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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Thorough discussion of the theory and application of the Volterra series for impairments compensation in RF circuits and systems
A Volterra Approach to Digital Predistortion: Sparse Identification and Estimation offers a comprehensive treatment of the Volterra series approach as a practical tool for the behavioral modeling and linearization of nonlinear wireless communication systems. Although several perspectives can be considered when analyzing nonlinear effects, this book focuses on the Volterra series to study systems with real-valued continuous time RF signals as well as complex-valued discrete-time baseband signals in the digital signal processing field.
A unified framework provides the reader with in-depth understanding of the available Volterra-based behavioral models; in particular, the book emphasizes those models derived by exploiting the knowledge of the physical phenomena that produce different types of nonlinear distortion. From these distinctive standpoints, this work remarkably contributes to theoretical issues of behavioral modeling.
The book contributes to practical state-of-the-art questions on linearization, granting the reader practical guidance in designing digital predistortion schemes and adopting up-to-date machine learning methods to exploit the sparsity of the identification problem and reducing computational complexity.
Later chapters include information on:
- Identification of Volterra-based models as a linear regression problem, allowing the adoption of sparse machine learning methods to reduce computational complexity while keeping rich model structures
- Deduction of Volterra models based on circuit model knowledge, offering pruned model structures that are better fitted for specific scenarios
- Wireless communication systems and the nonlinear effects produced by power amplifiers, mixers, frequency converters or IQ modulators
- Digital predistortion schemes and experimental results for both indirect and direct learning architectures
A Volterra Approach to Digital Predistortion: Sparse Identification and Estimation is an essential reference on the subject for engineers and technicians who develop new products for the linearization of wireless transmitters, as well as researchers and students in fields and programs of study related to wireless communications.
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Seitenzahl: 382
Veröffentlichungsjahr: 2024
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Notation Conventions
Acronyms
1 Overview of Nonlinear Effects in Wireless Communication Systems
1.1 Wireless Communication Systems
1.2 Modeling Power Amplifiers
1.3 Modeling Mixers and Modulators
1.4 Circuit Models of Nonlinear Devices
1.5 Experimental Evaluation of Nonlinear Circuits: Classical Methods
1.6 Behavioral Modeling and Linearization of Nonlinear Systems
1.7 Regression
1.8 Structure of the Book
Bibliography
Notes
2 Volterra Series Approach
2.1 Introduction
2.2 Volterra Series
2.3 Volterra Series Applied to RF Amplifier Modeling
2.4 Volterra Series in the Frequency Domain
2.5 Two-block Models: Wiener and Hammerstein
2.6 Double Volterra Series
2.7 Analysis of Intermodulation Distortion
2.8 Baseband Volterra Model
Bibliography
Notes
3 Discrete-time Volterra Models
3.1 Introduction
3.2 Discrete-time Volterra Models for Power Amplifiers
3.3 Reducing the Volterra Model Complexity
3.4 Discrete-time Double Volterra Model
3.5 Volterra–Parafac Model
3.6 Volterra Models in the Frequency Domain
3.7 Complex-valued Volterra Model
3.8 Figures of Merit for Experimental Methods in Modeling and Linearization
Bibliography
Notes
4 Volterra Models Pruning Based on Circuit Knowledge
4.1 Introduction
4.2 Heuristic Pruning of Volterra Models
4.3 Pruning Based on Equivalent Circuit Knowledge
4.4 Circuit Knowledge Model with Electrothermal Effects
4.5 Circuit Knowledge in Bivariate Volterra Models
4.6 Volterra Models for I/Q Modulators
Bibliography
Note
5 Regression of Volterra Models
5.1 Introduction
5.2 Least Squares Algorithm
5.3 Regularization
5.4 Adaptive Optimization and Iterative Regression
Bibliography
Note
6 Sparse Machine Learning
6.1 Introduction
6.2 Thresholding
6.3 Local Search: Hill Climbing
6.4 Greedy Pursuits
6.5 Stopping Criteria
6.6 Sparse Bayesian Learning
6.7 A Practical Sparse Regression
Bibliography
7 Transmitter Linearization with Digital Predistorters
7.1 Introduction
7.2 Digital Predistortion
7.3 Indirect Learning Architecture
7.4 Direct Learning Architecture
7.5 Some Practical Digital Predistortion Results
Bibliography
Note
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Amplifier modes of operation (normalized model).
Table 2.2 Nonlinear currents .
Table 2.3 Coefficient for some terms of the nonlinear currents.
Table 2.4 Different asymmetry situations in the context of two-tone tests wi...
Chapter 6
Table 6.1 Regressors selected by the OMP and DOMP techniques in the 30 first...
Chapter 7
Table 7.1 Linearization performance of a basic DPD following an indirect lea...
Table 7.2 Linearization performance of a DPD following an indirect learning ...
List of Illustrations
Chapter 1
Figure 1.1 Block diagram of a typical wireless communication transmitter–rec...
Figure 1.2 Basic circuit of an FET power amplifier.
Figure 1.3 Symbolic representation of a power amplifier.
Figure 1.4 FET transfer characteristics: ideal “strongly” (solid line) and r...
Figure 1.5 Symbol of a down-converter mixer.
Figure 1.6 Output frequency components of a second-order nonlinear system ac...
Figure 1.7 Schematic representation of an ideal I/Q modulator including the ...
Figure 1.8 Nonlinear circuit elements. (a) Nonlinear conductance. (b) Nonlin...
Figure 1.9 Compact equivalent three-node circuit of an FET, including the ex...
Figure 1.10 Simplified unilateral nonlinear model for an FET amplifier.
Figure 1.11 Distributed RC subcircuit with stages to model thermal effects...
Figure 1.12 Simplified layout of a charge-trapping subcircuit.
Figure 1.13 Spectral components at the output of a power amplifier, a nonlin...
Figure 1.14 AM–AM characteristic of a power amplifier and 1-dB compression p...
Figure 1.15 Spectral components at the output of a power amplifier for a two...
Figure 1.16 Power sweep with a two-tone test: third-order intercept point ....
Figure 1.17 Power of the third-order intermodulation products versus the ton...
Figure 1.18 DC bias networks in a power amplifier responsible for memory eff...
Figure 1.19 (a) Relative phase with respect to the tones of the third-order ...
Figure 1.20 Harmonic-balance technique. The nonlinear circuit is conceived w...
Figure 1.21 Normalized AM–AM characteristic of a memoryless power amplifier....
Figure 1.22 Model of a neuron.
Figure 1.23 Feedforward network with the input layer containing the source n...
Chapter 2
Figure 2.1 Nonlinearities with discontinuous derivatives (solid line) and sm...
Figure 2.2 (a) Elementary power amplifier circuit, (b) Simple FET model, (c)...
Figure 2.3 Amplitude response of an amplifier at fundamental and harmonic fr...
Figure 2.4 Dependence of the drain current (solid line) and parameters o...
Figure 2.5 Intput and output RF power for amplifiers operating in class-A an...
Figure 2.6 (a) Equivalent nonlinear circuit for an amplifier, (b) the associ...
Figure 2.7 Block nonlinear models with memory: (a) Wiener, (b) Hammerstein, ...
Figure 2.8 Schematic of a two-input nonlinear system.
Figure 2.9 Nonlinear circuit with two input ports.
Figure 2.10 (a) Predicted input/output nonlinear response and (b) conversion...
Figure 2.11 FET model used in the analysis.
Figure 2.12 (a) Wireless communication channel and (b) baseband equivalent c...
Chapter 3
Figure 3.1 Real-valued PA model (a) and complex-valued baseband behavioral m...
Figure 3.2 Grid of the third-order kernel for the baseband Volterra model. S...
Figure 3.3 Volterra models with aprioristic reduced complexity. (a) The univ...
Figure 3.4 Schematic representations of (a) the I/Q modulator modeled with b...
Figure 3.5 Block diagram of the proposed Volterra–Parafac model.
Figure 3.6 Experimental evaluation of the Volterra–Parafac parameters.
Figure 3.7 Frequency domain linearization in OFDM transmission.
Figure 3.8 Frequency-domain linearization of an OFDM signal in the DVB-T2 st...
Figure 3.9 Schematic of a complex-valued nonlinear system.
Figure 3.10 Framework of experimental equipment for the characterization of ...
Figure 3.11 Power spectral density of an output signal exhibiting spectral r...
Figure 3.12 Power spectral density for a signal with a notch in the center o...
Figure 3.13 Constellation of an output signal where nonlinear distortions pr...
Chapter 4
Figure 4.1 FET model used in the analysis.
Figure 4.2 Grid of the third-order kernel domain.
Figure 4.3 Number of coefficients for different fifth-order behavioral model...
Figure 4.4 NMSE as a function of the number of coefficients.
Figure 4.5 Nonlinear FET equivalent circuit with self-heating thermal subnet...
Figure 4.6 The envelope tracking power amplifier.
Figure 4.7 Simplified layout of the associated charge-trapping subnetwork in...
Figure 4.8 A bivariate Volterra approach to PAs with an internal variable ge...
Figure 4.9 Power spectral density of the dual-band output signal and compari...
Figure 4.10 Spectrum at the output of an I/Q modulator for the generation of...
Figure 4.11 Nonlinear block model of an I/Q modulator.
Figure 4.12 Asymmetries between the output components at and versus the ...
Figure 4.13 Equivalent model of an I/Q modulator with bivariate nonlinear (b...
Figure 4.14 Normalized power of the regressors with respect to the memoryles...
Chapter 5
Figure 5.1 Relation between the amplifier’s input signal , its output and...
Figure 5.2 Block diagram of Volterra series represented as a measurement pro...
Figure 5.3 Modeling scheme representation. The objective of least squares is...
Figure 5.4 Representation of the autocovariance matrix absolute value of a g...
Figure 5.5 Identification NMSE (a) and validation NMSE (b) for a memory poly...
Figure 5.6 Identification NMSE for a memory polynomial model versus number o...
Figure 5.7 Absolute value of the coefficients (a) and identification and val...
Figure 5.8 Block diagram of the steepest descent main variables.
Chapter 6
Figure 6.1 General scheme of coefficients selection techniques. Different sh...
Figure 6.2 Coefficients selection in a scenario with high correlation betwee...
Figure 6.3 Example of the BIC execution over the output of a greedy algorith...
Figure 6.4 Performance of the SBP identification procedure in NMSE versus nu...
Figure 6.5 Comparison of the execution time for the complete identification ...
Figure 6.6 Evolution of the NMSE in models with components taken randomly fr...
Figure 6.7 Evolution of NMSE in the OMP and DOMP techniques along with the o...
Chapter 7
Figure 7.1 General scheme of digital predistortion.
Figure 7.2 General scheme of indirect learning architecture.
Figure 7.3 Output versus input relationships with linear and compressed gain...
Figure 7.4 General scheme of direct learning architecture.
Figure 7.5 Framework of experimental equipment for the implementation of dig...
Figure 7.6 Normalized AM–AM characteristic of a commercial power amplifier, ...
Figure 7.7 Instantaneous gain of a commercial power amplifier versus the inp...
Figure 7.8 AM–PM characteristic of a commercial power amplifier, with and wi...
Figure 7.9 Power spectral density of the output signal of a commercial power...
Figure 7.10 Constellation of the output signal of a commercial power amplifi...
Figure 7.11 Linearization NMSE achieved for a commercial power amplifier by ...
Figure 7.12 Linearization ACPR, denoted as for the lower channel and for...
Figure 7.13 Linearization EVM achieved for a commercial power amplifier by e...
Figure 7.14 Power spectral density of the output signal of a commercial powe...
Figure 7.15 Linearization NMSE in a power sweep for a commercial power ampli...
Figure 7.16 Linearization ACPR, denoted as for the lower channel and for...
Figure 7.17 Linearization EVM in a power sweep for a commercial power amplif...
Figure 7.18 Linearization NMSE achieved for a commercial power amplifier by ...
Figure 7.19 Linearization ACPR, denoted as for the lower channel and for...
Figure 7.20 Linearization EVM achieved for a commercial power amplifier by e...
Figure 7.21 Power spectral density of the output signal of a commercial powe...
Figure 7.22 Normalized AM–AM characteristic of a commercial power amplifier,...
Figure 7.23 Linearization NMSE achieved for a commercial power amplifier by ...
Figure 7.24 Linearization ACPR, denoted as for the lower channel and for...
Figure 7.25 Linearization EVM achieved for a commercial power amplifier by e...
Figure 7.26 Power spectral density of the output signal of a commercial powe...
Figure 7.27 Constellation of the output signal of a commercial power amplifi...
Figure 7.28 Linearization NMSE in a power sweep for a commercial power ampli...
Figure 7.29 Linearization ACPR, denoted as for the lower channel and for...
Figure 7.30 Linearization EVM in a power sweep for a commercial power amplif...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Authors
Preface
Acknowledgments
Notation Conventions
Acronyms
Begin Reading
Index
End User License Agreement
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IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor-in-Chief
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James Duncan
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Brian Johnson
Hai Li
James Lyke
Joydeep Mitra
Desineni Subbaram Naidu
Tony Q. S. Quek
Behzad Razavi
Thomas Robertazzi
Diomidis Spinellis
A Volterra Approach to Digital Predistortion
Sparse Identification and Estimation
Carlos Crespo-Cadenas
Universidad de Sevilla
Sevilla
María José Madero-Ayora
Universidad de Sevilla
Sevilla
Juan A. Becerra
Universidad de Sevilla
Sevilla
Copyright © 2025 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
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About the Authors
Carlos Crespo-Cadenas was born in Madrid in December 1949 and he received an MSc degree in Physics at the Universidad de La Habana, Cuba, in 1974. In 1978, he performed R&D projects on topics such as piezoelectric quartz devices and RF engineering design in the Laboratorio Central de Telecomunicaciones leading several development projects of radio communications equipment. His work as an assistant researcher was positively valued, and he was appointed auxiliary researcher in 1988. In 1991, he started a four-year stay in the Universidad Politécnica de Madrid, thanks to an award from the Spanish National Board of Scientific and Technological Research for the return of doctors and technologists to the Spanish research system. His research was focused on the design of microwave monolithic integrated circuits (MMIC) ending with a PhD in 1995. In 1994, he became University Assistant in the Universidad Politécnica de Madrid and Associate Professor in the Universidad de Sevilla in 1995. There, he created the research Group of Radiocommunication Systems and led numerous research projects. In 1998, he was appointed Professor Titular de Universidad and in 2018, he was appointed Full Professor.
His research lines are nonlinear analysis of radio frequency and microwave devices, modeling and compensation of nonlinear impairments, and measurement techniques for nonlinear communication systems. He is author or coauthor of more than 80 papers published in refereed journals or international conference proceedings. He has participated in 12 research projects funded by competitive calls and in 6 research contracts with private companies of topics related to his research lines. Furthermore, he has supervised 4 PhD thesis, 2 of which were awarded the Premio Extraordinario de Doctorado (Outstanding PhD Award) of the Universidad de Sevilla.
He is a member of the Institute of Electrical and Electronics Engineers (IEEE) and the Microwave Theory and Techniques (MTT) Society. He has served as a reviewer for several research journals, such as IEEE Transactions on Microwave Theory and Techniques, IEEE Transactions on Signal Processing, and IEEE Transactions on Circuits and Systems. He participated as opponent in PhD thesis in Chalmers University of Technology, Sweden, and in University College Dublin, Ireland.
María José Madero-Ayora was born in Seville, Spain, in 1978. She received Telecommunications Engineering and a PhD from the Universidad de Sevilla, Spain, in 2002 and 2008, respectively. Since 2003, she has been with the Department of Signal Theory and Communications of the Universidad de Sevilla, where she has been an associate professor since 2012.
Her research activities are mainly focused on nonlinear analysis of RF and microwave devices, modeling and compensation of impairments in modulators and power amplifiers, and measurement techniques for nonlinear communications systems. This activity resulted in the development of novel Volterra-based behavioral models and the application of advanced signal processing techniques for model-order reduction in the design of digital predistorters, which have been presented in major conferences and published in international journals.
She is coauthor of more than 60 papers published in refereed journals or international conference proceedings. She is a member of the Institute of Electrical and Electronics Engineers (IEEE) and the Microwave Theory and Techniques (MTT) Society.
Juan A. Becerra was born in Seville, Spain, in 1986. He received the BS and MSc degrees in Telecommunication Engineering from the Universidad de Sevilla, Seville, Spain, in 2009 and 2012, respectively, a PhD in Electrical and Computer Engineering from the University of Delaware, Newark, DE, USA, in 2017, and a PhD in Telecommunication Engineering from the Universidad de Sevilla in 2019. Since 2017, he has been with the Department of Signal Theory and Communications, Universidad de Sevilla, and he has been an associate professor since 2023.
His main research areas include behavioral modeling and linearization of power amplifiers. He is specialized in compressed-sensing signal processing applied to the regression of Volterra series models, and the results of such activities have been presented at major conferences and published in international journals.
He is a member of the Institute of Electrical and Electronics Engineers (IEEE) and the Microwave Theory and Techniques (MTT) Society. He is a member of the Steering Committee of the Radio and Wireless Week (RWW) and the Technical Program Committee (TPC) of the IEEE Topical Conference on RF/Microwave Power Amplifiers for Radio and Wireless Applications (PAWR).
Preface
During the last decades, we have witnessed outstanding developments in wireless communications that have brought improved network capacity and coverage, increased energy efficiency, and reduced costs. Massive MIMO, millimeter-wave technology, and hybrid beamforming are being exploited with the deployment of 5G and 6G, and major breakthroughs are to come with future systems.
Energy-efficient base stations have been possible, thanks to the continuous development of transistor technologies and novel proposals of power amplifier schemes driven by highly varying envelope signals. On the other hand, increment of system capacity is related to the ability of the transmitter to strictly comply with the spectrum emission masks set by standardization and regulatory authorities. To meet both requirements, the use of linearization techniques is convenient, in particular those implemented with a digital predistorter in the baseband modules of the transmitter equipment to guarantee the linearity of the emitted radiofrequency signal.
The analysis and optimization of highly efficient power amplifiers have been addressed traditionally by several books. One of the most renowned, perhaps because it is tightly attached to practical objectives of the engineers, is the Steve C. Cripps book “RF Power Amplifiers for Wireless Communications.” Other recent texts, like “Behavioral Modeling and Predistortion of Wideband Wireless Transmitters” authored by Fadhel Ghannouchi, Oualid Hammi, and Mohamed Helaoui or “Behavioral Modeling and Linearization of RF Power Amplifiers” written by John Wood, have been devoted to transmitters linearization and have received a great deal of attention from the research community. Whereas the development of power amplifier architectures is feasible autonomously, it is reasonable to think that some knowledge on RF power amplifier schemes, although not strictly necessary, can be useful in efficient proposals of digital predistorter designs.
When we intended to compile this book, covering the field of behavioral modeling and linearization, our first thought was how we can provide something valuable and at the same time different from other publications. In second place, we also considered that fundamental concepts on digital predistortion can help RF engineers to redefine the design of a power amplifier that is to be used in any linearization system. More than bringing behavioral modeling and digital predistortion subjects up to date, one of the principal aspirations of this text is to expose our particular point of view following the Volterra series as the common thread on power amplifier and digital predistorter modeling, on one hand, and the use of pursuit techniques to determine a sparse set of regressors in transmitter linearization, on the other hand.
The book is organized in two parts with the hope that both can be regarded as closely related but also can be read independently. The first part, Chapters 1–4, establishes a connection between fundamental concepts of the microwave power amplifiers field and digital signal processing techniques, so that it can serve those not habituated engineers to familiarize with radio frequency amplifiers notions. The second part of the text, Chapters 5–7, examines statistical analysis for model identification and predistortion of communication transmitters and, although tightly related to the first part, can also be read separately.
In Chapter 2, we discuss how the RF power amplifier can be modeled under the perspective of Volterra series and how this view can explain theoretically classical nonlinear behavior of power amplifiers. Chapter 3 focuses on discrete-time Volterra models and how their complexity can be reduced. Some singular aspects in our approach are the consideration of the double Volterra, Volterra-Parafac, and complex-valued Volterra models. Chapter 4 is devoted to the discussion of model pruning based on the knowledge of the power amplifier equivalent circuit, relating the kernels structure to internal mechanisms associated with physical properties of transistors. Remarkable results of this chapter are model proposals based on power amplifier circuit knowledge and electrothermal or charge-trapping effects, in particular the bivariate model which includes the popular GMP model as a particular case. Following the spirit of this book, the same approach has been also extended to the proposal of an I/Q modulator model. Assuming that a predistorter is an bounded input-bounded output nonlinear system, we conclude that it can be also analyzed with the general Volterra structure used for power amplifier models.
The second part is more pragmatic and devoted to practical issues of regression, sparse models, and transmitter linearization with digital predistorters. Chapter 5 introduces the regression concept and the main statistics that allow to obtain an estimator of the power amplifier model. Chapter 6 explores the application of sparse signal processing to attain a reduced set of active coefficients in the model, including the doubly orthogonal matching pursuit (DOMP) and the sparse Bayesian pursuit (SBP) as techniques within this group. Finally, Chapter 7 reviews the most commonly used digital predistortion architectures and provides an extensive set of digital predistortion results based on the foundations set in previous chapters.
Sevilla, 27th November 2024
Acknowledgments
We want to manifest our gratitude to many people and institutions that supported us to carry out this task. Two researchers have had a remarkable influence on our work, and we owe them a special tribute. The first mention goes to Javier Reina, whose collaboration has been unique since the origin of our research team and throughout the time he belonged to the group. In the case of Sergio Cruces, his insightful knowledge on statistics has been decisive in the creation of our sparse model perspective.
Finally, we also acknowledge the financial and institutional support provided by the Spanish Ministerio de Ciencia, Innovación y Universidades, Junta de Andalucía, and Universidad de Sevilla.
Notation Conventions
Time signals are denoted by lower case letters and the Fourier transform of a time function is denoted by the same capital letter.
Vectors are denoted in boldface lower case letters , matrices in boldface uppercase letters , and is a -way array (or th-order tensor). By default, is a column vector.
is the transpose of and is the conjugate transpose of .
Normally, is reserved for the signal applied to the input of a system, for the output signal and for the coefficients vector in linear regression models.
When necessary, is used to represent the RF real-valued input signal and its complex envelope.
Index is kept to indicate the nonlinear order of a term, the sample index in a discrete-time signal.
In a linear regression model, the basis function or regressor is denoted as . In particular, is used to represent a th-order regressor of a Volterra model.
The vector of delays of the th-order kernel is denoted as .
In compact form, we write to indicate multiple sums.
Equally, we use
and denote the real and imaginary parts of , respectively.
The abbreviations AC and DC are used to mean simply alternating and direct, as when they modify current or voltage.
The ordinary frequency is denoted as and the angular frequency as .
The symmetrized nonlinear transfer function will be explicitly designated as only when necessary.
To simplify, we use the notation and the product is understood to have the value 1 when .
Acronyms
4G
fourth generation
5G
fifth generation
6G
sixth generation
AC
alternating current
ACEPR
adjacent channel error power ratio
ACPR
adjacent channel power ratio
ADC
analog-to-digital converter
AM
amplitude modulation
ARVTDNN
augmented real-valued time-delay neural network
BB
baseband
BIBO
bounded-input bounded-output
BIC
Bayesian information criterion
Bi-NL
bivariate nonlinear
BJT
bipolar junction transistor
BOTDNN
block-oriented time-delay neural network
CAD
computer-aided design
CKV
circuit knowledge Volterra
DAC
digital-to-analog converter
DC
direct current
DDR
dynamic deviation reduction
DLA
direct learning architecture
DOMP
doubly orthogonal matching pursuit
DPD
digital predistortion, digital predistorter
DSB-SC
double-sideband suppressed-carrier
DUT
device under test
DVB-T2
digital video broadcasting – terrestrial 2
ET
envelope tracking
EVBW
extended behavioral model for wideband amplifiers
EVM
error vector magnitude
FET
field-effect transistor
FFT
fast Fourier transform
FLOP
floating point operation
FV
full Volterra
GMP
generalized memory polynomial
GPIB
general-purpose instrumentation bus
HB
harmonic balance
HEMT
high-electron-mobility transistor
IF
intermediate frequency
IFFT
inverse fast Fourier transform
ILA
indirect learning architecture
IM3
third-order intermodulation
IMD
intermodulation distortion
third-order intercept point
fifth-order intercept point
seventh-order intercept point
I/Q
in-phase/quadrature
LASSO
least absolute shrinkage and selection operator
LDMOS
laterally-diffused metal-oxide semiconductor
LMS
least mean squares
LNA
low noise amplifier
LO
local oscillator
LS
least squares
LSI
linear shift-invariant
MESFET
metal-semiconductor field-effect transistor
MGS
modified Gram-Schmidt
MIMO
multiple-input multiple-output
ML
memoryless
MP
memory polynomial
NL
nonlinear, nonlinearity
NMSE
normalized mean squared error
NN
neural network
NPR
noise power ratio
NQS
non-quasi-static
NVNA
nonlinear vector network analyzer
OFDM
orthogonal frequency division multiplexing
OLS
orthogonal least squares
OMP
orthogonal matching pursuit
1 dB compression point
PA
power amplifier
PAE
power added efficiency
PAPR
peak-to-average power ratio
Parafac
parallel factor decomposition
PCA
principal component analysis
PM
phase modulation
PRSS
penalized residual sum of squares
RC
resistor-capacitor
RC-DOMP
reduced complexity doubly orthogonal matching pursuit
RF
radio frequency
RPV
radially pruned Volterra
RSS
residual sum of squares
RVM
relevance vector machine
RVTDNN
real-valued time-delay neural network
Rx
receiver
SBL
sparse Bayesian learning
SBP
sparse Bayesian pursuit
SCPI
standard commands for programmable instruments
SFDR
spurious free dynamic range
SRPV
simplified radially pruned Volterra
SVD
singular value decomposition
Tx
transmitter
UWB
ultra wideband
VBW
Volterra behavioral model for wideband amplifiers
VDTDNN
vector-decomposition time-delay neural network
VSA
vector signal analyzer
VSG
vector signal generator
1Overview of Nonlinear Effects in Wireless Communication Systems
This book is about the behavioral modeling of nonlinear communication circuits and their linearization based on a theoretical Volterra series approach. Nonlinear behavior is an inherent characteristic of electronic elements and devices, associated with the function they perform in a radio frequency (RF) communication system. To be successfully captured by a remote observer, the information-carrying signal must be strongly amplified at the cost of giving rise to nonlinear distortions. In the same way, the generation of carriers or the process of incorporating information into the carrier signal is realizable, thanks to the nonlinear operation of the different modules in the transmitter. The price for these valuable features is the generation of nonlinear imperfections in the signal sent, with the appearance of two adverse collateral consequences: provoking misinformation in the recipient of the signal and interfering with other users of the system. The central objectives of this book are the study of these nonlinear problems in wireless communication systems and the research of techniques to compensate for nonlinear impairments in order to ensure efficient, error-free transmission without interference to other users.
1.1 Wireless Communication Systems
1.1.1 Transmitters and Receivers
The scenario given by a typical wireless communication transmitter–receiver link is shown in Figure 1.1. The binary data with the information to be transmitted is converted to a baseband analog signal used to modulate an RF carrier and then amplified and radiated using the transmitter antenna. On the receiver side, the signal is captured by the receiver antenna and demodulated before its conversion to the discrete-time received sequence.
Figure 1.1 Block diagram of a typical wireless communication transmitter–receiver link.
The theoretical assumption of linear operation in wireless networks is only an approximation because the transmitters are built with several blocks, namely, modulators, mixers, power amplifiers, etc., whose electronic circuits are essentially nonlinear. In practice, undesired nonlinear effects produced in transmitters, mainly in the power amplifier, degrade the system performance and cause difficulties in meeting the stringent requirements set by the standardization entities, such as spectral masks and dispersion in the constellation.
Modern wireless communication systems are designed to operate with digitally modulated signals that have large bandwidths and high peak-to-average power ratios. Since the nonlinear behavior of the system is heavily dependent on the input signals employed, advanced knowledge of the digital world is more than advisable for the modern RF engineer.
Over the past few decades, wireless communication systems have been increasing data rates and capacity as a consequence of more sophisticated and efficient cellular networks. New generation systems employ highly spectrally efficient modulation schemes and solutions, such as orthogonal frequency division multiplexing (OFDM) and multiple-input multiple-output (MIMO). The fifth generation (5G) systems make use of both solutions, together with other technical developments that they have introduced in radio access networks. Furthermore, the trend in 5G systems to increase the use of frequency bands over 6 GHz, also forecasted for beyond 5G and 6G systems to satisfy the ever-increasing demand for connectivity, will also lead to signals more sensitive to nonlinear effects.
Noticeable nonlinear effects are generated in the circuits by the envelope variations of these signals. The nonlinear operation cannot be easily described in an analytical way; therefore, optimized designs are complex. In addition to the changes in amplitude and phase shifts typically observed in linear systems, spurious components are generated in nonlinear circuits, distorting the amplifier or mixer behavior. Among the effects of nonlinear distortions, intermodulation distortion and spectral regrowth should be taken into account since they cannot be eliminated by filtering and produce detrimental adjacent channel interference (Maas, 2003).
It is also worth noticing that, today, circuits operating in the RF range coexist with low-frequency baseband signal processing. This baseband signal processing has undergone a substantial evolution over the last few years that has led to complex modern systems. The study and prediction of the behavior of RF systems can benefit from the application of signal-processing techniques.
The present evolution toward more sustainable communications involves the search for energy-efficient transceivers, with the power amplifier being the most critical subsystem of the transmitter in terms of power consumption. In this context, the performance enhancement of a wireless communication system when its efficiency and power consumption are optimized is clear considering that these ubiquitous networks are constituted by a multitude of base stations, each one with a transmitter and a nonlinear power amplifier. However, we should recall that the power amplifier is a source of undesired nonlinear effects, more notable especially as its efficiency increases. Therefore, as RF engineers, we want to study and understand these nonlinear effects. We also want to compensate for nonlinearities and construct a linearized transmitter (predistortion). In other cases, we want to equalize the signal in the receiver (postdistortion) in the presence of a high-level noise.
1.1.2 Real-valued Continuous-time RF Signals and Complex-valued Discrete-time Baseband Signals
RF transmitters of communication systems based on modern wireless standards, like 4G, 5G, and beyond, generate real-valued continuous-time bandpass signals with a frequency response that occupies a limited bandwidth centered around the carrier frequency . The trend is to use a broad signal bandwidth, but in all cases, the condition is satisfied1. The transmitted bandpass signals are the result of modulation, i.e., the incorporation of the baseband signal information to an RF carrier.
A common representation of a bandpass signal is2
where is the signal’s complex envelope, an equivalent low-pass signal, and is the angular frequency of the carrier. The output of a bandpass linear system centered at , with a signal applied at the input, is given by the convolution:
where is the real-valued impulse response of the system. The bandpass radio communication channel is an example of a linear system described by (1.2), whereas this linear convolution is insufficient to explain the nonlinear behavior of the power amplifier, for example. The extension of the linear convolution (1.2) to the case of nonlinear systems is the Volterra series (Volterra, 1959), a major topic of discussion throughout the following chapters of this book.
Similar to the equivalent low-pass signal, the bandpass radio communication channel can be modeled with its equivalent low-pass channel impulse response to obtain the output complex envelope as the complex-valued convolution , formulated analogously to equation (1.2). It should not go unnoticed that this complex-valued convolution is essentially different to the real-valued convolution (1.2), because it involves the four convolutions of the real and imaginary parts of the equivalent low-pass signal and the impulse response .
All the interesting information is contained in the complex envelope, and as a consequence, the RF signal bandwidth can be controlled in the baseband blocks of the transmitter. This is why the low-pass equivalent representation offers not only a more convenient perspective to the analysis of a communication system, but also the possibility of counteracting in the baseband blocks for the alterations induced in the transmitted signal by the radio channel impairments. Practical techniques, e.g., OFDM, coding, equalization, and MIMO, are frequently implemented at baseband to maximize spectral efficiency in a reliable communication.
Formally, compensation of linear impairments in the radio channel can be managed with an RF linear block either in the transmitter (pre-compensation) or in the receiver (post-compensation). However, the benefits of low-frequency modules compared to high-frequency circuits suggest that a baseband filter involving a complex-valued linear convolution would be the best alternative. In this case, the pre-equalizer is the inverse module of the radio channel that introduces the necessary distortion to compensate for the transmission’s linear impairments. Therefore, the pre-equalizer output can be modeled with a linear convolution enunciated likewise in the baseband complex domain and in the RF real domain. The usefulness of a predistorter module to compensate for nonlinear impairments in a communication system can be substantiated similarly, giving also a justification for the baseband solution and bearing in mind an important reflection. Recalling that although convolutions for real-valued and complex-valued linear systems are strictly different because the complex-valued convolution involves four real-valued convolutions, both linear convolutions have the same form. This is not true for the case of nonlinear systems, and therefore, the reasoning behind the equivalence in the form of baseband and bandpass RF representations cannot be extended to nonlinear models. This is a crucial difference that the compensation of nonlinear impairments introduces in comparison to the linear case. Numerous publications have been devoted to the issue of baseband Volterra models, and it is also a matter of discussion in this text (Benedetto et al., 1979; Kim and Konstantinou, 2001; Morgan et al., 2006).
Digital modulation offers several advantages over analog modulation in terms of transmission capacity and robustness to channel impairments. With the same signal bandwidth, modern techniques such as OFDM allow much higher data rates as compared to other modulation schemes, and the advances in hardware and digital signal processing have led to much cheaper and more power efficient implementation of baseband modules in wireless transceivers. To profit from these aspects, modern wireless transceivers are currently built with a baseband section, where the information to be transmitted is processed digitally, converted to an analog signal, and then radiated by the RF front end. For this reason, modulation and coding as well as equalization and diversity techniques are implemented in the discrete-time digital domain, whereas frequency conversion and signal amplification in the transmitter front-end are designed within the continuous-time analog domain, creating a border between both fields marked by the digital-to-analog converter (DAC) and analog-to-digital converter (ADC).
An essential requirement for improving the performance of modern wireless systems is the cooperative development of comprehensive experimental and theoretical techniques in the two separate fields defined by RF and microwave modules involving real-valued continuous-time RF signals, on the one hand, and baseband modules where the discrete-time complex envelope is processed, on the other hand.
To begin the exposition of this book’s approach to nonlinear modeling, we will briefly discuss some basic concepts about the element that contributes the most to nonlinear distortions in a wireless communications system, the power amplifier.
1.2 Modeling Power Amplifiers
The unquestionable existence of a large number of base stations in modern wireless communication systems, together with the fact that each transmitter has a power amplifier, makes this nonlinear block a determining element in terms of energy consumption and cost. Since any a priori knowledge of this block structure can be helpful in building a mathematical representation, we first discuss some power amplifier topics to support its modeling.
Design of RF power amplifiers typically starts with procedures tightly associated with the A, B, and C classification, an orthodox subject treated in traditional books and also in more recent publications. In particular, Cripps advanced in Cripps (1999) the publication of a simple theory to explain the behavior of a power amplifier, which is a type of electronic circuit with an intrinsic nonlinear complexity. It will not go unnoticed by a clever reader that some parts of our perspective on this particular topic are inspired by that idealized approach and are also intended to be developed in the same spirit.
The search for energy-efficient communications has led today to the use of other efficient transmitter architectures (Qi and He, 2019), such as the introduction of switched and continuous-mode amplifying classes; the use of approaches based on dynamic bias such as envelope tracking, where the supply voltage to the amplifier changes in accordance with the signal envelope; or the use of active load adaptation techniques, in which the impedance presented to the core amplifier changes so that the instantaneous output power can be controlled. In this third group, we can mention outphasing and Doherty power amplifiers, being the latter of primary importance in base stations due to their optimized efficiency at the back-off region. It is worth mentioning at this point that the intention of the present book is to explain the Volterra-based techniques for linearization through digital predistortion, which can be applied to all types of power amplifiers irrespective of their architecture.
A classic power amplifier is composed of a transistor with input and output matching networks and a bias circuitry that supposedly does not perturb the RF response of the device. A typical schematic of a power amplifier with a transistor as active device is shown in Figure 1.2. The great effort made to develop nonlinear modeling of transistors and characterize power amplifiers with sophisticated computer-aided design (CAD) software is now available to RF engineers, providing acceptable amplifier design methods. More than in modeling issues and flawless CAD software analysis, we are interested in the nonlinear behavior of the final amplifier design, viewing the amplifier as a block with two ports: input and output. The behavioral characterization of the power amplifier is based on a more or less complex nonlinear approach with results fitted to the measured response.
Figure 1.2 Basic circuit of an FET power amplifier.
To correctly specify a well-designed amplifier, figures of merit such as power gain, power-added efficiency, or power consumption, as well as phenomena such as harmonic and intermodulation distortions, amplitude modulation to amplitude modulation (AM–AM) and amplitude modulation to phase modulation (AM–PM) conversions, play a critical role and are most relevant to RF power amplifier designers. Assuming the reader is familiar with these significant terms, an elementary discussion of them is summarized in Section 1.5 for the case of sinusoidal or multitonal input signals.
Representation of the power amplifier with a behavioral model, without explicit knowledge of the relation of its parameters to the corresponding equivalent circuit or the transistor physics, is impeccably conceivable. However, the line of reasoning on which our research has been based is a viewpoint that pursues the connection of the nonlinear system parameters to the equivalent circuit and the physical properties of the real device. Notwithstanding that comprehensive device modeling is outside the scope of this book, this subject is briefly discussed in a separate section.
An elementary analysis of nonlinear distortions in a quasi-linear amplifier can be proposed considering the response given by a power series expansion. Referring to the amplifier represented in Figure 1.3 as a black box, when the RF signal is applied to the input, the output signal can be expressed as
Figure 1.3 Symbolic representation of a power amplifier.
The amplifier response is a polynomial of consisting of a series of nonlinear terms up to
