On Radio Frequency Behavioral Modeling
Measurement Techniques, Devices and Validation Aspects
PER LANDIN
Licentiate Thesis
Signal Processing
School of Electrical Engineering
KTH
Stockholm, Sweden, 2009
TRITA-EE 2009:056
ISSN 1653-5146
ISBN 978-91-7415-526-6
KTH-Electrical Engineering
Signal Processing
SE-100 44 Stockholm, Sweden
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges
till offentlig granskning för avläggande av teknologie licentiatexamen torsdagen den
10 december 2009 kl. 10.00 i hörsal 99131, Kungsbäcksvägen 47, Gävle.
© Per Landin, December 2009
Tryck: Universitetsservice US AB
iii
Sammanfattning
Effektförstärkare för radiofrekvensapplikationer utgör fortfarande ett av
de största problemen i trådlösa kommunikationssystem. Detta beror på att
dessa förstärkare är ickelinjära, har låg energieffektivitet och ger mycket distortioner. Bättre verktyg för att förstå och korrigera dessa beteenden är nödvändiga. Ett sådant verktyg är beteendemodellering. En beteendemodell kan
ses som en svart låda med insignal(er) och utsignal(er). In detta fall är dessa
signaler samplade basbandssignaler och den svarta lådan är en matematisk
relation mellan en insignal och en utsignal.
Avhandlingen behandlar några krav för beteendemodellering av nämnda system genom att presentera metoder för utvärdering och förbättring av
modellernas prestanda. Detta åstadkoms genom att betrakta ett frekvensviktat felkriterium.
Ett högpresterande mätsystem är också nödvändigt för experimenten.
Prestandan hos det tillgängliga systemet jämförs med prestandan hos ett
allmänt erkänt mätsystem, en s.k. storsignalsnätverksanalysator, genom att
betrakta prestandan hos beteendemodellerna som extraheras och valideras
med data från respektive mätsystem. Resultatet visar att det existerande
mätsystemet har god prestanda.
Ett stort problem vid beteendemodellering är att kunna sampla med tillräckligt hög hastighet. Genom att använda Zhu-Franks generaliserade samplingsteorem vid beteendemodellering kan en del av detta problem undvikas.
Teoremet medför att man kan sampla med en väsentligt lägre samlingsfrekvens än vad Nyquistteoremet säger. Modeller extraheras och prestandan utvärderas genom att använda kriteriet normalized mean square error
(NMSE).
För stabil prediktion och korrektion av utsignalen måste robustheten
hos de använda modellerna verifieras. En sådan studie som berör robustheten mot variationer i lastimpedansen har genomförts. Prestandan på direkta modeller försämras med 7 dB mätt som adjacent channel error power ratio
(ACEPR). Prestanda på inversmodellen, implementerad som digital predistortion, försämras med upp till 13 dB mätt som adjacent channel power ratio
(ACPR).
iv
Abstract
Radio frequency (RF) power amplifiers (PA) are still the most troublesome part of a wireless system due to their inherent nonlinearity, low power
efficiency and high distortions. Better tools are needed to understand and
correct the undesirable behavior. Some of these tools are behavioral models.
A behavioral model is often thought of as a black box with some inputs and
some outputs. In the case here these inputs are sampled signals which means
that the modeling amounts to finding a mathematical relationship between
the input signal(s) and the output signal(s).
This thesis considers some requirements for behavioral modeling of said
systems by presenting methods for general performance evaluation and improvement by considering a frequency weighted error criterion. A high performance measurement system is also needed. The performance of the available
system is compared to the performance of a well recognized system, the large
signal network analyzer (LSNA). The results show that the existing measurement system can extract behavioral models with the same performance as the
LSNA and can give lower performance validation errors.
Still the need for higher bandwidths drives the measurement systems to
the limits, especially the digital parts. By utilizing the so called Zhu-Frank
generalized sampling theorem, behavioral modeling of a PA is done by using
data acquired at a sampling rate lower than the Nyquist rate. Models of a PA
are extracted and the performance is evaluated using the normalized mean
square error (NMSE) criterion.
For prediction and correction of the output signals the stability of the
models regarding changes must be investigated. One such study considering
controlled variations on the output load of the PA is done and both the predictive and corrective capabilities of the models are evaluated. The predictive
capability gets up to 7 dB worse measured as adjacent channel error power
ratio (ACEPR) and the corrective, as digital predistortion, gets up to 13 dB
worse measured as adjacent channel power ratio (ACPR).
v
Acknowledgements
Special thanks goes to my two supervisors Dr. Magnus Isaksson and Prof. Peter
Händel. Without good supervisors a thesis is never possible. Special thanks also
goes to Dr. Olof Bengtsson for inspiring discussions covering most subjects, ranging
from research, principles and critical thinking to drinks and food, especially the
“raggmunk”.
My colleagues Charles Nader, Sathyaveer Prasad, Javier Ferrer Coll, Carl Elofsson and Carl Karlsson that are always open for discussions on the most varying
subjects. All the colleagues at the TB/Electronics department for their contributions in creating a pleasant working atmosphere.
This thesis has been produced while working in the project “Radio Frequency
Measurement Technology for Future Power Amplifiers and Transmitters” managed
by Dr. Niclas Keskitalo and financed by the Knowledge Foundation in cooperation
with University of Gävle, Ericsson AB, Syntronic AB, Note AB, Rohde&Schwarz
Sverige AB, Freescale Semiconductor Nordic AB, Infineon Technologies Nordic AB.
My family that has supported me greatly throughout the whole journey of
completing this thesis.
Per Landin
Gävle, November 2009.
Contents
Contents
vi
I Introduction
1
1 Introduction
1.1 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . .
1.2 Related papers not included in the thesis . . . . . . . . . . . . . .
1.3 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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2 Radio Frequency Power Amplifiers and Behavioral
2.1 Power amplifiers and transmitters . . . . . . . . . . .
2.2 Behavioral modeling . . . . . . . . . . . . . . . . . .
2.3 Signal types . . . . . . . . . . . . . . . . . . . . . . .
Modeling
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3 Performance Evaluation Criteria
3.1 Model parameter extraction . .
3.2 Model validation . . . . . . . . .
3.3 Model and validation errors . . .
3.4 Model evaluation criteria . . . .
3.5 System identification . . . . . . .
3.6 Experimental verification . . . .
3.7 Statistical analysis . . . . . . . .
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4 Measurement systems, principles and performance
4.1 Measurement system . . . . . . . . . . . . . . . . . . . . . .
4.2 PA behavioral modeling based comparison of measurement
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4.3 Modeling and predistortion of single stage PAs . . . . . . .
4.4 Modeling, measurement principles, sampling theorems and
mance evaluation problems . . . . . . . . . . . . . . . . . .
4.5 Modeling under varying load conditions . . . . . . . . . . .
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CONTENTS
vii
Bibliography
50
II Included papers
63
Part I
Introduction
1
Chapter 1
Introduction
This thesis is devoted to the subject of behavioral modeling of nonlinear radio frequency (RF) systems. Some parts of the behavioral modeling of RF systems is
devoted to modeling of power amplifiers (PAs) [1]. This thesis will explore some
requirements and measurement techniques with the goal to improve model extraction.
Large amounts of both theoretical and practical work have gone into the modeling and measurements of linear systems at RF. A complete framework for the
behavioral description of linear time-invariant devices exists. The most fundamental form is probably that of considering the impulse response of a device; if this
property is known the complete response of the linear device is known [2, 3], i.e.
nothing is left to explore. At RF a representation different than the impulse response is normally used, namely the S-parameters [4]. The reason for this is mainly
related to the difficulty of extracting impulse responses directly from RF devices
and the much larger usefulness of the S-parameters. Special instruments for extracting S-parameters from measurements are also available, normally in the form
of vector network analyzers (VNAs).
However, when approaching the world of nonlinearities nothing is as clear as
in the linear world. Not even in theory is the equivalent concept of the impulse
response completely clear. Under certain assumptions the so called Volterra theory
of nonlinear systems can be considered as a multidimensional, nonlinear, impulse
response [5]. This theory has attracted considerable amounts of attention in nonlinear RF system modeling. In the nonlinear world the Volterra systems have not
gained the same status as the impulse response in the linear world, mainly due to
that it is harder to visualize the multidimensional impulse responses and the excessive amounts of parameters required for achieving good modeling properties. That
said, the Volterra theory has been extensively used in RF system modeling [1, 6, 7].
However, it is usually more intuitive and easier to work with suitable reductions of
Volterra models such as box-models whenever possible.
X-parameters or S-functions [8] are not considered in this thesis as they are
1
2
CHAPTER 1. INTRODUCTION
mainly intended for design and characterization of nonlinear components such as
transistors and mixers. The behavioral models considered here are limited to the
baseband region, that is the frequency range close to the carrier, and can thus
not give any direct information about the harmonic behavior even if the harmonic
influence on the baseband signal is incorporated in the baseband models. An explicit
description of this behavior is given by the X-parameters/S-functions [8].
That the harmonic behavior (typically harmonic impedance) influences the total
amplifier behavior is clear when considering the actual RF-waveforms [9]. Some
amplifier technologies are in fact using the harmonics of the RF signal for reducing
the signal level peaks by creating a RF voltage waveform more similar to a square
waveform instead of the more familiar (slightly) clipped sinewave. Actually creating
the “perfect” waveform with infinitely short transition time between on- and offstate would require an infinite amount of sinewaves and nothing less than that
due to the so called Gibbs’ phenomenon [10]. This “phenomenon” says that for a
repetitive waveform with discontinuities there will not be a uniform convergence
for a finite amount of basis functions (sinewaves) [10] but an overshoot will always
be present. This overshoot approaches an approximate value of 9% of the jump
size [10] as the number of basis functions grows.
The thesis will introduce the basic requirements for baseband behavioral modeling of RF PAs and a foundation for performance evaluation of nonlinear systems
will be given. The importance of knowing if one model has better performance
than another model is often neglected. This is also evident from the large number
of publications on behavioral modeling and models of RF PA and the vanishing
small number of publications on how to evaluate the performance. The situation
improved considerably with the publication of the book [1].
Another basic requirement for PA behavioral modeling is a high performance
measurement system. The system used in this thesis is based on a high performance
vector signal generator (VSG), the R&S SMU200A, and a high performance vector
signal analyzer (VSA), the R&S FSQ26 [11]. Although the bandwidth requirements for telecommunication testing has increased considerably with time, starting
from the channel bandwidths of a few kHz used in analog systems, such as the 25
kHz used in the Nordic Mobile Telephony (NMT) system, increasing to typically
a few hundred kHz in the digital second generation systems such as Global System for Mobile communications (GSM). The continuation is the third generation
with channel bandwidths in the order of a few MHz, such as the Universal Mobile
Telecommunications System (UMTS), with 5 MHz channel bandwidths, and continuing with the "almost fourth generation" systems such as Long Term Evolution
(LTE) with channel bandwidths up to 20 MHz. The basic principles of PA modeling do not change if the signal bandwidth is 5 MHz, 20 MHz or 1 GHz, even if
signals and system designs change with the time.
However, there are problems in the behavioral modeling field. The used bandwidths tend to require very high sampling frequencies, as Nyquist once stated [12].
It has turned out that it is possible to cheat this very harsh requirement somewhat when adding some knowledge about the nonlinear device. The theorem was
3
first given in [13] and used and extended in [14, 15]. The theorem is called the
Zhu-Frank generalized sampling theorem. These publications showed that for measurement based modeling of nonlinear devices it is not necessary to sample at the
Nyquist rate of the output signal but rather that it is sufficient to sample the output
signal at the Nyquist rate of the input signal under certain requirements.
A PA working with an input signal bandwidth of 100 MHz easily has an output
signal bandwidth of 500 MHz. Analog-to-digital converters (ADCs) with sampling
rates of 200 MHz and acceptable dynamic range is widely available at reasonable
prices. ADCs with similar dynamic range but sampling frequencies of 1-2 GHz are
not yet available at reasonable prices. Thus the use of the mentioned theorem can
ease the requirements on the equipment, and thus also on the available budget,
considerably.
Another measurement assumption in RF PA modeling seems to be that behavioral models are commonly extracted at (approximately) matched conditions,
as this is what most published measurement setups show [16]. However, this will
relatively seldom be the case when the PA is put into a system where it most likely
will not see the nominal impedance, commonly 50Ω, on the output port. In a real
system it is necessary to take these tolerances into account when constructing the
PA behavioral model. In the worst case the impedance can also vary over time.
From industry there is some interest in determining how easy it is to digitally
predistort components to evaluate the component performance before the components are put into a more complex system, such as a PA. Especially the so called
memory effects of the device is of interest as this is considered more difficult to
correct [17–19]. Behavioral modeling and digital predistortion are therefore used
on a single stage PA in the manufacturer’s test fixture.
That was a short review of the requirements explored in this thesis. The mentioned areas have been investigated and are presented in six papers.
Paper A P.N. Landin, M. Isaksson and P. Händel, Comparison of evaluation criteria for power amplifier behavioral modeling, in IEEE MTT-S International
Microwave Symposium Digest, Atlanta, GA, USA, June 2008, pp. 1441-1444.
Paper B P.N. Landin, M. Isaksson and P. Händel, Parameter extraction and performance evaluation method for increased performance in RF power amplifier
behavioral modeling, accepted for publication in International Journal of RF
and Microwave Computer-Aided Engineering
Paper C P.N. Landin, C. Fager, M. Isaksson and K. Andersson, Behavioral modeling performance comparison of the large signal network analyzer and the
modulation-domain system, in 72nd ARFTG Microwave Measurement Symposium Digest , Portland, OR, USA, December, 2008, pp. 73-78.
Paper D P.N. Landin, M. Isaksson, N. Keskitalo and O. Tornblad, A study of
memory effects in Si LDMOS single transistor amplifiers, in Proceedings 10th
CHAPTER 1. INTRODUCTION
4
Annual Wireless and Microwave Technology Conference, Clearwater Beach,
FL, USA, April 2009.
Paper E P.N. Landin, C. Nader, N. Björsell, M. Isaksson, D. Wisell, P. Händel,
O. Andersen and N. Keskitalo, Wideband characterization of power amplifiers
using undersampling, in IEEE MTT-S International Microwave Symposium
Digest, Boston, MA, USA, June 2009, pp. 1365-1368.
Paper F P.N. Landin, O. Bengtsson and M. Isaksson, Power amplifier behavioural
model mismatch sensitivity and the impact on digital predistortion performance, in Proceedings 39th European Microwave Conference, Rome, Italy,
October 2009, pp. 338-341.
1.1
Contributions of the thesis
The contributions of this thesis are given in six papers related to the area of measurement, extraction, evaluation and performance improvement of behavioral models for RF PAs.
The main purpose is to establish a performance evaluation criterion for behavioral modeling using models extracted on measured data. A second purpose is to
attempt behavioral modeling of nonlinear devices, i.e. PAs, using a digitally bandlimited measurement. This can basically be seen as cheating the Nyquist sampling
rate requirement by introducing extra knowledge in the modeling process, i.e. a
bandlimited assumption and the existence of an inverse function.
A third purpose is to compare measurement systems that can be used for baseband behavioral modeling in terms of the achieved behavioral model performance
based on the previously developed evaluation criteria.
Last, the reliability of baseband behavioral models and the performance of digital predistortion in changing RF environments is investigated. All of the above
mentioned subjects are parts of establishing and improving the knowledge of the
behavior of RF PAs when viewed from the baseband domain, also being the application domain.
The papers are summarized below.
Paper A and Paper B together presents a review of performance evaluation
criteria for behavioral modeling of RF PAs and proposes an improved parameter
extraction and evaluation method.
Paper C compares the performance of behavioral models extracted using a large
signal network analyzer manufactured by Maury Microwave and the modulation
domain system consisting of a vector signal generator (VSG) and a vector signal
analyzer (VSA).
Paper D is an experimental study of memory effects in behavioral models
extracted on single stage PAs using various signal types.
Paper E re-introduces the Zhu-Frank generalized sampling theorem for behavioral modeling of nonlinear systems and presents some specific issues related
1.2. RELATED PAPERS NOT INCLUDED IN THE THESIS
5
to performance evaluation of such behavioral models. A specially designed measurement setup based on a wideband downconverter and an ADC is used for the
measurements. A problem related to cross-correlation based synchronization of
wideband input and output signals is also discussed.
Paper F demonstrates possible problems with the usage of behavioral models
and digital predistortion with small changes in the RF environment. These changes
are controlled by inserting an automated tuner at the load-side of the PA, i.e.
basically behavioral modeling in a non-matched environment.
1.2
Related papers not included in the thesis
The following presentations have not been included in the thesis although related
to the subject of study.
P.N. Landin and M. Isaksson, "A review of validation criteria for behavioral power
amplifier models," presented at GigaHertz Symposium, Gothenburg, Sweden,
March, 2008.
P.N. Landin, C. Fager, M. Isaksson and K. Andersson, "Comparing two measurement systems for the purpose of PA behavioral modeling," presented at 6th
Radio & Microwave Measurement Workshop, Gavle, Sweden, February, 2009.
P.N. Landin, M. Isaksson, N. Keskitalo and O. Tornblad, "Sideband asymmetries
in RF power LDMOS before and after digital predistortion," presented at Radio Frequency Measurement Technology Conference, Gavle, Sweden, October,
2009.
CHAPTER 1. INTRODUCTION
6
1.3
Abbreviations
ACPR
ACEPR
ADC
AWG
BER
BLA
CDF
DPD
DUT
EVM
GSM
IF
IM
LSNA
LTE
MDS
MEMR
MER
NMSE
NMT
PA
PAPR
PDF
PISPO
RF
SNR
UMTS
VNA
VSA
VSG
VSWR
WCDMA
WESPR
adjacent channel power ratio
adjacent channel error power ratio
analog-to-digital converter
arbitrary waveform generator
bit error rate
best linear approximation
cumulative distribution function
digital predistortion
device under test
error vector magnitude
global system for mobile communications
intermediate frequency
intermodulation
large signal network analyzer
long term evolution
modulation domain system
memory error modeling ratio
memory error ratio
normalized mean-square-error
nordic mobile telephony
power amplifier
peak-to-average power ratio
probability density function
periodic input same period out
radio frequency
signal-to-noise ratio
universal mobile telecommunications system
vector network analyzer
vector signal analyzer
vector signal generator
voltage standing wave ratio
wideband code division multiple access
weighted error spectral power ratio
Chapter 2
Radio Frequency Power Amplifiers
and Behavioral Modeling
2.1
Power amplifiers and transmitters
One of the main problems in the transmitter chain is the PA due to the many
undesirable properties it has, i.e. nonlinearity, high manufacturing cost both due to
expensive parts and the subsequent need for tuning, and low power efficiency [18,19].
Some of these problems could be ignored if certain requirements such as those
posed by the regulatory authorities of respective country were less strict. The main
requirements relate to that of spectral regrowth, i.e. widening of the output signal
in the frequency domain due to nonlinearities. However, it is not likely that these
requirements will be substantially loosened anytime soon, if ever. Instead it is more
likely that the more undesirable signal properties, from the PA designer’s point of
view, will continue to get worse and worse.
These problems partly comes from the use of signals with wider bandwidths
and higher peak powers relative to the average power in combination with the
requirements on high efficiency operation [19]. High efficiency has traditionally
been a direct opposite of linearity as high efficiency operation has been a synonym
of compression when only considering the traditional classes A, AB, B and C.
Some change here is possible with “new” techniques such as Doherty [20],
Chireix [21], class E and F with inverses, switch-mode, envelope tracking and envelope elimination and restoration [22], to mention a few. Admittedly most of these
principles are quite old with the Doherty and Chireix coming from the 1930’s and
the envelope elimination and restoration from 1952. Neither have been extensively
used in modern telecommunication systems, mainly due to practical problems in
the construction.
According to Shannon’s well-known theorem a certain bandwidth is needed in a
noisy channel to transmit data at a given rate [23, 24]. The only variables that can
be changed are the signal-to-noise ratio (SNR) and the bandwidth. The transmitted
7
8
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
power, and thus the SNR, is below a certain level, partly due to economical and
technical difficulties in constructing highly efficient and linear PAs. This means
the only thing that can be changed is the bandwidth. At the same time the used
modulation formats tend to increase the peak-to-average-power ratio (PAPR) of
the used signals [19].
An example of increase in bandwidth and PAPR is the transition from the
UMTS channel bandwidth of 5 MHz to the LTE channel bandwidths of up to 20
MHz [25,26]. It should be remembered that PAs today are commonly not made for
handling a single channel but rather for handling all channels in a given telecom
system, or possibly in several different systems. This is taken more towards the
extreme in LTE with the support for multiple bandwidths.
So, the PA designers are left with a signal type that is everything they do
not wish for. What can be done to improve the situation? The system engineers
need improved models of the components of the system. This is one part where
behavioral modeling can aid by providing better models to work with. The second,
and probably largest field, is for usage in digital predistortion (DPD) algorithms.
The exact meaning of these terms will shortly be explained.
Static nonlinear effects
A common method of introducing the understanding of nonlinearities in RF PAs is
by considering the input-output relation
y(n) = au(n) + bu(n)2 + cu(n)3
(2.1)
where y(n) is the ideal output of the PA when subjected to the ideal input signal
u(n). Commonly u(n) is chosen to be a single or a two-tone signal.
The first case is connected to the classical amplitude-amplitude modulation
(AM/AM) curve. Expanding (2.1) for a two-tone signal gives
y(n) =
b
2
9
9
+ (a + c ) cos ( ω1 t ) + (a + c ) cos( ω2 t )
2
2
b
b
cos 2ω1 t + cos 2 ω2 t
+
2
2
+ b cos( ω1 − ω2 )t + b cos ( ω1 + ω2 )t
c
c
cos 3ω1 t + cos 3ω2 t
+
4
4
3
3
c cos( 2ω1 + ω2 )t + c cos( 2ω2 + ω1 )t
+
4
4
3
3
c cos ( 2ω1 − ω2 )t + c cos( 2ω2 − ω1 )t
+
4
4
.
(2.2)
2.1. POWER AMPLIFIERS AND TRANSMITTERS
9
The first line is a DC-term, second is the signal at the fundamental with the
compression terms from the 3rd order nonlinearity included, third line are the 2nd
harmonics, fourth line are the 2nd order difference and sum products, fifth line
are the 3rd order harmonics, sixth line are the 3rd order intermodulation (IM)
products close to the third harmonic and line seven contains the most troublesome
IM products close to the fundamental.
In the expression one can see the terms corresponding to the gain compression
(a + 92 c ). In this simple PA model this 3rd order contribution is responsible for the
gain reduction when going into compression.
AM/PM distortions are a little harder to explain and requires some measured or
simulated data. A good introductory explanation of this behavior is given in [19].
The basic results there says that AM/PM most likely contributes about as much
to IM products as AM/AM which means that any model has to take this term into
consideration.
The meaning of the term “nonlinear” is here defined to be a system that does
not obey the superposition principle, i.e. that does not obey the rule
y(u1 (t) + u2 (t)) = y(u1 (t)) + y(u2 (t))
(2.3)
for all choices of the signals u1 (t) and u2 (t).
Dynamic effects
Another undesired effect in PAs is known as dynamic effects, also termed memory
effects. In the simplest form these are just a gain and phase variation as a function of frequency. This is the well known and well studied problem of wideband
amplifier design [4, 27]. As such this is nothing new but when considering DPD
the requirement of flat gain becomes even more important due to the fact that the
inverse of a system with memory typically has long memory [28].
Memory effects are the time-domain view of frequency dependence, compare the
typical impulse response of a filter. The conceptually simplest memory effect is the
bandwidth limitation due to the design of the PA. For a wideband PA these will
be small compared to the direct linear term. Memory effects are of different orders
of magnitude, from the high frequency electrical effects in the matching networks
to low frequency effects due to effects of heating, so called thermal effects [17].
Not all memory effects give themselves to such easy explanations. Some other
commonly listed memory effects are bias modulation, thermal effects due to self
heating and trapping effects [19]. These are sometimes referred to as nonlinear
memory effects [29, 30].
According to [19] the major troublemaker, for certain PA classes, is the bias
modulation introduced by the variation in consumed current with envelope amplitude. This is supposedly also the easiest to cure by proper design of the bias
network [19].
Two other commonly mentioned effects are trapping in the semiconductor and
thermal effects [19]. Some efforts have gone into behavioral modeling of thermal
10
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
effects, an early attempt using sparse memory in memory polynomial structures
was made in [16]. Not as much effort, or progress, seems to have been done related
to the behavioral modeling of trapping effects [19].
2.2
Behavioral modeling
The previous sections introduced the basics of distortions in RF PAs. Here some
more advanced behavioral models are introduced, and also some less advanced
models.
It is assumed that all systems treated here are so called periodic input, same
period out (PISPO) systems [31]. Another distinction must also be made, namely
that of bandpass or lowpass models and systems. The section will start with giving
these definitions.
Bandpass and lowpass signals and systems
Most communication signals, with the exception of UWB-type signals and baseband
signaling, only utilize a relatively narrow frequency range around a main carrier
frequency. This can be expressed as
s(t) = Re[u(t)ejωc t ]
(2.4)
with s(t) being the real-valued time domain signal, u(t) the complex valued lowpass equivalent signal and ωc the chosen carrier or center frequency. Note that this
does not really put a restriction on u(t) but on the choice of ωc . s(t) is denoted
the bandpass signal since the information in the signal is found in a (narrow)
frequency region surrounding the carrier, whereas the, normally complex valued,
u(t) is denoted the lowpass equivalent or baseband signal.
u(t) is commonly divided into two parts, an in-phase si (t) and a quadraturephase sq (t) part by considering
u(t) = si (t) + jsq (t).
(2.5)
This has given yet another name for baseband data, namely IQ-data due to the
in-phase and quadrature-phase.
The practical approach of computing u(t) is to compute the complex valued
analytical signal sa (t) from s(t) by applying the Hilbert transform to s(t) and
perform the frequency translation. This is expressed by
u(t) = (s(t) + jH{s(t)})e−jωc t
(2.6)
where H{s(t)} denotes the Hilbert transform of the real valued signal s(t) [23, 24].
The Hilbert transform is defined as a filter with the impulse response
h(t) =
1
.
πt
(2.7)
2.2. BEHAVIORAL MODELING
11
The function of the Hilbert transform is to create a new signal where the negative
frequencies have shifted sign. Thus adding the original and the Hilbert-transformed
signals will cancel the negative frequencies present in a real-valued signal. The
resulting signal is called an analytical signal.
The main difference is that the required sampling frequency to avoid aliasing is
restricted by the baseband bandwidth and no longer by the RF bandwidth. The
same method can be applied to systems to get baseband systems [23, 24].
Some of the models below can be treated as either lowpass or bandpass models
depending on the preference one have. Each section specifies how the model is
normally used.
Linear models
All modeling of RF systems starts with the scattering-parameters (S-parameters)
[4]. These parameters relate the voltage waves coming into and going out of a device
by a complex number. Typically these parameters are extracted using VNAs by
sending a single sinewave into the device and measuring the transmitted and scattered waves. The S-parameters are then extracted by normalizing these measured
waves by the incident wave [4].
Because the S-parameters are nothing but the frequency response function, the
complete characteristics of a linear time-invariant system is known once the Sparameters are known. The S-parameter model is a bandpass model as it describes
what happens at a given frequency.
As said in [8]: "S-parameters are perhaps the most successful behavioral models
ever." Nevertheless, the S-parameters are not defined for nonlinear systems, i.e.
when the superposition principle is not valid.
Best linear approximation
In many cases the linear behavior is normally the desired behavior and everything
else can be regarded as disturbances. This fact is used in the best linear approximation (BLA), as described in [32]. There the final system output is considered
to consist of several different parts: the linear systematic part (the BLA), a noise
contribution and a stochastic nonlinear contribution that appears as another noise
source.
The BLA, commonly used with some special multisine signals, has been applied
in many different fields [32–34]. The main interest here is the application to RF
systems. Recently an extension to measure the BLA outside of the excited frequency
region was proposed in [35].
The BLA is more general than the S-parameters as the nonlinear contributions
are handled in a consistent and logical way. Nevertheless, the BLA cannot easily
be used for DPD since it is a linear model.
The BLA has the possibility of being used as both a lowpass and a bandpass
model without any significant change in the definition.
12
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
Polynomial models
One of the first approximation methods one usually learns is the Maclaurin-Taylor
polynomials [36] but then the target function is normally known. In the case here,
polynomial models, such as that in (2.1) are to be fitted to a measured transfer
function, using some suitable fitting method.
A simple example would be to approximate a nonlinear function with a characteristic quite typical for a saturated system. Such a system is the atan(x) seen in
Fig. 2.1 for x going from 0 to 2. There are also different methods of fitting a model
to this known transfer function. This example is limited to consider straight lines,
i.e. polynomials of order 1.
In Fig. 2.1 the blue line depicts the true transfer function. The red line is the
Maclaurin line, i.e. the fitting is based on the point x = 0 which gives a line with
the equation y = x. The black line shows the Taylor expansion around x = 0.5.
In reality the transfer function is only known by measurements so some kind of
model error minimizing fitting between the measured straight line and the measured
data is needed. Because this is only simulations the effect of noisy measurements
is ignored so the input data is chosen as a uniformly sampled ramp on the interval
0 to 1.
n
} with N
This means that the input data set is described by u(n) ∈ {x = N
being the total number of samples. This is an example of a uniform amplitude
distribution in one amplitude interval. The fitting method is chosen to minimize
the sum-square-error, which is described in more detail in a later chapter. It is
seen that all these methods give different results. The approach used in this thesis
is mainly based on the least-squares method. However, there is another detail in
this experiment that has not been explicitly pointed out, namely the amplitude
distribution of the values used to identify the model. This is considered in a later
section on stimuli signal choice.
Multiple variations on polynomial models are possible using orthogonalizations
but these do not add anything fundamental to the concept of modeling nonlinearities through polynomials. Although, sometimes it is highly recommended to
use some of the orthogonalizations to improve the condition numbers [37] of the
regression matrices in the parameter estimation process.
Polynomial models are used for describing both lowpass and bandpass behavior with the harmonics, although for modeling purposes the lowpass is probably
somewhat more common. However, this requires that the polynomial is rewritten
as
P
ap u(n)|u(n)|p−1 .
(2.8)
y[u(n)] =
p=1
The absolute value appears as a result of the bandpass to lowpass transform.
Back to the Volterra series mentioned in the Introduction. It was pointed out
that the Volterra series is a multidimensional nonlinear impulse response. It could
just as well be viewed as a polynomial with memory. If the dynamic parts are
2.2. BEHAVIORAL MODELING
13
2
Output
1.5
atan(x)
Maclaurin center=0
Taylor center=0.5
Least−squares U(0,1)
1
0.5
0
0
0.5
1
1.5
2
Input x
Fig. 2.1: Examples of fitting methods of a straight line to a known function. The
blue line is the true function, red is the Maclaurin approximation, black is the
Taylor centered at x = 0.5 and the green is the result of a least squares fitting
using uniformly distributed data in the interval [0, 1].
removed from the Volterra series what is left is a static polynomial, assuming that
the basis functions are chosen to be polynomials.
Hammerstein, Wiener and Hammerstein-Wiener models
In Section 2.1 it was argued that systems such as PAs are not only linear but also
exhibit dynamic effects. A straightforward approach would then be to create a
model, in an ad hoc manner, by taking a suitable static nonlinearity and placing a
dynamic block after it, as in the upper part of Fig. 2.2. This creates a so called
Hammerstein model [38]. Reversing the order of the blocks creates a Wiener model.
Combining the models yields a Wiener-Hammerstein model.
As the model structures in this section mainly are used for baseband modeling
the polynomials are modified to be written as in the lowpass manner introduced in
(2.8).
When choosing
the nonlinearity in the Hammerstein model to be a polynomial,
P
p
k
K[u(n)] =
p=1 p u(n)|u(n)| , and the dynamic part to be a linear FIR-filter,
M
H(q −1 )u(n) = m=0 hm u(n − m), q −1 denoting the unit delay operator q −1 u(n) =
u(n − 1) as in [2, 3], the relation between input and output signal can be written as
y(n) =
P
−1 M
p=0 m=0
kp hm u(n − m)|u(n − m)|p
(2.9)
14
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
with u(n) and y(n) being the sampled input and output signals respectively. M
and P are model orders for memory length respectively polynomial order.
For RF PA behavioral modeling some controversy seems to exist whether one
should discard the even order nonlinear terms as in [7] or keep them as in [39].
Some arguments speaking for keeping the even terms are given in [40]. Starting
from (2.2) no even order terms are expected to appear close to the baseband signal.
Nevertheless, it is seen that this yields a system that is nonlinear in the parameters. In [2,3] the suggested solution to this problem is to linearize the problem in the
parameters by considering the problem to be a multiple-input single-output system
with each nonlinear basis function, that is u(n)|u(n)|p or some other preferred basis
function, corresponding to one of the inputs.
This will indeed make the problem linear in the parameters but it will also
transform the model structure into another model, namely the parallel Hammerstein model. This model is defined by
yPH (n) =
P
p=1
−1
M
bm,2p−1 u(n − m)| u(n − m) |2(p−1)
(2.10)
m=0
where O = P corresponds to the polynomial order and M corresponds to the
memory length. Such a model is henceforth denoted PH(O, M ). This model is
linear in the parameters. However, it is also a new model structure which is best
illustrated by parallel connections of the nonlinear branches as in Fig. 2.3.
It is more common to see the parallel Hammerstein model with only odd polynomial orders included when considering RF PA behavioral modeling. In that case
(2.10) is rewritten as
yPH (n) =
P M
bm,2p−1 u(n − m)| u(n − m) |2(p−1)
(2.11)
p=1 m=0
with O = 2P − 1 being the nonlinear order and M being the memory length.
The parallel Hammerstein model or models closely resembling it have been
commonly used in behavioral modeling of RF PAs [41–52]. Some other commonly
used names for model structures similar to the parallel Hammerstein model are the
nonlinear tapped delay line [41] and memory polynomial [53] although these terms
sometime also refer to other model structures. A large number of variations on
the theme of parallel Hammerstein structures are possible, see [42–52, 54] for a few
examples. These model structures are interesting for RF PA behavioral modeling
due to the fact that there is a physical motivation behind using them [54].
As shown in Fig. 2.2, switching the order of the nonlinear and dynamic blocks
results in a Wiener model. These have also been used in PA modeling but seem
to have attracted less interest than the Hammerstein structures to judge from the
number of publications. A possible explanation for this is the increased complexity
in the model parameter extraction process due to the nonlinear way the parameters
appear, as compared to parallel Hammerstein-like structures.
2.2. BEHAVIORAL MODELING
15
u
y
K(·)
H(ω)
H(ω)
K(·)
H1 (ω)
K(·)
u
y
u
y
H2 (ω)
Fig. 2.2: From top to bottom: Hammerstein system with static nonlinearity followed by linear dynamics, Wiener system with dynamic linearity followed by static
nonlinearity, Wiener-Hammerstein system with linear dynamics-static nonlinearitylinear dynamics.
u(n)
{bn,1 }
2
u3 (n)
4
u5 (n)
{bn,3 }
{bn,5 }
y(n)
Fig. 2.3: An example of a complex-valued parallel Hammerstein model with O = 5
is shown. The length of the linear finite-impulse response filters {bn, } determine
the memory depth M . In the block diagram, the intermediate signals u3 (n) =
|u(n)|2 u(n) and u5 (n) = |u(n)|4 u(n) are indicated.
It is possible to get the same structure as the parallel Hammerstein by arranging
a parallel Wiener structure in a special way as in [53]. In some early publications the
chosen name has actually been the parallel Wiener model [16]. It is not necessary
to restrict oneself to using polynomials and FIR-filters in the blocks but it seems
to be the most common approach as this yields estimation problems that are linear
in the parameters.
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
16
Volterra series
The most general form of a nonlinear system, under certain restrictions, is the
Volterra series. As mentioned earlier the Volterra series can be seen as a generalization of the impulse response to nonlinear systems [5].
One way of expressing a discrete time, truncated Volterra model with memory
length M and nonlinear order O is [38, 55]
y(n) = H0 + H1 [s(n)] + H2 [s(n)] + . . . + HO [s(n)] =
H0 +
M−1
h1 (m)s(n − m)+
(2.12)
m=0
M−1
M−1
h2 (m1 , m2 )s(n − m1 )s(n − m2 ) + . . .
m1 =0 m2 =0
M−1
M−1
m1 =0 m2 =0
...
M−1
hO (m1 , m2 , . . . , mO )s(n − m1 )s(n − m2 ) . . . s(n − mO )
mO =0
with hk (m1 , . . . , mk ) being the kth order Volterra kernels and s(n) being the sampled passband signal. This means that the model is expressed as a real valued
passband model. It must be noted that the above description is the full expression
for the time-discrete Volterra series without reductions that accounts for symmetry
of the kernels [38,56] or other physically motivated reductions. Volterra models can
also be expressed as baseband models [6, 7, 38].
From (2.12) the earlier claims of Taylor expansion and memory polynomials are
easily seen. Also comparing the expression for the parallel Hammerstein in (2.10) to
(2.12) it is seen that the Volterra kernels of a parallel Hammerstein system typically
are “diagonal”. This means that the kernels can be described by
= 0 if m1 = m2 . . . = mk
(2.13)
hk (m1 , . . . , mk ) =
0
otherwise
This basically means that all terms that have combinations of nonlinear and linear
memory terms with different delays are zero.
Various orthogonalizations, physically and mathematically related simplifications and reductions of Volterra series have been done to give better modeling
properties with fewer parameters. Some examples of these are given in [57–61].
X-parameters and S-functions
Some fairly new models of microwave components are the X-parameters and Sfunctions [8,62,63]. These constitute an expansion of the S-parameters in that they
also describe nonlinear behavior but reduce to the classical S-parameters when the
devices operate in the linear region, if any such region exists [8].
2.2. BEHAVIORAL MODELING
17
The X-parameters and S-functions are based on the poly-harmonic distortion
model [8]. Basically the X-parameters and S-functions are mappings of the input
signal to the possible nonlinear components that can appear. Typically this is
illustrated using a sinewave with harmonics as input signal and the possible harmonics on the remaining waves as the mapped output values. The basic principle
is described in [8].
Recent updates to the modeling capability includes the possibility of describing
memory effects [64]. Using these methods it is claimed that typical fulll PA behavior
can be predicted [64] in a reliable way. The X-parameters/S-functions are, as are
the S-parameters, bandpass models of the system.
Other model structures
In [7,59] a conceptual feedback model representing the lowpass behavior using linear
dynamic and static nonlinear blocks to represent a PA is shown. In [7,54] it is shown
that making some assumptions about the models yields some other commonly used
block structures, such as an augmented form of the parallel Hammerstein model [54].
Neural network models have not been used in this thesis but it must be noted
that they have been extensively used in modeling of RF components and systems
[6, 7, 65].
Inverse modeling and digital predistortion
The main use of RF PA behavioral modeling is most likely that of digital predistortion. Since the PA is a system that introduces distortions into the signal it is
possible to pre-distort the input signal to remove undesired effects such as spectral
regrowth and increased error vector magnitude (EVM), of the PA [60, 66–68].
Many predistortion arrangements are possible, ranging from analog gain expansion on the RF-signal as exemplified in [17], to digital baseband schemes of varying
complexity. The simplest digital schemes are based on measured AM/AM- and
AM/PM-curves [69] going through medium complex model structures such as polynomials and memory polynomials [70] to the more complex Volterra models [68,71]
and neural networks [72–74].
The approach taken here is described in more detail in [72] and used in Paper
D and Paper F . The chosen method is to directly identify an (approximate)
inverse baseband model instead of going through the theoretical inverse of a direct
baseband model, such as that given in [75]. The used model structure in both
Paper D and Paper F is the parallel Hammerstein model since it gives good
performance in both direct and inverse modeling.
As mentioned is the main goal of the predistortion to reduce spectral regrowth
and improve the EVM of the output signal of the PA. From a simplistic point of
view this does not cause any problems if a sufficiently good model is available. However, one main problem is still present, namely that of increased signal amplitude
and larger occupied bandwidth. Both these goes outside of the ranges of the iden-
18
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
20
uDPD
15
u
y
Power [dBm]
10
5
0
−5
−10
−15
−20
93
93.5
Time [μs]
94
Fig. 2.4: Example of failed predistortion due to operation of the PA in compression.
Black line with diamond shapes shows the normalized output power envelope of the
PA and red line with stars the measured input power envelope. The blue line with
dots depicts the predistorted signal using a PH(9, 2) model and not taking any
precautions to limit the amplitude of the signal.
tification signal. Expressed in other terms; the predistortion is an extrapolation of
the measured values into previously unmeasured amplitude and frequency regions
of the PA.
The problem of gain compression cannot be fully overcome as there is a limit to
what output powers a PA can deliver. Thus driving the PA too hard and trying to
extract inverse models from data measured at such conditions tends to “explode”
as shown in Fig. 2.4 if no special precautions are taken.
In Fig. 2.4 a parallel Hammerstein model of order PH(9, 2) is used. The tested
PA is the PA intended for being used in the 3rd generation of mobile telecommunication manufactured by Ericsson AB, described in more detail in Section 4.1. The
input power level to the PA is -1.0 dBm and the PAPR of the used WCDMA-signal
is 7.7 dB, thus the peaks are well above the 1 dB-compression point of 3 dBm. One
can see the effects of extrapolation on the predistorted signal around 93.6 μs. The
input power peak is approximately 19 dB above the original instantaneous envelope power and would probably cause permanent failure of the PA. Note that the
exemplified situation is beyond what is normal for this PA.
2.3. SIGNAL TYPES
2.3
19
Signal types
Multiple signal types are used for testing PAs. Each of the different signals have
their pros and cons. Unfortunately, it has often been more of a question of what
one can afford to generate and measure than what is desired. For linear devices the
signal characteristics do not matter, if one ignores noise and measurement time.
However, for nonlinear devices the signal characteristics affect the final result, even
if there is no noise present. Some commonly used signal types for testing RF devices
are introduced and their characteristics are given.
Before the different signal types are introduced the important characteristics
will be introduced by extending the example of fitting a straight line to a nonlinear
function. Much of the presentation here is based on the ideas in [31,56]. Once again,
consider the function atan[u(n)] as shown in the upper part of Fig. 2.5. The lower
part shows what happens when one attempts a fitting of a straight line using two
different data sets, both being uniformly distributed but having the distributions
in different amplitude intervals. The first is uniformly distributed in the interval
[0, 2] while the second is uniformly distributed in [0.75, 1].
This illustrates a very important property of identification of nonlinear systems,
namely that of choosing the identification signal to have the same probability density function as that of the signal types that are going to be used! Of course, if an
exact model structure is available and only the parameters are sought for then the
identification data can be chosen arbitrarily, ignoring the effects of noise. However,
this is seldom the case.
For this particular reason a number of the commonly used test signals are not
always appropriate test signals for nonlinear systems. Nevertheless, these signals
continue to be used mainly because the instruments to use them are what is available
at affordable prices.
One could argue that the example of line fitting to the atan-function is not valid
as a too simple model was used. However, in the case of modeling a real world PA
there is no exactly known function that describes the transfer characteristics and
thus it is really difficult to tell what is “enough” and what is “too simple”. The
main lesson from the fitting example is to learn the major impact the amplitude
probability density function (PDF) of the used identification data has.
One more aspect of signal amplitude PDF is introduced by considering the
difference between lowpass and bandpass signals. As this thesis mainly is concerned
with lowpass signals and systems the signal types are presented from the perspective
of lowpass signals. This is not a cause of problems as long as it is defined if it is
the bandpass or lowpass definition that is given.
Single-tone excitation
In RF measurement the fundamental test signal is the sinewave, largely due to the
fact that this is what is easiest to generate at high frequencies. This has also lead
to the enormous success of the S-parameters [8], as noted earlier.
20
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
Output
2
1
0
0
atan(x)
Maclaurin a=0
Taylor a=0.5
0.5
1
1.5
2
Input x
Output
1.5
1
atan(x)
LSE U(0,1)
LSE U(0.75,1)
0.5
0
0
0.5
1
1.5
2
Input x
Fig. 2.5: a) The same figure as in Fig. 2.1 with atan, Maclaurin polynomial and
Taylor polynomial with center at 0.5. b) Least-square fittings of straight lines to
atan using uniformly distributed data in two different intervals, [0, 2] respectively
[0.75, 1].
For testing nonlinear devices it is not that suitable, unless the actual signal that
is used in the tested devices is a constant amplitude signal such as phase modulation, frequency modulation, phase-shift keying or frequency shift keying [23, 24].
Although all of these modulations are considered to be constant amplitude modulations there is a certain amount of amplitude change in the symbol transitions.
One exception in nonlinear testing is in radar systems based on continuous wave
operation but in this case the signal is normally pulsed still requiring some more
advanced test signal.
The low-pass equivalent amplitude PDF of a sinewave is a single fixed amplitude
value as the sinewave transforms into a complex exponential function when going
from bandpass to low-pass equivalent form.
With the single-tone excitation the only way in which one can get some more detailed information about nonlinearities is through the harmonics. The measurement
of signals several GHz apart poses some problems but also interesting solutions, as
described in Section 4.1. It should be noted that some insight into nonlinearities
can be gained in the form of AM/AM- and AM/PM-curves.
Two-tone excitation
The somewhat more sophisticated two-tone test signal overcomes the issue of information about the nonlinearity as the intermodulation products end up close to
the carrier if the two tones are relatively closely spaced as seen in (2.2). However,
2.3. SIGNAL TYPES
21
when considering identification of Volterra kernels, a two-tone is only sufficient to
fully characterize the second order kernel [29].
The amplitude PDF is still far from what is normally used in communication
systems. Nevertheless, it has been an easy signal to generate as it can be done using
two standard signal generators. The low-pass PDF is that of a bandpass sinewave,
i.e. high probability of finding signals at high amplitude values and low probability
of finding it at low amplitudes.
Multitone excitation
A generalization of the single and double sinewaves is the multisine approach [76,77].
Using special sets of phases many different amplitude PDFs can be synthesized.
This is used in Paper C where a multisine signal consisting of 62 tones has the
phases arranged in such a way as to have a PDF similar to that of a WCDMA-signal.
Furthermore, choosing the excited frequency components in a special way, the
so called odd-random-multisine [78], makes detection of nonlinear behavior very
easy [35]. This is based on the fact that if only odd frequency components are
present in the signal, the even order nonlinearities will only end up at even frequency
components [78].
Multisines have started to attract much attention due to the possibilities of
high SNR, periodicity of the excitation signal, ease of extracting frequency response
functions and the possibility of visualizing nonlinear effects in the frequency domain
in an easy way.
Communication test signals
These signals are generated according to some standard such as those in [25, 26]
using a pseudo-random number generator to generate the “information”. Different
performance criteria of the device under test (DUT) can then be evaluated by
sampling the signal. Example of these are EVM, bit error rate (BER), adjacent
channel power ratio (ACPR), occupied bandwidth, harmonic content and spurious
distortions [25, 26].
Test signals that are commonly used here are of the same type as the signals used
in the 3rd generation of mobile telecommunication. This signal type is henceforth
denoted wideband code division multiple access (WCDMA) signals. The signals
used here are not any of the standardized test cases in [25] but have similar statistical properties in terms of bandwidth and amplitude PDF. Normally, only the
PAPR of the signals are given. For the signals used here the PAPR ranges from 7
to 9 dB. The bandwidth of a channel in 3G is 5 MHz [25]. Combinations of multiple WCDMA-signals are made to get multicarrier WCDMA with different total
bandwidths. When these are used the spacing between the centers of the channels
are specified.
It is possible to view all these communication test signals as special cases of
multisines. Commonly a test signal of a certain time frame is generated. This
22
CHAPTER 2. RADIO FREQUENCY POWER AMPLIFIERS AND
BEHAVIORAL MODELING
signal is transmitted repeatedly from a VSG and measured using some kind of
receiver [25, 26], commonly a VSA. By applying the frequency and time discrete
Fourier transform to such a sampled signal it is realized that these signals indeed
are special cases of multisines.
Other types of excitation signals
Other types of signals such as pseudo-random binary sequences and white noise
can be used for identification of RF telecommunication systems but does not seem
to have gained much interest in recent years and are therefore not considered any
further. A plausible explanation is that the amplitude PDFs are not similar to
those of the signals that are being used when the system is operating.
Chapter 3
Performance Evaluation Criteria
As the purpose of behavioral modeling is to have a model capable of describing the
system in a “good” way a relevant performance evaluation criterion is needed. Not
all commonly used criteria are suitable for performance evaluation in this context,
as will be shown. This chapter goes through the basics of system identification,
gives the prerequisites for performance evaluation criteria and proposes a general
method for improving the performance of models when modeling weakly nonlinear
systems.
3.1
Model parameter extraction
The process to find a suitable model structure, model order, optimal parameters and
evaluation criteria are all parts of the system identification theory [2,3] and as such
is well-described in the literature. However, most of the classical literature tend to
focus on identification of linear systems and most often treat nonlinear effects as
noise disturbances. In many cases this is justified as nonlinear effects are usually
small compared to linear effects in system that were designed to be linear. In the
case of telecommunication systems this is not always a recommendable method as
the requirements on the components are strict regarding spectral regrowth.
As the system to be identified here mainly is a black box, i.e. unknown content,
the system identification starts from a set of N input/output measurements
{u(0), . . . , u(N − 1)}, {y(0), . . . , y(N − 1)}.
(3.1)
The identification process finds parameter values gathered in a vector θ that minimize the least squares difference between the actual PA output and a parameterized
model output f (u(n); θ). That means minimization of the least-squares criterion
C(θ) =
N
−1
(y(n) − f (u(n); θ))
n=0
23
2
(3.2)
24
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
where the minimization is obtained by varying the model parameters θ, and where
the solution (that is, the parameter values minimizing the criterion C(θ) in (3.2))
Note that the formal description of the actual amplifier f (·)
is denoted by θ.
and the user selected parametric model structure f (·; θ) are only distinguished by
the argument θ within the parentheses. How the actual minimization of (3.2) is
performed is not relevant for the current discussion, nor is the model structure; it
is sufficient to recall that the optimized set of parameters is denoted by θ.
3.2
Model validation
The second step of the parameter identification includes validation of the obtained
For that purpose, a new set of PA data is recorded – uval (n) and
model f (·; θ).
yval (n). From the new set of input data, a model output is calculated according to
n = 0, . . . , N − 1.
ymodel(n) = f (uval (n); θ),
(3.3)
Clearly, there is no need to have the same number of samples in the validation
process as in the identification process, but for the sake of simplicity N samples are
used.
Of general importance is the difference between the obtained model output
ymodel (n) in (3.3) and the actual response of the PA yval (n) = f (uval (n)), which is
the error
e(n) = yval (n) − ymodel(n), n = 0, . . . , N − 1.
(3.4)
A natural scalar quality measure is the estimated power of e(n), say Pe , normalized
by the estimated power of the validation signal, that is Pyval , which leads to the
well known normalized mean-square-error (NMSE) criterion.
NMSE =
Pe
.
Pyval
(3.5)
Based on a criterion like (3.5), one can compare two different (both in structure,
and number of parameters) parametric PA models: f1 (·; θ1 ) and f2 (·; θ2 ). Using
the NMSE, it is concluded that f1 (·; θ1 ) is a better (worse) behavioral model of the
PA if the NMSE is lower (higher) than the NMSE of f2 (·; θ2 ). Here, θk denotes the
output from the parameter estimation step, which is the minimizer of the criterion
(3.2) employing fk (·; θk ), for k = 1, 2.
3.3
Model and validation errors
Consider the case when the employed model structure is capable of perfectly describing the behavior of the PA up to some additive term v(n). This means that
the PA output y(n) can be described by the parametric model f (·; θ), that is
y(n) = f (u(n); θ0 ) + v(n)
(3.6)
3.4. MODEL EVALUATION CRITERIA
25
where θ0 is the underlying ’true’ parameters of the amplifier, and v(n) is the term
that describes noise sources and model errors, for example. Under the above assumption, the validation error in (3.4) is given by
e(n) = f (uval (n); θ0 ) + v(n) − f (uval (n); θ)
(3.7)
Any appropriate identification method provides, under some mild conditions, an
estimate θ close to the unknown θ0 . Thus, employing a first order Taylor series
around θ0 yields
expansion of f (·; θ)
e(n) =
d f (uval (n); θ) + v(n).
(θ0 − θ)
dθ
θ=θ0
(3.8)
Formally, we may write (3.8) as
+ v(n).
e(n) = g(n; θ)
(3.9)
is a formal notation for the first term in (3.8).
where g(n; θ)
The interpretation of (3.9) is as follows: the validation error e(n) depends both
on the quality of the estimation method employed in step one (through the first
term, which one may note equals zero for θ = θ0 ),and on the ability of the employed
model structure to capture the actual behavior of the amplifier.
A general feature of a scalar figure of merit (for example, the NMSE in (3.5))
is that a ’good’ value corresponds to a ’small’ validation error. For example, for
NMSE, a lower value is a measure of superior performance, compared with a higher
value. A ’small’ error may correspond to the full band properties of the model, but
also to out-of band or in-band properties.
3.4
Model evaluation criteria
In Paper A a number of common evaluation criteria were compared in terms of
performance for RF PA behavioral modeling. These criteria will be introduced and
comments on their performance characteristics are given.
The methods used to compare model performance found in Paper A are:
NMSE [6,59,79,80], AM/AM [79,81–83], AM/PM [81–83], time domain sample [59],
spectrum comparison [58, 79, 81, 83–86], IM3 product prediction [82], ACPR comparison [6,81,84,85,87], error spectrum [6,59,88], adjacent channel error power ratio
(ACEPR) [6, 79, 86], memory error ratio (MER) [16] and memory error modeling
ratio (MEMR) [16].
Some of these criteria are graphical: AM/AM, AM/PM, time domain sample,
spectrum comparison and error spectrum. Short descriptions of each method are
given in the following.
AM/AM is the deviation in linear gain in output amplitude as function of input
amplitude. AM/PM is the change in phase as function of the input signal amplitude.
26
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
The results are normally presented as graphs. The model performance is evaluated
by plotting measured and modeled AM/AM and AM/PM curves on top of each
other to see similarities.
Time domain sample comparison is done by viewing the time domain waveform
of the modeled output and the measured output in the same graph. One then looks
for deviations between the waveforms.
Spectrum comparison displays the estimated power spectral density of the modeled output signals and the measured output signal. The comparison is based on
which modeled output signal is “closest” to the power spectral density of the measured output signal.
Error spectrum plots the estimated power spectral density of the error signal
obtained by subtracting the modeled output signal from the measured output signal,
as in (3.4). The best model is then determined by considering the plotted graphs
of the estimated power spectral densities of the error signals.
The above mentioned criteria are all graphical and cannot easily be used for
performance comparison when the models have close to the same performance.
Another disadvantage is also that they cannot be used as a minimization goal
in the parameter extraction procedure. However, to illustrate behavior such as
compression the AM/AM and time domain plots are excellent, phase variations
are illustrated in the AM/PM plots. Spectral spread is illustrated in the spectrum
comparison and the parts of the spectral regrowth that is harder to model is seen
in the error spectrum plots.
The next category of criteria are classified as numerical because they all have
as the final criterion, a number.
In IM3 prediction the model performance is based on the prediction of the intermodulation products as function of a given input two-tone signal. The predicted
IM-products are then compared to the measured IM-products and the model that
predicts the IM-products closest in some sense, is labeled the best model. In some
cases the phases of the IM-products are included, in other cases they are not included.
On the same line as the IM3-prediction is the ACPR prediction. In this case the
ACPR is computed of the modeled and the measured output signals. The numbers
are then compared and the model with the ACPR closest to the measured output
signal is called the best model. This method neglects the importance of the phase
predictions needed in digital predistortion.
Both IM3- and ACPR-prediction considers quantities easily measurable using a
traditional spectrum analyzer and can thus be useful if there are no vector signal
analysis instruments are available.
The remaining performance evaluation criteria all operate on the model error
(3.4). For notational simplicity assume that the N samples of the model error e(n)
in (3.4) are gathered into a column vector e.
The first criteria are all related to the total model error defined in (3.4). The
standard criterion from linear system identification theory is the sampled normalized mean square error as defined in (3.5) with Pe and Pyval being the estimated
3.4. MODEL EVALUATION CRITERIA
27
power spectral densities of the model error in (3.4) respectively of the measured
output validation signal. By Parseval’s theorem we get
NMSE =
Pe
|| e ||22
=
Pyval
|| yval ||22
(3.10)
where yval is a vector containing the measured output signal and || · ||22 denotes the
normal vector 2-norm squared.
Two other criteria based on the NMSE are the MER and the MEMR where both
criteria originally were suggested for performance evaluation of memory polynomial
models [16]. For memoryless models the MER is
MER =
|| e0 ||2
|| yval ||2
(3.11)
where e0 is the error vector obtained when using a memoryless polynomial model
[16]. MEMR is defined by
MEMRm = 1 −
|| em ||2
|| e0 ||2
(3.12)
with em being the error vector when using the nonlinear order of the error polynomial found from evaluating the MER and using m as memory length. The idea is
that a value close to 0 indicates that most of the memory effects have been captured
whereas a value close to 1 indicates pronounced unmodeled memory effects.
However, as pointed out in Paper A there is no difference in performance
between the NMSE and the MER-MEMR criteria. Applying Parseval’s formula
yields
√
(3.13)
MER = NMSE.
That is, the MER is the square-root of the NMSE.
In a similar way the MEMR is rewritten as
NMSEm
MEMRm = 1 −
NMSE0
(3.14)
with NMSEm denoting the achieved NMSE for a memory polynomial model using
a memory length of m taps. Thus it is seen that the MEMR is another form of normalization and scaling of the NMSE. Other variations using different normalization
constants of the total model error power are possible.
In [6] the ACEPR was defined as
Φe (f ) df
adj ch
(3.15)
ACEPR = max ∈{1,2}
Φyval (f ) df
ch
28
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
where, in the denominator, the integration is over the in-band channel. Further, the
’1’ and ’2’ denote the lower and upper adjacent channels, respectively. Accordingly,
ACEPR is the ratio of the out-of-band error power in the ’worse’ adjacent channel
to the in-band validation signal power.
In a more general form (3.15) can be rewritten as
W (f ) Φe (f ) df
full band
(3.16)
V (f ) Φyval (f ) df
full band
where W (f ) and V (f ) are some appropriate frequency weighting functions. For the
ACEPR in (3.15)
1 f ∈ in-band channel
(3.17)
V (f ) =
0 otherwise
and
W (f ) =
1 f ∈ worse adjacent channel
0 otherwise
(3.18)
For some general weighting functions W (n) and V (n), (3.16) is known as weighted
error spectral power ratio (WESPR), as introduced in [89]. In this work, due to a
dominating in-band power, the WESPR is defined as
W (f ) Φe (f ) df
full
band
WESPR =
(3.19)
Φyval (f ) df
full band
where W (f ) is a non-negative weighting function. In practice, the integration in the
numerator of (3.19) is performed over the frequencies of interest. A rule-of-thumb
is that the integration is performed over the frequency regions where the spectral
regrowth is more pronounced than the noise floor; that is, the spectral region where
some signal can be found [89].
In WESPR the weighting function W (f ) needs to be defined to give the criterion
the appropriate properties. The soft weighting function
W (f ) =
max( Φe (f ) )
max( Φe (f ) ) + | Φu (f ) |
(3.20)
with Φe (f ) denoting the estimated power spectral density of the model error signal
and Φu (f ) denoting the estimated power spectral density of the measured input
signal.
This function will put more emphasis on the in-band error if the largest model
error in frequency domain, max( Φe (f ) ) is large and more on the out-of-band effects
if this maximum error is small. The disadvantage is that this error needs to be
3.5. SYSTEM IDENTIFICATION
29
known if the weighting function is to be used in the parameter extraction process.
This will shortly be described.
A weighting function of special interest is
W (f ) =
1
.
C + Φu (f )
(3.21)
where C controls the relative influence of the in- and out-of-band errors on the final
criterion. Hence, by proper choice of the constant C a desired trade of between
in-band and out-of-band modeling capabilities can be achieved. In one extreme
point the WESPR can be transformed to the ACEPR by taking a large C and limit
the area of error integration to the adjacent channel with maximum model error
power. The other extreme point is to consider the full available frequency band
as in the NMSE. The WESPR is thus a general description in where ACEPR and
NMSE are found as special cases.
3.5
System identification
Now that a criterion for performance evaluation, the WESPR with the proposed
weighting function in (3.21), has been established the question is how to find the
model structure, order and parameters to minimize the chosen error criterion.
Traditionally the model structure and model order for PA behavioral models
have been chosen based on graphs displaying the desired performance evaluation
criteria, such as those in Fig. 3.1. When the performance of the models is not
sufficiently increased in comparison to the increasing complexity of the model by
the increase in the number of parameters, it is considered that the model is "good
enough". Unfortunately, no general criteria for quantifying "good enough" or "suitably small" are available. Suggestions of model orders based on information theory
are available in the form of information criteria [2, 3] but have not been extensively
used in PA modeling. The method used here is more based on rule-of-thumb and
basically says that when the performance is increased by less than 0.5-1.0 dB for
an increase in the model order complexity of one unit, be it memory length or nonlinear order in some sense, it is considered that the model is "good enough". This
is an application of the parsimony principle [2, 3].
So far not much has been mentioned on the extraction of the model parameters,
only that normally a minimization of the sum-square-error in (3.2) is considered to
give the correct model parameters. In Paper A , Paper B and [1] it is argued
and shown that the NMSE, i.e. a normalized version of the sum-square-error,
is not the most suitable model performance evaluation criterion when considering
behavioral models of mildly nonlinear PAs. Still the model parameters are extracted
by minimization of (3.2).
A method for improving the performance, measured as WESPR, of all behavioral models would be to modify the parameter extraction criterion, also known as
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
30
−50
ACEPR (dB)
−52
−54
−56
PH(5,⋅)
PH(7,⋅)
PH(9,⋅)
PH(11,⋅)
PH(13,⋅)
PH(15,⋅)
−58
−60
−62
0
1
2
3
Memory length M
4
5
Fig. 3.1: Typical example of a graph used for selecting the model order based on
the model performance, in this case ACEPR.
the loss function, by consider a frequency weighted error
CF (θ) =
N
−1
n=0
2
|eF (n)| =
N
−1
|yF (n) − fF (u(n); θ)|
2
(3.22)
n=0
where eF (n) is the frequency weighted model error, yF (n) denotes the frequency
weighted output signal from the PA and fF (u(n); θ) is the frequency weighted
model output. eF (n) can be obtained by filtering the model error with a suitable
stable, linear filter.
If the goal is to minimize the WESPR with a given window function W (f ) the
filter w(n) should be chosen as the impulse response with the frequency characteristics given by W (f ). Taking the discretized version of the WESPR, i.e. replacing
the integration by a summation, and using Parseval’s formula for the denominator,
gives
CF (θ)
(3.23)
WESPR = N −1
N Fs
Φy (k)
k=0
where Φy (k) denotes the estimated frequency-discrete power spectral density of
the measured output signal y(n). When minimizing (3.22) it is seen that also the
WESPR is minimized for a given input-output signal pair {u(n), y(n)}.
The principles described above are most easily illustrated by considering a problem that is linear in the parameters, i.e. a problem that can be written as
y(n) = φ(u(n))T θ + v(n)
(3.24)
3.6. EXPERIMENTAL VERIFICATION
31
with φ(u(n))T being the regression vector [2, 3], θ being the parameter vector and
v(n) a term containing the noise and remaining model errors. If the least squares
solution in (3.2) is sought for the problem in (3.24) then (3.2) can be rewritten as
C(θ) = H(u)θ − y
(3.25)
where H(uu) is the regression matrix with every row corresponding to the φ(u(n))T
at a given time instant n, u and y being the measured input respectively output
signal.
The optimum θ in (3.25) is given by [2, 3]
−1 H
H y.
θ = HH H
(3.26)
Applying a linear filter to data gathered in a vector is equivalent to multiplying
said vector with a Toeplitz matrix. Using this method frequency weighted model
parameters can be identified using the weighted linear least-squares solution
−1 H H
H W Wy
(3.27)
θ = HH WH WH
with W denoting the frequency weighting matrix.
3.6
Experimental verification
The power spectra of a typical input-output signal pair when testing a PA can look
like what is found in Fig. 3.2. Considering the huge difference between the power
in-band and out-of-band of the output signal it is easy to see how a relatively small
model error in-band will contribute much more to the total error in (3.2) than large
model errors out-of-band.
The data in Fig. 3.2 is measured on the 3G LDMOS class AB PA described in a
later chapter, using a signal which is typical for single channel 3G communication.
Input powers were chosen to mimic typical working conditions.
In Paper A the performance of different evaluation criteria were evaluated, and
in Paper B the performance of models extracted using the frequency weighted least
squares criterion in (3.22) were compared using the previously found performance
evaluation criteria.
The results of the comparison between different performance evaluation criteria
in Paper A will be shown here by considering the graphs of the MEMR, NMSE,
ACEPR and WESPR using the soft window in (3.20).
Figs. 3.3 and 3.4 show the NMSE respectively the MEMR when evaluating
the behavioral modeling performance of the parallel Hammerstein model [41] for
varying memory lengths and nonlinear orders. As shown earlier, the two measures
are expected to have similar performance and that is also verified by these graphs.
Considering Fig. 3.5 it is seen that both these criteria require a higher nonlinear
order, 13 compared to 9, and a longer memory, 3 compared to 2 taps, to decrease
"sufficiently slow".
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
32
0
Input u(n)
Output y(n)
Power [dBx]
−20
−40
−60
−80
−15
−10
−5
0
5
10
15
Frequency [MHz]
Fig. 3.2: The measured power spectral densities of a typical input and output signal
when testing a PA.
−39.6
Poly7
Poly9
Poly11
Poly13
Poly15
−39.8
NMSE [dB]
−40
−40.2
−40.4
−40.6
−40.8
−41
1
2
3
4
5
6
7
8
Memory depth
Fig. 3.3: The NMSE for varying nonlinear order and memory lengths of the parallel
Hammerstein model when applied to data measured on a PA. Note that the memory
length starts on 1.
3.6. EXPERIMENTAL VERIFICATION
33
−5.5
−6
MEMR [dB]
Poly9
Poly11
Poly13
Poly15
−6.5
−7
1
2
3
4
5
6
7
8
Memory depth
Fig. 3.4: The MEMR plotted in the same way as in Fig. 3.3. Note that the memory
length starts on 1.
−48
WESPR PH(9,⋅)
WESPR PH(11,⋅)
WESPR PH(13,⋅)
WESPR PH(15,⋅)
ACEPR PH(9,⋅)
ACEPR PH(11,⋅)
ACEPR PH(13,⋅)
ACEPR PH(15,⋅)
ACEPR and WESPR [dB]
−50
WESPR + 3dB
−52
−54
ACEPR
−56
−58
−60
−62
−64
1
2
3
4
5
6
7
8
Memory depth
Fig. 3.5: The ACEPR and the WESPR plotted in the same way as in Fig. 3.3.
The WESPR is offset by 3 dB to give a clearer view. Note that the memory length
starts on 1.
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
34
Case #
1
2
3
4
Model
f1 (·; θ1 )
f2 (·; θ2 )
f2 (·; θ̄2 )
f1 (·; θ̄1 )
NMSE [dB]
-39.2
-38.7
-32.3
-35.5
WESPR [dB]
-39.9
-39.8
-41.4
-40.4
# parameters
24
16
16
24
Table 3.1: Evaluation criteria dependence on the loss functions and model parameters.
In Paper B the difference between using an frequency neutral loss function
(3.2) in the parameter extraction and a frequency weighted function (3.22) were
investigated. The PA under test was the same as in the previous cases but the
input signal was changed to a double-carrier WCDMA signal with channel center
spacing of 15 MHz.
It was shown that using the wrong cost function for parameter extraction can
result in an erroneous choice of model order, or possibly even of model structure.
In Table 3.1 the model results for two different parallel Hammerstein model orders
are compared. Cases #1 and #2 show the results when the model parameters, θ1
and θ2 , are extracted using the criterion in (3.2). It is seen that the first model
order, f1 (·; θ1 ), gives better performance when considering both the NMSE and
the WESPR than the second model order, f2 (·; θ2 ). This would label the model
f1 (·; θ1 )1 as the best model and the best parameter set.
As the main goal in this context is to get the best modeling performance in
terms of WESPR, is there something that can be done? As the model parameters
θi were extracted to optimize the NMSE it could be possible to change the cost
function. Consider Cases #3 and #4 where the loss function is given by (3.27) with
W (f ) given by (3.21). Case #3, for model f2 (·; θ2 ) which earlier was considered
to have inferior performance, suddenly has improved the WESPR by 1.6 dB by
changing the loss function to optimize the WESPR. Some performance has indeed
been lost in terms of NMSE but that loss is of less importance.
The next question is if it is possible to improve the performance of model order
1 in the same way? From Case #4 in the table it is seen that the performance
is improved but is not close to the performance of the second model structure.
Thus it is shown that it does not suffice to only evaluate the model performance
using a relevant criterion but it is also necessary to extract the parameters using a
relevant loss function. Failure to do so can result in the mistake of labeling a model
structure capable of giving a better description of out-of-band performance as an
inferior structure.
The data in Table 3.1 is extracted from measurements on the same 3G class AB
PA as was used earlier. The C in (3.21) was chosen to give an attenuation of 40
dB for the in-band model errors as compared to the out-of-band effects.
A comparison of the performance of parallel Hammerstein models extracted
3.7. STATISTICAL ANALYSIS
35
−34
NMSE [dB]
−35
−36
Method 1, PH(9,⋅)
Method 1, PH(11,⋅)
Method 1, PH(13,⋅)
Method 2, PH(9,⋅)
Method 2, PH(11,⋅)
Method 2, PH(13,⋅)
−37
−38
−39
−40
0
1
2
3
4
5
Memory length
Fig. 3.6: The NMSE dependence of the model parameter extraction criteria.
Method 1 is unweighted parameter extraction. Method 2 is frequency weighted
parameter extraction.
using the sum-square-error in (3.2) and the frequency weighted criterion in (3.22)
is shown in Figs. 3.6 for the NMSE and 3.7 for WESPR. These figures are for
the earlier mentioned double carrier WCDMA signal. It is seen that the NMSE
decreases as normal for the unweighted parameter extraction criterion whereas the
weighted varies considerably. The WESPR show increased performance when using
the frequency weighted criterion, as expected.
3.7
Statistical analysis
In the previous sections it is argued that for mildly nonlinear systems one should
consider the out-of-band effects more than the in-band effects for RF PA behavioral
modeling. An important question to answer is what happens to the model parameters due to the frequency weighting on a model capable of exactly describing the
PA under test.
The analysis in Paper B shows that the parameter variance increase due to
reduced SNR. Even if there are noises that are amplitude dependent, such as phase
noise, the SNR is higher in the channel where the original signal was transmitted.
As this frequency range is suppressed by the frequency weighting the total SNR is
reduced.
However, the major impact does not come from noise but from model errors.
Now assume that there is a perfect model of the PA under test, i.e. a model that
CHAPTER 3. PERFORMANCE EVALUATION CRITERIA
36
−33
Method 1, PH(9,⋅)
Method 1, PH(11,⋅)
Method 1, PH(13,⋅)
Method 2, PH(9,⋅)
Method 2, PH(11,⋅)
Method 2, PH(13,⋅)
WESPR [dB]
−34
−35
−36
−37
−38
−39
−40
0
1
2
3
4
5
Memory length
Fig. 3.7: The WESPR dependence of the model parameter extraction criteria.
Method 1 is unweighted parameter extraction. Method 2 is frequency weighted
parameter extraction.
is capable of describing the PA up to an additive noise term v(n):
y(n) = f (u(n); θ0 ) + v(n).
(3.28)
For simplicity, start by assuming a model that is linear in the parameters, i.e.,
a model that can be expressed as
y(n) = φ[u(n)]T θ0 + v(n)
(3.29)
where y(n) is the output, u(n) is the input, θ0 is the true parameter vector, v(n) is
additive zero-mean noise uncorrelated with u(n), and φ[u(n)]T is the linear regression vector for prediction of the output signal at one time instant. For the parallel
Hammerstein case, φ(u[n])T corresponds to one row of the H(u) matrix.
For a model linear in the parameters, it can be shown that the expected value
of the estimated parameter vector θ when applying the frequency weighting fulfills
= θ0 , i.e., the estimated parameters are unbiased, just as for the standard
E{θ}
linear least squares solution in (3.26).
In the above mentioned cases nothing has been said about noise in the input
data u(n). In the cases here the input signal u(n) is measured so it contains noise.
This results in a so called errors-in-variables problem [3]. The approach here is to
apply coherent averaging [90] to reduce the noise power and thus the bias.
For the more general case of arbitrary parameterizations, consider the normalization of the weighted cost function in (3.22), given the estimator output θN , which
3.7. STATISTICAL ANALYSIS
minimizes
VF (θ) =
N −1
1 2
|w(n) [y(n) − f (u(n); θ)]|
N n=0
37
(3.30)
where w(n) denotes the impulse response of the linear frequency weighting filter
with frequency function W (f ) as in (3.19), and denotes convolution. Under the
conditions of a wide sense stationary zero-mean noise v(n) and a sufficiently exciting
input signal u(n), it is possible to show that
VF (θN ) =
N −1
2
1 w(q −1 ) f (u(n); θ0 ) − f (u(n); θN ) N n=0
+
N −1
2
1 w(q −1 ) v(n)
N n=0
−
N −1
1 2 Re w(q −1 ) N n=0
f (u(n); θ0 ) − f (u(n); θN ) v(n) .
(3.31)
with q −1 denoting the unit delay operator q −1 u(n) = u(n − 1).
Furthermore, assuming that both the system output f (u(n); θ0 ) and estimated
model output f (u(n); θN ) are uncorrelated with v(n), then the last term in (3.31)
asymptotically goes to zero. Equation (3.31) is then simplified as
VF (θN ) =
N −1
2
1 w(q −1 ) f (u(n); θ0 ) − f (u(n); θN ) N n=0
+
N −1
2
1 w(q −1 ) v(n) .
N n=0
(3.32)
The first term in (3.32) is the frequency-weighted model error, and the second
term essentially corresponds to the variance of the frequency weighted noise. It is
seen that the minimizer of (3.32) is given by θN = θ0 if there is a unique parameter
vector θ that makes f (u(n); θ0 ) = f (u(n); θ) ∀ u(n). Because a large amount of
samples were assumed, the estimate is asymptotically unbiased. This is in line with
the theory given in [3].
In [3] it is shown that in cases where the noise model is to be identified the
parameters of the noise model should be identified using the cost function (3.2).
The estimation of the model parameters is then performed using the weighted loss
function (3.22).
Chapter 4
Measurement systems, principles
and performance
4.1
Measurement system
The measurement system used in most of the measurements is the modulation
domain system described in detail in [11]. Since earlier improvements to reduce
noise and increase bandwidth are implemented [11, 38, 90]. A review of the system
and the methods are given below. In Paper F an extension in the form of a
mechanical tuner is used to give the possibility of controlling the load impedance.
Modulation Domain System
The measurement system that has been used in most of the work is shown in
Fig. 4.1. The signal generation starts in a PC by making the low-pass equivalent
test signal in the form of IQ-data, see (2.5). This is uploaded to an arbitrary
waveform generator (AWG) inside a VSG, the R&S SMU200A. The IQ-signals
are then upconverted to RF using the superheterodyne principle [91], shown in a
simplified form in Fig. 4.1.
Two options are then offered. Either one can measure the output signal of the
VSG or the output signal of the DUT. This possibility is shown as a switch. The
reason for measuring both the input and the output signal with the same instrument
is to reduce the effects of nonideal behavior in the signal generator, the possible
driver PA and the VSA. In the case of Paper D a driver PA is needed to achieve
the necessary drive power level into the DUT.
The measurements are done by a VSA, the R&S FSQ26. The RF signal is downconverted to a low intermediate frequency (IF) and sampled by an ADC operating
at a sampling frequency of 81.6 MHz [92]. This arrangement avoids the common
problems of IQ-imbalance in the receiver architecture [93, 94].
The system has a dynamic range of around 80 dB ACPR when using
39
40
CHAPTER 4. MEASUREMENT SYSTEMS, PRINCIPLES AND
PERFORMANCE
X
DAC
LO1
IQ
PC
VSG
θ = 90◦
IQ
u(n)
y(n)
DAC
VSA
LO2
ADC
X
RF
X
DUT
Fig. 4.1: The measurement setup with simplified signal generation and analysis
showing the RF and the IQ parts. The digital baseband signal in the form of IQ-data
is loaded into the digital to analog converters (DACs) in the vector signal generator
(VSG). The baseband signal is then upconverted to the transmit frequency, in this
case 2.14 GHz. To facilitate measurement of both the input signal u and the output
signal y of the DUT a switch is used.
typical WCDMA signals and coherent averaging [90] to reduce noise.
A disadvantage with switching systems like the one presented here is
that there is no time reference when measuring input, u(n), and output,
y(n), signals. Thus a synchronization between these two are needed. This
is provided by a two-step method based on cross-correlation for a rough
sample-base adjustment, followed by a finer phase-based sub-sample delay
estimation [11, 38]. This method assumes that the input and output signals
are “close” to each other as it otherwise may fail. In fact in Paper E one
occasion when this method did not yield the expected results is shown.
The useful bandwidth is limited to around 28 MHz [90] but can be extended to nearly the full bandwidth of the used signal using the method
of frequency stitching [90]. The method is based on measuring the signal
at various center frequencies but keeping some frequency overlap between
4.1. MEASUREMENT SYSTEM
41
them. This overlap is then used to synchronize the signals in both frequencyand time-domain. The method is used in Paper B and Paper D to get
digital bandwidths up to 120 MHz.
The system is made for measurements of well-matched DUTs as there are
no possibilities of making simultaneous measurement of the incoming and
outgoing waves in the present form of the system. However, this is the case
for the devices that the system is used to measure. The system has also
been extended into a full 2-port VNA using directional couplers and a RF
switch [95–97] with capabilities of measuring incident and scattered waves.
Load-pull system
A load-pull system consisting of a mechanical tuner, the MT982 manufactured by Maury Microwave [98] is used to test a low power PA. The tuner
is capable of producing very high reflection coefficients, up to 0.95 in the
frequency ranges 1.8-2.8 and 4.9-6.0 GHz. Rated power handling capability
at maximum voltage standing wave ratio (VSWR) is 50 W input power.
In Paper F the tuner is used to subject a PA to varying reflection
coefficients on the output port. As this is of interest for wideband signals, the
variation of reflection and transmission coefficients over a frequency range
of 15 MHz is of interest. The method for determination of this variation is
explained in a later section.
The power amplifiers
As a major subject of the thesis has been to verify measurement systems
and modeling principles the device under test has been the same most of
the time. The PA is a commercially available LDMOS class AB PA with
rating of 53 dBm average output power. There are three main stages in
the PA, one high gain amplifier, one low power PA with maximum power
of approximately 10 W and the final stage with 200 W output power. The
gain is 52 dB, within 0.5 dB, over the frequency band 2100-2220 MHz.
The PA is manufactured by Ericsson AB and is intended for being used
in the third generation of mobile telecommunication systems. In commercial
operation the PA is supposed to be linearized by a feed-forward setup. This
stage is not in operation during the measurements presented in any of the
papers Paper A , Paper B , Paper C or Paper E . It has been verified
that this particular PA does not give rise to any significant nonlinear sourcepull effects [99] on the input port of the PA.
42
CHAPTER 4. MEASUREMENT SYSTEMS, PRINCIPLES AND
PERFORMANCE
In Paper D the DUT is a single transistor PA mounted in a test fixture,
the PTF210451 [100] manufactured by Infineon Technologies. Rated output
power is 47 dBm for P1dB and linear gain of 14 dB. Operating frequency
range is 2110-2170 MHz. To get sufficient input power to the PA a high
power driver amplifier is used.
For Paper F the Mini-Circuits ZVE-8G [101] wideband low power PA
is used. The bandwidth goes from 2 to 8 GHz with rated variations of less
than 2 dB within the full bandwidth. Output power at P1dB is 30 dBm.
In the present measurement case it is of interest to create a mismatch on
the output of the amplifier so the maximum allowable VSWR on the output
port is of interest to prevent amplifier failure. It is specified to be maximum
2.0, corresponding to a return loss of approximately 9.6 dB.
4.2
PA behavioral modeling based comparison of
measurement instruments
As the number of measurement systems that are capable of modulated measurements is relatively small it is of interest to compare the performance
of the MDS to an established measurement system, namely a large signal
network analyzer (LSNA). This is done in Paper C by considering the aim
of the MDS, to provide model parameter identification and validation data
for PA behavioral modeling.
The behavioral model evaluation criteria used in Paper C , the NMSE,
ACEPR and WESPR, are all strongly dependent not only on the model
performance, even if it would be desirable to have it that way, but also on
the noise and distortions levels in the measured input and output signals [6].
Thus the main impact on the final performance evaluation criterion comes
from the noise and distortions in the measured signals if the model structure
is of sufficiently high quality.
The above mentioned principle is used to evaluate the modeling performance of the LSNA MT4463A manufactured by Maury Microwave [102] in
Paper C . It is important to note that it is the performance of two measurement systems that are compared and not the principles which the respective
systems are based on.
The model choice is the parallel Hammerstein model and the PA is the
3G PA manufactured by Ericsson plus the power attenuator after it. As
this attenuator has an attenuation of 30 dB it provides good isolation from
possible influences the measurement system has on the PA on the load side.
The reason for choosing a well-known PA and a well-known model struc-
4.3. MODELING AND PREDISTORTION OF SINGLE STAGE PAS
43
ture with good performance is that it is the measurement systems that are
going to be tested and not the model or the PA. In other words, here it is the
PA and the model that constitute the “standard” used for the comparison.
As a comparison between the systems are done it is of importance not to
stay within one measurement system when doing the performance validation
but to also mix the identified model and the validation data between the
systems, i.e. to use data from one system to identify the model and data
from the other system to verify the performance.
The results are presented in detail in Paper C but in short it can be
said that both systems can extract parallel Hammerstein models of similar
performance whereas the validation capability is better for data measured
using the MDS. Because the MT4463A is a recognized measurement system
and the MDS has at least the same performance for this particular application it is concluded that the MDS is an adequate measurement system for
the purpose of measuring data for behavioral models of well matched PAs.
An extra investigation showed that modeling using a repetitive WCDMAlike signal give models with superior performance compared to the 62-tone
multisine signal used for the previous comparison.
4.3
Modeling and predistortion of single stage PAs
Paper D presents measurement and modeling results of a low-complexity
single stage PA, the Infineon 45W transistor mounted in a test fixture, described in 4.1. There is a growing interest in showing the predistortability of
lox-complexity devices, such as transistors in test fixtures. Especially characterization of memory effects are of interest as these makes predistortion
more complicated [17–19].
The PAs are tested by applying two-tone signals and double-carrier WCDMA
signals with varying tone spacings. The asymmetry between upper and lower
sidebands is normally an indication of the presence of memory effects [30].
The PA is measured using the mentioned signals and subsequently modeled
using the parallel Hammerstein model. It is found that a higher nonlinear
order is needed but only a limited amount of memory in the model. The
chosen model order is PH(13,1).
It is of interest to compare the performance of lox-complexity memoryless
predistorters to the performance of a more complicated model, such as a
PH(13, 1). Naturally it is found that the lox-complexity models do not
give the same performance as the more advanced models. It is noted that
the memoryless predistortion makes the asymmetries worse, indicating that
44
CHAPTER 4. MEASUREMENT SYSTEMS, PRINCIPLES AND
PERFORMANCE
the memory effects are more pronounced with predistortion. The plausible
explanation for this is the higher peak envelope powers that the predistorted
signals have. Predistorters with memory are much better at handling the
asymmetries and the PH(13, 1) predistortion reduces the asymmetry to a
much lower level.
4.4
Modeling, measurement principles, sampling theorems
and performance evaluation problems
The sampling theorem of Nyquist [12] is well known to everyone having taken
a basic course in signal theory. It states that for perfect reconstruction of
a sampled signal it is necessary to sample the signal at a rate higher than
twice the maximum frequency in the signal.
With some additional knowledge of the signal it is possible to perform
straightforward undersampling of signals, as is commonly done in many measurement instrument such as most VSAs [92]. It is also used in oscilloscopes
in different ways of which one commonly used method is the equivalent sampling method as described in [103, 104]. The LSNA MT4463A uses an ingenious sampler-based downmixing that essentially is a form a sophisticated
undersampling [105]. These are all more or less clever tricks that utilize
some knowledge about the spectra or the repetition time of the signals to
be measured.
There are other sampling theorems that uses other information than
knowledge about the signal. In [13] it is shown that if a bandlimited signal
is produced by a known mapping g(·) from a signal f (t) of possibly unlimited
bandwidth, then the signal f (t) can be restored by sampling the output of
the mapping g(f (t)) at the Nyquist rate. This was shown to also hold
for Volterra systems by [14] and is thus termed the Zhu-Frank generalized
sampling theorem.
Turning the whole theorem around it is shown that the same theorem
can be used for modeling [14,15]. If a system K(·) transforms a bandlimited
signal into a signal that is possibly unlimited in frequency, it is still enough
to sample the output signal at the Nyquist rate of the input signal [14, 15].
Experimental verifications are found in [14, 15, 106, 107].
The theorem is used in Paper E for baseband behavioral modeling of the
PA manufactured by Ericsson AB and described in Section 4.1. For the measurements a wideband downconverter together with an ADC is used. The
useful analog bandwidth of the system is approximately 550 MHz whereas
the highest sampling frequency of the ADC is 210 MHz. The measurement
4.4. MODELING, MEASUREMENT PRINCIPLES, SAMPLING THEOREMS
AND PERFORMANCE EVALUATION PROBLEMS
45
setup is described in detail in [108]. For the wideband signal generation a
baseband signal generator, the R&S AFQ100 [109], is used. The SMU200A
is used for modulation of the signals and upconversion to a center frequency
of 2160 MHz.
The used test signals are of WCDMA-type with three different bandwidths: 3.84, 50 and 96 MHz respectively. The power level of the signal
is approximately -7 dBm which is somewhat less than the power level normally used. This choice makes the output signal of the PA occupy around
500 MHz significantly above the noise floor for the 96 MHz input signal.
As suggested in [90] some parts of the nonidealities in the signal generation and measurement system are reduced by measuring both the input
and the output signal. The behavioral model chosen is the parallel Hammerstein model. As the only signal that is available is the undersampled
signal, where it is impossible to determine what comes from out-of-band
and in-band effects, only the NMSE can be used for evaluating the model
performance. In Paper A and Paper B it is argued that the NMSE is a
suboptimal criterion for model performance evaluation of weakly nonlinear
systems but as the error signal also is undersampled other frequency domain
based criteria cannot directly be used.
A new aspect in the behavioral modeling also entered with the high
bandwidths, that of synchronization. Earlier the cross-correlation based
approach in [38] has been used. Indeed for the low bandwidth of 3.84 MHz
this works satisfactorily.
However, when going to 50 and 96 MHz signal bandwidths the crosscorrelation based synchronization does not directly give the synchronization
that is desired. By first performing the synchronization as in [38] and then
shifting the input signal a number of samples to make it appear as if the
output signal is delayed compared to the input signal it is possible to increase
the model performance.
The found model orders are PH(9, 4), PH(9, 7) and PH(9, 9) for the 3.84,
50 and 96 MHz signals respectively. The shifts to delay the output signal
compared to the input signal is 0, 1 respectively 3 samples with performance
improvements of 0.4 dB in NMSE for the 50 MHz and 0.5 dB in NMSE for
the 96 MHz wide signal.
Due to the undersampling the signal and error spectra do not give much
useful information, as seen in Fig. 4.2. The figure shows the measured signal
and model error spectra of the PH(9, 9) model when using the 96 MHz wide
input signal and a sampling rate of 210 MHz in the ADC.
CHAPTER 4. MEASUREMENT SYSTEMS, PRINCIPLES AND
PERFORMANCE
46
0
Power [dBx]
−10
−20
−30
−40
Input u
Output y
Error PH(9,9)
−50
−60
−50
0
50
Frequency [MHz]
Fig. 4.2: The measured input, u, and output, y, spectra to respectively from the
PA when using the 96 MHz wide input signal. Used sampling rate in the ADC is
210 MHz.
As the only applicable evaluation criterion is the NMSE, and this has
been argued to be an inadequate criterion, a solution is needed. The developed criterion WESPR exclusively relays on having a full spectrum available
for the performance evaluation. This only seems to be possible by restoring
the original signal, sampled according to Nyquist, in some way.
Paper E suggested two solutions using the current measurement setup.
One is a multisine approach where it can be calculated where all the original
tones and IM-products end up when being in the sampling process.
Another method suggested in Paper E is a spectrum sweep by varying
the local oscillator and using the fact that the full 550 MHz bandwidth of
the measurement setup is available. This method is currently under investigation and results are expected to be published within short [110].
Neither of the suggested methods do solve the problem of using the
ZFGST for performance evaluation when considering baseband behavioral
modeling of RF PAs. They only focus on acquiring a full band signal to do
the performance evaluation on. No solution to the problem of out-of-band
performance verification of models identified using ZFGST has so far been
reported.
4.5. MODELING UNDER VARYING LOAD CONDITIONS
4.5
47
Modeling under varying load conditions
Paper F is a paper devoted to investigate the robustness of direct modeling performance and inverse modeling as digital predistortion performance
to variations in the output load. Only a few publications considering PA
behavioral modeling and digital predistortion in varying load impedances
are found [111–114]. In [111–113] the studies mainly consider AM/AM and
AM/PM characteristics whereas the main focus in [114] is the influence the
mismatch has on the designed Doherty amplifier.
Load-pull systems in combination with vector signal measurements have
been reported earlier [115] but up to the author’s knowledge it is the first
time a study of the performance of a more complicated baseband model, the
parallel Hammerstein model, as function of load impedance has been done.
When putting a PA into a system there is typically going to be some
mismatch on the output of the PA. In current generation basestations for
telecommunications this is typically on the order of 15 dB but in future
generations it is expected to reduce to about 10 dB [116].
The measurement system used for the testing is the MDS together with
a tuner from the load-pull system. The measurement setup is shown in Fig.
4.3. As the purpose is to vary only load impedance at the fundamental
frequency and not at the harmonics, a triplex filter is inserted between the
tuner and the DUT, as shown in Fig. 4.4.
The triplex filter introduces some variation of the reflected phase over
the bandwidth of interest by the delay τ . A variation in transmission gain
ΔG(f ) is also introduced by the combination of the tuner and the filter.
The phase variation does not cause any problems in the modeling process
provided the phase is “nearly” linear in the band of interest, approximately
2142-2153 MHz. This has been verified to be the case using a VNA. The
gain variations were found to be somewhat larger than desired, about 0.1
dB.
To quantify the impact the transmission gain and phase variations have
on the signals the input signal to the PA, u(n), is compared to the same signal when passing through the filter and tuner. The metric used to quantify
the distortion is a form of NMSE by considering the measured signal directly
from the signal generator as reference and the signal passing through the
tuner and filter as the test signal, i.e.
NMSE =
uref − u 22
.
uref 22
(4.1)
48
CHAPTER 4. MEASUREMENT SYSTEMS, PRINCIPLES AND
PERFORMANCE
DAC
X
LO1
IQ
PC
IQ
u(n)
y(n)
VSG
θ = 90◦
DAC
VSA
LO2
ADC
X
RF
X
DUT
Variable load impedance
Fig. 4.3: The measurement setup used in the load-pull measurements. The modification is the insertion of the variable load impedance on the output of the PA.
In (4.1) uref is the reference input signal and u is the signal passing through
the tuner both gathered into a vector. · 22 denotes the absolute-squaresum. Note that a gain and linear phase constant has been added to the
test signals u before the comparison. The NMSE thus gives a metric of the
deviation of gain flatness and linear phase, as these are compensated for in
other parts.
The experimental results show that the largest NMSE is around -46 dB.
As it turns out, the behavioral models have a NMSE on the order of -40 dB.
This means that the error of -46 dB is on the edge of what is acceptable
for the particular models used here. For wide signals the results would be
considerably worse as both the filter and the tuner show large deviations
from linear phase and constant gain in large bandwidths.
Detailed results are presented in the attached paper Paper F. Conclusions are that both direct and indirect model performance deteriorates by
varying the output load impedance of the PA. Larger magnitude of the reflec-
4.5. MODELING UNDER VARYING LOAD CONDITIONS
49
3f0 termination
Triplex
2f0 termination
DUT
Filter
f0 tuner
ΓSYS (f )
ΓT (f )
Δτ
ΔG(f )
Fig. 4.4: The DUT with variable baseband impedance. Due to the non-zero length
of the triplex filter some phase variation in the reflection coefficient over the frequency band of interest occurs. This is denoted by τ . The filter and the tuner
presents a variable transmission attenuation ΔG(f ) over the frequency band of
interest.
tion coefficient in general makes the performance worse but there are special
angles where the performance is the same as that of the well matched case.
This is in line with the observations in [114] although there it is referred to
the influence the reflections have on the peaking amplifier in the Doherty
configuration.
Furthermore, as the models show good direct modeling results in all cases
when the performance evaluation is performed with the data measured at
the same impedance as the data that was used to extract the model, it is
concluded that the choice of model structure is sufficiently good.
That some of the nonlinear behavior still is captured by the model is
also understood by the fact that even at the worst point the direct model
error is -37 dB measured as NMSE and -48 dB measured as ACEPR. The
compensation through predistortion gives 7 dB improvement in ACLR, still
an improvement but considerably worse than the achievable 20 dB improvement.
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Part II
Included papers
63
Paper A
Comparison of Evaluation Criteria for Power
Amplifier Behavioral Modeling
Per N. Landin, Magnus Isaksson and Peter Händel
Published in IEEE MTT-S International Microwave Symposium Digest 2008
©2008 IEEE
Comparison of Evaluation Criteria for Power Amplifier
Behavioral Modeling
Per Landin∗† , Magnus Isaksson∗ , and Peter Händel∗†
† Signal
∗ Center
for RF Measurement Technology, Gävle, SE-80176, Sweden
Processing Lab, Royal Institute of Technology, Stockholm, SE-10044, Sweden
Abstract— In this paper different evaluation criteria for
power amplifier behavioral modeling are studied and evaluated using measuremed data. The figure-of-merits are
calculated from complex-envelope data of a sampled power
amplifier intended for 3G. Both time- and frequency domain
methods are included in the study. It is found that a model
evaluation criterion should have ability to capture both the
linear and nonlinear distortion as well as the memory effects
in the power amplifier. The normalized mean square error
(NMSE) and the weighted error-to-signal power ratio (WESPR) are found to be the strongest candidates for capturing
the in-band and the out-of-band errors, respectively. Both are
also independent of power amplifier technology and stimuli
input.
Index Terms— Power amplifiers, nonlinear systems, radio
transmitters, memories, modeling.
I. I NTRODUCTION
In telecommunications, the base station power amplifier
(PA) is an important part. It is also nonlinear which gives
spectral regrowth. In the current 3G systems a certain
regrowth is handled by introducing guard bands. In future
systems, with increasing demands on efficient spectral
usage, these guard bands have to be decreased, which in
turn means that less spectral regrowth is tolerated. One
future technique of linearizing the PAs, i.e. decreasing
the regrowth, is baseband digital pre-distortion (DPD). In
order to use DPD a behavioral model, or black-box model,
to compensate for the distortions is needed.
The accuracy of the nonlinear behavior model is central and must be evaluated in some way so that strong
candidates for e.g. DPD can be found among a variety of
behavioral models. An attractive way of comparing two
models is to evaluate their performance on validation data
since it makes sense without probabilistic arguments and
assumptions about the true system. A major challenge is
to find an evaluation criteria that can be used for different
technologies, is independent of input stimuli and gives
results comparable with other models.
A search in the database IEEE Xplore was performed.
All scientific journal papers during the publishing period
2006 to 2007 matching all the keywords: power amplifier,
behavioral, and model, were investigated. The number of
papers found was 17 where 16 were applicable to the
TABLE I
VALIDATION METHODS USED IN INVESTIGATED PAPERS
Validation method
NMSE
AM/AM
AM/PM
Time domain sample
Comparison of spectra
IM3 product prediction
Comparison of ACPR
Error spectrum
ACEPR
No model validation
Citations
[3] [5] [11] [16]
[3] [10] [12] [14]
[10] [12] [14]
[16]
[1] [3] [7] [8] [10] [14] [15]
[12]
[1] [4] [5] [7] [10]
[5] [6] [16]
[3] [5] [8]
[2] [9]
below study [1]-[16]. These papers were (rather arbitrary)
selected among the large number of papers on the topic.
The authors are aware of the lack of completeness in
covering the area, and future work includes a thorough
literature search.
The procedures for the model validations were tracked
and the number of times each one of them is used are
tabulated in Table I. One paper can include more than one
validation method. In Table I we see on the one hand
a number of validation methods where the results are
shown as a figure, i.e. amplitude-to-amplitude distortion
(AM/AM), amplitude-to-phase distortion (AM/PM), time
domain sample, and error spectrum, and on the other hand
some figure-of-merits (FoMs) that can be calculated as a
number, i.e. normalized mean square error (NMSE), third
order intermodulation products (IM3), adjacent channel
power ratio (ACPR), and adjacent channel error power
ratio (ACEPR) [5]. The former category is about twice
as common as the latter.
All the methods have their pros and cons, the NMSE
is easy to calculate but will for all practical applications
only be affected by the error in-band while it is often the
error out-of-band that is of importance, see [5], [17]. The
authors therefore argue that the NMSE is an insufficient
FoM while other measures are better suited as evaluation
criteria.
In this paper a number of validation criteria for radio
frequency PA behavioral models are evaluated, showing
their pros and cons in a comparative way. The measures
that will be considered are the NMSE, the ACEPR, the
II. T HEORY
The used model structure is the parallel Hammerstein
(PH) model, also denoted the memory polynomial model
or the nonlinear tapped delay line, as described in [18]. The
model is widely used and acknowledged for its ability to
capture both linear and nonlinear memory effects with a
small number of parameters compared to other behavioral
models. Only odd-ordered polynomials are used for the
nonlinear part since only the first spectral zone is of
interest. Causal FIR-filters following each polynomial are
used to model memory effects. This model structure is
linear in the parameters and easily identifiable by the
method of least-squares [18].
For nonlinear models with memory the number of model
parameters becomes large even for short memories. Using
sparse delays in the filters similar to [19] will yield
the minimum NMSE for a given memory length. This
approach is tested using the same linear search as in [19].
Let uval (n) and yval (n) be the complex-envelope measured (sampled) validation input and output signals,
ymodel (n) the model output signal using uval (n) as input
and em (n) = yval (n) − ymodel (n) the model error when
using FIR-filters with m samples of memory to compute
ymodel (n), all in time domain. Let the corresponding uppercase letters denote the time-discrete Fourier transform of
said signal.
The NMSE is defined as
| E m (f ) |2 df
(1)
NMSE = BW
| Yval (f ) |2 df
BW
where BW is the normalized available bandwidth and f
is the normalized frequency.
Using the vector 2-norm · , MER is defined as [19]
N
0
| e0 (i) |2
e 2
MER =
= Ni=1
(2)
2
yval 2
i=1 | yval (i) |
−39.6
Poly7
Poly9
Poly11
Poly13
Poly15
−39.8
−40
NMSE [dB]
weighted error-to-signal power ratio (WESPR), the complementary memory measures, the memory effect ratio
(MER) and the memory effect modeling ratio (MEMR)
[18]. For illustration the power spectrum and the error
power spectrum are also included. All the criteria have
been evaluated using measured data.
Error vector magnitude (EVM) is also used [11] but
it is directly related to the NMSE. Accordingly, figures
for EVM will not be reported within this paper. IM3 is
another FoM but it is not defined for wideband signals
and does not directly consider memory effects. However,
a normalization by integrating the power difference in
upper and lower IM3 as in [17] gives a measure of
memory effects in the PA but is still not clearly defined
for wideband signals.
−40.2
−40.4
−40.6
−40.8
−41
1
2
3
4
5
6
7
8
Memory depth
Fig. 1. NMSE as function of nonlinear orders and memory length.
The NMSE without memory is considerably larger and is not plotted to
improve readability.
where the error is computed using the nonlinear order
that gives the smallest value and N is the number of
samples. Using the nonlinear order from MER, MEMR
with memory length m is [19]
MEMRm = 1 −
em 2
.
e0 2
(3)
where em is the error vector when m samples of memory
is used. ACEPR [20] is defined as
Wi+
| E m (f ) |2 df
W−
(4)
ACEPR = max i
i∈{1,2}
| Yval (f ) |2 df
ch
where 1 and 2 denote the lower and upper adjacent channel
and the integration limits are as in [20].
The WESPR [21] is defined through
| W (f ) E m (f ) |2 df
f ∈I
(5)
WESPR =
| Yval (f ) |2 df
f ∈J
where the integrations are performed over suitable frequency ranges using the weighting function W (f ). In [21]
two weighting functions W (f ) are suggested. Only the
soft thresholding window
W (f ) =
max( | E m (f ) | )
max( | E m (f ) | ) + | Uval (f ) |
(6)
is considered since tests have shown this window’s ability
to separate models with similar performance that the other
window could not.
III. E XPERIMENTAL
Measurements were made on a base station solid-state
power amplifier, manufactured by Ericsson AB, intended
for 3G. The gain is 52 dB and the input signal power
was -7.8 dBm at 2.14 GHz. The output power level was
TABLE II
MER AND MEMR PERFORMANCE
Nonlinear order
Possible MER [dB]
Memory length m
MEMRm [dB]
5
-37.43
1
-6.96
7
-37.92
2
-5.78
9
-38.04
3
-5.65
11
-38.07
4
-5.66
13
-38.08
5
-5.65
15
-38.08
6
-5.64
7
-5.61
8
-5.59
fs = 40.8 MHz, N = 10000
−5.5
−46
WESPR Poly9
WESPR Poly11
WESPR Poly13
WESPR Poly15
ACEPR Poly9
ACEPR Poly11
ACEPR Poly13
ACEPR Poly15
−48
ACEPR and WESPR [dB]
Non−sparse MEMR [dB]
Poly9
Poly11
Poly13
Poly15
−6
−6.5
−50
−52
−54
WESPR
−56
−58
−60
−62
ACEPR
−64
−7
1
2
3
4
5
6
7
8
Memory depth
−66
0
1
2
3
4
5
6
7
8
Memory depth
Fig. 2. MEMR in dB when considering different nonlinear orders and
memory lengths.
Fig. 3. WESPR and ACEPR for different nonlinear orders and memory
lengths.
comparable with normal operation conditions for the PA.
All the data were taken with the PA at thermal equilibrium.
The measurement set-up is described in [5]. During the
measurements the power amplifier showed considerable
memory effects and nonlinear behavior.
The number of samples in the identification data was
54400 and in the validation data 10880. Of these the last
N = 10000 samples were used for the crossvalidation.
The chosen IQ-sampling frequency was 40.8 MHz. The
sampling frequency was sufficiently large to analyze the
entire spectrum of the output signal aboce the noise floor
without aliasing. The bandwidth used to compute the
NMSE was limited to 26 MHz as suggested in [21]. Both
identification and validation signals were 3.84 MHz wide
WCDMA-like signals with peak-to-average ratios between
7 − 8 dB.
as NMSE. Further improvements are neglible.
In Fig. 3 the ACEPR and WESPR using the soft
thresholding window (6) are shown. The difference when
using nonlinear order 11 and 13 with memory length
exceeding 2, is approximately 0.5 dB in WESPR and is
not neglible. Nonlinear orders higher than 13 yield no
substantial difference compared to order 13. ACEPR and
WESPR show no substantial improvement when memory
lengths exceeding 4 are used. ACEPR has slightly larger
differences between model orders than WESPR.
Let PH(P, M ) denote the parallel Hammerstein model
using nonlinear order P and memory length M . The
spectra of the measured input and output signals, model
output PH(13,4) and PH(9,2) and the crossvalidation errors
when using PH(13,4), PH(13,0) and PH(9,2) models are
shown in Fig. 4.
IV. R ESULTS
The results of the FoMs are presented in Figs. 1-3 and
Table II. NMSE shows no further decrease when higher
model order than nonlinear order 9 and memory length 2
or 3 is used, depending on what is considered to be an
improvement.
The sparse filter structures have been tested using the
150 first delays and the results were sequentially ordered
up to at least 5 delays for this particular PA.
MER attains the minimum value for nonlinear order 9
in Table II. MEMR if hence found to give the same result
V. C ONCLUSIONS AND DISCUSSION
The reported study covering some of the most recent
works on PA behavioral modeling showed that model
performance evaluation is done in many different ways depending upon what is most important: in-band distortions
or spectral regrowth/out-of-band distortions. Generally, the
difference in power between these two is so prominent that
the out-of-band errors’ contribution to the total error is
neglible. In this study the spectral regrowth is considered
to be of highest importance as long as the in-band distor-
0
Meas input
Meas output
PH(13,4)
PH(9,2)
Error PH(13,4)
Error PH(13,0)
Error PH(9,2)
−10
Power [dB/Hz]
−20
−30
−40
−50
−60
−70
−80
−90
−15
−10
−5
0
5
10
15
Frequency [MHz]
Fig. 4. Spectra of the measured input and output signals, simulated
signals using PH(13,4) and PH(9,2), the crossvalidation error when using
PH(13,4), PH(13,0) and PH(9,2) models. The high order model PH(13,4)
is close to the measured output signal. The in-band performance of all
models are more or less the same, as seen from the model error. Outof-band performance varies more, with the highest model order showing
the best performance. The importance of memory effects is illustrated by
the performance difference between PH(13,0) and PH(9,2).
tion is sufficiently small as not to cause severe performance
degradation.
The most commonly used numerical FoMs for PA
behavioral modeling were compared. The NMSE and the
combination of MER and MEMR shows similar performance and chooses PH model orders as PH(11, 2-3). This
is explained by the similarity between MER and MEMR;
they are both scaled NMSEs.
ACEPR and WESPR, which both are weigthed to be
more influenced by the out-of-band errors, indicates higher
PH-model order, approx. PH(13,3-4). Advantages of the
WESPR over the ACEPR are its generality in stimuli
signals and its ability to capture effects both in- and outof-band. The performance difference between the models
is illustrated in Fig. 4. The out-of-band errors are clearly
smaller when using the higher order models according to
the weighted FoMs ACEPR and WESPR.
ACKNOWLEDGMENT
This work was supported by Ericsson AB, Freescale
Semiconductor Nordic AB, Infineon Technologies Nordic
AB, Knowledge Foundation, NOTE AB, Rohde&Schwarz
AB and Syntronic AB.
R EFERENCES
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Paper B
Parameter Extraction and Performance
Evaluation Method for increased performance
in RF Power Amplifier Behavioral Modeling
Per N. Landin, M. Isaksson and Peter Händel
Accepted for publication in International Journal of RF and Microwave
Computer-Aided Engineering
©2009 Wiley Blackwell
The layout has been revised
Parameter Extraction and Performance Evaluation Method for
Increased Performance in RF Power Amplifier Behavioral Modeling
Per N. Landin, Magnus Isaksson, and Peter Händel*
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The authors are with the University of Gävle, Center for
RF Measurement Technology, Gävle, SE-801 76 Gävle, Sweden.
P.N. Landin and P. Händel are also with the Royal Institute
of Technology, Signal Processing Lab, SE-100 44 Stockholm,
Sweden. Corresponding author: [email protected].
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Power Amplifier Behavioral Modeling
Performance Comparison of the LSNA and the
Modulation-Domain System
Per N. Landin, Christian Fager, Magnus Isaksson and Kristoffer Andersson
Published in 72nd ARFTG Microwave Measurement Symposium Digest,
2008
©2008 IEEE
Power Amplifier Behavioral Modeling Performance Comparison of the LSNA
and the Modulation-Domain System
Per Niklas Landin1,2, Christian Fager3, Magnus Isaksson1, and
Kristoffer Andersson3
1
Center for RF Measurement Technology, University of Gävle, SE-80176 Gävle, Sweden.
Signal Processing Lab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
3
Microwave Electronics Laboratory, Chalmers University of Technology, SE-41296 Göteborg,
Sweden
email: [email protected].
2
Abstract
The performance of power amplifier behavioral models depends strongly on the performance of the
system used to measure the amplifiers. In this study two different systems with nonlinear
measurement capability are used to model a commercially available PA. One system is a large-signal
network analyzer (LSNA) and the second system is a modulation-domain system (MDS) consisting
of a vector signal generator and a vector signal analyzer.
The PA was tested with multitone and WCDMA signals and behavioral models were extracted
from the measured data. The evaluation criteria normalized mean square error and weighted error
spectral power ratio or adjacent channel error power ratio were then computed to compare the
performance of the models from the two systems. Cross-validation between the systems, using data
from one system to obtain the model and validating its performance with data from the other system,
shows that the model performance is mainly affected by the used validation data. Validating the
performance of models from the LSNA with data from the MDS indicates that the identified models
have almost the same performance as the MDS-identified models, i.e. it does not matter which
system is used to identify the models. Cross-validation using a WCDMA-signal and multitone signal
from the different systems shows that the normalized mean square error is mainly affected by
modeling imperfections introduced by using another signal type. WESPR and ACEPR show a
certain difference in performance with somewhat lower values for the MDS. The behavior of the two
systems can be explained by different noise levels.
I. Introduction
In the present study two measurement setups
for measuring non-linear radio frequency
components and systems are compared when
considering behavioral modeling. The two
systems are the Maury Microwave MT4463A
large-signal network analyzer (LSNA) and the
modulation-domain
measurement
system
(MDS) consisting of a vector signal generator,
the R&S SMU200A, and a vector signal
analyzer, the R&S FSQ26.
The same device under test, a radio frequency
power amplifier (PA) intended for the 3rd
generation of mobile communications, has been
measured separately with the two systems. The
studied PA is well documented in previous
studies [1], [2]. The measured input and output
signals from the power amplifier with both the
MDS and the LSNA systems were compared.
The measured data was used to identify
behavioral models and the modeling results
were compared using accepted criteria for
behavioral modeling performance evaluation.
To the authors' knowledge no study of the
above character has previously been reported.
However, a study comparing a LSNA, a
microwave transition analyzer and a digital
sampling oscilloscope was done in [3]. This
study focused mainly on identifying error
sources due to possible mismatches in the
measurement systems.
II. Prerequisites
When comparing the two measurement
systems from a behavioral modeling perspective
the model is a critical choice. Note that it is the
systems that are under evaluation, not the
model. For the present study a well defined,
well documented model with excellent modeling
properties for the specific device under test, i.e.
the radio frequency PA, has been used: the
parallel Hammerstein (or the memory
polynomial) model [1], [2], [4]-[6]. The model
is described by the nonlinear order P and the
memory length M. Such a model is henceforth
denoted PH(P,M). Previously published reports
[1], [2] have shown that the modeling error both
in and out-of-band for the particular device
under test will not significantly decrease after a
nonlinear model order of 9 and a memory length
of 2 sample delays. That model order
complexity, i.e. PH(9,2), is therefore used in the
present study.
The test signals are two sets of 62-tone signals
and one WCDMA-signal with a peak-to-average
ratio of 6.5 dB. The 62-tone signals were
generated to have an amplitude probability
density function similar to the WCDMA-signal
by optimizing the amplitudes and phases with
the algorithm in [7]. The resulting 62-tone
identification signal was used in both systems to
obtain models and the 62-tone validation signal
was used to validate the performance. The
WCDMA-signal was also used in the MDS to
give a comparison of the modeling performance
when using different signal types.
The 62-tone modeling performance of the
systems is evaluated by using the two PA
behavioral modeling criteria normalized mean
square error (NMSE) and weighted error
spectral power ratio (WESPR) [8]. NMSE is
mainly affected by in-band model errors and
WESPR is designed to take account of the outof-band model errors. The weighting function in
WESPR is chosen to be the maximum of the
lower and upper adjacent channels to give a
criterion similar to the adjacent channel error
power ratio (ACEPR) but only the discrete
frequencies where IM-products can be found are
included. For the comparison when the
WCDMA-signal is used as input and output to
the model, the adjacent channel error power
ratio (ACEPR) and the NMSE are considered.
To compare the performance of the systems
for this particular purpose an identification
signal measured in one system is used to
identify a model. The performance of the model
is then evaluated using the validation signal
measured in the other system. The difference
between modeled and measured output, the
model error, is then computed and evaluated
using NMSE, WESPR and/or ACEPR. If one
system has an advantage in dynamic range this
will be shown by consistently having better
III. Experimental
The setups are similar in both cases. The
LSNA is described in detail in [9] and the MDS
in [1] and the references therein. To make the
comparison as fair as possible the considered
device under test included both the PA and
attenuator as shown in Fig. 1.
PA
Attenuator
Fig. 1. The PA setup without the measurement systems.
The reference planes are chosen at the input of the PA and
the output of the attenuator.
modeling process. Fig. 2 shows that the weakest
detectable signals are in the range -60 to -70
dBc. The MDS show lower measured powers
below and above the signal band. This could
indicate that the MDS is capable of
measurements with higher dynamic range than
the LSNA.
The output signals show the spectral regrowth
of the amplifier. The measured spectrum differs
somewhat between the two systems.
0
Normalized
power [dB]
performance when using that dataset for
validation.
Input MDS
Input LSNA
-20
-40
-60
-80
-10
IV. Results
Figs. 2 and 3 show the measured input and
output signals, respectively, for an input power
level of -9.2 dBm. The power is normalized to
make the signals from both systems have the
same power in-band.
For the input signal the measured differences
in the sidebands are large. The difference in
input signal is partly taken care of in the
0
5
10
Normalized
power [dB]
Frequency [MHz]
Input MDS
Input LSNA
-60
-70
-80
-90
-10
-9
-8
-7
-6
-5
-4
-3
Frequency [MHz]
Fig. 2. The measured input signal spectrum using both
systems at the power level -9.2 dBm. Blue line shows data
from the MDS and red stars show data from the LSNA.
The upper part shows data for the measured frequency
range of the LSNA and the lower shows the spectrum at
frequencies below the channel.
Normalized
power [dB]
0
Output MDS
Output LSNA
-20
-40
-60
-80
-10
-5
0
5
10
Frequency [MHz]
Normalized
power [dB]
To get the highest accuracy in the modeling of
the PA it is necessary to measure both the input
and the output signals. Using measured input
and output signals in the modeling process
reduces some of the distortions introduced by
the measurement setup.
The device under test (DUT) is a commercial
PA from Ericsson AB. It is a LDMOS-PA with
a gain of 52 dB and rated maximum input power
of 1 dBm. The used center frequency is 2.14
GHz. In earlier studies [1], [2] the PA has
shown significant memory effects when used at
high power levels. The power sweep covering
33 different power levels, from 1 W to 38 W
were chosen to keep the PA fairly close to
normal working conditions. Using higher power
levels will result in stronger nonlinearities but
also larger modeling errors. This is undesired
since this work focuses on comparing
performance of the measurement systems and
not the model itself.
-5
-40
Output MDS
Output LSNA
-50
-60
-70
-80
-10
-9
-8
-7
-6
-5
-4
-3
Frequency [MHz]
Fig. 3. Spectrum of the measured output signal in the
respective system. Upper and lower parts of the figure are
as in Fig. 2.
As has been shown earlier [1] the quality of
the validation data has the major impact on the
evaluation criteria. Fig. 4 shows the NMSE and
-40
WESPR
-50
NMSE
NMSE id-LSNA val-MDS PH(9,2)
NMSE id-MDS val-LSNA PH(9,2)
W ESPR id-LSNA val-MDS PH(9,2)
W ESPR id-MDS val-LSNA PH(9,2)
-40
-45
NMSE
WESPR
-50
-55
-60
-22
-20
-18
-16
-14
-12
-10
-8
Input power [dBm]
Fig. 5. The cross-system validation performance for
NMSE and WESPR is shown. "Id-LSNA val-MDS"
means that the model is identified using identification data
from the LSNA and the performance is validated using
the validation data form the MDS.
-55
-60
-30
-65
NMSE LSNA PH(9,2)
NMSE MDS PH(9,2)
W ESPR LSNA PH(9,2)
W ESPR MDS PH(9,2)
-70
-22
-20
-18
-16
-14
-12
-10
-8
Input power [dBm]
Fig. 4. The NMSE and the WESPR when using data
measured within the system. Lower power levels show
larger difference than higher powers. This is partly due to
the fact that the model cannot completely describe the
behavior of the PA.
NMSE and ACEPR [dB]
NMSE and WESPR [dB]
-45
models using data from the MDS and validating
with data from the LSNA gives the same
performance as when both identifying and
validating using data from the LSNA-system.
This behavior shows that the LSNA finds
approximately the same model parameters as the
MDS but is not quite able to validate the
performance of the model at the lower power
levels.
NMSE and WESPR [dB]
the WESPR when evaluated on data within the
respective system, i.e. 62-tone identification and
validation data are both measured in the same
system. PH(9,2)-models for each power level
are extracted for each system and then evaluated
using the validation data from the same system.
The data from the MDS has lower NMSE and
WESPR. The difference between overall model
performance decreases as the power increases
which mainly is explained by the model’s
inability to completely predict the PA behavior.
WESPR, which is an out-of-band modeling
error criterion, shows a behavior similar to
NMSE but with larger difference at the lower
power levels. The behavior at the lower power
levels is a clear indication of noise or distortions
in the measured data out-of-band. The power
levels out-of-band are naturally much lower
than the in-band and are thus more easily
affected by noise.
-35
NMSE
-40
NMSE id-LSNA val-W CDMA PH(9,2)
NMSE id-MDS val-W CDMA PH(9,2)
ACEPR id-LSNA val-W CDMA PH(9,2)
ACEPR id-MDS val-W CDMA PH(9,2)
-45
ACEPR
-50
-55
-60
-22
-20
-18
-16
-14
-12
-10
-8
Input power [dBm]
In Fig. 5 the NMSE and the WESPR are
shown when cross-validating the data between
the systems, i.e. the identification data comes
from one system and the validation data comes
from the other system. Fig. 5 shows that when
identifying with data from the LSNA and
validating using data from the MDS, the NMSE
and the WESPR are almost as good as when
staying within the MDS. However, identifying
Fig. 6. The NMSE and ACEPR when identifying using
62-tone data from both systems and validating the
performance using the WCDMA-signal measured in the
MDS. "id-MDS val-WCDMA" means that the model is
identified using the 62-tone and the performance is
validated using the WCDMA-signal.
Results based on different signal types are
shown in Fig. 6 where the models are identified
using the 62-tone signals from respective system
and validated using the WCDMA-signal
measured in the MDS. For NMSE the difference
in performance between the systems is
negligible whereas the ACEPR is 3-4 dB lower
when using the MDS. The negligible differences
in NMSE between the systems appear since the
models are not identified for the parts of the
spectra that fall between the discrete frequencies
defined by the 62-tone. A model error much
larger than the errors caused by the noise is thus
introduced. For the out-of-band effects,
measured as ACEPR, the difference between the
systems is significant. The out-of-band signal
power is lower than in-band and the ACEPR is
thus more easily affected by the noise.
-30
NMSE and WESPR [dB]
-35
-40
-45
WESPR
NMSE
the noise and distortion levels in the respective
systems.
In general, it is seen that the LSNA-system
finds models that give approximately the same
performance as the models found using data
from the MDS, both in-band, measured as
NMSE, and out-of-band, measured as WESPR
or ACEPR. It has earlier been shown [1] that the
main impact on the evaluation criteria is given
by the used validation data. The behavior is also
clear when considering that an approximate
model, like the PH-model, does not consider
every aspect of the behavior but only the major.
This property provides certain robustness
against noise when identifying the models.
When computing the model error all the noise in
the measured data is directly present and can
directly affect the final result.
Other model orders, both with and without
memory, have been tested with the same results.
One problem that arose in certain cases with
more complex models and the less-noisy data
from the MDS was over-modeling.
-50
-55
-60
NMSE id-W CDMA val-LSNA PH(9,2)
NMSE id-W CDMA val-MDS PH(9,2)
W ESPR id-W CDMA val-LSNA PH(9,2)
W ESPR id-W CDMA val-MDS PH(9,2)
-65
-70
-22
-20
-18
-16
-14
-12
-10
-8
Input power [dBm]
Fig. 7. NMSE and WESPR are shown when using the
WCDMA-data for model identification and then
validating the performance using the 62-tone signals from
the respective system.
The results when identifying the model using
the MDS-based WCDMA-signal and validating
the performance with the 62-tone signal for the
respective system are shown in Fig. 7. The
NMSE is almost the same for both systems, but
compared to the NMSE in Fig. 4 it is much
worse, at least 10 dB, for the MDS-data and
approximately 5 dB for the LSNA. For the
WESPR the difference between the systems is
large. When comparing the difference between
the WESPR in Fig. 4 and Fig. 7 there is almost
no difference. One possible explanation is again
V. Conclusions
The performance of two measurement systems
with nonlinear measurement capabilities has
been compared using PA behavioral modeling.
The performance is compared through the
accepted PA behavioral modeling criteria
NMSE, WESPR and ACEPR. The systems are a
LSNA and a MDS.
For 62-tone signals it was found that the
LSNA-setup gives behavioral models with
almost the same performance as the MDS both
in-band, measured as NMSE, and out-of-band,
measured as WESPR or ACEPR. However, the
tested LSNA could not validate the performance
of the models.
At the highest power levels the model
performance was the limiting factor. The
difference in performance between the systems
would have persisted also at higher powers if
another model structure better suited to describe
the PA had been used.
The results based on using different signal
types for identification and validation show that
it is not straightforward to use the models if the
best performance is desired. The differences
between the systems in-band, i.e. the NMSE,
were negligible and depended mainly on the
signal type. Out-of-band differences between
the systems, WESPR or ACEPR, were more
pronounced with lower values for the MDS.
It is important to remember that these results
are obtained under the assumption of well
matched systems. Measurements with the LSNA
indicate that this is the case for this particular
PA. For other PAs, this may not be the case and
the MDS system may then produce erroneous
results.
Acknowledgements
This work was supported by Ericsson AB,
Freescale Semiconductor Nordic AB, Infineon
Technologies
Nordic
AB,
Knowledge
Foundation, NOTE AB, Rohde&Schwarz AB
and Syntronic AB.
References
[1] M. Isaksson, D. Wisell, and D. Rönnow,
"A comparative analysis of behavioral
models for RF power amplifiers," IEEE
Trans. Microwave Theory Tech., vol. 54,
pp. 348-359, 2007.
[2] M. Isaksson and D. Wisell, "Extension of
the Hammerstein Model for Power
Amplifier Applications," in 63rd ARFTG
Conf. Dig., Fort Worth, TX, USA, 2004,
pp. 131-137.
[3] T. Williams, O. Mojón, S. Woodington, J.
Lees, M. F. Barciela, J. Benedict, and P. J.
Tasker, "A robust approach for
comparison and validation of large signal
measurement systems," in IEEE MTT-S
Int. Microwave Symposium Digest,
Atlanta, GA, 2008, pp. 257- 260.
[4] M. S. Heutmaker, E. Wu, and J. R. Welch,
"Envelope distortion models with memory
improve the prediction of spectral
regrowth for some RF Amplifiers," in
ARFTG 48, Clearwater, FL, USA, 1996.
[5] J. Kim and K. Konstantinou, "Digital
predistortion of wideband signals based on
power amplifier model with memory,"
Electron. Lett., vol. 37, Nov. 2001, pp.
1417-1418.
[6] H. Ku and J. S. Kenney, "Behavioral
modeling of RF power amplifiers
considering IMD and spectral regrowth
asymmetries," in IEEE MTT-S Int.
Microwave Symposium Dig., 2003, pp.
799-802.
[7] J. C. Pedro, and N. B. Carvalho,
"Designing multisine excitations for
nonlinear model testing," IEEE Trans.
Microwave Theory Tech., vol. 53, pp. 4554, 2005.
[8] D. Wisell, M. Isaksson, and N. Keskitalo,
"A general evaluation criteria for
behavioral PA models, " in 69th ARFTG
Conf. Dig., Honolulu, HI, 2007, pp. 251255.
[9] J. Verspecht, "Large signal Network
Analysis," IEEE Microwave Magazine,
vol. 6, no. 4, pp. 82-93, 2005.
Paper D
A Study of Memory Effects of RF Power
LDMOS before and after Digital Predistortion
Per N. Landin, Magnus Isaksson, Niclas Keskitalo and Olof Tornblad
Published in Proceedings IEEE 10th Annual Wireless and Microwave
Technology Conference
©2009 IEEE
A Study of Memory Effects of RF Power LDMOS
before and after Digital Predistortion
Per Niklas Landin1,2, Magnus Isaksson1, Niclas Keskitalo1,3, and Olof Tornblad4
1
Center for RF Measurement Technology, University of Gävle, SE-80176 Gävle, Sweden
2
Royal Institute of Technology, Signal Processing Lab, SE-10044 Stockholm, Sweden
3
Ericsson AB, SE-164 80 Stockholm, Sweden
4
Infineon Technologies North America Corp., 18275 Serene Drive, Morgan Hill, CA 95037, USA
[email protected]
Abstract
Sideband asymmetries in distortion products are
created due to electrical and thermal memory effects
and this can be difficult to correct for in a digital
predistortion algorithm. In this study, sideband
asymmetries in third-order intermodulation distortion
products before and after digital predistortion were
investigated using 2-tone and 2-carrier WCDMA
signals. The parallel Hammerstein model was used in
the digital predistortion algorithm. The sign of the
asymmetries before correction were found to depend
on power level. Memoryless correction lead to an
increase in asymmetries for some power ranges
whereas using a PH model of order 13 with only one
order of memory length lead to good correction over a
large power range.
1. Introduction
The traditional field of radio frequency engineering
has merged with the world of signal processing, one
example is the introduction of behavioral modeling
and digital linearization techniques in wireless
communication. The process of the two fields
approaching each other has been going on for at least
ten years. Despite this fact there is still much work
undone, also when it comes to fundamental
understanding of how device and circuit designs
should be optimized for best system performance. For
future modulation formats and amplifier technologies
this is even more unclear. There are a number of
characterization methods and key figures available for
power amplifiers; new standards need to be introduced
for benchmarking of device and circuit designs.
Sideband asymmetries in distortion products are
created due to electrical and thermal memory effects
[1-4] and this can be challenging to correct for in a
digital predistortion algorithm. In this paper, a study of
sideband asymmetries for radio frequency power
LDMOS before and after digital predistortion (DPD) is
presented. The device under test was a PTF210451
(2.1 GHz 45W part) from Infineon Technologies. The
study incorporated two-tone and 2-carrier wideband
code division multiple access (WCDMA) signals for
varying tone-spacing, bandwidth and swept over
power.
2. Experimental
The setup includes a modulation-domain system
which is described elsewhere [5] and previously used
for e.g. digital measurements on radio frequency
power amplifiers. Briefly, the setup consists of a vector
signal generator (R&S SMU200A, Rohde&Schwarz
GmbH), a vector signal analyzer (R&S FSQ26,
Rohde&Schwarz GmbH), and a personal computer.
The signal was fed to the device under test (DUT) via
a feed-forward linearized 3G power amplifier with
excellent linearity, here acting as a driver amplifier.
All data is gathered in the form of IQ-data and
processed in the personal computer. To achieve the
needed signal properties both frequency stitching and
coherent averaging were used to achieve high
measurement bandwidths (more than 50 MHz) and low
noise levels, respectively [6].
0
The DPD algorithm used throughout this paper is
the Parallel Hammerstein (PH) model. The model is
known under a few different names e.g. the nonlinear
tapped delay line and the memory polynomial model.
Let u ( n ) and y ( n ) be the input and output complexvalued signals given on complex-envelope form,
respectively. A radio frequency power amplifier is an
almost linear band-pass system and if only the first
spectral zone (near the carrier frequency) is considered
the output y ( n ) of the PH model is
Output signal
Input signal
-10
-20
Power [dB]
-30
-40
-50
-60
-70
0
Frequency [MHz]
50
Fig. 1: 2-carrier WCDMA measurement, input signal
and stitched output signal with 100 MHz digital
bandwidth.
The study incorporated two-tone and two-carrier
WCDMA signals for varying tone-spacing, bandwidth
and swept over power. The signals were defined in the
personal computer and stored in the signal generator.
The WCDMA signals had peak-to-average power
ratios of 8 dB.
The linearity of the measurement is an important
figure and will limit the ability to validate the model
and linearization performances [6]. In Fig. 1 an
example of a measured 2-carrier WCDMA with 10
MHz tone-spacing is shown. The input power level is
35 dBm. The output signal power is approximately 46
dBm and the digital bandwidth is 100 MHz. The
adjacent channel power ratio (ACPR) of the input
signal is approximately -72 dB.
The study includes digital predistortion (DPD)
measurements. In the DPD an inverse model of the
transistor's response function was used (here model
based). The signal was predistorted to optimize the
linearity of the output signal of the transistor. Here,
IM3 defined as
ò
ò
P
å H2p -1(q ) u(n) 2(p -1)u(n ) =
p =1
=
P
(2)
M
å å bm,2p -1
u(n - m )
2( p -1)
u(n - m )
p =1 m = 0
where P corresponds to the truncated nonlinear model
order, 2P - 1 , and M is the memory length
considered by the model. A PH model of nonlinear
order R with memory length M is henceforth referred
to as PH(R, M). The PH model has been manifested as
a very efficient parameter reduction of the full Volterra
model and the number of reports that uses or compare
with it is large. Some examples where the PH model is
used as an inverse model in an algorithm for digital
predistortion are [7-10].
3. Results
The third order intermodulation products (IM3) of a
power-swept two-tone are shown in Fig. 2 for tone
spacings 5, 10 and 15 MHz. The corresponding
asymmetries, IM 3l - IM 3u , are shown in Fig. 3 for
each case. The asymmetries for all tone-spacings start
out positive for low powers and then switch sign
around 33.5 dBm. The reason for this is currently
unclear and needs further investigation.
Y (F)d F
IM 3 channell / u
IM 3l / u =
y(n ) =
-35
Y (F)d F
(1)
Closest
channell / u
was used as the linearization measure. In (1) Y (Φ )
denotes the power spectrum of the 2-carrier WCDMAsignal with tone spacing F. "IM3-channel denotes the
two channels centered at B3F / 2 relative to the center
of the channels. "Closest channel" denotes which
channel is used to normalize the power in the IM3channel. This figure of merit is defined in analogy with
the standard IM3 definition for two-tone signals.
-40
IM3 power [dBc]
-80
-50
-45
-50
-55
5 MHz lower
5 MHz upper
10 MHz lower
10 MHz upper
15 MHz lower
15 MHz upper
-60
-65
-70
24
26
28
30
Average input power [dBm]
32
34
Fig. 2: Two tone measurement, IM3 swept over
power for tone spacings 5, 10 and 15 MHz.
T one-spacing 5 MHz
T one-spacing 10 MHz
T one-spacing 15 MHz
8
WCDMA-IM3 asymmetry [dB]
10
IM3 asymmetry [dB]
5
0
-5
-10
32
34
36
Output power [dBm]
38
0
20
IM3
IM3
IM3
IM3
IM3
IM3
WCDMA
WCDMA
WCDMA
WCDMA
WCDMA
WCDMA
5 MHz lower
5 MHz upper
10 MHz lower
10 MHz upper
15 MHz lower
15 MHz upper
-40
-45
30
35
Output power [dBm]
40
No DPD lower
No DPD upper
DPD PH(3,0) lower
DPD PH(3,0) upper
DPD PH(13,1) lower
IM3 DPD PH(13,1) upper
-50
-55
-60
-65
30
32
34
36
38
Output power [dBm]
Fig. 6: 2-carrier WCDMA measurement, IM3 versus
output power for tone-spacings 5, 10 and 15 MHz.
IM3 after using DPD for the 10 and 5 MHz tone
spacings are shown in Figs. 6 and 7 respectively.
Chosen model orders are PH(3,0) and PH(13,1).
-25
-30
-35
-45
-40
IM3 [dB]
-50
-55
-60
20
25
Fig. 5: 2-carrier WCDMA measurement,
asymmetries versus output power for tonespacings 5, 10 and 15 MHz.
IM3 [dB]
In Fig. 4 IM3 for the 2-carrier WCDMA
measurements without DPD are shown. At higher
powers the nonlinear behavior does not show much
memory.
Fig. 5 shows the asymmetries in IM3 for the 2carrier WCDMA measurement with tone-spacings 5,
10 and 15 MHz. The behavior is similar to the twotone case (cf. fig. 2). We note that the maximum
amplitudes of the asymmetries (both positive and
negative) are smaller for the 2-carrier WCDMA case
than the 2-tone case. This seems to make sense due to
cancellation effects for complex signals. Also, the
switch of signs occurs for lower power for the 2-carrier
WCDMA case (~30dBm versus ~33.5 dBm for the 2tone case). Further investigation is needed.
WCDMA-IM3 [dB]
2
40
Fig. 3: Two-tone measurement, asymmetries in IM3
versus output power for tone spacings 5, 10 and 15
MHz.
-40
4
-4
30
-35
6
-2
-15
-30
WCDMA spacing 5 MHz
WCDMA spacing 10 MHz
WCDMA spacing 15 MHz
-45
IM3
IM3
IM3
IM3
IM3
IM3
no DPD lower
no DPD upper
PH(3,0) lower
PH(3,0) upper
PH(13,1) lower
PH(13,1) upper
-50
-55
25
30
Output power [dBm]
35
40
Fig. 4: 2-carrier WCDMA measurement, IM3 swept
over power for tone-spacing 5, 10 and 15 MHz.
-60
-65
-70
30
32
34
36
Output power [dBm]
38
40
Fig. 7: 2-carrier WCDMA measurement IM3
versus output power for 10 MHz tone spacing
before and after DPD.
No DPD
DPD PH(3,0)
DPD PH(13,1)
Asymmetry (upper-lower) [dB]
4
3
2
1
4. Conclusions
0
-1
-2
-3
30
32
34
36
38
40
36
37
38
Ouput Power [dBm]
Fig. 8: The difference between the lower and upper
IM3 in Fig. 6.
3
IM3 asymmetry no DPD
IM3 asymmetry PH(3,0)
IM3 asymmetry PH(13,1)
2.5
2
IM3 asymmetry [dB]
manifests itself in steep increases in IM3s as the power
increases, which clearly is seen for the DPD using the
PH(13,1) on the 39-40 dBm output power levels in
Fig. 6.
1.5
1
0.5
0
-0.5
-1
26
28
30
32
34
Output power [dBm]
36
38
40
Fig. 9: The difference between the lower and upper
IM3 in Fig. 7.
Substantial improvements in IM3 are possible using
memoryless models. We note that the correction
degrades at higher powers where the transistor starts to
compress.
The asymmetries in IM3 after DPD for the two tone
spacings 5 and 10 MHz are shown in Figs. 8 and 9
respectively. For the 10 MHz comparing the
uncorrected case to the DPD PH(3,0) case, we note
that the asymmetries have different signs for output
power > 34dBm and that the magnitude of the
asymmetry for the PH(3,0) case is larger in the range
34-36 dBm. In other words, using memoryless DPD
can increase the asymmetries after correction.
The PH(13,1) model reduces the asymmetry
between the IM3s and the correction is very good over
the whole power range.. In Fig. 8, the asymmetry for
the 5 MHz tonespacing is shown. In this case the
asymmetry is reduced by both models. The result is
expected since a smaller bandwidth needs to be
considered by the model.
Higher power levels for the DPD were attempted
but the expansion given by the DPD algorithm could
not be properly handled at this point. This behavior
The device under test showed increased asymmetry
between upper and lower IM3 with respect to
increased tone-spacing for the two-tone and the 2carrier WCDMA-signals; the two-tone case shows a
larger asymmetry due to a lower degree of cancellation
effects.
For the 2-carrier WCDMA with a spacing of 10
MHz the difference between IM3 asymmetries before
and after memoryless DPD was approximately the
same but shifted in sign.
In the 5 MHz case the memoryless DPD increased
the asymmetry slightly for higher power levels. For
both cases the DPD using memory lead to a decrease
in the asymmetries. In both cases the IM3s were
reduced considerably, up to 20 dB for 10 MHz spacing
and 30 dB for 5 MHz spacing. The IM3-asymmetry
after DPD is also less sensitive to power level
compared to no correction.
To understand the memory effects and the influence
the DPD has on the asymmetries requires further
investigations.
Acknowledgment
This work was supported by Ericsson AB,
Freescale, Semiconductor Nordic AB, Infineon
Technologies Nordic AB, Knowledge Foundation,
NOTE AB, Rohde&Schwarz AB and Syntronic AB.
References
[1] N.B. Carvalho and J.C. Pedro, "A comprehensive
explanation of distortion sideband asymmetries," IEEE
Trans. Microwave Theory Tech., vol. 50, no. 9, Sept. 2002,
pp. 2090-2101.
[2] J. Vuolevi, T.Rahkonen and J. Manninen, "Measurement
technique for characterizing memory effects in RF power
amplifiers," IEEE Trans. Microwave Theory Tech., vol. 49,
no. 8, Aug. 2001, pp. 1383-1389.
[3] J. Brinkhoff and A. E. Parker, "Effect of baseband
impedance on FET intermodulation," IEEE Trans.
Microwave Theory and Tech., vol. 51, no. 3, March 2003,
pp. 1045-1051.
[4] O. Tornblad, B. Wu, W. Dai, C. Blair, G. Ma and R.W.
Dutton, "Modeling and measurements of electrical and
thermal memory effects for RF power LDMOS," IEEE MTTS Int. Microwave Symp .Dig, 2007, pp. 2015-2018.
[5] D. Wisell, "A baseband time domain measurement
system for dynamic characterization of power amplifiers
with high dynamic range over large bandwidths," Proc.
IMTC '03, 2003, vol. 2, pp. 1177-1180.
[6] D. Wisell, D. Ronnow, and P. Handel, "A technique to
extend the bandwidth of a RF power amplifier test bed,"
IEEE Trans. Instrum. Meas., vol. 56, no. 4, 2007, pp. 14881494.
[7] C. T. Burns, A. Chang, and D. W. Runton, "A 900 MHz,
500 W Doherty power amplifier using optimized output
matched Si LDMOS power transistors," in IEEE MTT-S Int.
Microwave Symp. Dig., 2007, pp. 1577-1580.
[8] L. Ding, G. T. Zhou, D. R. Morgan, Z. Ma, J. S. Kenney,
J. Kim, and C. R. Giardina, "A robust digital baseband
predistorter constructed using memory polynomials," IEEE
Trans. Commun., vol. 52, Jan. 2004, pp. 159-165.
[9] O. Hammi, F. M. Ghannouchi, S. Boumaiza, and B.
Vassilakis, "A data-based nested LUT model for RF power
amplifiers exhibiting memory effects," IEEE Microw.
Wireless Compon. Lett., vol. 17, 2007, pp. 712-714.
[10] M. Helaoui, S. Boumaiza, A. Ghazel, and F. M.
Ghannouchi, "Power and efficiency enhancement of 3G
multicarrier amplifiers using digital signal processing with
experimental validation," IEEE Trans. Microwave Theory
Tech., vol. 54, 2006, pp. 1396-1404.
Paper E
Wideband Characterization of Power
Amplifiers Using Undersampling
Per N. Landin, Charles Nader, Niclas Björsell, Magnus Isaksson, David
Wisell, Peter Händel, Olav Andersen and Niclas Keskitalo
Published in IEEE MTT-S International Microwave Symposium Digest
©2009 IEEE
Wideband Characterization of Power Amplifiers Using Undersampling
Per Niklas Landin1, 2, Charles Nader1, 2, Niclas Björsell1, Magnus Isaksson1, David Wisell3,
Peter Händel1, 2, Olav Andersen4, and Niclas Keskitalo1, 3
1
Center for RF Measurement Technology, University of Gävle, Gävle, SE-801 76, Sweden
2
Signal Processing Lab, Royal Institute of Technology, Stockholm, SE-100 44, Sweden
3
Ericsson AB, Stockholm, SE-164 80, Sweden
4
Ericsson AB, Gävle, Box 6206, SE-800 06, Sweden
Abstract — In this paper a radio frequency power amplifier is
measured and characterized by the use of undersampling based
on the generalized Zhu-Frank sampling theorem. A test system
has been designed allowing the bandwidth of the stimuli signal to
be 100 MHz in the characterization process. That would not be
possible with any vector signal analyzer on the market. One of
the more challenging problem within the proposed concept is the
model validation process. Here, two different techniques for
model validation are proposed, the multitone and the spectrum
scan validation methods.
Index Terms — Communication system performance,
modeling, nonlinear systems, power amplifiers, radio
transmitters.
I. INTRODUCTION
Contemporary wide band code division multiple access
(WCDMA) standards utilize bandwidth of 3.84 MHz. The
3GPP long term evolution (LTE) supports scalable
bandwidths up to 20 MHz, and its continuation LTE advanced
considerably more than that. Moreover, mobile worldwide
interoperability for microwave access is supporting up to 28
MHz. It is anticipated that future wireless communication
systems will be more and more broad-band. The power
amplifier (PA) is a key radio component in any wireless
communication system, and behavioral modeling of its inputoutput characteristics a developed research area. It is also
possible that signals from different bands and of different
standards may share the same PA.
Considering the PA as a nonlinear dynamic device, the
spectral support of its outputs does not only cover the spectral
support of the input, but also the adjacent channels. Relying
on the Nyquist-Shannon sampling theorem, sampling rates of
several hundreds of MHz are required to sample the amplifier
output according to the classical sampling theorem. For
bandwidths at this order of magnitude, there are no available
analog-digital converters (ADCs) with the required dynamic
range, and thus there is a strong need for alternative radio
frequency measurement technologies to circumvent this major
obstacle. Alternatives include the use of time-interleaved
ADCs, (frequency stitching employing repetitive amplifier
input [1], or sub-sampling based on the seminal work by Zhu
[2]. In short, Zhu’s work implies that if there is a static
invertable function that compresses the spectral support of an
analog signal, it is sufficient to sample it with a speed
corresponding to twice the bandwidth of the compressed
signal. After ideal pulse-modulation to obtain the
reconstructed signal, the inverse of the compressing function
is applied to reconstructed data to obtain a full-band signal.
The results of Zhu were later on generalized to nonlinear
dynamic systems of certain classes by [3] and [4].
Motivated by the fundamental results on sampling theory
for nonlinear system identification, under-sampling (in the
Nyquist-Shannon context) has been identified as an emerging
radio frequency measurement technology for wireless
communication PAs, mostly in terms of theoretical studies. In
the paper by Wisell [5], it was shown that by using real
measurements on a 3G WCDMA PA employing different
sampling rates spanning from some 4 MHz up to 40 MHz, the
amplifier model kept comparable performance using lower
sampling rates. The work in [5] was extended in [6].
In the current work, we take the work of Wisell a step
further by increasing the bandwidth of the signal to first 50
and then 96 MHz. This has been possible due to a specially
designed test setup described in Section II [7]. The theory of
the model identification is given in Section III and the results
are presented in Section IV. Zhu-Frank sampling can utilize
the whole Nyquist bandwidth of the ADC for model
identification. However, model validation requires special
solutions that are discussed in Section V, followed by the
conclusions in Section VI.
II. TEST SETUP
Zhu-Frank generalized sampling theorem (ZFGST) does not
decrease the bandwidth required by the measurement system;
only the sampling rate can be decreased. To utilize ZFGST
one cannot rely on the vector signal analyzers on the market
today since the intermediate frequency (IF) bandwidth
preceding the ADC sampler front-end is limited to increase
the dynamic range by avoiding noise folding from the
broadband noise. Thus, a specially designed test system has
been designed for PA characterization based on ZFGST.
In order to master the wide bandwidth requirements, the
test-bed has an ultra wideband radio frequency (RF) front-end.
The RF input frequency range is 500 – 2700 MHz and the
amplitude range is -10 to +10 dBm for dynamic range
depending on the signal. The output amplifier has been
designed with a frequency range of 20 - 1000 MHz and 14 dB
gain. In total this results in a front-end with exceptional
III. MODEL IDENTIFICATION
The model identification procedure has been described in
[7]. Sampled input and output data records were measured at
different time instants with the described measurement
system. Synchronization of the acquired time series was
needed before model identification.
Since the measurement system does not provide any
possibilities of precise triggering (on the order of tenths of a
sample interval) and the physical run-time through the system
is unknown, the synchronization has been done using crosscorrelation and phase-compensation [8] to obtain sub-sample
synchronization.
The synchronization was made in two steps. The first step
was a rough synchronization on sample basis using crosscorrelation to find the “delay” between the measured input and
output signals. The second step was a sub-sample
synchronization to find the linear phase offset in the frequency
domain.
The models were identified by minimizing the mean square
error (MSE) of the measured output and the model output. As
model structure, the commonly used parallel Hammerstein
(PH) model [9] is chosen. The PH is defined by its nonlinear
order P and the memory length M. Such a model is henceforth
denoted PH(P,M). The models were identified using the
measured quantities of the input and output signals, known at
the specific time instant, and formed to a model-specific
regression matrix ' . The non-linear model behavior is
absorbed by ' . It was described with the model predictor
yˆ(n ) RT '
Due to the non-flat gain in the signal bandwidth, memory
effects stronger than usual are expected. To obtain an accurate
model these variations must be considered. The linear part of
the PH model is simply a FIR-filter. With the variations within
the signal bandwidth it is not necessarily true that the first
coefficient of this filter is the largest coefficient. To check for
possible "small" initial coefficients in the linear FIR-filter, a
delay in the output signal was introduced with one sample at a
time and identification was done for each such delay. The
NMSE was then computed for comparison. In no case was
more than 10 samples delay tested.
For the normal 3.84 MHz WCDMA signal the model with
lowest normalized MSE (NMSE) was the PH(9,4) with a
NMSE of -40.2 dB and an adjacent channel error power ratio
(ACEPR) of -56.5 dB [10]. No additional delays were
required to obtain the latter model. These results with the
found model order and model errors are in line with the results
from [8] for this particular PA.
The model order with lowest NMSE for the 50 MHz wide
signal was the PH(9,7). In Fig. 1 the measured input, output
and the model error for the PH(9,7) are shown. The necessary
delay due to the wide bandwidth in this case was one sample
yielding an improvement of 0.4 dB in NMSE as compared to
using no extra delays.
0
Output
Input
Error PH(9,7)
-10
Normalized power [dB]
properties. It has a 1000 MHz bandwidth within ± 1.5 dB
amplitude variations. It can handle up to 30 dBm peak with
close to 50 dBm third interception point. That is well enough
for the subsequent 12-bit pipelined ADC intended for direct IF
sampling that operates up to 210 MSPS conversion rate with
analog bandwidth of 700 MHz. A frame grabber interfaces to
the ADC and in real-time records with a data length of
2 MSamples.
-20
-30
-40
-50
-60
-70
-5
(1)
which is linear in the parameters θ . The least-squares
estimation problem is then addressed as an over determined
set of equations, linear in the parameters. Powerful and simple
methods can then be used when determining those parameters.
IV. EXPERIMENTAL AND RESULTS
The tested amplifier is a LDMOS PA intended for being
used in base stations in the 3rd generation of mobile
communications. It has 52 dB gain and a maximum rated
input power of 1 dBm. This PA is designed for use in the
2110-2170 MHz band. In small-signal S21-measurements the
frequency range with variations less than 0.5 dB is 2100 –
2220 MHz; therefore the chosen center frequency is 2160
MHz.
-4
-3
-2
-1
0
1
Frequency [Hz]
2
3
4
5
7
x 10
Fig. 1. The power spectra of the input, output and the model error
of a PH(9,7) for the 50 MHz input signal.
For the 96 MHz wide signal the most suitable model based
on NMSE was found to be a PH(9, 9) with a NMSE of -32.8
dB and a delay of 3 samples compared to what the
synchronization found. Adding this delay improved the NMSE
by 0.5 dB compared to no delay and same model order.
V. MODEL VALIDATION
As shown in [6], [11]-[13] undersampling of the output
signal of the PA can be used for the purpose of PA modeling
with little or no loss in modeling performance. Common
model performance evaluation criteria for PA behavioral
IM products
aliased one
time
Fundamental
20
IM products
aliased two
times
10
0
Relative Amplitude
models are NMSE, ACEPR and weighted error-to-spectral
power ratio (WESPR). However, data sampled according to
the ZFGST does only allow direct evaluation of NMSE. The
NMSE has earlier [14]-[15] been shown to be an inadequate
metric for model performance evaluation. In fact, ACEPR was
in [16] found to be the best low-complexity metric to identify
nonlinear mismatches.
The different frequency components of the output signal of
the PA, due to aliasing, fall in the same frequency bins after
the Zhu-Frank sampling making it impossible to separate
linear from nonlinear model errors. In the following two
alternative solutions are discussed.
-10
IM products
aliased three
times
-20
-30
-40
-50
A. Multitone
-60
Here a multitone based approach to the model validation
problem is proposed. Multitones have been used extensively
for PA modeling purposes and their suitability for this task is
well established in numerous papers e.g. [17] and the
references therein. First, assume that a PA model has been
extracted using Zhu-Frank sampling. The model can now be
validated by using a multitone or a set of multitones according
to [17] and Zhu-Frank sampling under the condition that the
sampling frequency is set in such a manner that the
intermodulation (IM) products generated in the PA, after the
aliasing, fall on frequencies (with some margin) at which there
is no input signal. This is in practice a mild requirement that
still gives sufficient freedom to design the input signal and set
the sampling frequency. The principle is illustrated in Fig. 2.
A complex four-tone signal at baseband with a normalized
bandwidth of 60 is passed through a fifth order nonlinearity.
The output signal then has a normalized bandwidth of 300.
The output signal is sampled using a sampling clock of 79.
This will cause the IM products to alias multiple times as
shown and it is still possible to determine the amplitude and
phase of them and, thus, to compute the frequency domain
evaluation criteria ACEPR and WESPR.
The model validation is done by comparing the amplitudes
and phases of the different IM products of the measured
output of the PA and of the output from the model. Numerous
techniques to swiftly determine the amplitude and phase of
multitone signals of this kind exist. Here a method like the one
presented in [18] is recommended. In this manner, it is
possible to calculate the equivalent of adjacent channel
leakage ratio [19] by adding the power of the IM products that
fall in the adjacent channels. Further, for each IM product an
error vector is calculated. The power of these error vectors can
then be added and compared to the channel power, allowing
the computation of a WESPR similar to ACEPR.
Using a fine frequency grid in the multitone signal makes
the signal closely resemble a spectrum continuous signal and
the calculated validation criteria close to the ones that would
have been obtained with such a signal.
Fig. 2.
-30
-20
-10
0
10
Relative Frequency
20
30
Illustration of undersampling.
B. Spectrum Scan
The spectrum using Zhu-Frank sampling contains
information from the true frequencies and the frequencies
aliased back from higher Nyquist bands.
Ad X d
œ IA c,1 kfs QA c,2 2s
f
k 0
kfs
(2)
where Ad = [Ad(0) … Ad(π)]T is the sampled spectrum vector,
Ac,1( f ) is the IF spectrum for all frequencies within an odd
Nyquist band starting with the frequency f, Ac,2( f ) is the
corresponding spectrum for even Nyquist bands, fs is the
sampling frequency, I is the unity matrix and Q is a matrix
with zeros except on the sub-antidiagonal where it is -1.
In practice, a low pass filter on the ADC input will remove
frequencies outside the interesting frequency band and thus, in
the following discussion an ideal low pass filter with a cut-off
frequency at K·fs will be considered. The IF spectrum, AC,
cannot be recovered directly from the sampled spectrum.
However, by having a sequence of measurements, where the
frequency of the local oscillator (LO), fLO, is changed between
each measurements the complete spectrum can be recovered.
In order to preserve information within a frequency bin the
frequency step of the LO should be an integer, s, multiplied by
the distance in frequency between two adjacent bins in the
FFT. That is fs divided by the data length, N. The notation in
(2) will change to include different sets of measurements.
Ad X, l K

f ¬
f
œ IAc,1 žžžŸž kfs ls Ns ­­­®­ QAc,2 žžžŸž 2s
k 0
kfs ls
fs ¬­
­­ (3)
N ®­
where l is an index for the measurement series, l= [0...L-1].
Several measurements will form the set of equations:
¯
¡
°
A X, 0 ¯ I Q I " 0 ¯ ¡
°
Ac f0 ¡ d
° ¡
°¡
°
¡
°
¡
°
¡
°
#
#
¡
° ¡ % % % °¡
°
¡ A X, L 1 ° ¡ 0 " I Q I ° ¡  f
°
¬
f
±° ¡¢
¢¡ d ±° ¡¡ A žž s kf (L 1)s s ­­­ °°
cž
s
N ®­ ±°
¢¡ žŸ 2
(4)
Under the right conditions, (4) can be used to recover the IF
spectrum and to compute ACEPR and WESPR. The DC
components should be excluded and the matrix must be
quadratic and full rank.
The matrix can be quadratic by excluding the highest
frequency components and full rank can be achieved by using
proper step length, s. It can be shown that s = N/2-1 will fulfil
the requirement for full rank.
VI. CONCLUSIONS
The ZFGST for the purpose of PA behavioral modeling was
tested with different input signals of varying bandwidths,
going from 3.84 MHz through 50 MHz to 96 MHz. Main
difference in the models was the amount of required linear
memory due to gain variations. For the wider signals, the
normal cross-correlation based synchronization was no longer
sufficient to find the optimal linear FIR-filter in the model. It
was shown that introducing additional delays in the output
signal as compared to the input signal improved the model
performance with up to 0.5 dB for the same model order.
Validation of models extracted using the ZFGST was done
using the NMSE. As has been shown in [10] and [14]-[15], the
NMSE is not a well-suited criteria for PA behavioral model
performance evaluation. However, due to the aliasing, criteria
using out-of-band frequencies could not be used without
special methods. In this paper two different methods are
suggested. Multitone signals and frequency planning makes it
possible to characterize IM products in-band and use criteria
like WESPR. Another method is to achieve full-spectrum
coverage by variation of the local oscillator frequency in the
downconverter and thereby acquire signals for validation of
ZFGST-extracted models.
ACKNOWLEDGEMENT
This work was supported by Ericsson AB, Freescale
Semiconductor Nordic AB, Infineon Technologies Nordic AB,
Knowledge Foundation, NOTE AB, Rohde&Schwarz AB and
Syntronic AB.
REFERENCES
[1] D. Wisell, D. Rönnow, and P. Händel, “A technique to extend
the bandwidth of a power amplifier test-bed,” IEEE Trans.
Instrum. Meas., vol. 56, pp. 1488-1494, 2007.
[2] Y. M. Zhu, “Generalized sampling theorem,” IEEE Trans.
Circuits Syst., vol. 39, pp. 587-588, 1992.
[3] W. A. Frank, “Sampling requirements for Volterra system
identification,” IEEE, Signal Processing Lett., vol. 3, pp. 266268, 1996.
[4] J. Tsimbinos, and K. V. Lever, “Sampling frequency
requirements for identification and compensation of nonlinear
systems,” in Acoustics, Speech, and Signal Processing. ICASSP94., IEEE, vol. 3, pp. 513-516, 1994.
[5] D. Wisell, “Exploring the sample rate limitation for modeling of
power amplifiers,” presented at IMEKO 2006 Conf. Dig., Rio de
Janeiro, 2006.
[6] D. Wisell, and P. Händel, “Implementation considerations on
the use of Zhu's general sampling theorem for characterization
of power amplifiers,” in Instrumentation and Measurement
Technology Conf. Proc., 2007, pp. 1-4.
[7] O. Andersen, N. Björsell, and N. Keskitalo, “A test-bed
designed to utilize Zhu’s general sampling theorem to
characterize power amplifiers,” in I2MTC 2009, Singapore,
2007, to be published.
[8] M. Isaksson, D. Wisell, and D. Rönnow, “A comparative
analysis of behavioral models for RF power amplifiers,” IEEE
Trans. Microwave Theory Tech., vol. 54, no. 1, pp. 348-359,
Jan. 2006
[9] M.S. Heutmaker, E Wu, and J.R. Welch, “Envelope distortion
models with memory improve the prediction of spectral
regrowth for some RF amplifiers,” in ARFTG Conference
Digest-Fall, 48th, 1996, vol. 30, pp. 10-15.
[10] M. Isaksson, D. Wisell, and D. Rönnow, “Wideband dynamic
modeling of power amplifiers using radial-basis function neural
networks,” IEEE Trans. Microwave Theory Tech., vol. 53, pp.
3422-3428, 2005.
[11] D. Wisell, “A baseband time domain measurement system for
dynamic characterization of power amplifiers with high
dynamic range over large bandwidths,” in Instrumentation and
Measurement Technology Conf., 2003. Proc. of the 20th IEEE,
2003, vol. 2, pp. 1177-1180.
[12] P. Singerl and H. Koeppl, “A low-rate identification method for
digital predistorters based on Volterra kernel interpolation,” in
48th Midwest Symposium on Circuits and Systems, 2005, vol. 2,
pp.1533-1536.
[13] P. Singerl and H. Koeppl, “Volterra kernel interpolation for
system modeling and predistortion purposes,” in International
Symposium on Signals, Circuits and Systems, 2005, vol. 1, pp.
251-254.
[14] P. Landin, M. Isaksson and P. Händel, “Comparison of
evaluation criteria for power amplifier behavioral modeling”, in
IEEE MTT-S Int. Microwave Symp. Dig., Atlanta, GA, USA,
2008, pp. 1441-1444.
[15] D. Wisell, M. Isaksson and N. Keskitalo, “A general evaluation
criteria for behavioral power amplifier modeling,” in ARFTG
69, Honolulu, USA, 2007, pp. 251-255.
[16] D. Schreurs, M. O’ Broma, A. A. Goacher and M. Gadringer,
“RF Power Amplifier Behavioral Modeling,” Cambridge
University, Press 2009, 2008.
[17] N. B. Carvalho, K. A. Remley, D. Schreurs and K. C. Gard,
“Multisine signals for wireless system test and design”,
Microwave Magazine, vol. 9, no. 3, pp. 122–138, Jun. 2008.
[18] D. Wisell, B. Rudlund, and D. Rönnow, “Characterization of
memory effects in RF power amplifiers using digital two-tone
measurements,” IEEE Trans. Instrum. Meas., vol. 56, pp. 27572766, 2007.
[19] ETSI, “3GPP TS 25.141 V6.3.0.”
Paper F
Power Amplifier Behavioural Model Mismatch
Sensitivity and the Impact on Digital
Predistortion
Per N. Landin, Olof Bengtsson and Magnus Isaksson
Published in Proceedings 39th European Microwave Conference
©2009 IEEE
Power Amplifier Behavioural Model Mismatch
Sensitivity and the Impact on Digital Predistortion
Performance
Per Landin#1, Olof Bengtsson*2, Magnus Isaksson#3
#
Center for RF Measurement Technology, University of Gävle
Nobelvägen 1, 80276 Gävle, Sweden
1
[email protected]
3
[email protected]
*
Now with Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH)
Gustav-Kirchoff-Str. 4, D-12489 Berlin, Germany
2
[email protected]
Abstract— This paper presents an investigation of power
amplifier behavioural model performance sensitivity to in-band
reflections. The measurement system and model extraction
process is presented together with the results and an analysis of
the effect of parameter variations in a digitally predistorted
system. A load-pull system is used together with a digital
baseband model extraction system to identify the impact on
modelling performance. The results show that the model
performance varies greatly with in-band reflection magnitude
and phase on the amplifier output. It is also shown that a digital
predistortion based on a model extracted at matched conditions,
where it gives an excellent improvement of 20 dB in adjacent
channel leakage ratio (ACLR), gives a poor improvement of 7 dB
for miss-matched conditions with as low reflections as Γ=0.2
(return loss 14 dB). This indicates that in-band reflections need
to be considered and adaptive predistortion used also for low
VSWR system like base-stations for telecommunication.
amplifiers exhibiting memory effects, it is necessary to use
modulated signals in order to extract the model parameters.
This work has been conducted on a wideband general purpose
low-power PA with large memory effects. It is to the best of
our knowledge the first time such a complex model have been
analyzed with regards to mismatch i.e. load variations.
Γ SYS
Fig. 1 Reflection coefficient seen by the amplifier when connected in the
system.
I. INTRODUCTION
Digital predistortion (DPD) based on behavioural modeling
of power amplifiers (PAs) is used to improve system linearity
to fulfil the system specifications. The algorithms used in the
DPD are based on the parameters extracted when the model of
the amplifier is identified. In general the measurements
system used for model identification considers the amplifier to
work under perfect match [1]. This means that there are no inband or out-of-band reflections on the input or output of the
amplifier. From a system perspective this is seldom the case.
The power amplifier is normally connected to an antenna or
filter producing a system reflection ΓSYS or voltage standing
wave ratio (VSWR) on the amplifier output as shown in
Fig. 1.
This VSWR may be limited by system specifications
directly or by demands on overall spectral purity. For high
VSWR systems like low-power PAs in cellular handsets, it
has previously been shown that the great variation in output
VSWR due to variation in the hand-head interface requires a
model adaption for linearization algorithms to work properly
[2], [3]. That work was based on large signal S-parameter
AM-AM and AM-PM conversion data. For more complex
PREPRESS PROOF FILE
In part II the selected amplifier is described. Part III
outlines behavioural modeling, the model selection process
and a description of the selected model. The measurement
system and its limitations are described in Part IV. The results
of the parameter sensitivity analysis and the impact of on the
predistortion are presented in part V followed by conclusions
in part VI.
II. AMPLIFIER SELECTION
The Mini-Circuits ZVE-8G amplifier was selected for this
investigation. It has a 30 dB gain over a bandwidth of 2 – 8
GHz with a 1 dB compression point, P1dB, at minimum 30
dBm. It can handle a maximum VSWR of 2.0 on the output
which enable a sensitivity study up to a reflection coefficient
(Γ) of Γ=0.33. To prevent amplifier failure Γ was limited to
0.3. The centre frequency 2.15 GHz was selected for the study
to coincide with the Tx band for the modern universal mobile
telecommunications system (UMTS) 3G standard.
To motivate the current study, a load-pull two-tone,
constant input power measurement of the amplifier is shown
in Fig. 2.
1
CAUSAL PRODUCTIONS
inverse models are used for DPD to decrease spectral
regrowth and errors introduced by the PA imperfections.
A. The Parallel Hammerstein Model
For PA behavioural modeling a commonly used structure is
the parallel Hammerstein (PH), also know as the memory
polynomial model [5], [6]. The model can be seen as a
number of parallel odd-ordered monomials, each followed by
a finite impulse response (FIR) filter. This structure makes the
model linear in the parameters and allows separation of linear
and nonlinear memory.
A PH model is specified by the highest order monomial P
and the memory length M. Such a model is henceforth
denoted PH(P,M). The PH is a black-box model, i.e. a model
that does not have any well-known physical motivation but
nevertheless gives an adequate input-output relation [7]. The
model is used as a baseband model in this study, that is, the
parameters are extracted using complex sampled IQ-data.
Fig. 2 Detailed view of output power of the amplifier for the two-tone loadpull measurement when Γ < 0.3. Maximum average output power is 26.5
dBm. The contours indicate 0.2 dB decay gradients.
The output power is seen to be very sensitive to amplifier
load mismatch. Due to the wideband nature of the amplifier
this is expected. It is deliberately mismatched to provide a
more flat gain over a wide bandwidth. This is likely to be
detectable in the extracted linear gain model parameters.
Memory effects have been identified to be the cause of
many problems in modern systems with wideband signals and
high linearity demands [4]. Fig. 3 shows the IM3 measured in
the two-tone load-pull setup.
B. Model Order Selection
The method used to determine suitable nonlinear order P
and memory length M is described in detail in [7], [8] but will
be briefly repeated here. The aim of the model is to accurately
model the spectral regrowth, measured as ACLR. The model
error can be expressed as adjacent channel error power ratio
(ACEPR), the ratio of the model error in the adjacent channel
normalized by the power in the channel. The normalized mean
square error (NMSE), a measure of the total model error [8],
is included for reference.
Model order selection is done by comparing the model
error ACEPR of several identified PH models of varying
order. The model order where an increase in the number of
parameters does not give any reduction in the chosen error
criteria is considered to be the most suitable model.
Model order for DPD is chosen to be the same model order
as for the direct model. For systems with "short" linear
memory the pre-inverse is well approximated by a model of
the same order as the direct model. The DPD signal is
obtained by identifying the inverse model and applying the
original signal to the inverse model.
Fig. 3 Detailed view of two-tone load-pull measurement for Γ < 0.3. The
contours indicate 0.1 dB IM3 difference.
IV. THE MEASUREMENT SYSTEM
The model extraction system is based on a time-domain
(baseband) IQ sampling system as shown in Fig. 4 previously
described in [9]. A noise-like signal with properties similar to
the signals used in UMTS 3G was generated in the PC and
downloaded to the arbitrary waveform generator (AWG) in
the vector signal generator (VSG), a R&S SMU200A. The
vector signal analyzer (VSA), R&S FSQ26, is used to
measure the input and output signal to respectively from the
DUT in the form of IQ-data.
An automated tuner for load-pull measurements was
inserted to provide control of the in-band impedance at the
load side. To maintain control of the out-of-band impedance a
triplex filter was used which enabled constant individual
termination of the harmonics. The full circuit replacing the
DUT in Fig. 4 is shown in Fig. 5. The triplex filter provided
constant reflection coefficients at the centre frequency of the
The nonlinearities of the selected amplifier is showing high
sensitivity to output mismatch and an in imbalance in the
IM3-products are present also for matched conditions. It is
therefore likely that the selected amplifier will need a rather
complex model to properly describe its behaviour which
makes it a good candidate for this sensitivity study. The
choice of a wideband amplifier is furthermore motivated by
the increased interest in combining multiple transmission
bands in a single amplifier.
III. BEHAVIOURAL MODELING
Behavioural modeling of PAs is concerned with finding a
model that describes the relation between the input and the
output signals of the PA. In general, behavioural models can
be used for design and system level simulations whereas
2
bands
Γ(2 f 0 ) =0.10∠ − 43D
at
is valid only for fairly narrowband signals as the tuning
circuitry will show variations over wider bandwidths.
and
Γ(3 f 0 ) =0.23∠90D for all in-band impedances.
V. RESULTS
The variation of IM3 related to the respective position of
the tuner was presented in the introduction, Fig. 3. Each point
is presented as the imbalance of IM3 relative to the two-tone
power positioned on the reflection coefficient in the SmithChart, seen by the PA. Variations are seen with a "best"
position with both highest output power and lowest IM3, and
a worst position with lowest output power and highest IM3. It
should also be noted that the "best position" is not at zero
reflection.
ACEPR and NMSE for model order determination are
shown for one reflection case, =0, in Fig. 6. The ACEPR
shows no improvement for higher model orders than PH(9,3)
whereas NMSE indicates PH(5,1). Since ACEPR is the error
criteria of interest the model order is chosen as PH(9,3).
The significant decrease in ACEPR with increasing
memory together with the lack of improvement in NMSE
indicates large nonlinear memory effects. In Fig. 6 Only one
reflection case is shown but the results are analogous for all
the studied reflections for this particular set of PA and model
structure.
Fig. 4 Outline of the baseband IQ sampling system for baseband model
extraction.
3f0 Termination
Triplex
Filter
2f0 Termination
f0 Tuner
Γ SYS(ω)
τ
Γ T(ω)
-35
-35
ΔG(ω)
-40
-40
NMSE [dB]
Fig. 5 Impedance control circuitry for the model extraction system replacing
the DUT in Fig. 4.
To reduce the impact of imperfections in the measurement
system both the input and the output signals are measured.
These two signals are synchronized using a sub-sample
synchronization technique [7]. The data is then used to extract
a PH-model using the linear least-squares method.
The model extraction process assumes constant gain and
linear phase in the measurement system. For this reason the
in-band attenuation of the impedance tuning circuitry, ΔG(ω),
was measured. It was found to be within 0.1 dB in the band
(0.2 dB over the double bandwidth). The in-band linear phase
shift due to the added delay, τ, is considered in the system
calibration process.
An investigation of the gain variations and deviation from
linear phase of the impedance tuning circuitry was done by
comparing the input signal measured directly from the VSG to
the VSA, to the same signal passing through the circuitry of
Fig. 5. The deviation is measured as the vector sum square
error of the reference input signal without tuning circuitry and
the signal using the tuner to present varying loads. The value
is normalized by the total measured power.
The worst performance was about -53 dB in-band. It should
be noted that the best extracted PH-model has a NMSE of -40
dB, 13 dB worse than the tuning circuitry induced variation.
Thus the influence of the tuning circuitry can be considered to
be negligible for the present model. It must be noted that this
-45
NMSE
ACEPR
-50
-55
NMSE PH(3, ⋅ )
NMSE PH(5, ⋅ )
NMSE PH(7, ⋅ )
NMSE PH(9, ⋅ )
ACEPR PH(3, ⋅ )
ACEPR PH(5, ⋅ )
ACEPR PH(7, ⋅ )
ACEPR PH(9, ⋅ )
-60
0
-45
-50
ACEPR [dB]
harmonic
-55
-60
1
2
3
Memory length
4
5
Fig. 6 The evaluation criteria ACEPR and NMSE for model order selection
are shown as functions of different model orders for the case of no reflections.
To illustrate the system variation when varying the load,
the modelled output from a reference model identified in the
case of no reflections, is compared to the measured output
signal. The comparison is done using the evaluation criteria
ACEPR and NMSE. These results are shown as functions of
the reflection coefficient magnitude in Figs. 7 and 8 for
ACEPR respectively NMSE.
Recalling from earlier that for all reflections it was possible
to identify PH(9,3) models with ACEPR below -60 dB, a
deterioration of the modeling performance with as much as 7
dB is seen for the reflection Γ =0.30∠ − 178D . The larger
degradation of modelling performance measured as ACEPR
compared to NMSE could indicate that the memory effects of
3
the nonlinearities are more affected by the mismatch than the
linear memory.
Deterioration in the performance of the DPD computed
using the reference model for zero reflection is shown as
contours on the Smith-Chart in Fig. 9. The improvement in
ACLR using DPD is taken down from the best improvement
of 18 dB to, in the worst case, less than 7 dB improvement.
-53
-54
VI. CONCLUSIONS
A wideband low-power PA exhibiting nonlinear behaviour
and memory effects, was investigated with respect to
behavioural model performance sensitivity to load mismatch.
The used parallel Hammerstein model showed large
performance variations that are possible to correlate to
mismatch regions with large gain variations. These results
show the sensitivity of behavioral modeling to load mismatch.
In general, higher reflections increase the spread in the
model performance but a worst case angle was detected along
the largest gradient of gain reduction. The impact of the large
parameter variation showed a deterioration of DPD
improvement in ACLR from 18 dB to less than 7 dB. Even
moderate reflections with Γ<0.2 shows a deterioration in DPD
performance of 9 dB measured as ACLR. These findings
indicate that system mismatch is an important parameter and
has to be considered in DPD.
ACEPR [dB]
-55
-56
-57
-58
-59
-60
0
0.05
0.1
0.15
0.2
0.25
Magnitude of reflection coefficient Γ [1]
0.3
Fig. 7 The ACEPR is shown as function of the magnitude of the reflection
coefficient when comparing the measured output of the PA to the modelled
output from the reference model at zero reflection. Each symbol represents
one tuner position.
-38.8
ACKNOWLEDGEMENT
This work was supported by Ericsson AB, Freescale
Semiconductor Nordic AB, Infineon Technologies Nordic
AB, Knowledge Foundation, NOTE AB, Rohde&Schwarz AB
and Syntronic AB.
-39
-39.2
NMSE [dB]
-39.4
-39.6
REFERENCES
-39.8
[1]
-40
[2]
-40.2
-40.4
[3]
-40.6
0
0.05
0.1
0.15
0.2
0.25
Magnitude of reflection coefficient Γ [1]
0.3
[4]
Fig. 8 The NMSE extracted and shown in the same way as in Fig. 7.
[5]
[6]
[7]
[8]
[9]
Fig. 9 Detailed view of DPD deterioration for Γ < 0.3. The contours indicate 1
dB ACLR increase. Subplot shows position in Smith-Chart.
4
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