characterization and linearization of multi-band …1050839/...characterization and linearization of...

Characterization and Linearization of Multi-band

Multi-channel RF Power Amplifiers

SHOAIB AMIN

Doctoral Thesis

Electrical Engineering

School of Electrical engineering

KTH

Stockholm, Sweden, 2017

TRITA-EE 2016:185ISSN 1653-5146ISBN 978-91-7729-198-5

KTH, School of Electrical EngineeringDepartment of Signal Processing

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen fredag den 24februari 2017 klockan 10.15 i hörsal 99133, Kungsbäcksvägen 47, Gävle.

© Shoaib Amin, February 2017

Tryck: Universitetsservice US AB

iii

Abstract

The World today is deeply transformed by the advancement in wirelesstechnology. The envision of a smart society where interactions between phys-ical and virtual dimensions of life are intertwined and where human inter-action, more often than not, takes place with or is mediated by machines,e.g., smart phones, demands increasingly more data traffic. This continualincrease in data traffic requires re-designing of the wireless technologies whichcan accommodate multi-channel and multi-band scenarios for increased sys-tem capacity and flexibility. In this thesis, aspects related to behavioral mo-deling, characterization, and compensation of nonlinear distortions in the ra-dio frequency (RF) multiple-input-multiple-output (MIMO) and concurrentdual-band power amplifiers (PAs) are discussed.

When building a model of any system, it is advantageous to take intoaccount the knowledge of the physics of the system and incorporate this in-formation into the model. This approach could help to improve the modelperformance and might reduce the number of model parameters. In this con-text, three novel behavioral models and digital pre-distortion (DPD) schemesfor nonlinear MIMO transmitters are proposed. Different types of cross-talkin MIMO transmitter are investigated to derive simple and powerful modelingschemes. Effect of coherent and partially coherent signal generation on theperformance of DPD is also evaluated.

To model and compensate nonlinear distortions in gallium nitrite (GaN)based RF PAs in presence of long-term memory effects, two novel models forsingle-input-single-output (SISO) and three novel models for concurrent dual-band RF PAs are proposed. These models are based on a fixed pole expansiontechnique and have infinite impulse response. They show substantial perfor-mance improvement in comparison with the finite impulse response basedbehavioral models. A behavioral model based on the physical knowledge ofthe concurrent dual-band PA is derived, and its performance is investigatedboth for behavioral modeling and compensation of nonlinear distortions.

Two-tone characterization is a fingerprint method for the characterizationof memory effects in dynamic nonlinear systems. In this context, two noveltechniques are proposed for the characterization of concurrent dual-band andMIMO transmitters. The first technique is a dual two-tone characterizationtechnique to characterize the individual memory effects of self- and cross-modulation products in concurrent dual-band transmitter. The second tech-nique is for the characterization and analysis of self- and cross-Volterra kernelsof nonlinear 3×3 MIMO systems using three-tone signals. In the proposedtechnique, the self- and cross-Volterra kernels are analyzed along certain pathsin a frequency space to determine the block structures of the underlying sys-tem. By using these characterization techniques, it is shown that the knowl-edge extracted from the studied systems could be used to modify previouslypublished behavioral models.

iv

Sammanfattning

Världen av idag har omvandlats starkt av framsteg inom trådlos teknik.Framtidens smarta samhalle där samspelet mellan fysiska och virtuella di-mensioner i livet är sammanflätade och där mansklig växelverkan mer oftaän inte sker med eller formedlas av maskiner - exempelvis smarta telefoner -kräver allt mer datatrafik. Denna ständigt ökande datatrafik kraver i sin turutformning av trådlös teknik som kan rymma flera kanaler och flera frekvens-band för att öka systements kapactitet och flexibilitet. I denna avhandlingdiskuteras aspekter relaterade till beteendemodellering, karakterisering, ochkompensering av ickelinjär distortion i radiofrekvens (RF) effektforstarkare(PA) för multipla-insignaler-multipla-utsignaler (MIMO) och samtidiga sig-naler i dubbla frekvensband.

När man gör en modell av ett system är det fordelaktigt att ta hänsyn tillkunskap om det fysikaliska systemet och använda denna information i model-len. Denna strategi skulle kunna bidra till att forbattra modellens prestandaoch minska antalet modellparametrar. I detta sammanhang föreslås tre nyabeteende- och DPD-modeller för ickelinjära MIMO sändare. Olika typer avöverhörning i MIMO-sändare undersöks för att härleda enkla och kraftful-la modeller. Effekter av koherent och partiellt koherent signalgenerering påDPD-prestanda utvärderas också. För att modellera och kompensera ickelinjärdistortion i galliumnitridbaserade (GaN) RF PA med langtidsminneseffekterföreslås två nya modeller for RF PA med en in- och en utsignal (SISO) trenya modeller för RF PA för samtidiga signaler i olika frekvensband. Dessamodeller är baserade på teknik med expansion runt fasta polvärden och haroändligt minnesdjup. Modellerna visar betydande resultatförbättring i jäm-forelse med dem med ändliga impulssvar. En beteendemodell baserad på enfysikalisk modell av förstärkaren föreslås för en PA för samtidiga signaler iolika frekvensband. Dess prestanda undersöks som beteendemodell och förutjämning av ickelinjär distortion.

Tvåtonskarakterisering är som ett fingeravtryck vid karakterisering avminneseffekter i dynamiska ickelinjära system. I detta sammanhang föreslåstvå nya tekniker för karakterisering av sändare för samtidiga signaler medolika frekvens och och MIMO-sändare. Den första tekniken är en dubbeltvåtonskarakterisering för de individuella minneseffekterna i egen- och korsmo-duleringsprodukter i sändare med samtidiga signaler i olika frekvensband. Denandra tekniken är för karakterisering och analys av egen- och korsvolterrakär-nor hos ett ickelinjärt 3×3 MIMO-system med hjalp av tre-tonssignaler. I denföreslagna tekniken analyseras egen- och korskänor längs linjer i en frekvens-rymd för att fastställa blockstrukturer hos det underliggande systemet. Denkunskap som erhållits med hjälp av denna karakeriseringsteknik har kunnatanvändas för att modifiera tidigare publicerade beteendemodeller.

vi

Acknowledgments

First and foremost I want to thank my advisors Professor Peter Hän-del and Professor Daniel Rönnow. It is an honor to be working with them.Undoubtedly, there profound insights, knowledge, and support has been in-valuable, for that I will be forever grateful. I am also thankful to ProfessorWendy van Moer for sharing her knowledge and time. My gratitude to Dr.Per N. Landin for the fruitful discussions both on the subject of research andotherwise. I extend my gratitude to Professor Abdullah Eroğlu and KyleFlattery for the help during my stay at Indiana University-Purdue UniversityFort Wayne, Indiana, USA.

I take this opportunity to thank all the people at ATM department of theUniversity of Gävle and signal processing department at the KTH for pro-viding a comfortable working environment. I would like to thank the formerPh.D. students: Dr. Charles Nader, Dr. Prasad Sathyaveer, Dr. MohamedHamid, Dr. Javier Ferrer Coll and Dr. Efrain Zentino for light discussionsduring their time over at the University of Gävle, and the fellow Ph.D. stu-dents: Zain, Rakesh, Nauman, Mahmoud, Indrawibawa, Usman and Smrutiduring all my Ph.D. tenure.

This work would not have been possible without the unconditional supportand love from my parents Mian Mohammad Amin and Khalida Amin. Thankyou for being the anchor of my life, your support has enabled me to withstandthe roughest seas of life. I will always be in debt to my siblings Faisal, Aysha,Shumyila and Shumyil for their love and support. Thank you Shabai for thesupport and encouragement. Finally to my loving wife Yumna and adorabledaughter Mishaal for giving me the shoulder to relax on and love to keep megoing.

Shoaib AminGävle, February, 2017.

Contents

Contents vii

List of Tables ix

List of Figures x

List of Acronyms & Abbreviations xi

I Comprehensive summary 1

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Thesis contribution and outline . . . . . . . . . . . . . . . . . . . . . 5

2 Modeling Nonlinearities in Multi-channel and Muli-band RFAmplifiers 92.1 Time domain Volterra series . . . . . . . . . . . . . . . . . . . . . . . 102.2 Complex base-band Volterra series . . . . . . . . . . . . . . . . . . . 122.3 Cross-talk in MIMO Transmitters . . . . . . . . . . . . . . . . . . . . 142.4 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Measurement Setup, Power Amplifiers and Test Signals 173.1 Measurement Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Experimental Power Amplifiers . . . . . . . . . . . . . . . . . . . . . 203.3 Experimental Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Frequency Domain Characterization Techniques for RF Amplifiers 234.1 Characterization using Dual Two-tone Test . . . . . . . . . . . . . . 244.2 Characterization using Three-tone Test . . . . . . . . . . . . . . . . 28

5 Behavioral Modeling and Linearization of Multi-channel Multi-band Amplifiers 33

vii

viii CONTENTS

5.1 Behavioral Modeling and Linearization of MIMO PAs . . . . . . . . 345.2 Modeling and Linearization of Single and Multi-band GaN PAs . . . 385.3 Derivation of a Concurrent Dual-band Model . . . . . . . . . . . . . 425.4 Parameter Reduction Techniques . . . . . . . . . . . . . . . . . . . . 45

6 Conclusion 47

Bibliography 49

II Paper reprints 59

List of Tables

4.1 Comparison of the 2D-DPD . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 NMSE/ACEPR for given behavioral models . . . . . . . . . . . . . . . . 355.2 NMSE/ACPR for given behavioral models used as DPD . . . . . . . . . 365.3 Performance evaluation of given SISO . . . . . . . . . . . . . . . . . . . 405.4 Performance evaluation of given SISO DPD . . . . . . . . . . . . . . . . 415.5 Performance evaluation of concurrent . . . . . . . . . . . . . . . . . . . 425.6 Performance evaluation of the given models in . . . . . . . . . . . . . . 45

ix

List of Figures

2.1 General illustration of a MIMO system. . . . . . . . . . . . . . . . . . . 112.2 2×2 RF MIMO transmitter . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Experimental setup with . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Experimental setup for analyzing . . . . . . . . . . . . . . . . . . . . . 183.3 (Top) Measurement setup used . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Illustration of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Measured amplitude of upper . . . . . . . . . . . . . . . . . . . . . . . 264.3 Measured amplitude of CM . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Asymmetry energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 Linearized output spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Illustration of test signals . . . . . . . . . . . . . . . . . . . . . . . . . . 294.7 Block structure of the . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.8 Block structures of 2×1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.9 Block structures of 3×1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.10 Measured output of channel . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 DUT and corresponding DPD structure . . . . . . . . . . . . . . . . . . 365.2 Linearized output of the DUT . . . . . . . . . . . . . . . . . . . . . . . . 375.3 Measured NMSE and ACPR vs number of . . . . . . . . . . . . . . . . 385.4 Linearized output spectrum of the SISO . . . . . . . . . . . . . . . . . . 415.5 Linearized output spectrum of channel . . . . . . . . . . . . . . . . . . 425.6 The model structure proposed in. . . . . . . . . . . . . . . . . . . . . . 435.7 Linearized output spectra of ZVE8G+ . . . . . . . . . . . . . . . . . . . 455.8 Performance versus number of basis . . . . . . . . . . . . . . . . . . . . 46

x

List of Acronyms & Abbreviations

2D 2 Dimensional

3D 3 Dimensional

ACEPR Adjacent Channel Error Power Ratio

ACPR Adjacent Channel Power Ratio

CA Carrier Aggregated

CC Component Carrier

CM Cross Modulation

DA Drive Amplifier

dB Decibel

DNCV Down Converter

DPD Digital Pre-distortion

DSP Digital Signal Processing

DUT Device Under Test

FIR Finite Impulse Response

FPET Fixed Pole Expansion Technique

FLOPS Floating Point Operations Per Second

GaN Gallium Nitrite

GMP Generalized Memory Polynomial

IEEE Institute of Electrical and Electronics Engineers

IF Intermediate Frequency

xi

xii LIST OF ACRONYMS & ABBREVIATIONS

IIR Infinite Impulse Response

IM Inter-modulation

LASSO Least Absolute Shrinkage and Selection Operator

LDMOS Laterally Defused Metal-Oxide Semiconductors

LO Local Oscillator

LSE Least Square Estimation

LTE Long Term Evolution

MIMO Multiple-input Multiple-output

NMSE Normalized Mean Square Error

PAPR Peak to Average Power Ratio

PH Parallel Hammerstein

SISO Single Input Single Output

VS Volterra Series

VSG Vector Signal Generator

WGN White Gaussian Noise

Part I

Comprehensive summary

1

Chapter 1

Introduction

1.1 Background

The theoretical foundation laid down by James C. Maxwell [1] and Michael Faradaypaved the way for the development of wireless communication. This resulted inthe birth of wireless system in the summer of 1895 when Marconi successfullytransmitted the signal wirelessly over the distance of 2 miles. Since then, thewireless communication is evolving continuously and has transformed the humanlife. In the current era, voice and data communication through wireless is taken asgranted, with connectivity virtually ubiquitous. The quality of service of wirelesscommunications is inevitably high, and any loss in signal quality is noticed and isa source of concern [2]. The infrastructure that supports near error-free deliveryof wirelessly transmitted information is complex, costly and the demand for higherdata rate is growing exponentially [3]. To meet the demands of future wirelesssystems, extensive research is being conducted both in academia and industry onthe development of cost-effective, reconfigurable, and efficient radio frequency (RF)transmitters that can support multi-band multi-channel operations simultaneously.

In wireless communication systems, the coverage area of the transmitted signaldepends on the signal power and its frequency. For correct reception of the RFsignal at the receiver’s end, in transmitters, power amplifiers (PAs) are employedto increase the signal power [2]. In base-stations, the RF PAs are the main consumerof the power [4], and to improve the overall cost of operation; the PAs are operatedat high efficiency levels, in compression, close to saturation [2,5]. In this region, thePA causes nonlinear distortion [6], which is seen as in-band distortion and spectralleakage into the adjacent channels, thus, polluting the available frequency spectrumwhich is heavily regularized. To improve the linearity, the PA can be backed-off tooperate within its linear region but at the cost of low efficiency and increase powerconsumption. To meet the linearity and high-efficiency requirements, the systemdesigners prefer to operate the PAs at a high-efficiency level, and compensate for thedistortions by linearization techniques [7]. Both in academia and industry, much

3

4 CHAPTER 1. INTRODUCTION

attention has been given to the architecture for efficient PAs such as Doherty [8–10]and envelope tracking [11, 12], and to the compensation of the nonlinear behaviorof PAs [2].

Over the past decade, numerous techniques have been developed to modelthe nonlinear behavior of single-input-single-output (SISO) PAs. Among them,the behavioral models [13, 14] are widely used in system simulations. Behavioralmodels are cost effective solution for performance analysis compared to circuitbased simulations [2, 15]. Behavioral models are also used in digital pre-distortion(DPD) [2, 16–18] for compensating nonlinear distortions in RF PAs and in somemethods for the identification of the parameters in DPD algorithms [19].

In DPD, a nonlinear component is placed in front of the PA such that thecascaded system (DPD-PA) produces a linear scaled version of the input signal [20].In practice, the DPD technique incorporates a mathematical model that correctsthe distortion of the PA by applying the inverse distortion to the PA. Hence, thenecessity to have an accurate mathematical description of nonlinear distortion isof importance for the accurate cancelation of nonlinear distortion produced bythe PAs. Therefore, behavioral modeling is an important step in the formulationof DPD algorithm. However, this step is often ignored in practice, where, it isassumed that direct model is similar to the inverse model, which might not alwaysbe true.

For SISO PAs, numerous behavioral models have been proposed in the litera-ture, such as neural network [21,22], Volterra series [23–26], and memory polynomialmodels [27, 28] which are reduced form of the Volterra series. Reduced forms arepreferred over the complete Volterra series due to fewer parameters and the formereffectiveness in modeling and compensating nonlinear distortions in the SISO PAs.However, the Volterra series is complete in the sense that any time-invariant non-linear dynamic system with fading memory can be modeled. Dynamic behavioralmodels may give an insight into the memory effects of an RF PA and it shouldbe pointed out that memory effects are equivalent to frequency dependence of thePA’s transfer function within the signal bandwidth [29]. Thus, a commonly usedtechnique to quantify the memory effects in SISO nonlinear system is a scannedtwo-tone test [30–34]. Recently, generalized memory polynomial (GMP) like mo-dels have been proposed by nonlinear characterization of SISO PAs using two-tonetests [35, 36].

Over the past few years, efforts have been made to develop concurrent multi-band and multiple-input-multiple-output (MIMO) RF transmitters [37–41] for in-creased capacity and flexibility of wireless communication systems. The re-designof the RF front-end to support multi-band multi-channel operation imposes new ef-ficiency and linearity requirements. Starting from mature SISO modeling and DPDtechniques, researchers must extend these to MIMO and concurrent multi-band RFPAs.

The emphasis has been on the development of models and DPD schemes formodeling and compensating nonlinear distortion in MIMO and concurrent multi-band RF PAs [42–47]. In [48], the discrete-time complex low-pass equivalent SISO

1.2. THESIS CONTRIBUTION AND OUTLINE 5

Volterra series is extended to concurrent dual-band Volterra series. In [49] the SISOcontinuous time-domain Volterra series is extended to a MIMO Volterra series formodeling time-invariant nonlinear dynamic MIMO system with fading memory.Nevertheless, compared to the SISO techniques, the characterization and lineariza-tion of MIMO and concurrent multi-band RF PAs are in their early stages andfurther work is needed to develop useful behavioral models and mitigation tech-niques for concurrent multi-band and MIMO RF transmitters.

1.2 Thesis contribution and outline

The focus of this thesis is on behavioral modeling, characterization, and mitigationof concurrent dual-band and MIMO PAs. The thesis makes four contributionsto the field of behavioral modeling and linearization of concurrent dual-band andMIMO RF PAs and two distinct contributions to the field of characterization ofconcurrent dual-band and MIMO PAs.

Paper C presents novel behavioral models for modeling and compensating non-linear distortions in MIMO RF PAs in the presence of crosstalk. Relationshipsbetween the device under test and the corresponding DPD structures are also pre-sented. The effect of coherent and partially coherent signal generation [50] on theDPD performance is also investigated.

It is generally envisioned that gallium nitride (GaN) based PAs have the poten-tial to be the long-term replacement for laterally defused metal-oxide semiconduc-tors (LDMOS) PAs in the future base station applications [51,52] due to high-powerefficiency, operating temperature, and breakdown voltages [53,54]. In [53], it is re-ported that when excited with burst signals, GaN PAs introduce long-term memoryeffects such as bias circuit modulation, charge trapping, and self-heating. Paper Fpresents novel behavioral models based on an infinite impulse response (IIR) fixedpole expansion technique to model and compensate the nonlinear effects of SISOand concurrent dual-band GaN PAs in the presence of long-term memory effects.

In paper G, a mathematical model based upon the physical knowledge of aconcurrent dual-band RF PA is derived, and its performance is investigated bothas a direct and inverse model. MIMO and concurrent dual-band models sufferfrom high number of model parameters. In paper B, the least absolute shrinkageand selection operator (LASSO) is implemented to reduce the number of modelparameters of the MIMO Volterra series.

Traditionally, to characterize the memory effects of SISO PAs, two-tone cha-racterization techniques are often used. Paper D presents a dual two-tone cha-racterization technique for the characterization of memory effects of the inter- andcross-modulation products of concurrent dual-band PAs. In paper E a techniquefor the characterization of 3rd-order self- and cross-Volterra kernels of a 3×3 nonlin-ear MIMO system in the frequency domain is presented. The techniques presentedin papers D and E could be used to develop or modify previously published be-havioral models and DPD algorithms for concurrent and MIMO PAs. In paper A,


the characterization technique presented in paper D is used to modify previouslypublished concurrent dual-band model for DPD applications.

List of papers included in the Thesis

The papers included in this thesis are listed below:

Conferences

Part of the enclosed material in this thesis has been presented at the followingconferences:

[A] S. Amin, Z. A. Khan, M. Isaksson, P. Händel D. Rönnow, Concurrent Dual-band Power Amplifier Model Modification using Dual Two-Tone Test, Proc.46th European Microwave Conference (EuMC), London UK, Oct. 2016, pp.186-189.

The author of this thesis was the main contributor of this paper and performedthe experimental work. The other co-authors were involved in refining andpointing out the focus of the paper.

[B] E. Zenteno, S. Amin, M. Isaksson, D. Rönnow, P. Händel, Combating the Di-mensionality of Nonlinear MIMO Amplifier Predistortion by Basis Pursuit,Proc. 44th European Microwave Conf. (EuMC), Rome Italy, Oct. 2014, pp.833-836.

The author of this thesis was involved in experimental setup and manuscriptwriting. The major part of basis pursuit was done by Efrain Zenteno. Theother co-authors were involved in refining and pointing out the focus of thepaper.

Journals

The contents of this thesis have been based on the following journal publications:

[C] S. Amin, P. N. Landin, P. Händel and D. Rönnow, Behavioral Modeling andLinearization of Crosstalk and Memory Effects in RF MIMO Transmitters,IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 4, pp.810-823, Apr. 2014.

The author of this thesis was the main contributor of this paper and per-formed the experimental work, major part in developing models and writingthe manuscript. The measurement results were presented according to the in-sight given by the co-authors. The phase noise analysis was performed by PerN. Landin.

1.2. THESIS CONTRIBUTION AND OUTLINE 7

[D] S. Amin, W. Van Moer, P. Händel and D. Rönnow, Characterization of Concur-rent Dual-Band Power Amplifiers Using a Dual Two-Tone Excitation Signal,IEEE Transactions on Instrumentation and Measurement, vol. 64, no. 10,pp. 2781-2791, Oct. 2015.

The author of this thesis was the main contributor of this paper, performedthe experimental work and manuscript writing. The measurement results werepresented according to the insight given by the co-authors. The manuscriptwas reviewed and refined by the co-authors.

[E] M. Alizadeh, S. Amin and D. Rönnow, Measurement and Analysis of frequency-domain Volterra kernels of nonlinear dynamic 3×3 MIMO systems, IEEETransactions on Instrumentation and Measurement, accepted, Nov. 2016.

The author of this thesis was involved in parts of experimental setup andmanuscript writing. The major part of the paper was done by MahmoudAlizadeh. The measurement results were presented according to the insightgiven by Daniel Rönnow.

[F] S. Amin, P. Händel and D. Rönnow, Digital Predistortion of Single and Con-current Dual Band Radio Frequency GaN Amplifiers with Strong NonlinearMemory Effects, IEEE Transactions on Microwave Theory and Techniques,under revision, Sept. 2016.

The author of this thesis was the main contributor of this paper, performed theexperimental work, model formulation and manuscript writing. The measure-ment results were presented according to the insight given by the co-authors.The manuscript was reviewed and refined by the co-authors.

[G] S. Amin, P. N. Landin, P. Händel and D. Rönnow, 2D Extended EnvelopeMemory Polynomial model for Concurrent dual-band RF Transmitters, Inter-national Journal of Microwave and Wireless Technologies, submitted, Nov.2016.

The author of this thesis was the main contributor of this paper and performedthe experimental work, part in developing model and writing the manuscript.The measurement results are presented according to the insight given by theco-authors. Part of model formulation is performed by Per N. Landin.

List of papers not included in the Thesis

The author has been involved in the following publications that are not part ofthis thesis, either due to contents overlapping that of appended papers, or due tocontents not related to the thesis.


[H] S. Amin, E. Zenteno, P.N. Landin, D. Rönnow, M. Isaksson, P. Händel, Noiseimpact on the identification of digital predistorter parameters in the indirectlearning architecture, Communication Technologies Workshop (Swe-CTW),Lund Sweden, Oct. 2012, pp. 36-39.

[I] D. Rönnow, S. Amin, M. Alizadeh, E. Zenteno, Phase noise coherence of twocontinuous wave radio frequency signals of different frequency, accepted forpublication, IET Science, Measurement and Technology, 2016.

[J] K. J. Flattery and S. Amin and Y. Mahamat and A. Eroglu and D. Rön-now, High power combiner/divider design for dual band RF power amplifiers,IEEE International Conference on Electromagnetics in Advanced Applica-tions, Torino Italy, Sept. 2015, pp. 1036-1039.

[K] D. Rönnow, M. Alizadeh, S. Amin, and E. Zenteno, "Method and devices formeasuring phase noise and constructing a phase noise representation for a setof electromagnetic signals, Swedish Patent application 1550987-0, 2015.

[L] S. Amin, P. Händel, D. Rönnow, Characterization and modeling of RF ampli-fiers with multiple input signals, Swedish Microwave day, Linköping Sweden,Mar. 2016, pp. 33.

Thesis outline

The thesis is organized as follows. Chapter 2 presents an introduction to the Volt-erra series for SISO, MIMO, and concurrent dual-band RF PAs and discusses thecross-talk effects in MIMO transmitters. Chapter 3 presents the measurement setupused for obtaining the experimental data for papers A−G. In chapter 4 the sum-mary of the proposed characterization techniques in D and E are presented, alongwith the discussions on A.

Chapter 5 summarizes the behavioral models and DPD schemes for concurrentdual-band and MIMO PAs presented in B, C, F and G, and conclusions are madein Chapter 6.

Chapter 2

Modeling Nonlinearities in

Multi-channel and Muli-band RF

Amplifiers

Amplifiers play a vital role in wireless communication systems. The nonlinear be-havior of the PAs in the transmitter chain is a well-known phenomenon and issubjected to intensive research both in the academia and in the industry over thepast several decades [6,15,55,56]. Broadly speaking, there are two main approachestaken by the designers/researchers to model the behavior of transmitters and PAs:physical models and empirical models. Physical models are based on the physicaldescription of different components in the PA (or transmitter) and their interactionunder different conditions; they are generally formulated as equivalent circuit mo-dels [2,15]. Physical models can give insight into the nonlinear sources [57] and canprovide an accurate description of the amplifier’s performance, but at the cost ofhigh simulation time [2]. Therefore, the practical use of these models in simulatinga complete system or subsystems is limited.

Empirical models are commonly referred to as behavioral or black-box models,and attempt to model the system with little or no prior knowledge of the internalcircuitry of the device. Due to the advancements in the digital platforms, behavioralmodels have become popular and gained much attention over the past few decadesdue to the ease of implementation, and fast processing time [2]. In principle themodeling information is completely contained in the external stimuli and responseof the system. Both physical and behavioral models are useful, and combiningthem results in another class of models, and is generally referred to as gray-boxmodels [13, 58, 59]. The difference between the black-box and the gray-box modelsis that the latter are physically inspired models [13, 60]. The inclusion of priorknowledge may make the model much simpler and more accurate [2,13,60]. Behav-ioral models are also used to compensate nonlinearities in a system using indirectlearning architecture (ILA) [61] and direct learning architecture (DLA) [62]. In this

9

10CHAPTER 2. MODELING NONLINEARITIES IN MULTI-CHANNEL AND

MULI-BAND RF AMPLIFIERS

thesis, nonlinearities are compensated via ILA.Over the years, in electrical engineering, Volterra theory has been used exten-

sively to describe the behavior of time-invariant nonlinear dynamic systems withfading memory [63–65]. In wireless systems, particularly in the base stations, re-search emphasis was mainly put on the development of behavioral models to predictthe nonlinear behavior of RF transmitters and to compensate them. Many behav-ioral models for SISO RF transmitters have been proposed in the literature, e.g.,parallel Hammerstein (PH) [56, 66], GMP [27, 67], vector-switched memory poly-nomial (MP) [68] and truncated Volterra models [25, 69]. These developed modelsare reduced forms of the general Volterra series. Therefore, the Volterra seriesrepresents a mature technique for modeling and compensating the nonlinear SISOsystems.

In recent years, the research focus is directed toward the development of multi-channel and multi-band RF transmitters, e.g., MIMO and concurrent multi-bandtransmitters [37, 39, 40, 70, 71], to meet the increasing demand for wireless datatransmission. For both mobile and base stations, MIMO and concurrent multi-band RF PAs are desirable for high integration, and low equipment/operationalcost. In this thesis, studies are conducted to address the number of challengesassociated with the characterization and mitigation of nonlinear effects in MIMOand concurrent multi-band RF PAs. As mentioned earlier, the Volterra series is ageneral structure for describing the nonlinear behavior of the wide class of nonlineardynamic time-invariant systems. Therefore, to develop behavioral models for multi-channel and multi-band RF transmitters, it is natural to start with the generalVolterra series. In the following sections, time domain and complex baseband formof multi-channel/band Volterra models are presented.

2.1 Time domain Volterra series

To underline the difference between the nonlinear dynamic behavior of SISO, MIMO,and concurrent multi-band systems, in the following, we first outline the SISO Vol-terra series. Let us consider a nonlinear dynamic SISO system, whose input-outputrelationship can be described as

y(t) = f(x(t)), (2.1)

where x(t) and y(t) are the input and output signals, respectively, and are operatingat a single carrier frequency (ωc1), and f(·) denotes the nonlinear dynamic transferfunction. The function f(·) can be described by a time-domain SISO Volterra seriesas [23],

y(t) =∞

∑

p=0

∞∫

−∞

. . .

∞∫

−∞

hp(τ1, . . . τp) x(t − τ1) . . . x(t − τp) dτ1 . . . dτp, (2.2)

where hp(τ1, . . . τp) is the pth-order real-valued Volterra kernel.

2.1. TIME DOMAIN VOLTERRA SERIES 11

Figure 2.1: General illustration of a MIMO system.

Let us consider an M×M nonlinear dynamic MIMO system shown in Figure 2.1.The input-output relation of such a system can be represented as,

yj(t) = fj(x1(t), . . . , xm(t)) j ∈ [1 . . . m]. (2.3)

In (2.3), fj(·) operates simultaneously on the input signals operating at the samecarrier frequency, indicates that the nonlinear behavior of a MIMO system is differ-ent from the SISO system. The nonlinear transfer function fj(·) can be modeled bythe time domain MIMO Volterra series. An M×M MIMO system can be writtenas M MISO system [72,73]. For a 2×2 MIMO system, the output of channel 1 canbe represented as [49]

y1(t) =

∞∫

−∞

h(1,1,1)(τ) x1(t − τ) dτ +

∞∫

−∞

h(1,1,2)(τ) x2(t − τ)dτ (2.4a)

+

∞∫

−∞

∞∫

−∞

∞∫

−∞

h(3,1,111)(τ1, τ2, τ3) x1(t − τ1)x1(t − τ2)x1(t − τ3) dτ1dτ2dτ3 (2.4b)

+

∞∫

−∞

∞∫

−∞

∞∫

−∞

h(3,1,222)(τ1, τ2, τ3) x2(t − τ1)x2(t − τ2)x2(t − τ3) dτ1dτ2dτ3 (2.4c)

+

∞∫

−∞

∞∫

−∞

∞∫

−∞

h(3,1,112)(τ1, τ2, τ3) x1(t − τ1)x1(t − τ2)x2(t − τ3) dτ1dτ2dτ3 (2.4d)

+

∞∫

−∞

∞∫

−∞

∞∫

−∞

h(3,1,122)(τ1, τ2, τ3) x1(t − τ1)x2(t − τ2)x2(t − τ3) dτ1dτ2dτ3 (2.4e)

+ . . . +

∞∫

−∞

. . .

∞∫

−∞

h(p,1,11...1)(τ1, . . . , τp) x1(t − τ1) . . . x1(t − τp) dτ1 . . . dτp (2.4f)

+ . . . +

∞∫

−∞

. . .

∞∫

−∞

h(p,1,22...2)(τ1, . . . , τp) x2(t − τ1) . . . x2(t − τp) dτ1 . . . dτp, (2.4g)



where h(nonlinear order,output channel, input combination) denotes real-valued MIMO Vol-terra kernels for a specific nonlinear order, output channel and input signal(s)combination. In (2.4a), h(1,1,1) and h(1,1,2) are the linear Volterra kernels, respec-tively for the input signals x1(t) and x2(t). In (2.4b), and (2.4c), h(3,1,111), h(3,1,222)

denotes the 3rd-order self-kernels. In (2.4d) and (2.4e), h(3,1,112) and h(3,1,122) rep-resents the cross-kernels. The cross-kernels are due to the nonlinear interaction ofdifferent input signals. h(p,1,11...1) in (2.4f) is the pth-order self-kernel.

The self-kernels have the same symmetry properties as the SISO Volterra ker-nels, i.e., h(3,1,111)(τ1, τ2, τ3) = h(3,1,111)(τ2, τ1, τ3) for all the permutation of τ1, τ2

and τ3. However, the cross Volterra kernels have lower symmetry compared to theself-kernels [49], e.g., h(3,1,112)(τ1, τ2, τ3) = h(3,1,112)(τ2, τ1, τ3) only for the permu-tation of τ1, τ2, but not for other permutation of τ1, τ2, τ3, i.e., h(3,1,112)(τ1, τ2, τ3) 6=h(3,1,112)(τ1, τ3, τ2). Similarly h(3,1,122)(τ1, τ2, τ3) = h(3,1,122)(τ1, τ3, τ2) for the per-mutation of τ2, τ3 but not for the other permutation of τ1, τ2 and τ3.

2.2 Complex base-band Volterra series

PAs used in wireless communication systems are narrow-band in nature [19,74], andmodeling the PAs with band limited signals has been suggested [75]. Within thefield of wireless communication systems band-limited signals are commonly used [4],i.e., the bandwidth of the signal is in order of hundred of kHz to tens of MHz, whilethe operational carrier frequency is in order of GHz. Since the primary interestlies in the frequencies close to the operational frequencies, signals are commonlypresented in a complex-valued baseband form [76,77]. The discrete-time complex-valued baseband SISO Volterra model is presented as follow [67]:

y(n) =

P∑

p=1

Q∑

q1=0

. . .

Q∑

qp=qp−1

Q∑

qp+1=0

. . .

Q∑

q2p−1=q2p−2

h2p−1(q1, . . . , q2p−1)

·u(n − q1) . . . u(n − qp)u(n − qp+1)∗ . . . u(n − q2p−1)∗.

(2.5)

In (2.5), u(n) and y(n) are the complex-envelope discrete-time input and outputsignals, respectively, (·)∗ denotes the complex conjugate, Q represents the memorylength and h2p−1 denotes the pth-order complex Volterra kernel. In (2.5), e.g., for3rd nonlinear order, h3(q1, q2, q3) = h3(q2, q1, q3) due to kernel symmetry, thus theredundant terms are removed. Furthermore, even-order kernels are also removedsince their effect can be omitted in band-limited modeling [25]. In the following, weextend the SISO Volterra model (2.5) to a MIMO Volterra model for a 2×2 MIMO

2.2. COMPLEX BASE-BAND VOLTERRA SERIES 13

system. The output signal for channel 1 can be written as,

y1(n) =

Q∑

q=0

h(1,1,1)(q) u1(n − q) +

Q∑

q=0

h(1,1,2)(q) u2(n − q) (2.6a)

+

Q∑

q1=0

Q∑

q2=q1

Q∑

q3=0

h(3,1,111)(q1, q2, q3) u1(n − q1)u1(n − q2)u∗

1(n − q3) (2.6b)

+

Q∑

q1=0

Q∑

q2=q1

Q∑

q3=0

h(3,1,222)(q1, q2, q3) u2(n − q1)u2(n − q2)u∗

2(n − q3) (2.6c)

+

Q∑

q1=0

Q∑

q2=q1

Q∑

q3=0

h(3,1,112)(q1, q2, q3) u1(n − q1)u1(n − q2)u∗

2(n − q3) (2.6d)

+

Q∑

q1=0

Q∑

q2=q1

Q∑

q3=0

h(3,1,221)(q1, q2, q3) u2(n − q1)u2(n − q2)u∗

1(n − q3) (2.6e)

+

Q∑

q1=0

Q∑

q2=0

Q∑

q3=0

h(3,1,121)(q1, q2, q3) u1(n − q1)u2(n − q2)u∗

1(n − q3) (2.6f)

+

Q∑

q1=0

Q∑

q2=0

Q∑

q3=0

h(3,1,122)(q1, q2, q3) u1(n − q1)u2(n − q2)u∗

2(n − q3) (2.6g)

+...

In (2.6), u1(n) and u2(n) are the input signals for channel 1 and 2, respectively.The linear kernels are given in (2.6a) for each input signals. The self-kernels givenin (2.6b) and (2.6c) have same symmetry properties as the SISO Volterra series.Cross-kernels given by (2.6d) and (2.6e) have the same symmetry properties as thekernels in (2.4d) and (2.4e) i.e., h(3,1,112)(q1, q2, q3) = h(3,1,112)(q2, q1, q3) but notfor other permutations of q1, q2 and q3. Cross-kernels given in (2.6f) and (2.6g) donot have any symmetry properties for any permutations of q1, q2 and q3. Noticethat the kernel in (2.6f) and (2.6g) corresponds to the frequency domain Volterrakernel being excited in H(3,1,121)(ωc1 , ωc2 , −ωc1) and H(3,1,122)(ωc1 , ωc2 , −ωc2).

In a concurrent multi-band system, the PA is exited with a carrier aggregated(CA) signal, composed of multiple signals operating at multiple carrier frequencies.The frequency separation between these signals is in order of hundred of MHz toGHz [78]. If the CA input-output signals are considered, a concurrent multi-bandsystem can be viewed as a SISO system [73]. However, under this condition, thetraditional modeling and DPD would require larger bandwidths at the receiverand may become infeasible [73, 78]. Thus, a frequency selective scheme has beenproposed for modeling and DPD of concurrent multi-band systems, to avoid increasein analog and digital requirements [79]. The discrete-time complex-valued baseband



concurrent dual-band Volterra model is [48]:

y1(n) =

P∑

p=0

Q∑

q1=0

· · ·

Q∑

q2p+1=0

p+1∑

a=1

h1,2p+1(q1, q2, · · · , q2p+1, a)

a∏

i=1

u1(n − qi)

·

2a−1∏

i=a+1

u∗

1(n − qi)

a+p∏

i=2a

u2(n − qi)

2p+1∏

i=a+p+1

u∗

2(n − qi). (2.7)

In (2.7), u1(n) and u2(n) are the input signals operating at ωc1 and ωc2 , respectively,where ωc1 6= ωc2 , and y1(n) is the output signal at ωc1 . For 3rd nonlinear order,(2.7) results in the following terms

y1(n) =

Q∑

q1=0

h1,1(q1) u1(n − q1) (2.8a)

+

Q∑

q1=0

Q∑

q2=q1

Q∑

q3=0

h1,3(q1, q2, q3, 2) u1(n − q1)u1(n − q2)u∗

1(n − q3) (2.8b)

+

Q∑

q1=0

Q∑

q2=0

Q∑

q3=0

h1,3(q1, q2, q3, 1) u1(n − q1)u2(n − q2)u∗

2(n − q3), (2.8c)

where the self-kernel is given by (2.8b) and have the same symmetry propertiesas (2.6b), whereas the cross-kernel is given by (2.8c) and does not have symme-try properties. In comparison to MIMO Volterra series, the concurrent dual-bandVolterra series have a reduced number of model parameters.

2.3 Cross-talk in MIMO Transmitters

Coupling or cross-talk effects take place in electrical systems due to interferencebetween signals from different sources. In MIMO transmitters, cross-talk occursdue to the leakage of RF signals from one channel to another. These signals oftenhave the same carrier frequency and similar power level. Hence, cross-talk is likelyto occur and is difficult to avoid entirely, especially in integrated circuit designswhere the size is important [39, 40].

Cross-talk in MIMO transmitters can be characterized as input (nonlinear) oroutput (linear) cross-talk, depending on whether the cross-talk appears before orafter the nonlinear components [44]. Figure 2.2 shows a general architecture of a2×2 MIMO PA built on the same chip set. In Figure 2.2, α and β denote theimpulse response of the linear filters before and after the PAs, indicating the cross-talk level. In this thesis, it is assumed that the cross-talk is frequency independent,i.e., memoryless. In the rest of the thesis, input and output cross-talks will bereferred to as nonlinear cross-talk and linear cross-talk, respectively.

2.3. CROSS-TALK IN MIMO TRANSMITTERS 15

Figure 2.2: 2×2 RF MIMO transmitter, α and β represents input and outputcross-talk, respectively.

In the presence of nonlinear cross-talk only, the relationship between the input-output signals of a 2×2 MIMO transmitter can be presented as follows

y1(t) = f1(x1(t) , α12 ∗ x2(t))

y2(t) = f2(α21 ∗ x1(t) , x2(t)), (2.9)

where α21 is the amount of cross-talk from channel 1 to 2, and ∗ denotes theconvolution. In (2.9), f1(·) and f2(·) are the nonlinear dynamic transfer functionsof channel 1 and 2, respectively. These nonlinear transfer functions can be modeledby (2.6) since f1(·) and f2(·) operate simultaneously on both input signals. Inthe presence of linear cross-talk only, the output of a 2×2 MIMO transmitter canrepresented as

y1(t) = f1(x1(t)) + β12 ∗ f2(x2(t))

y2(t) = β21 ∗ f1(x1(t)) + f2(x2(t)), (2.10)

where β21 and β12 denote the amount of cross-talk at the output. Equation (2.10)indicates that the output of each channel is a linear combination of f1(·) and f2(·),respectively, and can be modeled by the SISO Volterra series (2.5). In the presenceof no cross-talk i.e., α and β are equal to zero, and Figure 2.2 simplifies to twoindependent SISO PAs and can be modeled by (2.5).

The models discussed in papers B and C do not require the prior knowledge ofthe cross-talk, i.e., cross-talk is not an input parameter to these models. In systemidentification, the model parameters take into account the amount of cross-talk, anymismatch in the level of will also be taken into account by the model parameters.

In practice, it is difficult to know the type of cross-talk present in a MIMO PAbefore hand, however, recently in [80], a technique is proposed to identify the typeof cross-talk in MIMO PAs. Hence, by utilizing the technique proposed in [80],a relevant model among those presented in paper C could be used to model andlinearize the 2×2 MIMO PA.



2.4 System Identification

The estimation techniques used depends on the structure of the model. For modelsthat are linear in parameters, least square estimation (LSE) technique is usuallyused [81]. In order to estimate the model parameters of 2×2 MIMO and concurrentdual-band models that are linear in parameters [cf (2.6) and (2.7)], the output canbe written as

[

y1

y2

]

=

[

H1 00 H2

] [

θ1

θ2

]

+

[

v1

v2

]

, (2.11)

where y1 and y2 are the column vectors containing the samples of the measuredoutputs for channel 1 and 2, respectively, and H1 and H2 are the regression ma-trices. The signals v1(n) and v2(n) are noise in channel 1 and 2, respectively, andare assumed to be mutually uncorrelated zero-mean circular symmetric complexGaussian noise. However, in case of a non zero-mean, LSE will result in biasedestimates [82, 83]. The measurement noise can come from the DUT or the mea-surement systems. To reduce the effect of noise, coherent averaging is used in thisthesis and is described in the next chapter. In (2.11), θ1 and θ2 are the complexvalued parameter vectors. Note that the outputs are decoupled in (2.11), i.e., theparameters for channel 1 and 2 are independently estimated. The regression matrixH1 is

H1 =

φ1(1) φ2(1) . . . φO(1)φ1(2) φ2(2) . . . φO(2)

......

. . ....

φ1(N) φ2(N) . . . φO(N)

, (2.12)

where φi(·) are the basis functions of the model. In (2.12), O is the total numberof basis functions and N is the number of samples.

As mentioned earlier, the discrete-time complex-valued baseband Volterra mo-dels presented in (2.6) and (2.7) are linear in the parameters, hence, LSE is usedto estimate the model parameters by minimizing the cost function

θ̂j = argθj

min ‖yj − Hjθj‖22 j ∈ [1, 2]. (2.13)

The parameters, θj , in (2.13) can be determined as

θ̂j = (H∗

j Hj)−1H∗

j yj , (2.14)

where θ̂j are the estimated parameters. In papers A-E, and G, LSE is employed toidentify the model parameters except in paper F. In paper F, the models are linearin parameters and nonlinear in poles. Thus, the identification is a separable leastsquares problem [84]. To identify the pole positions, we used an iterative techniqueas in [26, 85, 86]. Real valued poles were used, and the algorithm was initializedwith different pole values but the identified poles were the same.

Chapter 3

Measurement Setup, Power

Amplifiers and Test Signals

This chapter gives a brief introduction to the measurement setups used for obtainingexperimental data in papers A-G. In this thesis, the complex baseband signals werecreated in Matlab and uploaded to Rohde & Schwarz SMBV 100A vector signalgenerators (VSGs). The VSGs have a maximum sampling frequency of 150 MHzand can generate signals with a maximum bandwidth of 120 MHz. For MIMOand concurrent dual-band measurements, the VSGs were operated in a master-slave configuration and were baseband synchronized, i.e., the baseband generatorsin the VSGs are triggered at the same time. The RF coherence was achievedby providing the local oscillator (LO) signals to the VSGs externally by usingHolzworth HS9003A RF synthesizer. The RF output(s) of the devices under test(DUTs) were down-converted to an intermediate frequency (IF) and the IF signalswere digitized using SP-devices ADQ-214 analog-to-digital converter (ADC). TheADC has a resolution of 14-bit with a maximum sampling rate of 400 MHz and hastwo channels. The total achievable bandwidth for the used ADC is 200 MHz.

Coherent averaging was used to improve the dynamic range of the measurementsystems. Repetitive signals are used in coherent averaging and sampling frequencyand number of samples are chosen such that an integer number of repeated periodsare captured [87].

3.1 Measurement Setups

MIMO Transmitter Setup

The experimental setup used in paper C is shown in Figure 3.1. The setup consistsof two VSGs, operating at the carrier frequencies of 2.14 GHz. Proposed modelswere evaluated against three different cross-talk scenarios. In the first scenario,the cross-talk was introduced at the inputs of the PAs, in the second scenario, the

17

18CHAPTER 3. MEASUREMENT SETUP, POWER AMPLIFIERS AND TEST

SIGNALS

cross-talk was introduced at the outputs of the PAs and in the last scenario, cross-talk was introduced both at the inputs and outputs of the PAs. The cross-talkwas introduced with the use of commercially available directional couplers (modelNarda 4243B-10). Two identical1 PAs were used as DUTs. To study the effectof partially coherent RF signals on the performance of DPD algorithm, the VSGswere only sharing a common 10 MHz reference clock and the common LO wasremoved. The RF outputs of the DUT were down-converted using a wide-banddown-converter and the IF signals were digitized using the above mentioned ADC.

Figure 3.1: Experimental setup with a DUT with input and output cross-talk (paperC). For measuring nonlinear cross-talk effects only, the directional couplers at theoutput were removed.

Paper B uses the same experimental setup with the changes in the down-conversion stage, where the down-conversion from RF to IF was performed withthe use of two mixers and bandpass filters (BPFs). The VSGs were operating atthe carrier frequencies of 1.8 GHz. In paper B, the cross-talk level was −20 dB,whereas in paper C cross-talk levels of −20 and −30 dB were analyzed.

Figure 3.2: Experimental setup for analyzing frequency-domain Volterra kernels ofa 3×3 MIMO system.

Figure 3.2 shows the experimental setup used in paper E for determining thefrequency-domain Volterra kernels of a 3×3 MIMO system. Three VSGs operating

1by identical it means that the PAs were from the same manufacturer and of the same type

with the same specifications.

3.1. MEASUREMENT SETUPS 19

at the carrier frequency of 2.14 GHz are used. The DUT consists of three identicalPAs with input cross-talk. The cross-talk was introduced using locally designedthree channel coupler, where the mutual coupling between the inner and outerchannel is −13.5 dB and −21.5 dB between the two outermost channels. Thedown-converter chain was composed of two mixers and two BPFs. Due to thelimited number of available channels in the ADC, two of the three output channelsof the 3×3 MIMO system were connected with the coaxial switch and were sharingone of the down-converter chains. The coaxial switch (RLC Electronics SR-2 min-H) used has a minimum isolation of 80-dB between the channels (off state) andmaximum insertion loss of 0.1 dB with switching time of 15 ms.

Concurrent Dual-band Transmitter Setup

The experimental setup used in papers A and D is shown in Figure 3.3 (Top), andin paper G measurement setup used is shown in Figure 3.3 (Top/bottom). TwoVSGs operating at the carrier frequencies of 2.0 and 2.3 GHz, respectively, wereused. The RF outputs of the VSGs were combined using a wide-band combinerand the combined RF signal was fed to the DUT. The output of the DUT wasdown-converted to an intermediate frequency (IF) and digitized using the abovementioned ADC. Since the RF signal operates at two carrier frequencies, the LOfrequency of the down-converter is tuned for down-converting the lower and upperband signals one at a time.

Figure 3.3: (Top) Measurement setup used in papers A, D and G. (Bottom) Mea-surement setup used in paper F.

The experimental setup used in paper F is shown in Figure 3.3 (Bottom), andthe VSG were operating at the carrier frequencies of 1.9 and 2.2 GHz. The drive


SIGNALS

amplifier (DA) used is a Mini-circuits ZHL42W wide-band amplifiers. For SISOmeasurements in paper F, only one of the two VSGs was used.

3.2 Experimental Power Amplifiers

In papers A−D, the PAs used are Mini-circuits ZVE8G+ general purpose wide-band amplifier with a small signal gain of 30 dB and an output 1 dB compressionpoint of 30 dBm. The PA has an operational frequency range of 2.0 to 8.0 GHz. Inpaper D, a Freescale Doherty PA with transistor model MRF8S21120HS-Dohertywas used to validate the proposed technique. The Freescale Doherty PA has a gainof 15 dB and an output 1 dB compression point of 46 dBm.

In paper E, the PAs used are Mini-circuits ZHL42W wide-band amplifier witha typical small signal gain of 34 dB (with ±1.3 dB gain flatness) and output 1dB compression point of 30 dBm. The PA has an operational frequency range of10−4200 MHz. In paper F, Mini-circuits ZHL42W amplifiers were used as DA andthe DUT was a Cree GaN amplifier (transistor model Cree CGH21120F-AMP) withsingle carrier modulated gain of 15 dB and an operating frequency band of 1.8−2.3GHz. In paper G, the DUTs were Mini-circuits ZVE8G+, ZHL42W general purposewide-band amplifiers and an Infineon high power RF LDMOS PA with transistormode PTFA210601E with a gain of 16 dB and an output 1 dB compression point of48.3 dBm were used. In paper G, Mini-circuits ZVE8G+ amplifier was used as DAwhen Infineon PA was used as a DUT. The PAs were warmed up for an hour, andthen operated at a steady stage, thus, the assumption of a time-invariant systemshould, hence, be valid.

3.3 Experimental Signals

Different excitation signals were used for characterization, behavioral modeling andlinearization of DUTs. In paper A, the DUT was characterized using two two-tone signals with varying input power and frequency spacing; the power sweep wasperformed between −10 to 0 dBm and the frequency sweep between 1−10 MHz. Theperformance of the proposed model was evaluated by exciting the DUT with twoseparate set of wide-code-division-multiple-access (WCDMA) signals with peak-to-average-power-ratio (PAPR) of more than 7 dB and a sampling rate of 30.72 MHz.A similar set of WCDMA signals were used in paper C. In paper B, multi-tonesignals with a bandwidth of 4.8 MHz and PAPR of more than 9 dB were used toevaluate the performance of the sparse DPD model. The total number of samplesused were 20000 with the sampling rate of 80 MHz.

In paper D and E, two two-tone and three-tone test signals were used, respec-tively, to characterize the DUT. In paper D, the signals were symmetric aroundtheir respective carrier frequencies and were swept both in power and frequencyto characterize the memory effects of concurrent dual-band PAs. The input powerlevel was swept between −16 to 1 dBm and the frequency swept is performed from

3.4. PERFORMANCE METRICS 21

100 kHz to 10 MHz. In paper E, to analyze the 3rd-order frequency domain Volt-erra kernels of the 3×3 MIMO system, the frequencies of the three tones are sweptalong different paths in a three-dimensional (3D) frequency volume.

In paper F, to evaluate the SISO DUT, three different CA signals were used.Each CA signal was composed of two non-contiguous component carriers (CCs) witha frequency separation of ± 25 MHz from the center frequency. The first CA signalwas composed of two orthogonal frequency-division multiplexing (OFDM) signalswith the bandwidth of 10 MHz each, and with a PAPR of 11.42 dB. The secondCA signal was composed of two four-carrier GSM signals, each with a bandwidthof 5 MHz and a PAPR of 8.89 dB. The third CA signal was composed of an OFDMsignal and a GSM signal, with a bandwidth of 10 MHz and 5 MHz, respectively andwith a PAPR of 10.78 dB. For concurrent dual-band measurements in paper F, thefrequency bands were excited with two different 10 MHz wide OFDM signals witha PAPR of 10.71 dB and 10.54 dB, respectively. In paper G, two sets of signalswere used; first set was composed of two WCDMA signals and the second set wascomposed of two 5 MHz wide OFDM signals.

3.4 Performance Metrics

In order to evaluate the performances of the proposed models, the metrics used areas follows. The NMSE is defined as [14]

NMSE =

∫

Φe(f) df∫

Φy(f) df, (3.1)

where Φy(f) is the power spectrum of the measured output signal and Φe(f) isthe power spectrum of the difference between measured and the desired signal;integration is carried out across the available bandwidth. The ACEPR is definedas [14]

ACEPR =

∫

adj. ch. Φe(f) df∫

ch. Φy(f) df, (3.2)

where the integration in the numerator is performed over the adjacent channel withmaximum error power and in the denominator, integration is performed over theinput channel. The ACPR is defined as [14]

ACPR =

∫

adj. ch. Φy(f) df∫

ch. Φy(f) df, (3.3)

where in the numerator integration is performed over the adjacent channel withthe largest amount of power; in the denominator, integration is performed over theinput channel band.

To evaluate the frequency dependency in paper D, 3D asymmetric energy sur-faces were plotted by subtracting the upper intermodulation (IM) and cross-modulation


SIGNALS

(CM) products from their corresponding lower IM and CM products, respectively.As such the amount of asymmetry between the IM and CM products can be de-termined, respectively. The asymmetry between the IM products can be definedas [88]

IMasymmetry = 20 × log10(IMU

IML

) = IMUdB− IMLdB

, (3.4)

where IMU and IML indicates upper and lower IM products. The 2D plots for bothfrequency and input power sweep were also used. Furthermore, for the power sweep,the measured IM and CM amplitudes were also evaluated against a 3:1 amplitudeslope. The 3:1 amplitude slope indicates whether or not the amplitudes of IM andCM products are proportional to the third power of the input amplitude [15].

Chapter 4

Frequency Domain

Characterization Techniques for

RF Amplifiers

Traditionally the nonlinear RF amplifiers are characterized by power-sweepingcontinuous-wave (CW) signals to measure amplitude-amplitude (AM-AM) and amp-litude-phase (AM-PM) distortions. For narrow band systems e.g., global system formobile communications (GSM) the above characterization technique is enough [30].However, for systems with pronounced memory effects, e.g., wideband code-divisionmultiple-access (WCDMA) or long-term evaluation (LTE) 4G, with fast changingsignal envelope, two-tone characterization is usually used [30–34,89]. In a two-tonetest, the amplitude of upper and lower 3rd-order IM products (or 5th-order IMproducts) are measured and used as a measure of nonlinearity [30]. By measuringthe amplitude of the IM products vs. power sweep, the transition from small toa large signal regime of a device can be identified [15, 90]. Asymmetry betweenthe amplitudes of upper and lower IM products and frequency dependency of theIM products versus tone spacing is used as a metric for determining the memoryeffects in the devices’ nonlinear transfer function [30–34]. In recent years, severalmodifications to the two-tone test have been reported. In [91], a two-tone signalis combined with a CW signal for the characterization of AM-AM and AM-PMbehavior of the amplifier. In [89], a two-tone signal is superimposed on a CW sig-nals of different amplitude to characterize different amplitude regimes of PAs. Ina two-tone test, only a limited part of the Volterra kernel is excited. To determinethe 3rd-order Volterra kernel, one has to perform a three-tone test and separate thethird-order products from higher order products [29].

With the continual drive toward increasing data rates and the use of multi-band multi-channel RF transmitters, the characterization techniques for concurrentmulti-band and MIMO transmitters becomes more important due to the presenceof cross-kernels as mentioned in Chapter 2. These multi-tone characterization tech-

23

24CHAPTER 4. FREQUENCY DOMAIN CHARACTERIZATION

TECHNIQUES FOR RF AMPLIFIERS

niques for the above-mentioned systems will not only help to analyze the individualbehavior of self- and cross-distortion products, but the information could also beused to develop or modify previously published behavioral models. To characterizethe individual memory effects of self- and cross-distortion products, a dual two-tone characterization technique is proposed in paper D for concurrent dual-bandRF PAs. In paper E, a three-tone technique to analyze frequency-domain self- andcross-Volterra kernels of a 3×3 MIMO system based upon measurement data ispresented. Along with the analysis of 3rd-order self- and cross-Volterra kernels, thedominating block structures of the underlying system were also determined.

In behavioral modeling of RF PAs, generally the models used are entirely math-ematical, i.e., models are usually built on the knowledge of its input and outputsignals [2] and may have no or little relationship to the phenomena taking placeinside the PAs. The use of techniques such as in paper D and E as a prior, maylead to behavioral models with an insight about the DUT. In paper A, the tech-nique developed in D is applied to modify an existing model for the linearizationof a concurrent dual-band PA and the results are shown in the following; moreover,application of technique presented in paper E is also highlighted at the end of thisChapter. More recently, SISO two-tone measurement are being used and SISOGMP like models are proposed in [35, 36].

4.1 Characterization using Dual Two-tone Test

The technique presented in paper D could be used not only to characterize theindividual memory effects of IM and CM distortion products but also to analyzethe differences or similarities between distortion products at multiple carrier fre-quencies. To analyze the distortion products in a concurrent dual-band nonlinearsystem, the DUT is excited with dual two-tone test signals; and these signals aresymmetric around their respective carrier frequencies. To avoid overlapping of the3rd-order IM and CM products, the dual two-tone signals are design such that∆ωL > ∆ωU ; where ∆ωL and ∆ωU are the lower and upper angular frequencies ofthe two-tone signals operating at the carrier frequencies of ωc1 and ωc2 , respectively.Figure 4.1 illustrates the frequency locations of IM and CM products at ωc1 .

In paper D, power sweep were performed with a step size of 0.09 dB and theinput power was swept from −16 dBm to 1 dBm, and the frequency sweep was bet-ween 100 kHz and 10 MHz. The distortion products were analyzed graphically using2D plots; for the power sweep, the IM and CM products were evaluated against3:1 slope. Any deviation from this behavior indicates that the 3rd-order distortionproducts are not purely due to 3rd nonlinear order [15]. 3D asymmetric energysurfaces [88] were also introduced to identify the regions in power and frequencywhere the memory effects that contributes to asymmetry are more pronounced. Ifthe 3D asymmetric surface is flat, it indicates symmetric memory effects.

Figure 4.2 (left) shows the amplitude of the upper and lower 3rd-order IM prod-ucts versus power sweep. As mentioned earlier, the difference in the amplitude

4.1. CHARACTERIZATION USING DUAL TWO-TONE TEST 25

Figure 4.1: Illustration of frequency location of IM and CM products at a carrierfrequency of ωc1

levels of the IM products indicates the presence of memory effects that contributesto the asymmetry. The 3:1 slope show that the IM products are not purely dueto the 3rd degree nonlinearity and the notch in the lower IM product is due to theopposite phases of 3rd and 5th degree coefficients [15]. Figure 4.2 (right) show theamplitudes of IM products versus tone-spacing. The rapid change in the amplitudeof the IM products with an increase in frequency spacing indicates the presence ofsignificant memory effects, and the different in the amplitude indicates the presenceof memory effects that contributes to the asymmetry.

Figure 4.3 (left) and (right) shows the amplitude of the inner and outer CMproducts versus power and tone spacing, respectively. A comparison between Fig-ures 4.2 and 4.3 shows that for the power sweep, the amplitude of the CM productsincrease proportionally to the 3rd nonlinear order and the memory effects that con-tributes to the asymmetry are negligible compared to those of the IM products.Similar observation can be drawn when amplitude of IM and CM products versustone spacing are compared (cf. Figures 4.2 (right) and 4.3 (right)), where the CMproducts does not exhibit significant memory effects. A comparison between innerand outer CM products indicates that for both power and frequency sweep, theseproducts have approximately the same behavior.

Application of Dual Two-tone Technique

As the individual memory effects of self- and cross-distortion products could beanalyzed using the technique presented in paper D; it is also possible to analyze thedifferences/similarities in distortion products at two different carrier frequencies.As shown in Figure 4.4, the behavior of IM and CM products is approximatelythe same over the different power and frequency regions at two different frequencybands. This technique can be used to get an insight about the nonlinear behaviorof the DUT and the information can be used to modify the behavioral and DPDmodels.

Paper A proposes a 2D modified DPD model (2D-MDPD) where the modifica-tion has been made to a previously published 2D-DPD model [78]. The proposed



−16 −14 −12 −10 −8 −6 −4 −2 0−65

−60

−55

−50

−45

−40

−35

−30

−25

Pin

[dBm]

IM3 [

dB

c]

IMU

IML

−15 −10 −5 00

5

10

15

Asym

metr

y [

dB

]

3:1

106

107

−46

−44

−42

−40

−38

−36

Tone spacing [Hz]

As

ym

me

try

IM

3 [

dB

]

IMU

IML

106

−1

0

1

2

3

Asym

metr

y [

dB

]

Figure 4.2: Measured amplitude of upper and lower IM products versus power (left)and frequency (right). 3:1 line indicates that with the increase in 1 dB input power,distortion products increase by 3 dB. The inset shows the asymmetry between upperand lower IM products.

−16 −14 −12 −10 −8 −6 −4 −2 0−80

−60

−40

−20

CM

inn

er [

dB

c]

CMU

CML

−15 −10 −5 0

−3

−2

−1

0

1

As

ym

me

try

[d

B]

−16 −14 −12 −10 −8 −6 −4 −2 0−80

−60

−40

−20

Pin

[dBx]

CM

ou

ter [

dB

c]

CMU

CML

−15 −10 −5 0−4

−2

0

As

ym

me

try

[d

B]

106

107

−36

−34

−32

CM

inn

er [

dB

c]

106

107

−36

−34

−32

Tone spacing [Hz]

CM

ou

ter [

dB

c]

106

0

0.5

1

As

ym

me

try

[d

B]

106

−0.5

0

0.5

As

ym

me

try

[d

B]

CMU

CML

CMU

CML

Figure 4.3: Measured amplitude of CM products versus power (left) and frequency(right).

Table 4.1: Comparison of the 2D-DPD and 2D-MDPD pre-distorters

Model # of FLOPs NMSE ACPR

(# of coefficients) CH1 / CH2 CH1 / CH2

No DPD − −30.3 / −29.7 dB −40.1 / −39.8 dB

2D-DPD 1162 (180) −42.6 / −42.7 dB −59.6 / −56.5 dB

2D-MDPD 457 (69) −44.3 / −45.1 dB −58.8 / −56.2 dB

model not only results in the same linearization performance, but also results ina number of FLOPs value that is 2.5 times lower than the FLOPs value of the2D-DPD model. The modification is done by characterizing the individual memoryeffects of IM and CM products, and analyzing the behavior of the distortion prod-

4.1. CHARACTERIZATION USING DUAL TWO-TONE TEST 27

107

−10

−5

0−2

0

2

4

6

Freq−sweep (Hz)Pin

IM3

CH

1 A

sym

metr

y [

dB

]

−1

0

1

2

3

4

5

107

−10

−5

0−1

−0.5

0

0.5

1

1.5


CM

CH

1 A

sym

metr

y [

dB

]

−0.5

0

0.5

1

1.5

107

−10

−5

0−2

0

2

4

6


IM3

CH

2 A

sym

metr

y [

dB

]

−1

0

1

2

3

4

5

6

107

−10

−5

0

−1

−0.5

0

0.5

1

1.5


CM

CH

2 A

sym

metr

y [

dB

]

−1

−0.5

0

0.5

1

Figure 4.4: Asymmetry energy surface of upper and lower intermodulation (IM)(top-left) and cross-modulation (CM) (top-right) products at the carrier frequencyof 2.0 GHz. Asymmetry energy surface of upper and lower intermodulation (IM)(bottom-left) and cross-modulation (CM) (bottom-right) products at the carrierfrequency of 2.3 GHz.

ucts at two different carrier frequencies. Table 4.1 summarizes the performance ofthe two models and Figure 4.5 shows the linearized output spectrum of channel 1and 2.

The results obtained in paper A are encouraging, they show that the charac-terization techniques could be of importance, especially for the cases where morethan two frequency bands are concurrently used to amplify the RF signals and MPmodels are used for the linearization.



−10 −5 0 5 10

−60

−40

−20

0

Freqency (MHz)

PS

D [

dB

x/H

z]

−10 −5 0 5 10

−60

−40

−20

0

Freqency (MHz)

PS

D [

dB

x/H

z]

Ych2

2D−DPD

2D−MDPD

Ych1

2D−DPD

2D−MDPD

Figure 4.5: Linearized output spectra of channel 1 and 2 with 2D-DPD and 2D-MDPD models.

4.2 Characterization using Three-tone Test

Analysis of Volterra kernels in frequency-domain for nonlinear dynamic systemsdates back at least to 1983, where in [92], a 2nd-order Volterra kernel was analyzed.Similarly, in [93] and [94], multi-sine signals were used to analyze the 2nd-order Volt-erra kernels. To determine the nth-order Volterra kernel, which is an n-dimensionalfunctions, an input signal with at least n frequencies is required [95]. Thus, toanalyze a 3rd-order Volterra kernel, a minimum of three-tones are required [29].By analyzing Volterra kernels along certain frequency paths, symmetry proper-ties can be obtained and could be used to determine the dominating block struc-tures [29]. In [95], by analyzing the 2nd-order Volterra kernel and its’ symmetryproperties along certain frequency paths, it is shown that the ADC follows a PHblock-structure. Similar approach has been made in [29], where 3rd-order Volterrakernel of a nonlinear dynamic RF PA was analyzed and it was concluded that theunderlying system shows the properties of a Hammerstein-Wiener system.

In the last few years, much interest has been shown in the use of MIMO and con-current multi-band PAs in wireless communication systems due to their potentialto increase the system capacity [43–46]. The amplifiers’ performance, as mentionedearlier, is not only affected by self-distortion products, but also by cross-distortionproducts. Therefore, from the analysis of frequency domain self- and cross-Volterrakernels of MIMO/concurrent multi-band system, the symmetry properties and thedominating block structures can be obtained. The knowledge of the dominatingblock structures could be used to develop or modify behavioral and DPD models.

4.2. CHARACTERIZATION USING THREE-TONE TEST 29

Figure 4.6: Illustration of test signals for determining self- and cross-Volterra ker-nels: (a) a three-tone signal is used for determining the self-kernel H(2:2,2,2) (b) atwo-plus-one signal is used for determining 2×1 cross-kernel H(2:2,2,3) and (c) threesingle-tone signals are used for determining 3×1 cross-kernel H(2:1,2,3).

Figure 4.7: Block structure of the self-kernel H(2:2,2,2)3 .

Characterization of 3×3 MIMO System

A general framework for analyzing the self- and cross-Volterra kernels of MIMOsystems is proposed in [49,96–98]. However, the frequency-domain Volterra kernelsfor nonlinear dynamic MIMO systems has not been analyzed experimentally. PaperE propose a method for the analysis of self- and cross-Volterra kernels of a 3×3nonlinear dynamic MIMO amplifiers based on experimental data. In paper E, theinput channels were excited by a combination of one-, two-, and three-tone signals,the frequencies of which were stepped. The Volterra kernels were analyzed alongcertain paths in the 3D frequency space and the dominating block structures weredetermined. Figure 4.6 illustrates how the test signals were used to determine theVolterra kernels of a 3×3 MIMO system.

To determine the self-kernel, the 2nd-channel of a 3×3 MIMO system was excitedwith a three-tone signal and the output was measured at the corresponding outputchannel. The system consisted of three wideband RF amplifiers; cross-talk wasintroduced at the amplifiers’ input by a microstrip coupler. Such input cross-talkcould occur, e.g., if the amplifiers were integrated on the same chip set [39, 40].

In paper E, the analysis of the self-kernel is restricted to H(2:2,2,2)3 (−ω

′

, ω′′

, ω′′′

)1,

this is because, the self-Volterra kernel is symmetric under all permutations of ω′

,ω

′′

and ω′′′

. The analysis of the self-Volterra kernel revealed the following block

structure of the self-kernel H(2:,2,2,2)3 as shown in Figure 4.7.

1Note that in H(2:r1,r2,r3), 2 corresponds to 2nd output channel and r1, r2, and r3 corresponds

to input channels. If r1 = r2 = · · · = rn, then the kernel is a self-kernel.



Figure 4.8: Block structures of the 2×1 cross-kernel H(2:2,2,3)3 . Hc,1(·) and Ha,1(·)

encompasses encompasses the frequency dependency of the cross-talk.

By using the standard techniques given in [18,63], a discrete time-domain com-plex baseband terms are derived from the block structure shown in Figure 4.7. Thecorresponding complex baseband third-order terms are,

y(2:2,2,2)3 (n) = u(2)(n)

Q∑

q=0

h3(q)|u(2)(n − q)|, (4.1)

where h3(q) are the parameters, and Q is the memory length of Hd,1(·). Equation(4.1) corresponds to the third-order terms in an envelope memory polynomial model[99]. The above analysis indicates that for the behavioral modeling of the studiedsystem, the inclusion of this term is of importance.

To determine the 2×1 cross-kernel H(2:2,2,3)3 , the system was excited with a

two-plus-one signal as shown in Figure 4.6 (b), i.e., the 2nd and 3rd input channelswere excited respectively with a two-tone signal and a single-tone signal. The 3rd-order 2 × 1 cross-kernel is symmetric under the permutations of ω

′

and ω′′

, i.e.,

H(2:2,2,3)3 (−ω

′

, ω′′

, ω′′′

) = H(2:2,2,3)3 (ω

′′

, −ω′′

, ω′′′

), but not for the other permuta-

tions of ω′

, ω′′

and ω′′′

. Hence, the analysis of the 2×1 cross-kernel was restricted to

H(2:2,2,3)3 (−ω

′

, ω′′

, ω′′′

) and H(2:2,2,3)3 (ω

′

, ω′′

, −ω′′′

). The analysis of the 2×1 cross-kernel revealed that two dominating block structures as shown in Figure 4.8 arerequired to model 2×1 cross-kernel, where the upper block corresponds to the kernel

H(2:2,2,3)3 (−ω

′

, ω′′

, ω′′′

), and the lower block corresponds to H(2:2,2,3)3 (ω

′

, ω′′

, −ω′′′

).In Figure 4.8, Hc,1(·) and Ha,2(·) model the effect of cross-talk.

The corresponding discrete time-domain complex baseband equivalent terms of

4.2. CHARACTERIZATION USING THREE-TONE TEST 31

Figure 4.9: Block structures of the 3×1 cross-kernel H(2:1,2,3)3 . Ha−c,1−c(·) encom-

passes the frequency dependency of the cross-talk.

the block structure in Figure 4.8 are

y(2:2,2,3)3 (n) =

Q1∑

q1=0

Q2∑

q2=0

h3,1(q1, q2)u(3)(n − q2)|u(2)(n − q1)|2

+u(2)(n)

Q1∑

q1=0

Q2∑

q2=0

h3,2(q1, q2)u(2)(n − q1)u∗(3)(n − q1 − q2)

+u(2)(n)

Q1∑

q1=0

Q2∑

q2=0

h3,2(q1, q2)u∗(2)(n − q1)u(3)(n − q1 − q2).

(4.2)

In (4.2), Q1 corresponds to the memory length of Hd,1(·) and Hd,2(·), and Q2 is thememory length of Hc,1(·) and Ha,2(·). The first term in (4.2) corresponds to the2×2 PH model [43] if q1 = q2. The second and third term in (4.2) correspond to thethird-order terms of extended generalized memory polynomial model for nonlinearcross-talk (EGMPNLC) presented in paper C. The above analysis for a self and2×1 cross-kernels indicates that, for a behavioral model to give accurate descriptionof the underlying system, the terms presented in (4.1) and (4.2) are of importance.

To analyze the 3×1 cross-kernel H(2:1,2,3)3 , each channel was excited with a

single-tone signal as shown in Figure 4.6 (c). It was found that the 3×1 cross-kernel can be described by the block structure shown in Figure 4.9.

The characterization technique presented in paper E can be used to identifythe dominating block structures of M×M nonlinear dynamic MIMO system. Theinformation extracted from this technique could be used to modify, or to develop



−20 −15 −10 −5 0 5 10 15 20

−70

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

PS

D (

dB

x/H

z)

OutputCH1

2×2 PH

2×2 PH mod

−20 −15 −10 −5 0 5 10 15 20

−70

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

PS

D (

dB

x/H

z)

OutputCH2

2×2 PH

2×2 PH mod

Figure 4.10: Measured output of Channel 1 and 2 of a 2×2 MIMO PA in presenceof cross-talk. The error spectrum of 2×2 PH model with (red) and without additionof terms given in (4.1) and (4.2).

novel behavioral models with improved performance. In behavioral models of theRF PAs, higher nonlinear orders are seldom required e.g., up to 9th, thus, to analyzeand visualize the nonlinear properties of higher orders terms, it is possible to extendthe technique. However, the analysis of higher order terms would be tedious as thefrequency space becomes multidimensional.

Application of Three-tone Technique

To illustrate the importance of the technique proposed in paper E, in the followinga modification is made to the 2×2 PH model by adding the terms given in (4.1)and (4.2). Without the modification, the 2×2 PH model resulted in NMSE valuesof −39.5 and −41.6 dB, respectively, for channel 1 and 2, and the correspondingACEPR values are −43.5 and −45.3 dB. With the modification, the modified modelresults in NMSE values of −44.9 and −47.2 dB respectively, for channel 1 and 2,with the corresponding ACEPR values of −49.6 and −51.3 dB. Figure 4.10 showsthe error spectrum of the models with and without the terms given by (4.1) and(4.2).

The above result indicates that the performance of the behavioral model im-proves with the inclusion of dominating terms. The envision of capacity increasewith the increase in the number of channels in MIMO and concurrent multi-bandtransmitters, these characterization techniques become more important. The tech-niques proposed in papers D and E could be used to extract information of thesystem and the knowledge could be incorporated into the models to achieve betterperformance for behavioral modeling and DPD.

Chapter 5

Behavioral Modeling and

Linearization of Multi-channel

Multi-band Amplifiers

This chapter summarizes the behavioral models for 2×2 MIMO, concurrent dual-band and SISO PAs proposed in papers C, F and G. Paper C proposes threemodels for behavioral modeling and DPD of 2×2 MIMO transmitters in presenceof linear, nonlinear, and nonlinear-&-linear crosstalk. The DPD structure andits relationship to the DUT under different cross-talk scenarios is also discussed.Furthermore, the impact of coherent and partially non-coherent signal generationon DPD performance has also been studied. The performance of the proposedmodels is compared with the 2×2 PH model [44].

Paper F present models based on a fixed pole expansion technique (FPET) forbehavioral modeling and DPD of SISO and concurrent dual-band PAs with long-term memory effects. The proposed models are reduced forms of the FPET basedVolterra series for SISO [26] and concurrent dual-band PAs. Paper G presentsa 2D extended envelope memory polynomial (2D-EEMP) model that is derivedbased on physical knowledge of a concurrent dual-band PA. The 2D-EEMP modelis an extension of the EEMP model proposed in [13] for SISO PAs. Behavioralmodels (and DPD) of multi-channel multi-band transmitter suffer from a largenumber of model parameters, making it difficult to identify the most significantmodel parameters. At the end of this chapter, the parameter reduction techniquepresented in paper B is summarized.

The chapter is structured as follows; in the first sections, the models proposedin paper C are presented and the results are summarized, followed by the summaryof papers F and G. Results from the parameter reduction techniques are presentedat the end.

33

34CHAPTER 5. BEHAVIORAL MODELING AND LINEARIZATION OF

MULTI-CHANNEL MULTI-BAND AMPLIFIERS

5.1 Behavioral Modeling and Linearization of MIMO PAs

In MIMO transmitters, ideally, each channel has an independent and isolated path.However, with the integration of MIMO transmitters on the same chip-set [40]and towards efficient hardware implementation, the hardware is associated withleakage (coupling) between different paths. This coupling together with nonlineardistortion of the PAs in MIMO transmitters can cause significant degradation if nothandled properly [44]. Paper C proposes three models for modeling and mitigationof nonlinearities in 2×2 MIMO transmitters in presence of cross-talk. Three typeof cross-talk effects were studied, namely nonlinear cross-talk, linear cross-talk,and nonlinear-&-linear cross-talk. In presence of linear cross-talk only, the outputof the 2×2 MIMO PA can be represented as in (2.10). The nonlinear transferfunctions, f1(·) and f2(·) in (2.10) can be modeled by SISO MP models availablein the literature. In paper C, we used the SISO GMP model, since it is basedon the physical knowledge of the PAs [67] and has been used extensively for bothbehavioral modeling and DPD of SISO PAs. By replacing the nonlinear transferfunctions f1(·) and f2(·) in (2.10) with a GMP model, the resulting model is ageneralized memory polynomial model for linear cross-talk (GMPLC)

yi(n) =

P∑

p=1

Q1∑

q1=0

Q2∑

q2=0

gip,(q1,q2)ui(n − q1)|ui(n − q1 − q2)|2(p−1)+

P∑

p=1

Q1∑

q1=0

Q2∑

q2=0

gjp,(q1,q2)uj(n − q1)|uj(n − q1 − q2)|2(p−1),

(5.1)

where ui(n) and yi(n) are the ith channel input and output signals, respectively. In(5.1) gip,(q1,q2)

are complex-value model parameters, P is the nonlinear order, andQ1 and Q2 are the memory depths.

In presence of nonlinear cross-talk only, the output of the 2×2 MIMO PA canbe represented as in (2.9), where the nonlinear transfer functions f1(·) and f2(·)operate simultaneously on both input signals, and the output signals contain boththe self- and cross-distortion products. In order to compensate for these effects, themodel should include both the self- and cross-distortion products. Hence, by intro-ducing the cross-terms in the GMPLC model, the resulting models are the extendedgeneralized memory polynomial model for nonlinear cross-talk (EGMPNLC) [cf Ta-ble I in paper C] and generalized memory polynomial model for nonlinear cross-talk

5.1. BEHAVIORAL MODELING AND LINEARIZATION OF MIMO PAS 35

(GMPNLC). The GMPNLC model is written as

yi(n) =P

∑

p=1

P −p+1∑

r=1

Q1∑

q1=0

Q2∑

q2=0

gi,p,r,(q1,q2)ui(n − q1)|ui(n − q1 − q2)|2(p−1)

·|uj(n − q1 − q2)|2(r−1) +

P∑

p=1

P −p+1∑

r=1

Q1∑

q1=0

Q2∑

q2=0

gj,p,r,(q1,q2)uj(n − q1)

·|uj(n − q1 − q2)|2(p−1)|ui(n − q1 − q2)|2(r−1).

(5.2)

The difference between the EGMPNLC model and the GMPNLC model is that theformer contains more cross-terms. Furthermore, it is shown in paper C that theEGMPNLC is a subset of the 2×2 MIMO Volterra model (2.6), and GMPLC ⊂GMPNLC ⊂ EGMPNLC, where ⊂ denotes subset.

Behavioral Modeling of MIMO PAs

Table 5.1 summarizes the performance of the proposed models. For all studiedcross-talk scenarios, the 2×2 PH model resulted in largest model error than theproposed models. These results indicate that the 2×2 PH model is not as suited asthe proposed models. In presence of linear cross-talk, the proposed models resultedin approximately the same model performance in terms of NMSE, whereas in termsof ACEPR, the GMPLC resulted in an ACEPR value that is 0.3-1.3 dB highercompared to the GMPNLC and EGMPNLC models. In presence of nonlinear,and nonlinear-&-linear cross-talk, among the proposed models, the GMPLC modelresulted in the largest model error, and EGMPNLC resulted in the lowest modelerror. The increase in model error for the GMPLC model is due to the missingcross-terms. In comparison, the GMPNLC resulted in NMSE and ACEPR valuesthat are approximately 5- and 1.2 − 2 dB higher, respectively, than those of theEGMPNLC model.

Table 5.1: NMSE/ACEPR [dB] for given behavioral models and cross-talk type,the cross-talk level was -20 dB.

Model Linear Nonlinear Nonlinear-&-Linear No ofNMSE/ACEPR NMSE/ACEPR NMSE/ACEPR coeff

2×2 PH -41.5/-56.6 -40.1/-54.7 -40/-54.3 180GMPLC -50.1/-58.9 -42.3/-52.7 -40.1/-51.9 126

GMPNLC -50.3/-59.2 -45.4/-58.9 -45.1/-58.3 242EGMPNLC -50.2/-60.2 -50.4/-60.1 -50.3/-60.3 486

As mentioned in Chapter 2, the technique proposed in [80] to identify cross-talktype can be used in advance for pre-selection of the proposed models. For example,in case of linear cross-talk, the GMPLC model can be used since it has the samemodel performance as the other proposed models but with fewer model parameters.



Figure 5.1: DUT and corresponding DPD structure

Linearization of MIMO PAs

In the ILA, the post-inverse of the DUT is identified and is used as a pre-inverse(DPD); to identify the DPD parameters, the input and output signals are inter-changed in (2.11) and (2.13) [61]. Figure 5.1 shows the relationship between theDUT under different cross-talk conditions and the corresponding DPD structures.For the DUT in presence of nonlinear cross-talk, the corresponding DPD structurecan be described as the inverse of nonlinear transfer function f−1(·) followed bythe cross-talk, as shown in Figure 5.1-(a). To linearize the DUT with nonlinearcross-talk, the DPD structure can be modeled by the GMPLC model. Figure 5.2shows the linearized output spectrum of a DUT in presence of nonlinear cross-talkand Table 5.2 summarizes the performance of the given models. In Table 5.2, itcan be observed that the proposed models, when used for the linearization of DUTwith nonlinear cross-talk, resulted in approximately the same performance.

Table 5.2: NMSE/ACPR [dB] for given behavioral models used as DPD for differentcross-talk type, the cross-talk level was -20 dB.

Model Linear Nonlinear Nonlinear-&-Linear No ofNMSE/ACPR NMSE/ACPR NMSE/ACPR coeff

2×2 PH -39.7/-54.3 -40.1/-52.1 -38.1/-51.3 180GMPLC -41.2/-52.1 -45.3/-58.6 -37.3/-48.3 126

GMPNLC -45.3/-58.2 -45.1/-58.5 -45.8/-58.7 242EGMPNLC -45.1/-58.4 -45.4/-58.8 -45.6/-57.9 486

Similarly, in presence of linear cross-talk, the DPD structure can be described asshown in Figure 5.1-(b), and to linearize the DUT with linear cross-talk, the DPDstructure can be described by the GMPNLC and EGMPNLC models. Table 5.2shows the performance of GMPNLC and EGMPNLC models when used as DPDwhich resulted in approximately the same NMSE and ACPR values. The GMPLC

5.1. BEHAVIORAL MODELING AND LINEARIZATION OF MIMO PAS 37

−6 −4 −2 0 2 4 6

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

PS

D (

dB

x/H

z)

No DPD

DPD SISO GMP

DPD GMPLC

DPD GMPNLC

DPD EGMPNLC

DPD 2x2 PH

Figure 5.2: Linearized output of the DUT in presence of nonlinear cross-talk

model resulted in NMSE and ACPR values that are approximately 4 − 6 dB higherthan those of the GMPNLC and EGMPNLC models. A similar observation can bemade for DUT in presence of nonlinear-&-linear cross-talk, where the GMPNLCand EGMPNLC resulted in approximately the same performance. In paper C,effects of coherent and partially coherent signal generation on the performance ofDPD was also evaluated. Figure 5.3 shows the impact of coherent and partiallycoherent transmitter on the performance of DPD, where partially coherent signalgeneration degrades the DPD performance.



10 20 30 40 50 60 70 80 90 100−50

−48

−46

−44

−42

NM

SE

(d

B)

10 20 30 40 50 60 70 80 90 100−60

−58

−56

−54

−52

Number of Averages

AC

PR

(d

B)

Partially coherent

Coherent

Partially coherent

Coherent

Figure 5.3: Measured NMSE and ACPR vs number of coherent averaging of thelinearized output signal with coherent and partially coherent signal generation. TheDUT has -30 dB nonlinear cross-talk and the inverse model for DPD was GMPNLC.

5.2 Modeling and Linearization of Single and Multi-band

GaN PAs

In cellular base stations, LDMOS PAs have been employed as the technology ofchoice for high power operations [100]. The LDMOS PAs have the advantagesof high power and high operating temperatures [101], but may reach there limi-tation in operational frequency and power efficiency for future base stations. It isenvisioned that GaN transistor based PAs have the potential to be the long-term re-placement for LDMOS PAs in future base station applications [51,52] due to higherpower efficiency, operating temperature, and breakdown voltages [53, 54]. In [53],it is reported that when excited with burst signals, GaN PAs introduce long-termmemory effects such as bias circuit modulation, charge trapping, and self-heating.Therefore, behavioral and DPD models are required that incorporate such effects.

Paper F presents behavioral models that are based on the FPET (hereafterreferred to as FP-models) for SISO and concurrent dual-band PAs. The formulatedFP-models are reduced forms of the FP-Volterra series (FP-VS) [26] in the sameway as the FIR based MP [56] and GMP [67,102] models are reduced forms of theFIR-VS. In these models, the input signal is filtered by an infinite impulse response(IIR) filter (defined by Kautz functions [103]) and used in the FP-models. The IIRmakes them potential candidates for modeling and mitigating nonlinear distortionwith long-term memory effects.

A discrete-time, odd-order SISO FP-VS with input and output signals u(n) and

5.2. MODELING AND LINEARIZATION OF SINGLE AND MULTI-BAND

GAN PAS 39

y(n), respectively, can be described as [26],

y(n) =

P∑

p=1p:odd

Q1−1∑

q1=0

· · ·

Qp−1∑

qp=0

kp,q1,...,qp

p+12

∏

d=1

vp,qd(n)

p∏

d= p+12 +1

v∗

p,qd(n), (5.3)

where Q1 is the number of basis functions for order 1, and kp,q1,...,qpare the model

parameters. In (5.3), (·)∗ denotes the complex-conjugate operator, and vp,qd(n) is a

filtered version of u(n) as the pth output of the IIR filter [for filter definition cf (2)in F]. To reduce the number of parameters of the FP-VS in (5.3) and to formulatethe FP-MP model, we set q1 = q2 = · · · = qp = q, and Q1 = · · · = Qp = Q in (5.3),and obtain the following

y(n) =

P∑

p=1p:odd

Q−1∑

q=0

kp,qvp,q(n)|vp,q(n)|(p−1). (5.4)

The FP-MP model only contains the main diagonal terms of the FP-VS. The FP-MP model can, hence, model the long term memory effects in a system with sig-nificantly reduced number of model parameters. Moreover, by setting the pole atthe origin, (5.4) becomes an FIR-MP model [2,56]. Similarly, to formulate the FP-GMP model, the diagonal and off-diagonal terms of the SISO FP-VS are considered[cf (5) in paper F]. For concurrent dual-band PAs, 2D-FP-VS can be described as

y(i)(n) =

P∑

p=0

Q1−1∑

q1=0

· · ·

Q2p+1−1∑

q2p+1=0

p+1∑

p′ =1

k(i)

(2p+1,q1,··· ,q2k+1,p′ )

p′

∏

d=1

v(i)2p+1,qd

(n)

·

2p′

−1∏

d=p′ +1

v(i),∗2p+1,qd

(n)

p′

+p∏

d=2p′

v(j)2p+1,qd

(n)

2p+1∏

d=p′ +p+1

v(j),∗2p+1,qd

(n), (5.5)

where v(i)2p+1,qd

(n) and v(j)2p+1,qd

(n) are the outputs of IIR filters excited by the input

signals u(i)(n) and u(j)(n), respectively, from the ith and jth channel and y(i)(n) isthe ith channel output. The IIR filters excited by u(i)(n) and u(j)(n) may have dif-ferent poles per nonlinear order for each channel because the matching network(s)of concurrent dual-band PAs are optimized for the different carrier frequencies.As compared to the SISO FP-VS, the 2D-FP-VS has more number of model pa-rameters. Thus, to reduce the number of model parameters, paper F proposesthe 2D-FP-MP and 2D-FP-GMP models. The 2D-FP-MP model given in (5.6) isformulated by setting q1 = · · · = q2p+1 = q and Q1 = · · · = Q2p+1 = Q in (5.5).

y(i)(n) =

P −1∑

p=0p:even

p∑

r=0r:even

Q−1∑

q=0

k(i)p,r,qv(i)

p,q(n)|v(i)p,q(n)|p−r|v(j)

r,q (n)|r . (5.6)



Table 5.3: Performance evaluation of given SISO behavioral models in terms ofNMSE [dB] and ACEPR [dB]. CC1 and CC2 consists of 10-MHz OFDM signalsand are at the offset frequencies of ∓ 25 MHz from the carrier frequency.

Models OFDM-OFDMNMSE ACEPR(CC1/CC2)

FIR-MP −36.4 −42.1/ − 41.9FIR-GMP −38.9 −46.1/ − 42.5FIR-VS −41.3 −48.7/ − 50.1FP-MP −45.2 −55.2/ − 56.8

FP-GMP −47.8 −57.3/ − 58.4FP-VS −49.3 −59.8/ − 59.3

As in the case of the SISO FP-GMP model, the 2D-FP-GMP model is formulatedby considering the diagonal and off-diagonal terms of the 2D-FP-VS [cf (9) in paperF].

SISO and Concurrent Fixed Pole Model Performance

Table 5.3 summarizes the modeling performance of the SISO FP-models when theGaN PA was excited with intra-band non-contiguous CA signal; the componentcarriers (CCs) consisted of two OFDM signals at an offset frequencies of ± 25 MHzfrom the carrier frequency. One to one comparison between the FP- and FIR-modelsindicates that the FP models resulted in NMSE values that are approximately8 − 10 dB lower than those of the FIR-models. Similarly, the FP-models resultedin ACEPR values that are approximately 9 − 15 dB lower than those of the FIR-models. These results indicate that the FIR-models are not adequate models forthe DUTs with long-term memory effects. Moreover, the proposed FP-models havethe same number of model parameters as the FIR-models.

Table 5.4 summarizes the performance of the proposed SISO FP-models whenused as DPD algorithms and Figure 5.4 shows the linearized output spectra. Whenused for DPD, the FIR-models resulted in NMSE values that are in the rangeof 13 − 17 dB than when no DPD was used, whereas, for the FP-models, DPDresulted in NMSE values within the range of 21 − 25 dB. One to one comparisonbetween the FIR and FP-models shows that the FP-model have NMSE values thatare approximately 8 dB lower than the FIR-models. Similar observation can bemade when the performance of FP and FIR-models is evaluated in term of ACPR;the FP-models show an improvement of approximately 8 − 15 dB in ACPR valuescompared to the ACPR values of the FIR-models. In terms of total number ofFLOPs, the FP-models give slightly higher FLOPs values than the FIR-models.However, if the maximum spectral emission limit of −45 dB is applied [104], themodel complexity of the FP-models [cf Table 5.4] can be reduced to meet theminimum required spectrum limit, which is not the case for the FIR-models.

5.2. MODELING AND LINEARIZATION OF SINGLE AND MULTI-BAND

GAN PAS 41

Table 5.4: Performance evaluation of given SISO DPD models in terms of NMSE[dB], ACPR [dB] and total number of floating point operations (FLOPs). CC1 andCC2 are at the offset frequencies at ∓ 25 MHz from the carrier frequency.

Models OFDM-OFDM FLOPsNMSE ACPR(CC1/CC2)

No DPD −23.1 −34.3/ − 34.9 −FIR-MP −36.6 −42.7/ − 42.2 209

FIR-GMP −38.9 −46.0/ − 44.6 529FIR-VS −40.6 −49.6/ − 51.7 7471FP-MP −44.9 −56.5/ − 56.6 230

FP-GMP −47.6 −58.7/ − 59.5 550FP-VS −48.2 −59.1/ − 60.1 7511

−40 −30 −20 −10 0 10 20 30 40

−70

−60

−50

−40

−30

−20

−10

0

Frequency [MHz]

PS

D [

dB

x/H

z]

Output

FIR−MP

FIR−GMP

FIR−VS

FP−MP

FP−GMP

FP−VS

Figure 5.4: Linearized output spectra of the SISO DUT excited with non-contiguousCCs consisting of OFDM signals at an offset frequencies of ± 25 MHz from the CF.The DPD algorithms are described in the legend.

Figure 5.5 shows the measured and linearized output spectra of concurrent dual-band GaN PA excited with two OFDM signals operating at the carrier frequenciesof 1.9 and 2.2 GHz. Table 5.5 summarizes the performance of different models.The improvement in NMSE values of the 2D-FP models compared to the 2D-FIRmodels is in the range of approximately 5 − 8 dB. Similarly in terms of ACPR, the2D-FP models give ACPR values that are 9−14 dB lower than those of the 2D-FIRmodels.

The FP-models presented in paper F achieve excellent performance both formodeling and linearization of SISO and concurrent dual-band PAs. Moreover, theproposed models have approximately the same model complexity as the correspond-ing FIR-models. However, the only disadvantage is that the FP-models are nonlin-ear functions of the poles and thus nonlinear estimation techniques are required for



−20 −15 −10 −5 0 5 10 15 20−70

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

PS

D (

dB

x/H

z)

Output

2D−FIR−MP

2D−FIR−GMP

2D−FIR−VS

2D−FP−MP

2D−FP−GMP

2D−FP−VS

−20 −15 −10 −5 0 5 10 15 20−70

−60

−50

−40

−30

−20

−10

0

Frequency (MHz)

PS

D (

dB

x/H

z)

Output

2D−FIR−MP

2D−FIR−GMP

2D−FIR−VS

2D−FP−MP

2D−FP−GMP

2D−FP−VS

Figure 5.5: Linearized output spectra of channel 1 (left) and channel 2 (right) whenthe concurrent dual-band PA was excited with 10 MHz wide OFDM signals.

Table 5.5: Performance evaluation of given models in terms of NMSE [dB] andACPR [dB] when concurrent dual-band PA was excited with two 10 MHz wideOFDM signals.

Models Channel 1 Channel 2 FLOPsNMSE ACPR NMSE ACPR

No-DPD −23.6 −32.6 −23.1 −33.4 -2D-FIR-MP −37.4 −42.2 −37.9 −45.6 752

2D-FIR-GMP −40.7 −45.0 −40.5 −47.7 14082D-FIR-VS −42.4 −49.8 −42.9 −48.2 133632D-FP-MP −44.1 −56.4 −44.4 −55.3 804

2D-FP-GMP −48.5 −58.9 −47.0 −56.8 14602D-FP-VS −48.6 −59.6 −47.8 −58.0 13519

pole estimation. Nevertheless, once the poles are identified, linear LSE techniquescan be employed to identify the model parameters.

5.3 Derivation of a Concurrent Dual-band Model

Paper G present a 2D extended envelope memory polynomial (2D-EEMP) model,which is derived based on the physical knowledge of dual-band PAs. The blockstructure considered is shown in Figure 5.6 and is proposed in [58]. It has beenused previously to derive PA models [13, 16, 60, 105] for SISO PAs. In paper G,the derivation of the SISO EEMP model [13] is extended to concurrent dual-bandPAs. In the following, we discuss the assumptions made in paper G and presentthe derived 2D-EEMP model.

In Figure 5.6, HI(ω) and HO(ω) represent the time-invariant filters that des-cribe the input and output matching networks that are optimized for two carrier

5.3. DERIVATION OF A CONCURRENT DUAL-BAND MODEL 43

+

−

x(t) u(t) e(t) v(t) y(t)

HI(ω) K(·) HO(ω)

F (ω)

Figure 5.6: The model structure proposed in [58] which represents the bandpassbehavior of a RF PA. HI(ω) and HO(ω) are linear filters representing the inputand output matching networks, F (ω) is a linear filter for the feedback representingbias modulation and thermal memory effects, and K(·) is a static nonlinearity [16].x(t) is the input signal, and y(t) is the output signal.

frequencies, e.g., ωc1and ωc2

. The feedback filter F (ω) is a low-pass filter and des-cribes thermal, bias, and trapping effects [58]. K(·) is a nonlinear current source,generating nonlinear distortions [13, 58].

The input signal x(t) in Figure 5.6 is composed of two independently modulatedsignals operating at ωc1

and ωc2

x(t) = x1(t) + x2(t) = A1(t) cos(ωc1t + φ1(t)) + A2(t) cos(ωc2

t + φ2(t)) (5.7)

where Ai(t) and φi(t) are the amplitude and phase modulation of carrier ci. Toderive the discrete-time complex base-band 2D-EEMP model the following assump-tions were made.

• The frequency dependence of the input matching network HI(ω) is negligible,i.e., u(t) is a delayed and scaled version of x(t). Presence of strong memoryeffects at the input matching network would degrade the error vector mag-nitude of the amplified signal [13]. Moreover, without this assumption, theresulting system would be a Wiener-type system.

• K(·) is static [13,58] and its output can be approximated by a polynomial oforder P, such that

v(t) = K[e(t)] = K[u(t) − f ∗ v(t)] =P∑

p=1

ap[u(t) − f ∗ v(t)]p, (5.8)

where f is the impulse response of the feedback filter F (ω) and v(t) is theoutput signal from the static nonlinearity. The feedback loop is broken afterone loop as in [13]. This is a well founded approximation since the looprepresents effects that are small relative to the output signals [58], and it isimplicitly assumed in all memory polynomial type models.



• F (ω) is of low-pass characteristics and does not pass any signal componentsat ωc1

or above.

• To derive the 2D-EEMP model, two further assumptions related to the rel-ative magnitude of linear and nonlinear components, and to the frequencydependence of HO(ω) and F (ω) were made; without these assumptions, theresulting model would be a 2D-GMP like model. The first assumption is thatthe power of the linear signal components is significantly larger than the non-linear components; this is equivalent to the common assumption that the PAis weakly nonlinear. The second assumption is that the frequency dependenceof F (ω) is considerably larger than the HO(ω), within the excited bandwidth.The above mentioned assumption means that the filtering by HO(ω) can beneglected for nonlinear terms but not for the linear terms.

Based upon these assumptions and approximations, the resulting 2D-EEMP modelbecomes

yi(n) =Q1∑

q1=0

gi,1q1

ui(n − q1) +P −1∑

p=2p:even

p∑

r=0r:even

Q2∑

q2=0

gi,2p,r,q2

ui(n)|ui(n − q2)|(p−r)

·|uj(n − q2)|(r) +P −3∑

p=2p:even

r∑

r=0r:even

2∑

s=0s:even

Q2∑

q2=1

gi,3p,r,s,q2

ui(n)|ui(n)|(p−r)|uj(n)|(r)

·|ui(n − q2)|(2−s)|uj(n − q2)|(s) +2∑

r=0r:even

P −3∑

γ=4γ:even

γ∑

s=0s:even

Q2∑

q2=1

gi,4r,γ,s,q2

ui(n)|ui(n)|(2−r)

·|uj(n)|(r)|ui(n − q2)|(γ−s)|uj(n − m2)|(s) +P −3∑

p=0p:even

p∑

r=0r:even

Q2∑

q2=1

gi,5p,r,q2

uj(n)|ui(n)|p−r

·|uj(n)|rui(n − q2)u∗

j (n − q2),(5.9)

where ui(n) and yi(n) are the input and output signals at ωci, respectively.

In the following, the DPD performance of the derived model is compared withthe 2D-DPD model [78] and the dual-band GMP model (DB-GMP) [106]. Figure5.7 shows the linearized output spectra and Table 5.6 summarizes the performanceof the used models. The proposed model resulted in the NMSE values of −46.7dB and −48.0 dB for channel 1 and 2, respectively, and in ACPR values of −55.8dB and −54.6 dB for channel 1 and 2. In comparison, the 2D-DPD resulted inthe largest error. In terms of number of model parameters, the proposed modelhas 107 parameters, whereas, the 2D-DPD and DB-GMP have 112 and 320 modelparameters, respectively. In Figure 5.7 it can be seen that the linearized spectra ofthe 2D-DPD and DB-GMP model flatten out at an offset frequency of ± 5 MHz

5.4. PARAMETER REDUCTION TECHNIQUES 45

Table 5.6: Performance evaluation of the given models in terms of NMSE (dB) andACPR (dB) for the ZVE8G+ amplifier.

Models Channel 1 Channel 2(P, M1, M2) NMSE ACPR NMSE ACPR

No DPD −28.4 −33.4 −29.3 −36.82D-DPD (7,4) −39.5 −45.6 −38.7 −47.4

DB-GMP (7,4,3) −45.9 −53.6 −41.7 −50.32D-EEMP (7,4,3) −46.7 −55.8 −48.0 −54.6

−15 −10 −5 0 5 10 15

−60

−40

−20

0

Frequency (MHz)

PS

D (

dB

x/H

z)

−15 −10 −5 0 5 10 15

−60

−40

−20

0

PS

D (

dB

x/H

z)

OutputCH2

2D−DPD

DB−GMP

2D−EEMP

OutputCH1

2D−DPD

DB−GMP

2D−EEMP

Figure 5.7: Linearized output spectra of ZVE8G+ amplifier. The amplifier wasexcited with two 5 MHz wide OFDM signals operating at the carrier frequencies of2.0 and 2.3 GHz.

and beyond, whereas, for the proposed model this does not happen. Presently wedo not have any explanation for this behavior.

5.4 Parameter Reduction Techniques

For behavioral modeling and DPD of multi-channel and multi-band transmitters,the MIMO and concurrent dual-band models often have large number of parame-ters. In paper B a sparse estimation technique is used to find model with reducedparameters. Paper B discusses the MIMO transmitter.

In paper B, the least absolute shrinkage and selection operator (LASSO) is usedto reduce the parameters of a 2×2 MIMO Volterra series. The MIMO Volterra serieswith 1400 basis functions was used, and by implementing the LASSO technique, thenumber of basis functions is reduced to 220 basis functions at the cost of a slightlyincreased model error. The number of basis functions versus NMSE is shown inFigure 5.8.



−50 −45 −40 −35 −30 −25 −20 −150

250

500

750

1000

1250

1500

NMSE (dB)

Num

ber

of basis

functions

Output 1

Output 2 Full Volterra LS post−distorter

Linear LS post−distorter

Figure 5.8: Performance versus number of basis functions used in the 2×2 MIMOVolterra series.

In Figure 5.8, it can be seen that large reduction in model parameters can beachieved at a moderate cost of model performance. For example, by setting thedesired NMSE value to −42 dB, the model parameter reduced to 220 that areapproximately 7 times lower than the number of parameters of the full MIMOVolterra.

Chapter 6

Conclusion

The main goal of this thesis is to develop new behavioral models, characterizationtechniques, and compensation methods of nonlinear distortions in concurrent dual-band and MIMO transmitters. In this context, two topics have been studied. Thefirst topic is the characterization techniques, and the second topic addresses the be-havior modeling and linearization of the concurrent dual-band and MIMO transmi-tters.

A technique for the characterization of individual memory effects of 3rd-orderIM and CM product in concurrent dual-band PAs is proposed in paper D, and caneasily be adapted to MIMO PAs. It is an extension of a conventional two-tone testwhich is widely used for the characterization of SISO PAs. The analysis made indi-cates that for the studied system, the IM products show significantly larger memoryeffects than the CM products. Moreover, the memory effects that contribute to theasymmetry are more pronounced in the IM products than the CM products. Theinformation extracted from the characterization of the memory effects in concur-rent dual-band amplifiers could be used to develop new behavioral models and DPDschemes or to modify existing ones. The application of the characterization tech-nique proposed in paper D is highlighted in paper A, where a previously publishedmodel is modified after the characterization of the DUT. The resulting modifiedmodel not only compensated the nonlinear distortions in a concurrent dual-bandPA but also results in lower model complexity with approximately the same per-formance.

To characterize the 3rd-order self and cross-Volterra kernels a three tone cha-racterization technique is also proposed. It is shown that from the analysis of self-and cross-Volterra kernels along certain frequency paths and by analyzing the sym-metry properties, the dominating block structure of a 3×3 MIMO system can bedetermined. This knowledge can be used for the development of new behavioralmodels or to modify previously published models. In this thesis, it is demonstratedthat by including the terms extracted from the analysis of the self and cross Vol-terra kernels into the previously published model, the modified model shows an

47

48 CHAPTER 6. CONCLUSION

improvement of more than 4 dB in NMSE and 6 dB in ACEPR.For the modeling and compensation of nonlinear distortions in MIMO trans-

mitter, paper C proposes three novel behavioral models. The proposed modelsare formulated based on the type of cross-talk present in the MIMO transmitter.The DUT structures in presence of cross-talk and their corresponding DPD struc-tures are also studied. It is shown that for a DUT with nonlinear cross-talk, theGMPLC model could be used for the mitigation of nonlinear distortion since theDPD structure for such a DUT can be described by the inverse of the nonlineartransfer function followed by the cross-talk. Therefore, for different cross-talk cases,the selection of a correct model is essential in order to achieve desired behavioralmodeling or linearization performance.

To model and compensate nonlinear distortions in SISO and concurrent dual-band transmitters in presence of long-term memory effects, two novel models forSISO PAs and three models for concurrent PAs are proposed. In these models,the input signals are filtered by IIR filters, hence the models have infinite memorydepth. The models are, thus, potential candidates for modeling and compensatingnonlinearities in DUTs with long-term memory effects. For the behavioral modelingof SISO GaN PA, the proposed models resulted in NMSE values that are 8 − 10 dBlower than those of the corresponding FIR models. Similar observations were madewhen the DUT was used in concurrent dual-band application. The complexity ofthe proposed models is approximately the same as that of the FIR models in termsof the FLOPs. In this thesis, a 2D-EEMP model is derived. The derivation is basedon the physical knowledge of a dual-band PA and the performance of the derivedmodel is evaluated both as a direct and inverse model. The results indicate thatthe proposed model achieve improved performance compared to the 2D-DPD andDB-GMP models with a reduced number of model parameters.

The behavioral models for MIMO and concurrent dual-band transmitters resultsin large number of model parameters compared to SISO models. The increase innumber of parameters is due to the presence of cross-terms and reduced kernelsymmetry, therefore, the implementation complexity increases. Hence, the DPD ofthese amplifiers further increase the need for model reduction techniques. In paperB, such a technique is implemented on MIMO Volterra series and the measurementresults discussed in chapter 5. The results show that with the implementation ofsparse techniques, a working sparse DPD with comparable performance to that ofcorresponding dense model could be achieved.

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