for the characterization of nonlinear devices. Application to high efficiency power amplifiers
and frequency multipliers optimization". IEEE Trans. on Instrum. and Measurement, vol. 47, n°5,Oct. 1998, pp.1259-1264.
[5] M. Zoyo, N. Cartier  & Al.,"X-Band 22W  SSPA for Earth observation satellites", GAAS 1999 Digest pp 190-193
20
25
30
35
40
45
50
55
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
L
in
e
a
r 
A
m
p
li
tu
d
e
 )
Fig. 6 : Amplitude of Parametric Gain for
modulation frequency of +/- 10 MHz around pump
130
140
150
160
170
180
190
200
210
220
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
P
h
a
s
e
 i
n
 D
e
g
re
e
 )
Fig. 7 : Phase of Parametric Gain for modulation
frequency of +/- 10 MHz around pump carrier
0
2
4
6
8
10
12
14
16
18
20
-20 -15 -10 -5 0 5
Input power (dBm) 
F
re
q
u
e
n
c
y
 C
o
n
v
e
rs
io
n
 G
a
in
 (
L
in
e
a
r 
A
m
p
li
tu
d
e
 )
Fig. 8 : Amplitude of conversion Gain for modulation
frequency of +/- 10 MHz around pump carrier
-120
-100
-80
-60
-40
-20
0
20
40
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
P
h
a
s
e
 i
n
 D
e
g
re
e
 )
Unknown Phase starting point K 
for each frequency
Fig. 9 : Phase of Conversion Gain for modulation
frequency of +/- 10 MHz around pump carrier
Fig. 10 : Distribution of Small signal values (K)  for phases
of Frequency conversion Gain at each modulation frequency
55
60
65
70
75
80
85
90
95
100
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Modulation frequency  ( MHz )
S
m
a
l 
S
ig
n
a
l 
P
h
a
s
e
 o
f 
fr
e
q
u
e
n
c
y 
C
o
n
v
e
rs
io
n
 G
a
in
  
( 
D
e
g
re
e
)


A NOVEL MEASUREMENT METHOD FOR THE EXTRACTION
OF DYNAMIC VOLTERRA KERNELS OF MICROWAVE POWER AMPLIFIERS
N. Le Gallou*, D. Barataud**, H. Buret*, J.M. Nebus **, E. Ngoya**,
* ALCATEL SPACE INDUSTRIES, 26 Av. J.F. Champollion, 31037 Toulouse cedex (France)
** IRCOM, , UMR CNRS 6615, University of Limoges, 123 av Albert Thomas , 87060 Limoges cedex (France)
e-mail : Nicolas.Le-Gallou@space.alcatel.fr
ABSTRACT
A novel measurement method for the behavioral modeling of power amplifiers is presented. The proposed method is
based on the use of a specific configuration of a vector network analyzer. The principle consists in measuring under two
excitation, a parametric and a conversion gain of an amplifiers versus the magnitude of the large input signal carrier.
INTRODUCTION
With increasing  complexity of signals travelling through RF and microwave analog devices, accurate measurement
based system level (behavioral) models are rewarded. The difficulty for modeling nonlinear systems or subsystems like
power amplifiers [5], linearizers or mixers, is to take into account the nonlinear memory exhibited in such equipments.
Classical behavioral memoryless models made with AMAM and AMPM characteristics stay very poor, namely for
wide band applications with complex modulated signals.  Recently a new rigorous modified Volterra model [1] has
been presented for accurate modeling of nonlinear distributed memory in subsystems like power amplifiers. This model
is based on both single tone and two tones measurements. In this paper we present the required characteristics for
building the model and the measurement setup used to practically extract the model. A particular highlight is made on
the solution found to measure the two tones nonlinear complex frequency conversion gain which is one of the basis of
the model. The solution proposed, based on the use of a vector network analyzer (VNA) doesn't require specific test
equipment and is easy to introduce in an industrial environment.
THEORICAL CONSIDERATIONS ON DYNAMIC VOLTERRA APPROACH
At system level low pass equivalent signals are used and
real signals are replaced by associated input/output
complex envelopes )(tX

 and )(ˆ tY  (Fig. 1.) .
As formerly described in [1] the output of a nonlinear
system with memory could be expressed with sufficient
accuracy as the first order development as following :
ΩΩΩ−
π
+ΩΩΩ
π
+=
−
Ω−Φ
−
ΩΦ de)(Xe),)t(X(H
2
1
  de)(X),)t(X(H
2
1
  e))t(X(Y)t(Y
2/BW
2/BW
tj*2
2
2/BW
2/BW
tj
1
j
dc
)t(X)t(X

  (1)
where :   BW is the modulation bandwidth,  )(ΩX

 the
spectrum of input signal.
)())(( tX
j
dc etXY
Φ  represents the purely static response of
the system : this is the response calculated with
Memoryless Model.  ),)((1 ΩtXH

 and ),)((2 Ω−tXH

are the Dynamic Volterra Kernels accounting  for
nonlinear memory effects, expressing the distance from
the real response to the purely static one (without
memory).
Measurement of static part of the model ))(( tXYdc

 is
easily done with classical AMAM and AMPM
measurements performed on a nonlinear vector network
analyzer at the center frequency ω0 of amplifier, with an
input sweept power. The typical shape of such characteristics is as shown in Fig.2 .
[ ]tjetXetx 0)()( ωℜ= [ ]tjetYety 0)()( ωℜ=
Fig.1: Nonlinear system with memory
5
10
15
20
25
30
35
40
-28 -23 -18 -13 -8 -3 2 7
Input Power  ( dBm )
O
u
tp
o
u
t 
P
o
w
e
r 
 (
 d
B
m
 )
130
140
150
160
170
180
190
200
210
220
230
In
p
u
t 
/ 
O
u
tp
u
t 
p
h
a
s
e
 s
h
if
t 
( 
° 
)
AM AM Characteristic 
AM PM Characteristic 
Fig.2 : Power Amplifier AMAM and AMPM Characteristics
Nonlinear transfer functions 
1H

 and 
2H

 are however
extracted from specific characterization as shown in Fig 3.
The signal at the input of the device is constituted of two
tones at different frequencies ω0 and ω0 +Ω having
different amplitudes.
The first tone, depicted as a pump, is fixed at the mid band
frequency ω0(carrier frequency). The pump power 0X̂  is
swept from small signal region of amplifier to strongly
nonlinear region. The second tone, depicted as a probing
signal at frequency ω0 +Ω, has a constant amplitude fixed
in linear region of amplifier. The modulation frequency Ω
is swept for covering the entire bandwidth of the amplifier.
The resulting signal at the output of the device is composed of three tones. We are especially interested in the two tones
+
ΩŶδ  and 
−
ΩŶδ . Indeed building nonlinear transfer functions 1H

 and 
2H

 consists of calculating the following ratios :
( )
0
ˆ)0(
)0(
01
ˆˆ
,ˆ
XPOWERINPUTFOR
X
Y
X
Y
XH





−=Ω
=Ω
+
=Ω
Ω
+
Ω
δ
δ
δ
δ      ( )
0
ˆ)0(
)0(
02
ˆˆ
,ˆ
XPOWERINPUTFOR
X
Y
X
Y
XH





−=Ω−
=Ω
−
=Ω
Ω
−
Ω
δ
δ
δ
δ (2)
In facts 
Ω
+
Ω
X
Y
δ
δ ˆ
is a small signal parametric gain Gp at the frequency ω0 +Ω where the parameter is the input power of
pump 
0X̂ .  In the same time 
Ω
−
Ω
X
Y
δ
δ ˆ
 is a frequency conversion gain Gc from the input frequency ω0+Ω to the output
frequency ω0-Ω also depending of input power of pump  0X̂ . Hence equation (2) becomes :
( ) ( ) ( )
00
0,ˆ 01 XpXp GGXH −Ω=Ω      ( ) ( ) ( ) 00 0,ˆ 02 XcXc GGXH −Ω−=Ω−  (3)
Measuring complex parametric gain Gp could be done without any problem with a vector analyzer, as it is done for
AMAM and AMPM characteristics, because input and output signals are at the same frequency.
However directly measuring Gc is difficult with a VNA, because input and output signal are not at the same frequency.
The extraction of such a characteristic necessitates an intermediate reference signal as it will be shown in the following.
A close look to eq. (3) could provoke some interrogation. Of course it is impossible into a practical measurement to
determine Gp(0) and Gc(0), because in the case where Ω=0, the pump signal and the probing signal are at the same
frequency at the input of the device and result on a unique single tone signal.
In facts this is not a problem. Indeed Gp(0) and Gc(0) are the nonlinear parametric gain and the conversion gain of the
amplifier considered as memoryless. For this reason these two nonlinear transfer function could be easily obtained in
simulation by applying the setup depicted in Fig.3 to the measurement based memoryless static part of the model
obtained at center frequency ω0 with AMAM and AMPM characteristics (Fig.2).
MEASUREMENT OF NONLINEAR FREQUENCY CONVERSION GAIN
The problem faced for measurement of complex frequency conversion gain is that tones to be characterized are not at
the same frequencies. This problem has found some solutions presented in [3] for signals constituted of harmonic tones
and in [2] for any kind of signal with  low frequency conversion and a time domain measurements. However solutions
proposed require specific equipment or rather complicated calibration procedures.
Here the measurement procedure proposed for model extraction is not a fully and absolute determination of all phase
relations between tones at input or output of the device, but a more simple characterization allowing recording of phase
evolution as well as amplitude evolution of the different tones at input  and output ports of the device with swept input
power. Distance between tones is not limited to any size. The few unknown relations not directly obtained by
measurements are resolved both by general knowledge of phase relations in two tones fed nonlinear devices as well as
optimization.
The proposed principle is based on the use of a modified vector network analyzer, used in receiver mode [4] , calibrated
with a classical SOLT procedure and power calibration. In this operation mode, the analyzer is capable of measuring
power of any tone constituting input or output device spectrum by using an internal power reference, as it could be done
by a spectrum analyzer. For this reason determination of input output power relation between tones for procedure
depicted in Fig. 3  is not a problem. In other words  Gp(Ω)  and  Gc(-Ω)  are directly determined by vector analyzer
without any difficulty.
0
)( XcG Ω−
+
ΩY

δ
0ω
0Y

0ω
0X

ΩX

δ
Ω
−
ΩY

δ
0
)( XpG Ω
Fig. 3 : Signal setup for model extraction
Furthermore determination of phase Φ(Gp(Ω)) is  easy to
do because it is a simple S21 measurement, tones being at
the same frequency at the input and output of the device.
The difficulty takes place in the extraction of the phase
Φ(Gc(Ω)) because input and output tones are not at the
same frequency.  The principle used for this measurement
is to use a reference fixed signal constituted of all
necessary spectral frequencies. This signal is placed on a
reference port of network analyzer, and input or output
tones of devices are measured relatively to the reference
signal, as shown in Fig. 4 . The key for this measurement
is to force the reference to stay constant in amplitude and
phase during extraction. In this case we have eq. (4) .
The ideal reference generator would be a three tones
generator with perfectly known amplitudes and phase
relationships and having the capability of selecting any
frequency spacing : for example from 100 Hz to 1 GHz.
Hence determination of Gc(-Ω) would be  complete and
absolute. However such a standard signal is difficult to
obtain. We decided to use as reference signal a more simple to obtain signal constituted of intermodulation of the two
tones at ω0 and ω0+Ω passed through a nonlinear amplifier.
Therefore exact phase relation between ( )Ω+0ωRef  and ( )Ω−0ωRef  is not known, but reference staying constant,
measurement of relative drift of Φ(Gc(Ω)) versus input power 0X̂  is known using both measurements M1 and M2
with an undetermined constant K :      ( ) ( )( ) KGMM
XcX
+Ω−Φ=Φ
0021
*   (5)
On the other hand we are interested in evolution of Φ(Gc(-Ω)) with input power 
0X , and K is independent of 0X . So
the evolution of measurement (5) with 
0X  gives us the evolution of ( )( )
0Xc
G Ω−Φ  with 0X . In fact the constant K
corresponds to the unknown value of  Φ(Gc(-Ω)) for very small input power.
Determination of K is done afterwards by a numerical treatment. Indeed, given the model (1), we know that this model
is a very good representation of amplifier response at least for the quasi linear region of amplifier. Thus, a model is
build for a given modulation frequency Ω, with an arbitrary small signal phase K on Gc. Then an optimization is
performed for minimizing model error compared to single tone power measurement at frequency Ω in quasi linear
region of the amplifier. The phase corresponding to this minimum is found to be always unique and gives us the value
of K.
MEASUREMENT SETUP
The bench corresponding to the extraction methodology
described above is finally relatively simple, as shown in Fig 5.
It is important to note that in this bench, reference signal,
pump and probing signals are coming from same sources. This
method avoids any problem of phase coherence, or jitter
between reference signal and input signals used to feed the
DUT.
The method described in the previous paragraph is applied to
characterize the amplifier under test  with f1 placed at the
center bandwidth of amplifier and f2 covering step by step the
entire bandwidth of amplifier. For each step of f2, variable
attenuator allows to sweep input power of the device from
linear to strongly nonlinear region of DUT.
APPLICATION
The behavioral modeling of a L band (1.5 GHz) Hybrid 4 Watt power amplifier for space applications has been
achieved with this method. Static part of the model has been extracted with AMAM AMPM measurements at center
frequency of amplifier bandwidth (Fig. 2). Dynamic kernels are obtained by measurements of nonlinear parametric gain
and frequency conversion gain by the described technique. The next figures show the obtained characteristics for Gp
+
ΩY

δ
0ω
0Y

0ω
0X

ΩX

δ
−
ΩY

δ
ω0-Ω      ω0      ω0+Ω     ω0+2Ω
Reference
Fig. 4 : Input Output relations using reference signal
50 Ω
f2
Microwave
Source 2
Coupler
Vector Network Analyzer W360B
RB IN RA IN TA IN TB
IN
Ampli.
.
f1
10 MHz
Ampli
10 MHz1
0
 M
H
z
Var. Atten.
Coupler
f1 +f2
f1  f2 f1 f2
DUT
f1 f2f1 f2
Fixed frequency, 
Variable amplitude
Microwave 
Source 1
Reference signal
N.L Amp.
Fig. 5 : Measurement setup
      ( ) ( )
( )
Ω
−
Ω
Ω
−
Ω Ω+
Ω−
==Ω−
X
Ref
Ref
Y
X
Y
G
Xc δ
ω
ω
δ
δ
δ 0
0
*
ˆˆ
0
   (4)
2 Ratios measured by VNA:  M1         M2
and Gc for a swept input power. For each curve the probing signal has been swept in frequency from -10 MHz to +10
MHz around the carrier pump signal at 1.53 GHz.
After the previously described numerical optimization on
the model build with parametric gain and conversion gain,
the small signal phases K for conversion gain are recovered
for each modulation frequency as shown in the next graph .
With this information Φ(Gc(-Ω)) is now completely known,
and behavioral completely determined.
CONCLUSION
A simple and easy to carry out measurement principle has
been exposed for nonlinear behavioral model extraction.
The particularity of the method proposed is to allow
nonlinear frequency conversion gain measurement both in
phase and amplitude. Application to dynamical
characterization of a power amplifier has shown that method could give rather accurate results. Based on a nonlinear
vector network analyzer this measurement setup doesn't necessitate complicated calibration procedure or specific
equipment. Hence it could easily be introduced in an industrial environment.
REFERENCES
[1] E Ngoya, N. Le Gallou & Al , "Accurate RF and microwave system level modeling of wide band nonlinear
circuits",  IEEE MTT-S International Micowave Symposium  2000 Digest, Boston, Vol 1 pp 79-82
[2] C.J. Clark , AA Moulthrop, M.S. Muha and C.P. Silva, "Transmission response measurements of frequency
translating devices using a vector network analyzer", IEEE Trans. Microw. Theory Tech., vol 44,pp 2724-2737, dec 96
[3] J. Verspecht, P. Debie, A. Barel and L. Martens,  "Accurate on-wafer measurement of phase and amplitude of the
spectral components of incident and scattered voltage waves at the signal ports of a non-linear microwave device",
IEEE MTT-S IMS Digest Orlando May 1995, TH1C-1, pp. 1029-1032.
[4] D. Barataud, & Al, "Measurements of time domain voltage/current waveforms at R.F. and microwave frequencies, based on
the use of a Vector Network Analyzer, for the characterization of nonlinear devices. Application to high efficiency power amplifiers
and frequency multipliers optimization". IEEE Trans. on Instrum. and Measurement, vol. 47, n°5,Oct. 1998, pp.1259-1264.
[5] M. Zoyo, N. Cartier  & Al.,"X-Band 22W  SSPA for Earth observation satellites", GAAS 1999 Digest pp 190-193
20
25
30
35
40
45
50
55
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
L
in
e
a
r 
A
m
p
li
tu
d
e
 )
Fig. 6 : Amplitude of Parametric Gain for
modulation frequency of +/- 10 MHz around pump
130
140
150
160
170
180
190
200
210
220
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
P
h
a
s
e
 i
n
 D
e
g
re
e
 )
Fig. 7 : Phase of Parametric Gain for modulation
frequency of +/- 10 MHz around pump carrier
0
2
4
6
8
10
12
14
16
18
20
-20 -15 -10 -5 0 5
Input power (dBm) 
F
re
q
u
e
n
c
y
 C
o
n
v
e
rs
io
n
 G
a
in
 (
L
in
e
a
r 
A
m
p
li
tu
d
e
 )
Fig. 8 : Amplitude of conversion Gain for modulation
frequency of +/- 10 MHz around pump carrier
-120
-100
-80
-60
-40
-20
0
20
40
-20 -15 -10 -5 0 5
Input power (dBm) 
P
a
ra
m
e
tr
ic
 G
a
in
 (
P
h
a
s
e
 i
n
 D
e
g
re
e
 )
Unknown Phase starting point K 
for each frequency
Fig. 9 : Phase of Conversion Gain for modulation
frequency of +/- 10 MHz around pump carrier
Fig. 10 : Distribution of Small signal values (K)  for phases
of Frequency conversion Gain at each modulation frequency
55
60
65
70
75
80
85
90
95
100
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Modulation frequency  ( MHz )
S
m
a
l 
S
ig
n
a
l 
P
h
a
s
e
 o
f 
fr
e
q
u
e
n
c
y 
C
o
n
v
e
rs
io
n
 G
a
in
  
( 
D
e
g
re
e
)


Le Gallou, Nicolas

Barataud, Denis

Burêt, Hervé

Nebus, Jean-Michel

Ngoya, Edouard

HAL-UNILIM

A novel measurement method for the extraction of dynamic Volterra kernels of microwave power amplifiers

https://hal-unilim.archives-ouvertes.fr/hal-00924935

A novel measurement method for the extraction of dynamic Volterra Kernels of microwave power amplifiers

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