LINEARIZATION OF RF POWER AMPLIFIERS BY USING MEMORY

LINEARIZATION OF RF POWER AMPLIFIERS BY USING MEMORY POLYNOMIALDIGITAL PREDISTORTION TECHNIQUE

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

GOZDE ERDOGDU

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

ELECTRICAL AND ELECTRONICS ENGINEERING

JUNE 2012

Approval of the thesis:

LINEARIZATION OF RF POWER AMPLIFIERS BY USING MEMORY

POLYNOMIAL DIGITAL PREDISTORTION TECHNIQUE

submitted by GOZDE ERDOGDU in partial fulfillment of the requirements for the degree ofMaster of Science in Electrical and Electronics Engineering Department, Middle EastTechnical University by,

Prof. Dr. Canan OzgenDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Ismet ErkmenHead of Department, Electrical and Electronics Engineering

Assoc. Prof. Dr. Simsek DemirSupervisor, Electrical and Electronics Engineering Dept., METU

Assist. Prof. Dr. A. Hayrettin YuzerCo-supervisor, Engineering Dept., Karabuk University

Examining Committee Members:

Prof. Dr. Canan TokerElectrical and Electronics Engineering Dept., METU

Assoc. Prof. Dr. Simsek DemirElectrical and Electronics Engineering Dept., METU

Prof. Dr. Nevzat YıldırımElectrical and Electronics Engineering Dept., METU

Prof. Dr. Sencer KocElectrical and Electronics Engineering Dept., METU

Necip SahanSenior Expert Engineer, ASELSAN

Date:

I hereby declare that all information in this document has been obtained and presentedin accordance with academic rules and ethical conduct. I also declare that, as requiredby these rules and conduct, I have fully cited and referenced all material and results thatare not original to this work.

Name, Last Name: GOZDE ERDOGDU

Signature :

iii

ABSTRACT

LINEARIZATION OF RF POWER AMPLIFIERS BY USING MEMORY POLYNOMIALDIGITAL PREDISTORTION TECHNIQUE

Erdogdu, Gozde

M.Sc., Department of Electrical and Electronics Engineering

Supervisor : Assoc. Prof. Dr. Simsek Demir

Co-Supervisor : Assist. Prof. Dr. A. Hayrettin Yuzer

June 2012, 88 pages

In modern wireless communication systems, new modulation types are introduced in order to

support more users by considering spectral efficiency. These new signals are sensitive to non-

linearity when they have high peak to average ratio. The main part in the system that causes

nonlinearity is the power amplifier. For power amplifiers, between linearity and efficiency,

there is a trade-off. However, by using predistortion techniques, both linearity and efficiency

can be obtained. In this thesis, various predistortion methods are explained and memory

polynomial digital predistortion is studied because of its great advantages. The results are ob-

tained by simulations through MATLAB and experiments. An open loop test bench is built up

with real amplifier. During experimental procedure, as input two tone signal, 8psk modulated

signal and π/2 bpsk modulated signal are used. Predistortion with memory and memoryless

predistortion performances are compared and superiority of the predistortion with memory

is shown. Predistortion performance with respect to memory depth and polynomial order is

also studied. Moreover, predistortion model range is investigated through evaluation of per-

formance by applying predistorter function estimated at a specific bandwidth and power to

other signals having different bandwidth and power. Besides these works, the details of

iv

predistortion algorithm and the problems that can be countered in practice are explained.

Keywords: Digital predistortion, Linearization, Power amplifier, Memory polynomial

v

OZ

RF GUC YUKSELTECLERIN HAFIZA ETKILI POLINOMSAL TEMELBANTONBOZUNUM TEKNIGIYLE DOGRUSALLASTIRILMASI

Erdogdu, Gozde

Yuksek Lisans, Elektrik ve Elektronik Muhendisligi Bolumu

Tez Yoneticisi : Doc. Dr. Simsek Demir

Ortak Tez Yoneticisi : Yrd. Doc. Dr. A. Hayrettin Yuzer

Haziran 2012, 88 sayfa

Modern kablosuz iletisim sistemlerinde, daha cok kullanıcıyı desteklemek icin spektral ver-

imlilik goz onunde bulundurularak, yeni kiplenim turleri ortaya atılmıstır. Bu yeni isaretler

yuksek tepe-ortalama guc oranına sahip olduklarından dogrusal olmayan bozunumlara karsı

cok hassastırlar. Sistemde dogrusal olmayan bozunumu yaratan ana kısım guc yukseltectir.

Guc yukselteclerde, dogrusallık ile verimlilik arasında bir odunlesim vardır. Ancak, onbozu-

num teknikleriyle hem dogrusallık hem verimlilik elde edilebilir. Bu tezde, cesitli onbozunum

metodları anlatılmıstır ve yuksek avantajlarından oturu hafıza etkili polinomsal temelbant

onbozunum calısılmıstır. Sonuclar MATLAB’taki simulasyonlar ve deneyler ile elde edilmistir.

Gercek bir guc yukseltec ile acık dongu test duzenegi kurulmustur. Deneylerde, iki ton isaret,

8psk isaret ve π/2 bpsk isaret kullanılmıstır. Hafıza etkisi dikkate alınarak ve alınmayarak

bulunan onbozunum performansları karsılastırılmıs ve hafıza etkili onbozunumun ustunlugu

gosterilmistir. Hafıza derinligi ve polinom derecesine gore onbozunum performansı da calısıl-

mıstır. Ayrıca, bir bant genisligi ve gucte modellenen bir onbozunum fonksiyonunun baska

bant genislikleri ve guclerdeki performansı degerlendirilerek onbozunum model aralıgı belir-

lenmistir. Bu calısmaların yanı sıra, onbozunum algoritmasının detayları ve pratikte karsılasa-

vi

bilecek sorunlar anlatılmıstır.

Anahtar Kelimeler: Temelbant onbozunum, Dogrusallastırma, Guc yukseltec, Hafıza etkili

polinom

vii

to the memory of my beloved father

viii

ACKNOWLEDGMENTS

I would like to express my greatest thanks to my supervisor Assoc. Prof. Dr. Simsek Demir

for his guidance, suggestions and encouragement throughout this thesis. I have benefited from

his deep knowledge and discipline on research. Without his support and encouragement, this

work would not been finished.

I would like to convey thanks to my cosupervisor A. Hayrettin Yuzer for his support, guidance

and help. I have benefited from his deep knowledge on research.

I am deeply grateful to my design leader Atak Ozkan for his precious guidance and help that

he provided during my M.S. study.

I would also like to express my appreciation to Ertugrul Kolagasıoglu for his support and

advices in my thesis.

I would like to thank jury members for their valuable comments on this thesis.

I am thankful to my company ASELSAN Inc. for letting and supporting of my thesis study.

I would like to thank Selim Ozcukurlu and all of my friends for their support.

I would like to extend my special appreciation and gratitude to my mother for her love and

support. I feel her support always with me throughout my life.

ix

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 POWER AMPLIFIER NONLINEARITIES AND DIGITAL PREDISTOR-TION TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Parameters to Characterize Power Amplifier Nonlinearities . . . . . 4

2.1.1 Gain Saturation . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Intermodulation Distortion . . . . . . . . . . . . . . . . . 5

2.1.3 AM/AM and AM/PM Nonlinearity . . . . . . . . . . . . . 8

2.2 Effects of Power Amplifier Nonlinearity . . . . . . . . . . . . . . . 10

2.2.1 Harmonic Generation . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Intermodulation Distortion . . . . . . . . . . . . . . . . . 10

2.2.3 Spectral Regrowth . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Cross Modulation . . . . . . . . . . . . . . . . . . . . . . 11

2.2.5 Desensitization . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Memory Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

x

2.3.1 Electrical Memory Effects . . . . . . . . . . . . . . . . . 14

2.3.2 Thermal Memory Effects . . . . . . . . . . . . . . . . . . 15

2.4 Linearization Techniques . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.3 Predistortion . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Baseband Predistortion . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.2 Classification of Digital Predistortion Techniques . . . . . 22

2.5.2.1 Look Up Table Model Predistortion . . . . . . 23

2.5.2.2 Nested Look Up Table Model Predistortion . . 24

2.5.2.3 Volterra Model Predistortion . . . . . . . . . 24

2.5.2.4 Memory Polynomial Digital Predistortion . . 24

2.5.2.5 Envelope Memory Polynomial Predistortion . 25

2.5.2.6 Wiener Model Predistortion . . . . . . . . . . 25

2.5.2.7 Hammerstein Model Predistortion . . . . . . 27

2.5.2.8 Augmented Wiener Model Predistortion . . . 28

2.5.2.9 Augmented Hammerstein Model Predistortion 30

2.5.2.10 Twin Nonlinear Two Box Model Predistortions 30

2.5.3 Comparison of the Model Structure Performances of thePredistorters . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Memory Polynomial Digital Predistortion . . . . . . . . . . . . . . 32

2.6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6.2 Inverse Structures . . . . . . . . . . . . . . . . . . . . . . 33

2.6.2.1 pth-order Inverse Method . . . . . . . . . . . 33

2.6.2.2 Indirect Learning Architecture . . . . . . . . 34

2.6.3 Predistortion algorithm using indirect learning architecture 35

2.6.4 Estimation Algorithms . . . . . . . . . . . . . . . . . . . 36

2.6.4.1 Least Mean-Square Algorithm (LMS) . . . . 36

2.6.4.2 Recursive Least-Squares Algorithm (RLS) . . 36

2.6.5 Delay Alignment . . . . . . . . . . . . . . . . . . . . . . 36

xi

2.6.6 Power Alignment . . . . . . . . . . . . . . . . . . . . . . 39

3 SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Memory Polynomial Amplifier Model (Forward Model) . . . . . . . 42

3.3 Memory Polynomial Predistorter Model (Inverse Model) . . . . . . 43

3.4 Memory Depth Effect on Predistortion Performance . . . . . . . . . 45

3.5 Polynomial Order Effect on Predistortion Performance . . . . . . . . 48

3.6 Model Validity Range . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.1 DPD Performance For Different Data Sets of Same Mod-ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.2 DPD Performance For Different Data Sets of Different Mod-ulations and Bandwidth . . . . . . . . . . . . . . . . . . . 53

4 HARDWARE IMPLEMENTATION . . . . . . . . . . . . . . . . . . . . . . 56

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Test Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.1 Power Amplifier . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Signal Generator . . . . . . . . . . . . . . . . . . . . . . 57

4.2.3 Vector Signal Analyzer . . . . . . . . . . . . . . . . . . . 58

4.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Comparison of Memoryless Predistortion and Predistor-tion With Memory . . . . . . . . . . . . . . . . . . . . . 59

4.3.1.1 Two-Tone Input Data . . . . . . . . . . . . . 59

4.3.1.2 Wideband Input Data . . . . . . . . . . . . . 61

4.4 Model Validity Range . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.1 Predistortion Performance With Different Data of SameModulation . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.2 Predistortion Performance With Different Modulations . . 71

4.4.3 Predistortion Performance Change by Power Sweep . . . . 72



4.4.4 Predistortion Performance Change by Bandwidth Sweep . 75


xii


5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

xiii

LIST OF TABLES

TABLES

Table 3.1 Predistorter performance for two-tone CW having 20 kHz spacing w.r.t.

memory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table 3.2 Predistorter performance for two-tone CW having 1 kHz spacing w.r.t. mem-

ory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table 3.3 Predistorter performance for 8psk input having 40 kHz spacing w.r.t. mem-

ory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 3.4 Predistorter performance for 8psk input having 80 kHz spacing w.r.t. mem-

ory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 3.5 Predistorter performance for π/2 bpsk input having 40 kHz spacing w.r.t.

memory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


memory depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Table 3.7 Predistorter performance for two-tone CW having 20 kHz spacing w.r.t.

polynomial order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Table 3.8 Predistorter performance for two-tone CW having 1 kHz spacing w.r.t. poly-

nomial order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Table 3.9 Predistorter performance for 8psk input having 40 kHz spacing w.r.t. poly-

nomial order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Table 3.10 Predistorter performance for 8psk input having 80 kHz spacing w.r.t. poly-

nomial order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51



xiv



Table 3.13 Normalized 40kHz 8psk input predistortion with different sets . . . . . . . 53

Table 3.14 Normalized 20kHz two-tone input predistortion with predistorter modeled

by 1 khz two-tone input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Table 3.15 Normalized 80kHz 8psk input predistortion with predistorter modeled with

40 kHz 8psk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Table 3.16 Normalized 80kHz π/2 bpsk input predistortion with predistorter modeled

with 40 kHz π/2 bpsk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


with 40 kHz 8psk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


with 80 kHz 8psk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Table 4.1 35kHz 8psk input predistortion with different sets . . . . . . . . . . . . . . 70

Table 4.2 35kHz 8psk input predistortion with different sets . . . . . . . . . . . . . . 71

Table 4.3 35kHz π/2 bpsk input predistorted with model trained by 35 kHz 8psk

(units:dBm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Table 4.4 80kHz π/2 bpsk input predistorted with model trained by 80 kHz 8psk

(units:dBm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xv

LIST OF FIGURES

FIGURES

Figure 2.1 The 1dB compression point [1] . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 2.2 Spectrum of intermodulation products [1] . . . . . . . . . . . . . . . . . . 6

Figure 2.3 Growth of harmonics as a function of input power [1] . . . . . . . . . . . 7

Figure 2.4 Normalized Input - Output amplitude versus time [3] . . . . . . . . . . . . 8

Figure 2.5 AM/PM characteristics of an amplifier . . . . . . . . . . . . . . . . . . . 9

Figure 2.6 Spectral regrowth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.7 Upper and Lower adjacent channel bands [3] . . . . . . . . . . . . . . . . 12

Figure 2.8 Decomposition of Memory Effects [6] . . . . . . . . . . . . . . . . . . . . 14

Figure 2.9 Feedback Linearization [7] . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 2.10 Feedforward Linearization [7] . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 2.11 Predistortion method of Linearization [8] . . . . . . . . . . . . . . . . . . 18

Figure 2.12 a)RF predistortion, b)IF predistortion, c)Baseband predistortion [7] . . . . 19

Figure 2.13 Block diagram of Digital Predistortion . . . . . . . . . . . . . . . . . . . 20

Figure 2.14 Predistortion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 2.15 Detailed block diagram of Digital Predistortion . . . . . . . . . . . . . . . 22

Figure 2.16 Block Diagram of Look Up Table Model [11] . . . . . . . . . . . . . . . . 23

Figure 2.17 Block Diagram of a)Memory polynomial model b)Envelope memory poly-

nomial model [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figure 2.18 Block Diagram of Wiener Model [9] . . . . . . . . . . . . . . . . . . . . 26

Figure 2.19 Wiener model block diagram [17] . . . . . . . . . . . . . . . . . . . . . . 27

Figure 2.20 Block Diagram of Hammerstein Model [9] . . . . . . . . . . . . . . . . . 28

Figure 2.21 Hammerstein model block diagram [19] . . . . . . . . . . . . . . . . . . . 28

xvi

Figure 2.22 Block Diagram of Augmented Wiener Model [17] . . . . . . . . . . . . . 29

Figure 2.23 Augmented Wiener Model block diagram [17] . . . . . . . . . . . . . . . 29

Figure 2.24 Augmented Hammerstein Model block diagram [19] . . . . . . . . . . . . 30

Figure 2.25 Digital Predistortion Scheme [15] . . . . . . . . . . . . . . . . . . . . . . 32

Figure 2.26 Indirect Learning Architecture Scheme [23] . . . . . . . . . . . . . . . . . 34

Figure 2.27 AM/AM dispersion without delay alignment [25] . . . . . . . . . . . . . . 37

Figure 2.28 AM/PM dispersion without delay alignment [25] . . . . . . . . . . . . . . 37

Figure 2.29 The gain change with respect to peak power [10] . . . . . . . . . . . . . . 40

Figure 2.30 The gain change with respect to peak power [27] . . . . . . . . . . . . . . 41

Figure 3.1 Simulation result of AM/AM and AM/PM curves of the amplifier model

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 3.2 Simulation result of AM/AM and AM/PM curves of the predistorter model

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 3.3 Simulation result of AM/AM and AM/PM curves of the cascade of predis-

torter and amplifier models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Figure 3.4 Simulation result of Power Spectral Density of input, output with predis-

tortion, output without predistortion . . . . . . . . . . . . . . . . . . . . . . . . . 45

Figure 4.1 Digital Predistortion Scheme [27] . . . . . . . . . . . . . . . . . . . . . . 57

Figure 4.2 Digital Predistortion Test bench . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 4.3 The output of the amplifier without predistortion with 1 kHz two-tone input 60

Figure 4.4 The output of the amplifier with memoryless predistortion with 1 kHz two-

tone input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 4.5 The output of the amplifier with memory polynomial predistortion, mem-

ory depth=1, PEP=54.58dBm with 1 kHz two-tone input . . . . . . . . . . . . . 61

Figure 4.6 The output of the amplifier without predistortion with 20 kHz two-tone input 62

Figure 4.7 The output of the amplifier with memoryless predistortion with 20 kHz

two-tone input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 4.8 The output with memory polynomial predistortion, memory depth=4, PEP=54,45dBm

with 20 kHz two-tone input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

xvii

Figure 4.9 Spectrum of 80 kHz 8psk input . . . . . . . . . . . . . . . . . . . . . . . 64

Figure 4.10 Spectrum of 80 kHz π/2 bpsk input . . . . . . . . . . . . . . . . . . . . . 64

Figure 4.11 Constellation diagram of 80 kHz 8psk input . . . . . . . . . . . . . . . . . 65

Figure 4.12 Constellation diagram of 80 kHz π/2 bpsk input . . . . . . . . . . . . . . 65

Figure 4.13 The output of the amplifier without predistortion with 35 kHz 8psk input . 66


8psk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 4.15 The output with memory polynomial predistortion, memory depth=2, PEP=54,66dBm

with 35 kHz 8psk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 4.16 The output of the amplifier without predistortion with 80 kHz 8psk input . 67


8psk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 4.18 The output with memory polynomial predistortion, memory depth=6, PEP=54.02dBm

with 80 kHz 8psk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Figure 4.19 The output of the amplifier without predistortion with 80 kHz π/2 bpsk input 69

Figure 4.20 The output of the amplifier with memoryless predistortion with 80 kHz π/2

bpsk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Figure 4.21 The output with memory polynomial predistortion, memory depth=4, PEP=54.12dBm

with 80 kHz π/2 bpsk input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Figure 4.22 Linearization of 20 kHz two-tone input by power sweep . . . . . . . . . . 73

Figure 4.23 Linearization of 35 kHz 8psk input by power sweep (Lower side) . . . . . 73

Figure 4.24 Linearization of 35 kHz 8psk input by power sweep (Upper side) . . . . . 74

Figure 4.25 Linearization of 35 kHz π/2 bpsk input by power sweep (Lower side) . . . 74

Figure 4.26 Linearization of 35 kHz π/2 bpsk input by power sweep (Upper side) . . . 75

Figure 4.27 Linearization of 1 kHz two-tone input by bandwidth sweep . . . . . . . . 76

Figure 4.28 Linearization of 20 kHz two-tone input by bandwidth sweep . . . . . . . . 77

Figure 4.29 Linearization of 35 kHz 8psk input by bandwidth sweep (Lower side) . . . 77

Figure 4.30 Linearization of 35 kHz 8psk input by bandwidth sweep (Upper side) . . . 78

Figure 4.31 Linearization of 80 kHz 8psk input by bandwidth sweep (Lower side) . . . 78

xviii

Figure 4.32 Linearization of 80 kHz 8psk input by bandwidth sweep (Upper side) . . . 79

Figure 4.33 Linearization of 35 kHz π/2 bpsk input by bandwidth sweep (Lower side) . 79

Figure 4.34 Linearization of 35 kHz π/2 bpsk input by bandwidth sweep (Upper side) . 80

Figure 4.35 Linearization of 80 kHz π/2 bpsk input by bandwidth sweep (Lower side) . 81

Figure 4.36 Linearization of 80 kHz π/2 bpsk input by bandwidth sweep (Upper side) . 81

xix

CHAPTER 1

INTRODUCTION

1.1 Motivation

Nowadays, the wireless communication systems provide service for more and more users.

This increase in number of users forces to use the limited radio frequency spectrum efficiently.

In order to overcome this limitation, new modulation techniques are used like WCDMA,

OFDM which have intense constellation diagrams and are obtained using multiple access

techniques. These new signals have high peak to average ratio which causes the system to

work inefficiently. If the system is desired to work with high efficiency, then the nonlinearity

problem appears. The main source of the nonlinearity in communication systems is power

amplifiers. If the signals with high peak to average ratios pass through the nonlinear amplifier,

constellation diagram is distorted and adjacent channel power level increases. These are not

desired situations for communication systems. Therefore, there are linearization requirements

on RF front ends. Moreover, the power amplifiers have trade-off between the linearity and

efficiency. When the amplifier works linearly at back-off, efficiency decreases. Low efficiency

means loss of power; signals which have high peak to average ratio results in low efficiency

operatiton. Besides, this power loss causes over-heating problem and shortens the battery

life. To overcome the nonlinearity problem of the amplifier, many linearization techniques

are developed like feedback, feedforward, predistortion etc. Digital predistortion has become

popular lately since it is easy to apply, it has high performance of linearization and it is

cost efficient. In digital predistortion, there is no need for deep knowledge about the power

amplifier and its functionality. The amplifier is represented as a black box. The behaviour

of the amplifier is modeled and inverse of this model is used as a predistorter stage in front

of the amplifier. Firstly, input excitation signal is distorted by using this inverse model at

the predistorter stage, and then amplifier is excited with this distorted signal. By this view,

1

the signal at the output of amplifier becomes linear, AM/AM and AM/PM distortions are

eliminated, the spectral regrowth is prevented and constellation of signals is given without

deformation.

1.2 Objectives

In this thesis, the objectives are;

• To understand the nonlinearities of the power amplifier,

• To learn the linearization techniques, especially predistortion in detail,

• To have a deep knowledge of digital memory polynomial predistortion,

• To gain experience of memory polynomial digital predistortion by simulations in MAT-

LAB and to gain experience of memory polynomial digital predistortion by hardware

implementation.

1.3 Outline

In this thesis, the main goal is to investigate the linearization method, memory polynomial

digital predistortion. As a starting point, the nonlinearities of the power amplifier are studied

in the second chapter. The parameters to characterize the nonlinearity and the consequences

of the nonlinearity are introduced. Memory effects are explained since the memory polyno-

mial digital predistortion is expected to linearize the distortions coming from these. Then,

the linearization methods are mentioned shortly but baseband predistortion technique is ex-

plained in detail. Baseband predistortion techniques are compared and memory polynomial

digital predistortion is chosen to study because of its advantages. Furthermore, in the same

chapter, the memory polynomial digital predistortion is studied in detail. The theory and the

predistortion with indirect learning method are introduced. Then, delay alignment and power

alignment are mentioned. In the third chapter, the simulations in MATLAB are conducted us-

ing the knowledge about memory polynomial amplifier and predistortion models as explained

in the second chapter. In simulations, the memory depth and polynomial order effects on

linearization performance are investigated using measurement results having different modu-

lation types. Finally, the validity range of the model is studied by looking at the linearization

performance when a data set of one modulation is used for estimating the predistorter func-

tion and this function is used for different data sets of the same modulation and also it is used

2

for data sets of other types of modulation. In the fourth chapter, the hardware implemen-

tation of the digital predistortion is explained. First, the test bench is described. Then, the

conducted experiments are explained. In the experimental procedure, firstly, the linearization

performances of the memoryless and memory polynomial predistortion are compared with

different type of inputs. After that, in order to check the model validity range, the predistor-

tion performances for different data sets of the same modulation and cross validity of different

modulation types are mentioned. Moreover, peak power and bandwidth of excitation signal

are swept in between predefined region in order to investigate model performance dependency

on power level and bandwidth. In the last chapter, the results of the works conducted through-

out this thesis are summarized and the works that can be done in the future are mentioned.

3

CHAPTER 2

POWER AMPLIFIER NONLINEARITIES AND DIGITAL

PREDISTORTION TECHNIQUES

The linearity of the communications systems is mainly limited by the nonlinear RF power

amplifier component used in the transmitter. The nonlinear characteristic of the RF power

amplifier is a fundamental problem in a wireless communication system. Operating the power

amplifier in linear region with the signals used in modern wireless communication, which

have high peak to average ratios, leads to low power efficiencies. Therefore, there is a tradeoff

between the linearity and the power efficiency of power amplifiers. In order to get more power

from the amplifier, which means operating power amplifier efficiently, the nonlinearities up

to a point is defined with special mask such as IEEE802.11 a,b,g etc. In order to handle the

power amplifier nonlinearity, it is necessary to know its reasons first.

2.1 Parameters to Characterize Power Amplifier Nonlinearities

2.1.1 Gain Saturation

When an input voltage, V(t), is applied to the power amplifier, the output voltage of the

amplifier, V0(t), can be represented as follows:

V0(t) = α0 + α1V(t) + α2V(t)2 + α3V(t)3 + ... + αnV(t)n (2.1)

When the input voltage is small, which means the amplifier works linearly, the output

power can be represented as:

V0(t) ≈ α1V(t) (2.2)

4

So, the power amplifier has linear gain α1. However, the power amplifier doesn’t always

work linearly. As the input voltage increases, higher order terms rather than α1 cannot be

ignored and the relation between the input and output becomes nonlinear. Since α2 term is

negative, as input increases, this term becomes significant and causes saturation.

One of the critical parameters in order to define the nonlinearity of the power amplifier

is 1dB compression point. It is the point at which the power gain of the power amplifier is

reduced by 1dB from its small signal linear power gain value. In other words, it is the output

power at which the output power of the amplifier is 1dB less when it has the linear relation

between the input and the output.

Figure 2.1: The 1dB compression point [1]

A power amplifier has a maximum output power that it can reach as seen in Figure 2.1.

This is the saturation power Psat beyond the 1dB compression point which is reached as the

input of the amplifier is increased.

2.1.2 Intermodulation Distortion

Operating a power amplifier under large signal conditions, which leads the amplifier work

in nonlinear region, causes distortions in the output signal. This deviation from the linearity

produces additional frequencies at the output when an amplifier is excited with n-tone signal

where n is greater than one. These additional frequency products are called Intermodulation

products.

Let a two tone sinusoidal signal to be applied to the input as follows;

5

Vi(t) = cos(2π f1t) + cos(2π f2t) (2.3)

where ( f2 > f1).

The output of the amplifier can be found by substituting Equation (2.3) into Equation

(2.1). It can be seen that when two tone Vi(t) is applied to the power amplifier with this

input output characteristics, the intermodulation products appear at 2 f1, 2 f2, 3 f1, 3 f2, f1 ± f2,

2 f1 ± f2, f1 ± 2 f2 as can be seen in Figure 2.2.

Figure 2.2: Spectrum of intermodulation products [1]

2 f1, 2 f2, 3 f1 and 3 f2 are the second and the third harmonics of the power amplifier re-

spectively. They can be filtered out easily so the distortion caused by them can be minimized.

Second order intermodulation products, f1 ± f2, can also be filtered out since they are far

enough from the tones f1 and f2 at the spectrum. However, third order intermodulation prod-

ucts, 2 f1 − f2 and 2 f2 − f1, fall within the amplifier bandwidth and cannot be filtered out.

Thus, they can cause distortions at the output. These third order intermodulation products are

important because they are the parameters which limit the dynamic range and the bandwidth

of the amplifier. A mathematical concept is defined as third order intercept point in order to

define the nonlinearity with these parameters [2].

As seen in Figure 2.3, the third order intercept point, P3, is the output power level at

which the extended third order harmonic slope meets that of the fundamental. At this output

power, the fundamental and the third order harmonic levels are equal even though operation

6

Figure 2.3: Growth of harmonics as a function of input power [1]

at the third order intercept point is impossible since the output power usually saturates below

this level. 1dB decrease in input signal level results in 1dB decrease in fundamental tone level

and 3dB decrease in all third order product levels. This means if the input power is decreased

by one-third of the distance in decibels from P3 to noise floor, the third harmonic drops to

the noise level. This output power range for fundamental is called the spurious-free dynamic

range (SFDR). It is calculated by [1];

S FDR(dB) =23

[P3−PNOIS E] =23

[P3 +174dBm−10 log BW(dB)−NF(dB)−G(dB)] (2.4)

where P3 = third order intercept point (dBm), BW = amplifier bandwidth (Hz)), NF = ampli-

fier noise figure (dB), G = amplifier gain (dB)

This shows that third order intercept power is a figure of merit for intermodulation sup-

pression. If the third order intercept power is high, then the spurious-free dynamic range

is high which means undesired intermodulation products are suppressed. Thus, this is an

7

important figure of merit for linearity of a power amplifier.

2.1.3 AM/AM and AM/PM Nonlinearity

In ideal case, the power amplifier has constant gain as defined in Equation (2.2).

Input and output has a linear relation. When the amplifier works with small signals, it is

in linear region, so input-output relation remains linear. However, as mentioned before, as the

input amplitude of the amplifier increases, the output cannot increase linearly and it saturates

at a level as seen in Figure 2.1. This is the AM/AM characteristics of the power amplifier and

this distortion is called AM/AM distortion.

As shown in Figure 2.4, a sinusoidal signal is given to the amplifier that makes the am-

plifier work nonlinearly, thus the output signal is clipped since it doesn’t have linear gain. As

the input power increases, the power amplifier begins to work in nonlinear region. This leads

amplifier to have less gain than the small signal gain. So, the output signal is clipped when

the power amplifier goes into saturation.

Figure 2.4: Normalized Input - Output amplitude versus time [3]

Again, in ideal case, the power amplifier gain has no phase or has constant phase. How-

8

ever, in reality this is not the absolute case. The phase of the gain changes with the input

amplitude. This relation of input amplitude and the phase shift of the amplifier is the AM/PM

characteristics of the power amplifier. The distortion it causes is called AM/PM distortion. In

order to have a deep look at this condition, let the input of the amplifier be:

x(t) = A(t) cos(ω0t + φ(t)) = Re(A(t) exp( jφ(t)) exp( jω0t)) (2.5)

The gain of the amplifier is not just constant with constant phase. So the output can be

represented as;

y(t) = G(|A(t)|) cos(ω0t + φ(t) + θ(|A(t)|)) = Re(G(|A(t)|) exp(θ(|A(t)|)) exp( jφ(t)) exp( jω0t) (2.6)

Here, φ(t) is the phase modulation of the input and θ(t) is the phase of the amplifier gain.

In order to get the input phase without distortion, θ(t) must be zero or constant. As seen in the

Equation (2.6), the gain has a phase term depending on the amplitude of the input and can be

non-constant which causes AM/PM distortion (Figure 2.5).

Figure 2.5: AM/PM characteristics of an amplifier

9

2.2 Effects of Power Amplifier Nonlinearity

2.2.1 Harmonic Generation

In order to characterize the nonlinear behavior of the power amplifier, Equation (2.1) is used

for amplifier behaviour. Let the input is a single tone, Vi(t) = cos(2π f t). According to the

Equation (2.1), applying Vi(t) as input, the output V(t) has frequency components not only at

f but also nf where n is integer. This nth frequency component is called nth harmonic of the

amplifier. These harmonic components causes distortion if they are not suppressed enough

but since they are far away from the fundamental tone on the spectrum, it is easy to filter them

out.

2.2.2 Intermodulation Distortion

Given the Equation (2.1), that shows the input-output nonlinear relation, if a two tone input,

Vi(t) = cos(2π f1t) + cos(2π f2t), is applied to the amplifier, the output contains frequency

components as ±m f1±n f2. These components are called intermodulation products. Given k =

|m|+ |n|, ±m f1±n f2 products are called the kth intermodulation product of the power amplifier.

If these components are not suppressed enough, they cause distortion. Since the frequency of

intermodulation components are close to the fundamental components, they cannot be filtered

out.

2.2.3 Spectral Regrowth

Spectral regrowth is similar to the intermodulation distortion. Intermodulation distortion is

caused by nonlinearity of the amplifier when multi-tone input is applied. Spectral regrowth

can be observed when a modulated signal is given as input. When a modulated signal passes

through the nonlinear amplifier, its bandwidth is broadened by the nonlinearities. This is

caused by the generation of the mixing products between the individual frequency compo-

nents of the spectrum.

Spectral regrowth leads to adjacent channel interference which is caused by the unwanted

leakage of the adjacent channel. So, adjacent power ratio (ACPR) is kept low in order not to

cause adjacent channel interference. The adjacent channel power ratio (ACPR) is a commonly

used figure-of-merit to describe linearity in modern telecommunication systems. ACPR is the

ratio between the main channel’s power to the total adjacent channel power measured over

the signal band [4]. ACPR can be calculated as total or either upper ACPR or lower ACPR

10

Figure 2.6: Spectral regrowth

according to the Equations (2.7), (2.8), (2.9) [4]. The main and adjacent channel powers are

also shown in Figure 2.7.

ACPRT =

(Pmainchannel

Pupperad jacentchannel + Plowerad jacentchannel

)(2.7)

ACPRlower =

(Pmainchannel

Plowerad jacentchannel

)(2.8)

ACPTupper =

(Pmainchannel

Pupperad jacentchannel

)(2.9)

2.2.4 Cross Modulation

The nonlinear power amplifier causes to transfer the modulation of one carrier of frequency

ω1, to another carrier of frequency ω2. Assume that the input given in Equation (2.10) with

modulation on carrier of frequency ω1 and unmodulated carrier of frequency ω2 is given to

the power amplifier.

11

Figure 2.7: Upper and Lower adjacent channel bands [3]

Vin(t) = V1 cos(ω1t) + (1 + m(t)) cos(ω2t) (2.10)

where m(t) is the modulating waveform.

Let the power amplifier nonlinear model up to 3rd order is used as;

Vout(t) = α1Vin(t) + α2V2in(t) + α3V3

in(t) (2.11)

The output of the amplifier contains the frequency component of ω2 as(α1V1 + 3

4α3V31 + 3

2α3V1(1 + m(t))2)

cos(ω1t). This shows that the unmodulated carrier of fre-

quency ω2 of the input is transferred the modulation of ω1 after passing through the nonlinear

power amplifier. This is called cross modulation.

2.2.5 Desensitization

Desensitization is blocking the desired weak signals by the strong unwanted signals of differ-

ent frequencies by the nonlinear power amplifier. Let the input be;

Vi(t) = V1 cos(ω1t) + V2 cos(ω2t) (2.12)

where the desired signal is with carrier ω1, the unwanted signal is with carrier ω2 and V2 �

V1. If the power amplifier nonlinear model with 3rd order is used as in Equation (2.11), the

12

output contains the frequency of wanted signal of ω1 as;

(α1V1 + 3

4α3V31 + 3

2α3V1V22

)cos(ω1t)

Here α3 is negative which gives the property of the power amplifier saturation. Therefore,

the ω1 component is dominated by the ω2 component. This is the desensitization.

2.3 Memory Effects

Memory effect of a power amplifier means that the amplifier output is not only determined

by the current input, but also by the previous inputs. Another definition for memory effect is

given as the changes in amplitude and phase of distortion components caused by changes in

modulation frequency [5].

If the power amplifier is taken as memoryless, its input-output characteristic can be de-

fined with Equation (2.11). If a two tone signal defined as in Equation (2.12) ,where V1 =

V2 = A, is applied to this amplifier model, the third order intermodulation products can be

found as;

IM3 =34α3A3. (2.13)

According to this Equation, IM3 product depends on the input amplitude A. For memory-

less amplifier, it only depends on the input amplitude but in reality this is not the case. Power

amplifier IM3 products also depend on the spacing of the two tone. Thus, it can be said that

any nonconstant distortion behavior at different tone spacings (modulation frequencies for

wideband signal) is regarded as memory effect.

The power amplifier nonlinearity is composed of two types, static nonlinearity and dy-

namic nonlinearity as mentioned in [6]. The static nonlinearity is the AM/AM and AM/PM

curves of the current input of the amplifier. It is a strong nonlinearity. The dynamic nonlinear-

ity corresponds to the memory effect. It is a weak nonlinearity and can be represented with a

weakly nonlinear filter. Moreover, the dynamic nonlinearity can also be categorized into two

as the linear memory effects and the nonlinear memory effects (Figure 2.8).

The linear memory effect is caused by the non-ideal fundamental frequency response. It

can be represented with a linear filter. The nonlinear memory effect is caused by trapping

effects, impact ionization, matching conditions at harmonic frequencies, bias circuit design

13

Figure 2.8: Decomposition of Memory Effects [6]

etc. It can be represented with a weak nonlinear filter. The memory effect is addition of the

two. The linear filter response is;

y1(n) =

M1−1∑i=0

ai0x(n − i) (2.14)

The weak nonlinear filter response is;

y2(n) =

M2−1∑i=0

ai1|x(n − i)|x(n − i) + ... +

Mp−1∑i=0

ai,p−1|x(n − i)|(p−1)x(n − i) (2.15)

So, total memory effect is defined as;

y(n) = y1(n) + y2(n) (2.16)

The memory effects cause spectral regrowth and as the input level increases, the memory

effect gets stronger. Furthermore, as the input increases, the linear memory effect gets weaker

and the nonlinear memory effect becomes more important [6]. The nonlinear memory effects

are dominant for the power amplifier dynamic nonlinearity.

Memory effects are categorized in two sections;

• Electrical memory effects

• Thermal memory effects

2.3.1 Electrical Memory Effects

The electrical memory effects are mainly caused by trapping effects, impact ionization, match-

ing conditions at harmonic frequencies, the impedance-matching conditions at the envelope

14

frequency and bias circuit design [6]. IM3 sidebands are affected by the fundamental voltage

waveforms, the voltage waveforms of the different nodes at the envelope frequency and the

second harmonics. Since these voltage waveforms are affected by node impedances and the

node impedances change with respect to the frequency and the input, the memory effects are

unavoidable. Node impedances can be categorized as internal impedance of the transistor and

the external impedance. External impedance consists of the impedances of bias networks and

the matching network. The effects of these are combined and cause electrical memory effects.

Electrical memory effects are caused by varying envelope, fundamental or second harmonic

impedances at different modulation frequencies [5]. The fundamental impedance has minor

deviation and the second harmonic impedance is narrow and matching is simple. So, they

don’t have dramatic effects on memory. However, the envelope impedance is not like that.

Envelope frequency varies from dc to the maximum modulation frequency which can be up

to a few megahertz. Since it is difficult to keep the envelope impedance constant over this fre-

quency band, the memory effects occur mostly because of the changing envelope impedance

because it changes the output impedance. Memory effects are also caused by the bias net-

works as mentioned. The bias networks are designed with large time constants and those

cause memory effects. In order to avoid this, bias networks must be designed considering

memory effects.

2.3.2 Thermal Memory Effects

Thermal memory effects are caused by electro-thermal couplings and this is effective up to

a few megahertz modulation frequencies [5]. Temperature change caused by the power dis-

sipated on amplifier is determined by the thermal impedance of the transistors. The thermal

impedance does not act like a resistor; it acts like a low pass filter. Thus, the dissipated power

leads temperature change with respect to its frequency. The dissipated power is calculated as

the multiplication of the voltage and the current on the transistor. This means the fundamental

two frequencies are multiplied and their product consists of second order signal components

which are envelope and second harmonics. Since the thermal impedance acts like a low pass

filter, the DC and envelope components cause temperature variation that can be defined with

the Equation [5];

T = Tamb + Rth.pdiss(dc) + Zth(ω1 − ω2).pdiss(ω1 − ω2) (2.17)

As seen from Equation (2.17), the temperature change depends on the frequency of the

15

envelope. This means it changes with the applied signal, so the electrical parameters changes

with the signal applied. The gain, output conductance and capacitance of the amplifier depend

on the temperature change. This dependence causes thermal memory effects. The thermal

memory effects are dominant at low modulation frequencies.

2.4 Linearization Techniques

To get power efficiency from power amplifier, which means operating it close to saturation,

causes the amplifier work nonlinearly. Using linearization techniques, the nonlinearity of the

amplifier can be avoided. The linearization techniques can be classified as;

• Feedback

• Feedforward

• Predistortion

Linearization techniques other than predistortion are not explained in detail since predis-

tortion is studied in detail in this work.

2.4.1 Feedback

Feedback is a linearization technique that can be applied either rf signal or baseband signal.

The block diagram is given in Figure 2.9. Vi(t) is the input signal, Vo(t) is the output signal.

The output signal is fed back with a gain β and Vr(t) signal is obtained. The difference of

Vi(t) and Vr(t) gives the error signal Ve(t). The error signal is given to the amplifier and linear

output is obtained.

Figure 2.9: Feedback Linearization [7]

The main advantage of feedback linearization technique is that the closed loop gain gets

less sensitive to the gain variation of the power amplifier. Furthermore, it is easy to add the

16

necessary circuitry to the power amplifier for feedback [7]. The main disadvantage is the delay

between the input and copied output signal which causes a limitation of gain and bandwidth

product. This affects the stability issue [7].

2.4.2 Feedforward

It is first suggested in 1928 by H.S. Black. The block diagram of Feedforward method is

given in Figure 2.10. In this method, the input is divided into two, one is for the input of

the main amplifier, and the other is for the error estimation. The input divided for the error

estimation is delayed to compensate the delay coming from the main amplifier. The output

of the main amplifier and the delayed input is compared and the difference is given to the

error amplifier. The main output is delayed for compensation of delay coming from the error

amplifier. Finally, this error signal is subtracted from the delayed output of the main amplifier

to get a linear signal.

Figure 2.10: Feedforward Linearization [7]

The advantages of the Feedforward technique over feedback technique are that the gain

of the amplifier is not reduced, the system is unconditionally stable and the bandwidth can be

higher. The main disadvantage is that the system is not adaptive. It doesn’t compensate the

change of characteristics of power amplifier with respect to ageing, temperature or any other

change in circuitry. Moreover, it makes the bandwidth sensitive to the matching circuit [7].

2.4.3 Predistortion

Predistortion is one of the most commonly used linearization technique of the various other

techniques. The idea behind the predistorter is that to insert a nonlinear model between the

input and the power amplifier which has the inverse transfer characteristics of the amplifier,

17

so that the cascade of the predistorter and the amplifier gives a linear response.

Figure 2.11: Predistortion method of Linearization [8]

As seen from Figure 2.11, the amplifier has a saturation region and has a compressive

input-output characteristic. However, the predistorter has an expanding input-output charac-

teristic so that it can compensate the nonlinearity of the amplifier and the output becomes

linear theoretically. The predistortion can be categorized in three parts according to the fre-

quency at which it is implemented;

• RF predistortion

• IF Predistortion

• Baseband Predistortion

The RF predistortion works directly on the power amplifier input signal. Its implementa-

tion is simple but since it works on high frequency, it suffers from relatively poor performance

and constrained adaptability [7]. It usually compensates the odd order distortion. The IF pre-

distortion works on intermediate frequency. This allows the IF predistorter to work at different

RF carrier frequencies [7]. So, it is more adaptable than RF predistorter but still limited. It is

narrower bandwidth predistortion than RF. Besides, DSP techniques also cannot be efficiently

used since it needs high sampling frequency for digitization.

To identify the advantages and disadvantages of the RF/IF predistortion, it can be stated

that, the simplicity of implementation, using few components with low cost, no stability prob-

18

Figure 2.12: a)RF predistortion, b)IF predistortion, c)Baseband predistortion [7]

lems compared to feedback linearization technique, usable in microwave frequencies, wide

linearization bandwidth are the advantages. The disadvantages are that it has modest linear-

ity improvement, high order distortion cannot be get rid of and if the transfer characteristics

change, the circuit has to be fabricated again which means it is not adaptable to different

amplifier characteristics.

The baseband predistortion is applied to the signal after it is downconverted from RF to

baseband. After the predistortion and passing through the amplifier, it is again upconverted

to the carrier frequency. It is easier and more adaptable predistortion technique than the other

ones. It will be discussed in detail in the following section.

The predistortion can also be categorized due to its adaptiveness to the changing charac-

teristics of the amplifier as [7];

• Adaptive Predistortion

• Non-adaptive Predistortion

The power amplifier characteristics can change due to the changes with time like temper-

ature change, ageing of the device, output antenna matching etc. In adaptive predistortion,

the current condition information of the amplifier is always used with the input while adjust-

ing. However in non-adaptive predistortion does not use the amplifier’s current condition by

assuming the amplifier characteristics do not change quickly and once it is adapted, then it is

not changed.

19

Finally, the predistortion can be categorized in terms of handling the memory effects of

the power amplifier;

• Memoryless Predistortion

• Predistortion with Memory

In the following section, the baseband predistortion is discussed.

2.5 Baseband Predistortion

Baseband predistortion, in other words digital predistortion, has a widespread popularity for

linearization. The digital predistortion solution has the potential for significantly reducing size

and cost of linearization circuit over that of other methods [9]. Since software defined radios

and new network topologies contain digital parts, the infrastructure already exists, so digital

predistortion does not need so many extra components. It seems the most promising one

among other linearization techniques because of its excellent linearization performance and

easiness of hardware implementation. It can be applied typically up to 20 MHz bandwidth.

2.5.1 Theory

The idea of the predistortion is to apply a complementary nonlinearity of the power amplifier

so that the cascade of the predistorter and the amplifier gives a linear response. In order to

have that, the behavioral modeling is crucial to predict the nonlinearity of the amplifier [10].

Besides, the predistortion is also a behavioral modeling problem itself since it is necessary to

get the reverse function of the amplifier.

Figure 2.13: Block diagram of Digital Predistortion

The block diagram of the digital predistortion is given in Figure 2.13. Since the ideal small

signal gain of the amplifier is constant G, the cascade of the predistorter and the nonlinear

power amplifier gives the ideal small signal gain G. Therefore, it can be stated that the input

of the predistorter is normalized with G as in Equation (2.18).

20

xinDPD(n) =xout(n)

G(2.18)

where G is small signal gain, xout is output of the amplifier.

After the predistorter, the output is given in Equation (2.19) as;

fDPD(xout(n)/G) = xin(n) (2.19)

The output of the predistorter is input to the amplifier as in Equation (2.21);

fDUT (xin(n)) = xout(n) (2.20)

fDUT

(fDPD(

xout(n)G

))

= xout(n) (2.21)

Here, the function of the predistorter fDPD is equivalent to the behavioral modeling of

the power amplifier’s reverse function obtained by swapping the amplifier’s input and output

signals with appropriate small signal gain normalization [10]. After the cascade system of

predistorter and the amplifier, the input output has a linear relation and the power does not

saturate anymore at the level it used to (Figure 2.14).

Figure 2.14: Predistortion Theory

The detailed block diagram of the digital predistortion is given in Figure 2.15. The base-

band input is given to digital to analog converter (DAC) and after passing through the low pass

filter (LPF), it is upconverted to the RF and filtered again. It is now input of the amplifier.

21

The output of the power amplifier is downconverted and it is digitized with analog to digital

converter (ADC). This baseband output is used for predistortion algorithm with the baseband

input.

Figure 2.15: Detailed block diagram of Digital Predistortion

2.5.2 Classification of Digital Predistortion Techniques

The digital predistorters can be classified as following [10];

• Look Up Table Model

• Nested Look Up Table Model

• Volterra Model

• Memory Polynomial Model

• Envelope Memory Polynomial Model

• Wiener Model

• Hammerstein Model

• Augmented Hammerstein Model

• Augmented Wiener Model

• Twin Nonlinear Two-Box Models

22

2.5.2.1 Look Up Table Model Predistortion

The look up table predistortion is memoryless predistortion. It is used for the memoryless

AM/AM and AM/PM nonlinearities. AM/AM and AM/PM characteristics of the power am-

plifier are stored in two look up tables, one is the magnitude, and the other is the phase

information of both input and output as given in the Equation (2.22) [11].

Vout = VinG(|Vin|2) (2.22)

where Vout is output, Vin is input and G is instantaneous complex gain of the amplifier.

Also, the predistorter coefficients which are multiplied by the input of the predistorter to

give the output are kept in two look up table in the same way (Equation 2.23).

Vdpd = VindpdF(|Vindpd |2) (2.23)

where Vdpd is output, Vindpd is input and F is instantaneous complex gain of the predistorter

[11].

The predistorter coefficients are calculated from the input-output characteristics of the

power amplifier, in other words from the instantaneous complex gain of the amplifier. Since

the amplifier shows compressive input-output characteristics, the predistorter coefficients are

calculated as it shows expanding input-output characteristics. They are calculated as the cas-

cade gives the constant small signal gain as seen in the general predistortion formula given in

Equation (2.21). The block diagram of look up table predistortion is given in Figure 2.16.

Figure 2.16: Block Diagram of Look Up Table Model [11]

23

2.5.2.2 Nested Look Up Table Model Predistortion

Nested look up table model is proposed to include the memory effect compensation to the

conventional memoryless look up table model. The model is based on nested look-up tables

that compute the estimated amplifier output signal, based on the current input sample and the

preceding ones [12].

2.5.2.3 Volterra Model Predistortion

The Volterra model is the most comprehensive model in terms of memory effects for predis-

tortion. A Volterra series is a combination of linear convolution of inputs and a nonlinear

power series. Therefore, the input-output relationship of a general nonlinear, causal, and

time-invariant system with fading memory can be described with it [13]. In the discrete time

domain, a Volterra series can be written as in Equation (2.24).

y(n) =

P∑p=1

M∑i1=0

...

M∑ip=0

hp(i1, ..., ip)p∏

j=1

x(n − i j) (2.24)

In this series, x[n] and y[n] correspond to the input and output of the DUT respectively. hp

represents the pth order Volterra kernel. p is the nonlinearity order of the model and M is the

memory depth. In Volterra series, all nonlinearities and memory effects are treated in the same

way. If the system is highly nonlinear and has high memory depth, the complexity increases

since the number of the coefficients increases exponentially. So, the predistortion technique

with Volterra series modeling is used only for the weakly nonlinear systems [13]. There are

also simplifying techniques of Volterra series like dynamic reduction deviation technique [13]

and pruning techniques [14]. Although these methods are not complex as the conventional

Volterra Series, they are not used for highly nonlinear systems.

2.5.2.4 Memory Polynomial Digital Predistortion

Memory polynomial model is one of the most popular predistortion technique and is widely

used. It is the simple form of the Volterra series [15]. The coefficients of the memory polyno-

mial are the diagonal coefficients of the Volterra series. The memory polynomial predistorter

can be described as;

xout(n) =

M∑j=0

N∑i=1

a ji.xin(n − j).|xin(n − j)|i−1 (2.25)

24

where M is memory depth, N is polynomial order and ai j is polynomial coefficients. This

predistortion technique is one of the easiest one to apply which takes the memory effects into

account and gives considerable amount of linearization. It will be explained in detailed in the

following sections.

2.5.2.5 Envelope Memory Polynomial Predistortion

This predistortion model is implemented in a complex gain based architecture and takes ad-

vantage of the dependency of power amplifier nonlinearity on the magnitude of the input

signal [16]. In this model, the nonlinearity of the amplifier is a function of only the magnitude

of the input signal and not its complex value. It is successful in modeling and linearizing

weakly nonlinear amplifiers having memory effects. This model is a combination between

the nested look up table model and the memory polynomial model [10]. The polynomial of

the envelope memory polynomial model can be defined with equation

yEMPM(n) = x(n).M∑

i=0

N∑j=0

ai j.|x(n − i)| j (2.26)

where x(n) is input, M is memory depth, N is polynomial order and ai j is polynomial coeffi-

cients. As seen in the formula, it is similar to the memory polynomial but in this model, only

the magnitude information of the {xin(n − 1), xin(n − 2), ..., xin(n − M)} terms are needed.

The block diagrams of both memory polynomial and envelope memory polynomial mod-

els are given in Figure 2.17. A complex gain value is first computed using the magnitude of

the input samples x(n)...x(n−M). Then, to generate the baseband complex output sample, this

complex gain is applied to the baseband complex input signal, x(n). Since just magnitude in-

formation of the input signal is used, this model is straightforward for rf digital predistorters.

Besides, its performance is comparable to that of the conventional memory polynomial pre-

distortion technique for weakly nonlinear amplifiers.

2.5.2.6 Wiener Model Predistortion

The Wiener model is a two box model which consists of a linear finite impulse response filter

and a following memoryless nonlinear function that can be considered as power series. The

block diagram is given in Figure 2.18.

25

Figure 2.17: Block Diagram of a)Memory polynomial model b)Envelope memory polynomialmodel [16]

Figure 2.18: Block Diagram of Wiener Model [9]

The detailed diagram of the Wiener model is given as in Figure 2.19.As seen in the figure,

the FIR filter represents the dynamic nonlinearity and the nonlinear function part represents

the static nonlinearity which can be characterized by the look up tables based on the AM/AM

and AM/PM curves [17]. Therefore, the polynomial of the Wiener model can be defined as in

Equation (2.27) and Equation (2.28).

v(k) =

N∑n=0

βn.x(k − n) (2.27)

26

Figure 2.19: Wiener model block diagram [17]

y(k) =

P∑p=0

γ(k).|v(k)|p (2.28)

Combining Equations (2.27) and (2.28) [18];

y(k) =

P∑p=0

γ(k)

N∑n=1

βn.x(k − n)

.| N∑n=1

βn.x(k − n)|p (2.29)

The Wiener model is one of the simplest ways to combine memory effects with nonlinear-

ity. However, it has limited effectiveness in modeling most of the power amplifiers. Moreover,

in Wiener model, the output depends on the coefficients nonlinearly which makes coefficient

estimation more difficult than for the models that are linear in parameters [9].

2.5.2.7 Hammerstein Model Predistortion

The Hammerstein model is a two box model which consists of a memoryless nonlinear func-

tion that can be considered as power series and a following linear finite impulse response

filter. The block diagram is given in Figure 2.20. The detailed diagram of the Wiener model

is given as in Figure 2.21. As seen in the figure, like Wiener model, the FIR filter represents

the dynamic nonlinearity and the nonlinear function part represents the static nonlinearity

which can be characterized by the look up tables based on the AM/AM and AM/PM curves.

Therefore, the polynomial of the Hammerstein model can be defined as in Equation (2.31).

27

Figure 2.20: Block Diagram of Hammerstein Model [9]

Figure 2.21: Hammerstein model block diagram [19]

v(k) =

P∑p=0

.γ(k).|v(k)|p (2.30)

y(k) =

D∑d=1

αd.y(k − d)+ =

N∑n=0

βn.v(k − n) (2.31)

y(k) =

D∑d=1

.y(k − d)+ =

N∑n=0

βn

= P∑p=1

γp.x(k − n).|x(k − n)|p

(2.32)

Like the Wiener model, this is a very simple memory nonlinearity but is of limited effec-

tiveness for predistortion. However, it does have the desirable property of being linear in the

parameters considering the estimation of the parameters [9].

2.5.2.8 Augmented Wiener Model Predistortion

In the conventional Wiener model, the linear filter is for correcting the nonlinearity that comes

from the electrical memory effects attributed to the frequency response of the amplifier around

the carrier frequency. However, it does not take care of the nonlinearity coming from the

impedance variation of the bias circuits and harmonic loading [10]. Therefore, a second order

branch containing FIR filter is added to the conventional Wiener model for the second order

nonlinearities coming from these reasons seen on Figure 2.22. The detailed block diagram is

given in Figure 2.23.

28

Figure 2.22: Block Diagram of Augmented Wiener Model [17]

Figure 2.23: Augmented Wiener Model block diagram [17]

The augmented Wiener model is a cascade of a dynamic weak nonlinear model and a

strong nonlinear static model. The strong nonlinear model is based on the AM/AM and

AM/PM characteristics of the amplifier and can be implemented by using look up tables.

The dynamic weak nonlinear model is composed of the new inclusive memory-effect model,

which takes into account the dynamic properties of the transmitters in the presence of the

modulated communications signal [17].

Thus, the polynomial for the Augmented Wiener model is defined in Equation (2.33).

x1(n) =

M1∑j1=0

h1( j1).xin(n − j1) +

M2∑j2=0

h2( j2).xin(n − j2).|xin(n − j2)| (2.33)

where h1( j1 and h2( j2 are the impulse responses of the filters FIR1 and FIR2, M1 and M2 are

the memory depths of the FIR1 and FIR2 [10].

29

2.5.2.9 Augmented Hammerstein Model Predistortion

The Augmented Hammerstein model is the dual of the Augmented Wiener model. In order

to take care of the second order nonlinearities, an FIR filter is added to the conventional

Hammerstein model as seen in Figure 2.24.

Figure 2.24: Augmented Hammerstein Model block diagram [19]

So, the polynomial for the Augmented Hammerstein model is defined as in Equation

(2.34).

xout(n) =

M1∑j1=0

h1( j1).x1(n − j1) +

M2∑j2=0

h2( j2).xin(n − j2).|xin(n − j2)| (2.34)

where h1( j1) and h2( j2) are the impulse responses of the filters FIR1 and FIR2, M1 and M2

are the memory depths of the FIR1 and FIR2, x1(n) is the output of the look up table box [10].

2.5.2.10 Twin Nonlinear Two Box Model Predistortions

These types of models consist of a static look up table based nonlinear function part and

a dynamic nonlinear function part. They are designed for dynamic nonlinear behaviors, in

other words memory effects. They can be categorized in three sections [10];

• Wiener and Hammerstein based models

• Volterra based models

• Memory polynomial based models

2.5.3 Comparison of the Model Structure Performances of the Predistorters

Several predistortion models are introduced in previous sections. In order to compare their

performance, the linearity parameters are used such as adjacent channel power ratio (ACPR)

30

which is used in the frequency domain and the error vector magnitude (EVM) in the time do-

main. Furthermore, the complexity of the models is an important parameter to compare since

the predistorters are implemented in real systems. Therefore, the linearity performance and

complexity of the model are used for comparison of the models. The predistortion is catego-

rized according to take care of the dynamic nonlinearity, in other words memory effects, as

memoryless predistortion and memory predistortion as mentioned before. The performance

of the digital predistortion algorithms that do not take the memory effects into account is

severely degraded as the bandwidth of the input signal increases [15]. Therefore, the linear-

ity performance of the memoryless predistorters, such as look up table model predistorter,

is lower than the polynomial ones that take the memory effects into account. Among the

memory predistorters, the Volterra series and its simplified version memory polynomial pre-

distorters are better in linearization than the other ones except twin nonlinear two box model

predistorters [10]. They are better because Volterra series is a more comprehensive model, the

others are originated by this model but their performances are poorer. Moreover, the Volterra

series model and memory polynomial model have comparable performance on linearity but

the Volterra series is much more complex as a disadvantage. Although the Volterra series is

comprehensive, twin nonlinear two box model predistorters’ performance is higher because

the dynamic and static nonlinearities are taken separately in this model. For the Volterra series

and memory polynomial models, these nonlinearities are taken simultaneously so the estima-

tion of the parameters is dominated by the static nonlinearity. The memory effect nonlinearity

is cared separately in twin nonlinear two box class. Moreover, the twin nonlinear two box

models have better performance than the augmented Wiener and Hammerstein models. They

both model the static and dynamic nonlinearity separately but the twin nonlinear class models

the dynamic nonlinearity with a memory polynomial function whereas the augmented Wiener

and Hammerstein models it with a first and second order nonlinearity. So, in twin nonlinear

two box case, it is better modeled, although it is more complex. Nested look up table model

also take the memory effects into account. Its performance is comparable to other polyno-

mial predistortion models; however, its complexity is very high, so it is not easy to apply.

Among these predistortion models, memory polynomial predistortion technique is an opti-

mum method considering linearization and complexity. Memory polynomial predistortion

will be discussed in detail in the following section.

31

2.6 Memory Polynomial Digital Predistortion

After the comparison of the model structure performances of predistortion which is discussed

in the previous section, memory polynomial digital predistortion is decided to study.

2.6.1 Theory

In memory polynomial digital predistortion, like other predistortion techniques, the input is

intentionally distorted in order to compensate the nonlinearity of the power amplifier. The

predistortion algorithm is applied in baseband. The structure for the digital predistortion is in

Figure 4.1 [15].

Figure 2.25: Digital Predistortion Scheme [15]

As defined in Figure 4.1, the predistortion algorithm is run in baseband. un and xn are

the baseband input and output of the predistorter respectively. xn is upconverted to the carrier

and passes through the amplifier. The output of the amplifier is downconverted; digitized,

normalized by the small signal gain of the amplifier and yn is obtained. For predistortion

algorithm, xn and yn are used. The calculated predistortion function is used in predistortion

part. This approach is divided into two steps. First, the power amplifier is modeled by esti-

mating its characteristics properly. This modeling is called forward modeling, so the amplifier

model is called forward model. After that, the predistorter function is obtained by getting the

inverse characteristics of the amplifier [15]. This is called inverse modeling and the predis-

torter model is inverse model. Therefore, modeling the amplifier and finding its parameters

are crucial for the predistortion. The amplifier and the predistorter are modeled with the same

function named memory polynomial in this type of predistortion. Memory polynomial model

is a truncation of the general Volterra model in which only diagonal terms in Volterra kernels

are considered. So, the number of coefficients is significantly decreased [20]. The memory

polynomial is defined as in Equation (2.35).

32

z(n) =

K∑k=1

Q∑q=0

akqx(n − q)|x(n − q)|k−1 (2.35)

where x(n) is the baseband complex input signal, z(n) is the baseband complex output signal,

K is the polynomial (nonlinearity) order, Q is the memory depth.

As seen in the formula, the output is not determined just with the instantaneous input; it

also depends on the previous input samples which means the memory effects are taken into

account. Furthermore, the even order nonlinear terms are included in modeling. Including

even order terms in the predistorter makes the predistorter linearization even better [21]. Be-

sides, x(n-q) term contains phase information, the input term is not phase blind, which makes

the model more comprehensive. While modeling both of the amplifier and predistorter, the

input and the output of DUT is known. The problem is to find the coefficients akq. Their

characteristics are determined by finding these coefficients. As mentioned before, the ampli-

fier is modeled with the memory polynomial, the forward model is found. The predistorter is

modeled with the inverse of the amplifier characteristics, so the inverse model is found. The

inverse structures for inverse modeling are described in the following section.

2.6.2 Inverse Structures

There are two main methods;

• pth-order inverse method

• indirect learning architecture

2.6.2.1 pth-order Inverse Method

It is first mentioned by Schetzen [22]. This method is suggested in order to compensate the

nonlinearities with memory which is characterized by the Volterra series. The approach is

straightforward. The parameters of the amplifier are initially estimated, and then the inverse

of this polynomial model is found for the nonlinear memory polynomial predistorter function

[9]. The design of this method is very complicated and must be based on a known Volterra

series model of the nonlinear system. Besides, as the nonlinearity order increases, the design

complexity becomes unrealistically high [23].

33

2.6.2.2 Indirect Learning Architecture

It is first suggested by C. Eun and E.J. Powers for an alternative easy method for finding the

inverse characteristics of the amplifier for predistortion [23]. This model is independent of a

specific nonlinear model for the system to be compensated as an advantage compared to the

pth-order inverse model which first requires a Volterra series model of the system. Therefore,

if this method is used in predistortion, the procedure is reduced to one step by skipping the

modeling the characteristics of the amplifier step. Indirect learning architecture uses the input

and output of the amplifier. The output of the amplifier is used to predict the input of the

amplifier. The polynomial is set so and the coefficients are estimated. In other words, to model

the characteristics of the amplifier, the input is used to predict the output and coefficients are

estimated, in the same way, to model the predistorter, the output of the amplifier is used to

predict input of the amplifier and coefficients are estimated. So, the inverse characteristic of

the amplifier is modeled. Then the polynomial with the estimated parameters of the inverse

amplifier is copied to the predistorter part. (Figure 2.26)

Figure 2.26: Indirect Learning Architecture Scheme [23]

There is a worry about this technique [9]. Will the response of an amplifier followed by

postdistortion, means the part with parameters estimated for predistorter with this technique,

be the same as the response to the same model applied before the amplifier? The answer

is yes. It is theoretically proven by Schetzen if one has pth-order postinverse of a general

Volterra system, then the pth-order preinverse is identical [22]. Thus, with this technique,

there is no need to model the amplifier and then take the inverse of the polynomial. By using

the input and output of the amplifier, the inverse model can be estimated. It is an indirect

method and much simpler than the pth-order inverse model. This method is commonly used

34

for the predistortion structures.

2.6.3 Predistortion algorithm using indirect learning architecture

By using indirect learning architecture, the coefficients of the memory polynomial predistorter

given in Equation (2.35) can be found with the following steps [9];

• The coefficients akq are collected in a column vector Jx1, w, where J is the total number

of the coefficients, J=Kx(Q+1).

• The input signal samples x(n) are collected in a Nx1 vector where N is the total number

of the signal samples. Then, the coefficients in Jx1 vector and signal samples in Nx1

vector are used to arrange input sample vector in such a way that each coefficient is

associated with a signal sample as in Equation (2.36). For example, a31 is associated

with x(n − 1)|x(n − 1)|2 where q=1 and k=3.

Xkq(n) = x(n − q)|x(n − q)|k−1 (2.36)

So, an NxJ matrix, X, is formed containing input samples.

• The output samples of the amplifier are assembled in an Nx1 vector, y as;

y = Xw (2.37)

• For inverse modeling, there is the analogous setup as

x = Yw (2.38)

Now, the input is estimated from the output samples. The coefficient vector is multiplied

with the output samples.

• After setting up the matrix with known input and output, the next step is to estimate the

polynomial coefficients, w. It is found with Equation (2.39).

w =(YHY

)−1YH x (2.39)

where YH is the conjugate transpose of matrix Y.

35

Since this method has difficulties to solve like finding the inverse of the matrix, some

estimation algorithms are used which are mentioned in the following section.

2.6.4 Estimation Algorithms

2.6.4.1 Least Mean-Square Algorithm (LMS)

This algorithm is applied sample by sample. This sample by sample estimation decreases the

storage requirements comparing the matrix solving requirements. The error feedback is used

while estimating the coefficients. Least mean square algorithm has low complexity so it is

efficient to apply [24]. However, in predistortion, the high order nonlinearity coefficients are

so small that LMS algorithm shows extremely slow convergence. Therefore, this estimation

algorithm is not practical for predistortion [9].

2.6.4.2 Recursive Least-Squares Algorithm (RLS)

This is also a sample by sample algorithm having low storage requirements comparing matrix

solution. This algorithm updates the inverse covariance matrix as new sample arrives by using

a Kalman like coefficient update [24]. It has an advantage comparing LMS, its convergence is

not affected by small coefficients, poorly conditioned matrixes and it converges fast. However,

RLS algorithm has high complexity so it is computationally intensive [9].

2.6.5 Delay Alignment

In predistortion scheme, there exists a delay between the input and the output since there are

analog circuits between them. The modeling approach based on input and output waveforms

is sensitive to the delay alignment between these data streams. If the delay between the two

paths is not cancelled, the modeling becomes inaccurate since the delay causes additional

dispersion in the AM/AM and AM/PM characteristics of the amplifier as shown in Figure

2.27 and Figure 2.28.

There are some methods for delay alignment. One method is simple correlation method

claimed by the authors of the US patent [26]. The correlation is defined in Equation (2.40) as

[7];

R(m) =1N

N−1∑n=0

z(n)z∗pa(n + m) (2.40)

36

Figure 2.27: AM/AM dispersion without delay alignment [25]

Figure 2.28: AM/PM dispersion without delay alignment [25]

where z and zpa are the input and output of the DUT respectively, m is the delay variable,

N is the length of the sequences. The correlation R(m) reaches its maximum value when m

corresponds to the delay.

The other method is to apply cross-covariance between the input and output sequences

[17]. Let the input sequence be u(n) and the output sequence be y(n). The cross covariance of

37

them is defined as;

Cxy(m) =

N−m−1∑n=0

(u(n + m) − u)(y∗(n) − y∗

),m ≥ 0 (2.41)

Cxy(m) =

N+m−1∑n=0

(u∗(n + m) − u∗

)(y(n) − y) ,m < 0 (2.42)

where N is the length of the sequences, u and y are the average values of u(n) and y(n)

respectively which are represented as in Equations (2.44) and (2.44).

u =1N

N−1∑i=0

u(i) (2.43)

y =1N

N−1∑i=0

y(i) (2.44)

So, the time delay between input and output data streams are determined as τ = mmaxx 1fs

,

where fs is the sampling frequency and mmax is the index of maximum covariance. As seen

in the formula, the delay accuracy is dependent on the sampling rate. Its resolution is defined

by the sampling rate. The sampling rate depends on the analog to digital converter which

is used in predistortion scheme. One way to increase the sampling rate is to upgrade the

hardware but it also increases the cost. Besides, the resolution given by the current ADC speed

cannot provide enough resolution. Another solution to this is to use digital signal processing

techniques like interpolation. By using interpolation, there is no need for high cost hardware

and any higher resolution can be achieved without depending on any limit. Estimating the

delay with high accuracy avoids the AM/AM and AM/PM dispersion but it uses resources

more and the process takes long. In order to avoid this problem, coarse delay is found with

low resolution, then by interpolation high resolution is get with lower number of data stream

and fine delay estimation is done [17].

Another issue about the delay alignment is the sensitivity of the memory polynomial to

fine time delay alignment. Most of the time, it is not possible to align the input and output

by finding the exact value of delay because of the resolution. It is claimed that the memory

polynomial model is sensitive to the delay overestimation. However, if the delay is underes-

timated up to one sampling period, the performance of the model is not affected [25]. This

solves the problem about the need for the high resolution for delay alignment. If the delay is

estimated coarsely and underestimated up to one sampling period, the model works correctly.

38

For delay alignment, the input signal waveform can be sent itself and the corresponding

output is taken, they can be used while estimating the delay. Another way is to use special

sequences, like Barker sequence. This sequence has high autocorrelation value and is used

for synchronization since it is easy to find the time delay. In predistortion scheme, instead of

the actual input waveform, a Barker sequence can be sent as input and with the corresponding

output, the time delay can be found before sending the actual input. However, the delay can

vary with time or temperature change, if previously found delay will be used for predistortion,

it must be taken into account.

2.6.6 Power Alignment

In predistortion algorithm, the output is used as it is normalized by the small signal gain as

mentioned before. This step is done in order to make sure that there exists a power alignment

between the predistorter and the amplifier. In predistortion scheme, the input of the amplifier

differs when it passes through the amplifier directly and when it passes through the amplifier

after the predistorter. So the responses of the amplifier can be different with respect to the

input average power. The power amplifier response is not dependent on the peak to average

of the signal so much but it is dependent on the operating power as can be seen in Figure 2.29.

Power alignment is done in order to avoid this. By power alignment, the average power of the

input of the amplifier is kept the same; therefore the amplifier doesn’t have different responses

between the characterization and linearization steps [27].

For weakly nonlinear amplifiers, to normalize the power amplifier output with small signal

gain ensures the power alignment between the amplifier and the predistorter. However, for

highly nonlinear amplifiers, if the output of the amplifier is normalized with small signal

gain, the average power variation cannot be avoided. This variation is caused by the power

amplifier nonlinearity shape and the output signal properties.

If power amplifier having high nonlinearities is not normalized properly, the input average

power of the amplifier varies after passing through the predistorter which causes the variation

on amplifier’s response. Therefore, the response of the amplifier changes and the parameters

found for the predistortion become useless because they are estimated with previous response

of the amplifier. They need to be estimated again. In order to solve this problem, a new ap-

proach is suggested [27]. Instead of turning back to the amplifier characterization step because

of the insufficient linearization caused by power misalignment, the proper normalization gain

can be tried in digital predistortion algorithm step as shown in Figure 2.30.

39

Figure 2.29: The gain change with respect to peak power [10]

In conclusion, if the power amplifier nonlinearity is weak, then the power alignment be-

tween the predistorter and the amplifier can be done with the small signal gain. However, if

the power amplifier is highly nonlinear, the gain normalization depends on the input’s aver-

age power and the statistics, and also the shape of the amplifier nonlinearity. Thus, optimal

normalization gain must be found in this situation for better linearization.

40

Figure 2.30: The gain change with respect to peak power [27]

41

CHAPTER 3

SIMULATIONS

3.1 Introduction

In this chapter, memory polynomial digital predistortion algorithm simulations are presented.

MATLAB is used as the simulation tool. Firstly, amplifier and predistorter models are ex-

plained. In preceding sections, the polynomial order and memory depth effects on predis-

tortion performance are discussed in predistortion performance using different kinds of input

modulations. Finally, the model adaptability is observed. This simulation part is a kind of ini-

tial step in order to have an idea about predistortion performance of the inputs having different

modulation types. In simulations, memory polynomial model is used for the amplifier model.

This model is extracted from the real amplifier that is used for the hardware implementations.

For each kind of input having different modulation, bandwidth and power, the model is rebuilt

since the amplifier characteristics change with respect to these parameters. For predistortion

model, again memory polynomial model is used.

3.2 Memory Polynomial Amplifier Model (Forward Model)

Memory polynomial model is used the as amplifier model. This forward model is extracted

from the real power amplifier that will be explained in hardware chapter. The model is ex-

tracted by using the way explained in the previous chapter. To solve the matrix for the co-

efficients, estimations algorithms could be used. However, another easy solution is used for

solving matrix, pinv (Moore-Penrose pseudoinverse of matrix) function of MATLAB.

AM/AM and AM/PM curves of the amplifier, forward model, are given below in Figure

3.1.

42

Figure 3.1: Simulation result of AM/AM and AM/PM curves of the amplifier model simula-tion

3.3 Memory Polynomial Predistorter Model (Inverse Model)

Memory polynomial model is used as predistorter model. Indirect learning architecture is

used in order to find the inverse model as explained in previous chapter. To solve the matrix

for the coefficients, pinv function of MATLAB is used. AM/AM and AM/PM curves of the

predistorter, inverse model, are given below in Figure 3.2.

Figure 3.2: Simulation result of AM/AM and AM/PM curves of the predistorter model simu-lation

As seen in Figure 3.2, the AM/AM and AM/PM characteristics of the predistorter is the

43

inverse of the amplifier characteristics in order to compensate the distortions.

In predistortion scheme, the input passes through the predistorter and it expands. Then the

expanded output of the predistorter passes through the amplifier and it is saturated. The sat-

uration of the expanded curve gives the linear response. The AM/AM and AM/PM response

of the cascade of the predistorter and the amplifier is given in Figure 3.3.

Figure 3.3: Simulation result of AM/AM and AM/PM curves of the cascade of predistorterand amplifier models

The power spectral density of the input of the amplifier, output of the amplifier without

predistortion and output of the amplifier with predistortion is given in Figure 3.4.

44

Figure 3.4: Simulation result of Power Spectral Density of input, output with predistortion,output without predistortion

3.4 Memory Depth Effect on Predistortion Performance

Memory depth effect on performance is studied by using different kind of inputs. Polynomial

order is kept constant in order to eliminate its effect. The inputs used on this work are the

following:

• Two-tone input with different spacing

• 8psk modulated input with different bandwidths

• π/2 bpsk modulated input with different bandwidths

Two-tone continuous wave input with 20 kHz frequency spacing between the tones are

used as training input and the results are given in 3.1. As seen in the table, the best lineariza-

tion is obtained when the memory depth is 1. When the memory depth is 1, linearization is

about 32dBc in IMD3, 25dBc in IMD5 and 38dBc in IMD7. The linearization performance

is close to the best one when the memory depth is up to 4.

45

Table 3.1: Predistorter performance for two-tone CW having 20 kHz spacing w.r.t. memorydepth

Memory Depth Fundamental Tone (dBc) IMD3 (dBc) IMD5 (dBc) IMD7 (dBc)No pred 0 -18,60 -34,04 -33,43

0 0 -46,24 -54,83 -73,731 0 -50,61 -59,53 -71,942 0 -47,77 -50,42 -56,503 0 -40,52 -46,067 -60,044 0 -41,43 -42,96 -51,425 0 -34,68 -39,04 -48,826 0 -36,95 -41,21 -49,40

Table 3.2: Predistorter performance for two-tone CW having 1 kHz spacing w.r.t. memorydepth

Memory Depth Fundamental Tone (dBc) IMD3 (dBc) IMD5 (dBc) IMD7 (dBc)No pred 0 -18,01 -34,88 -36,38

0 0 -68,56 -75,07 -76,101 0 -62,67 -70,34 -77,502 0 -36,43 -46,62 -55,563 0 -34,79 -47,26 -47,354 0 -36,22 -48,79 -47,855 0 -28,29 -39,24 -45,816 0 -39,29 -39,71 -42,25

Two-tone input with 1 kHz spacing between the tones are used as training input and the

results are given in Table 3.2. As seen in the table, the linearization performance is best

in memoryless case by obtaining 50dB linearization in IMD3, 40dB in IMD5 and 39dB in

IMD7. The linearization when memory depth is 1 is also close to the memoryless case. The

highest linearization of memoryless case shows that the amplifier doesn’t have much memory

effects with 1 kHz input. Considering two-tone input with 20 kHz spacing, it is shown that

optimal memory depth is 1. It is expected that the amplifier has higher memory effect when

the frequency spacing increases. This makes sense why the memoryless case is optimal for

the two-tone input signal having 1 kHz spacing.

8psk input with 40 kHz bandwidth is used as training input and the results are given

in Table 3.3. The highest linearization is obtained when the memory depth is 1 by 32dB

linearization in 1st, 23dB in 2nd and 2dB in 3rd adjacent channels. The linearization perfor-

mances up to memory depth 2 are close. However, as the memory depth increases more than

necessary, the linearization gets worse.

46

Table 3.3: Predistorter performance for 8psk input having 40 kHz spacing w.r.t. memorydepth

Memory Depth Main channel (dBc) 1st adj ch (dBc) 2nd adj ch (dBc) 3rd adj ch (dBc)No pred 0 -31,32 -44,54 -59,69

0 0 -58,97 -66,06 -62,391 0 -63,19 -67,55 -61,772 0 -61,07 -63,37 -67,193 0 -54,34 -60,08 -62,534 0 -54,15 -60,80 -60,605 0 -55,30 -62,52 -59,026 0 -54,66 -64,77 -56,387 0 -53,58 -62,23 -53,11

Table 3.4: Predistorter performance for 8psk input having 80 kHz spacing w.r.t. memorydepth


0 0 -53,86 -65,18 -65,921 0 -56,89 -60,76 -60,452 0 -57,68 -60,66 -63,823 0 -61,43 -61,60 -64,734 0 -60,68 -62,18 -64,285 0 -61,73 -63,43 -65,506 0 -55,54 -60,87 -64,187 0 -48,36 -57,91 -59,04

8psk input with 80 kHz bandwidth is used as training input and the results are given in

Table 3.4. As seen from the table, when the input is 80 kHz 8psk data, the optimal memory

depth is 5 by obtaining 31dB linearization in 1st, 19dB in 2nd and 9dB in 3rd adjacent chan-

nels. The linearization performances when the memory depth is 3 and 4 are also close to the

best linearization. However, as in previous cases, when the memory depth increases more

than necessary, the linearization performance decreases seriously. Comparing 8psk inputs of

bandwidth 40 kHz and 80 kHz, it is seen that the optimal memory depth of input of 80 kHz

bandwidth is higher. It is expected because the input signal bandwidth has a direct impact

on the memory effect of the power amplifier. As the bandwidth of the input increases, the

memory effect becomes more important on amplifier nonlinearity.

π/2 bpsk input with 40 kHz bandwidth is used as training input and the results are given in

Table 3.5. The highest linearization performance is obtained in memoryless case by obtaining

33dB linearization in 1st, 18dB in 2nd and 10dB in 3rd adjacent channels. When the memory

depth is 1, the performance is close to the memoryless case, but after that memory depth

the linearization decreases seriously. Since the memoryless case is the optimal case for this

47

Table 3.5: Predistorter performance for π/2 bpsk input having 40 kHz spacing w.r.t. memorydepth


0 0 -66,56 -71,77 -70,481 0 -60,89 -65,15 -66,222 0 -50,02 -58,78 -61,983 0 -49,31 -57,42 -60,354 0 -48,76 -56,55 -58,425 0 -47,33 -56,68 -57,936 0 -46,22 -55,62 -56,707 0 -44,76 -55,05 -57,15

input, it can be said that the power amplifier doesn’t have much memory effect with this

input. Comparing 40 kHz input of 8psk and π/2 bpsk, the amplifier has higher memory effect

when the input is 8psk. It makes sense because the amplifier can show different memory

effect response with different input types. Input bandwidth, modulation etc can change this

response as stated before in Memory Effects section.

π/2 bpsk input with 80 kHz bandwidth is used as training input and the results are given in

Table 3.6. As seen in the table, the optimal memory depth is 1 by obtaining 30dB linearization

in 1st, 12dB in 2nd, 8dB in 3rd adjacent channels. The linearization performances are close

when the memory depth is up to 4. When the input is π/2 bpsk of 40 kHz bandwidth, it is

shown that the optimal memory depth is zero. Since the amplifier has higher memory effect

as the bandwidth increases, the results are consistent.

Comparing 80 kHz π/2 bpsk and 80 kHz 8psk input data considering memory depths;

the 8psk data causes the amplifier to have more memory effect. This is observed in 40 kHz

π/2 bpsk and 40 kHz 8psk cases. So, the results are consistent. The amplifier shows higher

memory effect when the input is 8psk rather than π/2 bpsk. The difference of the amplifier

response to these signals are caused by the difference in power distribution of them. For π/2

bpsk signal, the power is more evenly distributed over bandwidth than 8psk signal. Thus, the

amplifier shows lower memory effect with this input.

3.5 Polynomial Order Effect on Predistortion Performance

Polynomial order effect on predistortion performance is studied by using different kind of

inputs. Memory depth is kept constant at its optimal value as estimated in previous section in

order to eliminate its effect. The inputs used on this work are the following:

48

Table 3.6: Predistorter performance for π/2 bpsk input having 80 kHz spacing w.r.t. memorydepth


0 0 -61,60 -63,24 -67,251 0 -63,13 -63,73 -68,172 0 -60,55 -62,37 -60,953 0 -58,97 -60,44 -55,764 0 -57,31 -58,33 -52,155 0 -55,72 -55,27 -48,816 0 -51,05 -50,32 -46,157 0 -47,91 -47,30 -44,66

Table 3.7: Predistorter performance for two-tone CW having 20 kHz spacing w.r.t. polyno-mial order

Polynomial Order Fundamental Tone (dBc) IMD3 (dBc) IMD5 (dBc) IMD7 (dBc)No pred 0 -18,60 -34,04 -33,43

5 0 -37,60 -46,89 -53,247 0 -45,42 -52,06 -66,239 0 -49,32 -58,08 -74,41

11 0 -50,62 -59,53 -71,9413 0 -50,75 -59,63 -71,3315 0 -50,72 -59,62 -71,2817 0 -50,70 -59,61 -71,27

• two-tone input with different spacing

• 8psk modulated input with different bandwidths

• π/2 bpsk modulated input with different bandwidths

Two-tone input with 20 kHz spacing between the tones are used as training input and

the results are given in Table 3.7. As seen in the table, the linearization in IMD3 increases

8dB as the predistorter polynomial order is increased from 5 to 7. Similarly, linearization in

IMD3 increases 4dB as order is increased from 7 to 9, and 1dB as order is increased from 9

to 11. Maximum linearization is reached when the polynomial order is 11 and as it increases,

the linearization doesn’t change. It is expected because as polynomial order increases, the

unnecessary orders have very small values that it doesn’t affect the modeling.

Two-tone input with 1 kHz spacing between the tones are used as training input and the

results are given in Table 3.8. In linearization of two-tone input with 1 kHz spacing, the

optimal polynomial order is found as 9. The linearization in IMD3 increases 16dB if the

polynomial order increases from 5 to 7 and 8dB if order increases from 7 to 9. Once it

49

Table 3.8: Predistorter performance for two-tone CW having 1 kHz spacing w.r.t. polynomialorder

Polynomial Order Fundamental Tone (dBc) IMD3 (dBc) IMD5 (dBc) IMD7 (dBc)No pred 0 -18,23 -35,96 -36,60

5 0 -37,90 -48,12 -67,227 0 -54,29 -62,64 -64,719 0 -62,49 -73,86 -76,49

11 0 -62,43 -70,11 -77,7913 0 -62,66 -70,32 -77,4815 0 -62,45 -70,12 -77,4417 0 -62,40 -70,07 -77,43

Table 3.9: Predistorter performance for 8psk input having 40 kHz spacing w.r.t. polynomialorder

Polynomial Order Main channel (dBc) 1st adj ch (dBc) 2nd adj ch (dBc) 3rd adj ch (dBc)No pred 0 -31,32 -44,53 -59,69

5 0 -38,93 -55,22 -67,517 0 -44,03 -55,65 -58,889 0 -55,71 -60,59 -58,5511 0 -63,23 -66,38 -61,4713 0 -63,19 -67,55 -61,7715 0 -63,18 -67,55 -61,8617 0 -63,18 -67,54 -61,88

reaches its optimal polynomial order, the increase in order doesn’t affect the linearization

performance as in 20 kHz case. Although there is a little decrease in linearization of 1st

adjacent channel, this can be caused by the mathematical error since the matrix gets greater

as the order increases. Also, it is close to the noise level, so it can be ignored. Comparing

the two-tone input signals of spacing 1 kHz and 20 kHz, the input with 20 kHz spacing needs

higher polynomial order to reach its maximum linearization value.


Table 3.9. As seen in the table, maximum linearization is reached when the polynomial order

is 11. After the optimal polynomial order, increasing order doesn’t change the linearization

performance. Up to polynomial order 11, 5dB from order 5 to 7, 12dB from 7 to 9 and 8dB

increase in linearization from 9 to 11 in 1st adjacent channel is observed.


Table 3.10. In linearization of 8psk input with 80 kHz bandwidth, the optimal polynomial

order is realized as 13 since as it increases, the linearization performance remains the same.

There is a serious linearization enhancement when the order of 5 or 7 increases to 9. Com-

paring the optimal polynomial order in linearization of 8psk inputs of bandwidth 40 kHz and

50

Table 3.10: Predistorter performance for 8psk input having 80 kHz spacing w.r.t. polynomialorder


5 0 -37,80 -52,6 -63,377 0 -39,92 -52,48 -55,399 0 -57,67 -60,61 -61,8211 0 -57,93 -59,99 -62,7613 0 -61,43 -61,60 -64,7315 0 -61,02 -61,24 -64,8917 0 -61,33 -61,63 -64,70

Table 3.11: Predistorter performance for π/2 bpsk input having 40 kHz spacing w.r.t. polyno-mial order


5 0 -41,74 -55,30 -70,007 0 -48,62 -56,85 -60,149 0 -61,55 -62,25 -63,9611 0 -62,60 -65,24 -67,2513 0 -61,99 -64,88 -66,9515 0 -61,99 -64,91 -66,9317 0 -61,99 -64,91 -66,93

80 kHz, it is observed that 80 kHz input needs higher polynomial order to reach its maximum

linearization performance.

π/2 bpsk input with 40 kHz bandwidth is used as training input and the results are given

in Table 3.11. As seen in the table, the minimum polynomial order of the maximum lineariza-

tion reached is 9. After that value, the linearization remains constant and up to that point,

linearization continues to increase like previous kinds of inputs. Comparing 40 kHz - π/2

bpsk input with 40 kHz - 8psk input linearization, π/2 bpsk input needs lower polynomial

order for maximum linearization.

π/2 bpsk input with 80 kHz bandwidth is used as training input and the results are given

in Table 3.12. In linearization of 80 kHz π/2 bpsk input, the optimal polynomial order is 11.

The polynomial order increase follows the same pattern as previous input cases.

Comparing 80 kHz π/2 bpsk input with 80 kHz 8psk input linearization, 8psk input needs

higher polynomial order for its maximum linearization as it was experienced in 40 kHz band-

width case. Furthermore, comparing 40 kHz and 80 kHz π/2 bpsk input linearization, 80 kHz

signal’s optimal polynomial order is higher, again as happened in 8psk case. The optimal

polynomial order increase depends on the increase of the complexity of the input signal.

51

Table 3.12: Predistorter performance for π/2 bpsk input having 80 kHz spacing w.r.t. polyno-mial order


5 0 -41,38 -54,59 -67,507 0 -47,63 -56,20 -60,059 0 -56,98 -61,18 -66,7111 0 -61,53 -64,21 -65,2913 0 -60,89 -65,15 -66,2215 0 -60,73 -65,26 -66,3617 0 -60,71 -65,27 -66,39

3.6 Model Validity Range

Model adaptability in different cases is examined in this section. This is studied in two

branches. First one is estimating a memory polynomial for the predistorter with a random

set of wideband input data of specific modulation, and checking the linearization with other

random sets of input data of that specific modulation. Second one is again estimating a mem-

ory polynomial for the predistorter with a random set of wideband input data of specific

modulation, and checking the linearization with other random sets of input data of not that

specific modulation but another modulation types. The approach will be the following:

• Trial of polynomial for predistorter estimated on one set of random 8psk data to another

random sets of 8psk data

• Trial of polynomial for predistorter estimated on one modulation to another types mod-

ulation

– Predistorter polynomial modeled with two-tone input of one spacing on two-tone

input of different spacing

– Predistorter polynomial modeled with 8psk and π/2 bpsk modulated input data of

one bandwidth on another bandwidth

– Predistorter polynomial modeled with 8psk modulated input data of one band-

width on π/2 bpsk modulated input data of the same bandwidth

3.6.1 DPD Performance For Different Data Sets of Same Modulation

A set of random 8psk modulated 40 kHz input data (8psk set 1) is trained and a memory

polynomial predistortion model is obtained. Using this model, other sets of random 8psk

52

Table 3.13: Normalized 40kHz 8psk input predistortion with different sets

8psk set Main channel (dBc) 1st adj ch (dBc) 2nd adj ch (dBc) 3rd adj ch (dBc)No pred 0 -31,32 -44,53 -31,21

set1 0 -65,66 -68,4 -63,51set2 0 -64,94 -68,37 -63,09set3 0 -64,42 -67,40 -62,73set4 0 -65,59 -68,72 -63,30

Table 3.14: Normalized 20kHz two-tone input predistortion with predistorter modeled by 1khz two-tone input

Predistorter Fundamental Tone (dBc) IMD3 (dBc) IMD5 (dBc) IMD7 (dBc)No pred 0 -19,20 -34,02 -53,46

Trained by 20 kHz spacing 0 -50,12 -59,03 -71,44Trained by 1 kHz spacing 0 -30,24 -50,25 -43,01

modulated 40 kHz inputs (8psk set 2, 8psk set 3, and 8psk set 4) are predistorted. The lin-

earization results are given in Table 3.13.

As seen in the table, random 8psk inputs of one bandwidth are linearized with the predis-

torter polynomial which is trained by a set of random 8psk data stream of same bandwidth.

This shows that the model is accurate such that it consists of all randomization of the input

response. This adaptability is crucial because once the predistorter is trained with one set

of input, and then there is no need to update the model if the incoming inputs are in same

modulation and bandwidth. This provides an important easiness in implementation process.

3.6.2 DPD Performance For Different Data Sets of Different Modulations and Band-

width

First, a polynomial predistorter model is trained by using a two-tone input of one spacing and

this model is used to linearize two-tone inputs with other spacings. A polynomial predistorter

model is obtained by using a two-tone input data of spacing 1 kHz. Then, two-tone input data

of spacing 20 kHz is predistorted using this polynomial model trained with 1 kHz spacing

data. The linearization is compared with the linearization by the polynomial model trained

with the data itself and the results are tabulated in Table 3.14. As seen in the table, the input

with different spacing is also linearized by the model trained with 1 kHz. However, this

linearization is lower than the linearization by using the model trained by the input data itself

as expected. The inputs have different spacing so the amplifier memory effect response is

different when these inputs are applied.

53

Table 3.15: Normalized 80kHz 8psk input predistortion with predistorter modeled with 40kHz 8psk

Predistorter Main channel (dBc) 1st adj ch (dBc) 2nd adj ch (dBc) 3rd adj ch (dBc)No pred 0 -30,14 -44,23 -56,35

Trained by 80khz 8psk 0 -61,43 -61,60 -64,73Trained by 40khz 8psk 0 -45,79 -57,82 -61,65

Table 3.16: Normalized 80kHz π/2 bpsk input predistortion with predistorter modeled with40 kHz π/2 bpsk

Predistorter Main channel (dBc)1st adj ch (dBc)2nd adj ch (dBc)3rd adj ch (dBc)No pred 0 -33,47 -53,40 -60,64

Trained by 80khz π/2 bpsk 0 -61,53 -64,21 -65,29Trained by 40khz π/2 bpsk 0 -47,30 -56,88 -71,23

Secondly, a polynomial predistorter model is obtained using a modulated signal of one

bandwidth and it is used to linearize another signal with same modulation but different band-

width. This experiment is tried by models obtained by 40 kHz 8psk and 40 kHz π/2 bpsk on

80 kHz 8psk and 80 kHz π/2 bpsk respectively. The results are given in Table 3.15 and Table

3.16.

In 8psk input case, the model trained by the data itself provides 31dB linearization in

the 1st adjacent channel, whereas, the model trained by 40 kHz 8psk input provides 15dB

linearization. Besides, in π/2 bpsk input case, the model trained by the data itself provides

28dB linearization in 1st adjacent channel, whereas, the model trained by 40 kHz π/2 bpsk

input provides 14dB linearization. This shows that the model is still valid in simulations

as bandwidth changes since it provides linearization but it is not accurate as the predistorter

trained by the data itself. This result is expected because as bandwidth increases, the memory

effect of the amplifier increases. Since the response of the amplifier to the inputs having same

modulation but different bandwidths is different, the models of the predistorter is also altered.

Finally, a polynomial predistorter model is obtained by training an input data with a mod-

ulation type and bandwidth, then use this predistorter model to linearize again a wideband

input having another modulation type but same bandwidth. In order to do this, π/2 bpsk input

data is attempted to be linearized with a polynomial model obtained by 8psk signal of same

bandwidth. For this work, 40 kHz and 80 kHz bandwidth signals are used. The comparisons

are given in the Table 3.17 and Table 3.18.

As seen in the table 3.17, 40 kHz π/2 bpsk input is linearized 28dB in 1st adjacent channel

when the predistorter model is trained by itself, whereas, it is linearized 18dB if the predis-

54

Table 3.17: Normalized 40kHz π/2 bpsk input predistortion with predistorter modeled with40 kHz 8psk


Trained by 40khz π/2 bpsk 0 -61,98 -64,88 -66,95Trained by 40khz 8psk 0 -51,26 -55,36 -66,16

Table 3.18: Normalized 80kHz π/2 bpsk input predistortion with predistorter modeled with80 kHz 8psk


Trained by 80khz π/2 bpsk 0 -61,53 -64,21 -65,29Trained by 80khz 8psk 0 -45,70 -58,33 -66,14

torter model is obtained by using 40 kHz 8psk signal. Similarly, 80 kHz π/2 bpsk input

is linearized 28dB in 1st adjacent channel when the predistorter model is trained by itself,

whereas, it is linearized 12dB if the predistorter is obtained by using 80 kHz 8psk data. As

expected, higher linearization is achieved when the predistorter model is obtained by the in-

put itself. This is because the statistics of 8psk and π/2 bpsk modulation are different. So,

the amplifier gives different responses to these inputs. For high linearization, the amplifier

response of the input which is to be linearized must be used. However, since a linearization

is achieved with different modulations, it can be said that the model is still valid in simula-

tions. This result is expected because the amplifier response changes with respect to the input.

Since the input modulation type changes, the response of the amplifier also changes. If the

forward and inverse models of the amplifier are estimated by one modulation type input, then

its linearization performance is not good with another type modulation input. Similar result

is obtained in [28] when cross validation with new signal type is studied.

55

CHAPTER 4

HARDWARE IMPLEMENTATION

4.1 Introduction

In this chapter memory polynomial digital predistortion implementation is presented. After

verifying the linearization of the amplifier by memory polynomial method in simulation in

Chapter 3, the predistorter is implemented. First of all, the test bench is explained, and then

several conducted experiments are mentioned. In the experiment part, memory effect on

predistortion performance is shown. After that, predistortion performance with different data

of the same modulation is presented. Finally, the change in predistortion performance is

observed by sweeping the power and bandwidth.

4.2 Test Bench

The test bench built for the digital predistortion is to implement the proposed digital predistor-

tion scheme given in Figure 4.1. The test bench consists of power amplifier, signal generator,

vector signal analyzer which is an option of signal analyzer as shown in Figure 4.2. The

corresponding devices will be explained in detail.

4.2.1 Power Amplifier

An HF power amplifier is used as DUT for the predistortion. It is designed to work between

the frequencies 1MHz to 30Mhz. The amplifier is designed with a push pull structure with a

driver stage. The saturation power is 56dBm and 1 dB compression point (P1dB) is 53.3dBm.

The gain of the amplifier is 45.7dBm. The operation frequency is chosen as 10,Hz randomly.

The operation frequency doesn’t affect the predistortion algorithm since the algorithm is run

in baseband (i.e. carrier frequency is suppressed in baseband). It can only affect the power

56

Figure 4.1: Digital Predistortion Scheme [27]

Figure 4.2: Digital Predistortion Test bench

amplifier characteristics which the predistorter is adapted.

4.2.2 Signal Generator

The signal generator used in test bench is HP ESG-D Series E4433B signal generator. It

has arbitrary waveform generator option. This option provides to load baseband signal to

the generator and it carries that baseband signal to the desired frequency. So, the signal

generator mission in the test bench is to take the baseband signal and convert it to analog like

57

digital to analog converter and upconvert it to the desired frequency. In arbitrary waveform

generator option, the signal generator has sampling clock, reconstruction filter, and similar

settings. The baseband input data is generated in MATLAB, these I-Q signals are loaded to

the signal generator and with the corresponding settings. Then, the power amplifier is excited

with the signal generated by the signal generator. There is a critical issue about the signal

generator. The upper limit of the linearization is determined by the system’s ACPR. Therefore

the signal generator’s ACPR is important for the predistortion performance. Considering this

important point, the signal generator’s ACPR is kept as low as it can be in order not to affect

the predistortion performance. Moreover, the signal generator’s sampling clock of arbitrary

waveform generator option determines the resolution of the input signal. The upper limit of

the sampling clock of it limits the sampling frequency of the input which is critical for delay

alignment process.

4.2.3 Vector Signal Analyzer

The vector signal analyzer in the test bench is the option of the Signal Analyzer of Agilent

Technologies EXA signal analyzer N9010A. The vector signal analyzer takes the signal at

rf frequency downconverts and digitizes it like an analog to digital converter. It gives I-Q

signal of the given input, in other words it demodulates the given signal. Thus, by using

the vector signal analyzer, the output of the amplifier is taken in baseband and it is ready

for the predistortion algorithm which uses the baseband input and output signals. There is

also a critical point about the vector signal analyzer. The signal acquisition path, here espe-

cially signal analyzer, bandwidth should be at least five times of the bandwidth of the input

in order to characterize the amplifier accurately including the nth orders of intermodulation

[10]. Therefore, the bandwidth of the signal analyzer limits the characterization accuracy of

the amplifier, so the predistorter. Furthermore, since the vector analyzer does the mission of

DAC, the output resolution depends on it. As stated in Signal Generator section, the output

resolution also affects the delay alignment process. If the output is not sampled with enough

resolution, an upsampling operation is done in MATLAB in order to find the delay accurately

which is an important step in estimating the behavioral models.

58

4.3 Experimental Procedure

Using the test bench explained in previous sections, the baseband input and output signals

are obtained. These signals are used to estimate the forward model of the amplifier and the

inverse model of the amplifier in MATLAB in order to use as the predistorter function. Firstly,

the delay alignment is done by using correlation technique explained in Delay Alignment

section. Before the delay alignment process, the input and output signals are upsampled

and interpolated in order to find the delay finely. After alignment, the inverse model of the

amplifier is estimated as stated in Predistortion algorithm using indirect learning architecture

section. The input signals for the experiments are two-tone input with different frequency

spacing and wideband signals. Wideband signals are 8psk modulated signal with 6dB crest

factor and π/2 bpsk modulated signal with 4.5dB crest factor.

4.3.1 Comparison of Memoryless Predistortion and Predistortion With Memory

The memory effects of the power amplifier cause nonlinearity and spectral regrowth. It also

depends on the signal statistics, which means as the input signal changes, the memory ef-

fects of the amplifier change as explained in Memory Effects section in detail. The strong

static AM/AM and AM/PM nonlinearity can be modeled and handled with memoryless pre-

distortion, however, the dynamic nonlinearity caused by memory effects still cause spectral

regrowth. In order to linearize further, the memory effects should be considered. The memory

polynomial predistortion takes the memory effect nonlinearity into account, so the lineariza-

tion performance is expected higher than the memoryless predistortion on which the amplifier

with high memory effects. In this section, the performances of the memoryless predistortion

and predistortion with memory are compared. To do this comparison, two-tone input signal

and wideband signals are used. In order to get the memoryless predistortion, the memory

depth used in memory polynomial is set to zero.

4.3.1.1 Two-Tone Input Data

Two-tone input signals with 1 kHz and 20 kHz frequency spacing are given to the system. The

adjacent channel powers of the power amplifier output without predistortion, with memoryless

predistortion and predistortion with memory are measured. The peak powers kept constant in

59

all three cases. For two-tone input having 1kHz spacing, the polynomial order is taken as 11

and memory depth is 1, whereas, for two-tone input having 20kHz spacing, the polynomial

order is taken as 11 and memory depth is 2.

For two-tone input with 1 kHz spacing, the spectrum is observed in these three cases.

When memoryless predistortion is applied, 17.2dB linearization in IMD3, 5.7dB in IMD5,

16.7dB in IM7 are obtained as seen in Figure 4.4. However, when predistortion with memory

is applied, 25.9dB linearization in IMD3, 5.9dB in IMD5, 12.6dB in IMD7 are obtained as

seen in Figure 4.5. So, memory polynomial predistortion gives better results than memoryless

predistortion with 1 kHz two-tone input.

Figure 4.3: The output of the amplifier without predistortion with 1 kHz two-tone input

Two-tone input with 20 kHz frequency spacing is also used. When memoryless predistor-

tion is applied, 17.5dB linearization in IMD3, 25.5dB in IMD5, 4.5dB in IM7 are obtained as

seen in Figure 4.7. However, when predistortion with memory is applied, 27.1dB linearization

in IMD3, 24.9dB in IMD5, 7.4dB in IMD7 are obtained (Figure 4.8). So, memory polynomial

predistortion gives better results than memoryless predistortion with 20 kHz two-tone input

as in 1 kHz case.

60

Figure 4.4: The output of the amplifier with memoryless predistortion with 1 kHz two-toneinput

Figure 4.5: The output of the amplifier with memory polynomial predistortion, memorydepth=1, PEP=54.58dBm with 1 kHz two-tone input

4.3.1.2 Wideband Input Data

Wideband input signals with 8psk and π/2 bpsk modulation are given to the system. The

adjacent channel powers of the power amplifier output without predistortion, with memoryless

61

Figure 4.6: The output of the amplifier without predistortion with 20 kHz two-tone input

Figure 4.7: The output of the amplifier with memoryless predistortion with 20 kHz two-toneinput

predistortion and predistortion with memory are measured. The peak powers kept constant in

all three cases.

8psk and π/2 bpsk inputs having 80 kHz bandwidth have symbol rate as 64 kHz, whereas,

62

Figure 4.8: The output with memory polynomial predistortion, memory depth=4,PEP=54,45dBm with 20 kHz two-tone input

35 kHz inputs with same modulations have symbol rate as 28 kHz. The roll-off factor of

the raised cosine filter is 0.25 while generating the data in all cases. The spectrum without

any amplifier of 80 kHz 8psk and π/2 bpsk modulated input signal are measured. 8psk input

has 1st adjacent channel about 50dBc, 2nd and 3rd adjacent channels about 56dBc (Figure

4.9), whereas, π/2 bpsk input has 1st channel about 53dBc, 2nd and 3rd channels about 56dBc

(Figure 4.10). Furthermore, the constellation diagrams are also given in Figure 4.11 and

Figure 4.12.

For 8psk modulated input signal of 35 kHz bandwidth, when the linearization is applied

using memoryless predistortion, 12.5dB linearization in 1st adjacent channel, 11.8dB in 2nd

adjacent channel and 3.6 dB in 3rd adjacent channel are seen in Figure 4.14. However,

when memory polynomial predistortion is used, 15.3dB linearization in 1st adjacent chan-

nel, 12.3dB in 2nd adjacent channel and 2.5dB in 3rd adjacent channel are obtained (Figure

4.15. Therefore, when memory effects are taken into account, the linearization performance

increases with 35 kHz 8psk input signal.

When 8psk modulated signal of 80 kHz bandwidth is applied as input, with memoryless

predistortion, 7dB linearization in 1st adjacent channel and 7dB in 2nd adjacent channel are

observed (Figure 4.17). However, in memory polynomial predistortion, 11dB linearization

in 1st adjacent channel and 8.5dB in 2nd are obtained (Figure 4.18). So, by taking memory

63

Figure 4.9: Spectrum of 80 kHz 8psk input

Figure 4.10: Spectrum of 80 kHz π/2 bpsk input

effects into account, up to 4 dB improvement is obtained in 1st adjacent channel when the

input is 80 kHz 8psk signal.

When π/2 bpsk modulated signal of 80 kHz bandwidth is applied as input, with memory-

64

Figure 4.11: Constellation diagram of 80 kHz 8psk input

Figure 4.12: Constellation diagram of 80 kHz π/2 bpsk input

less predistortion, 5dB linearization in 1st adjacent channel and 2.2dB in 2nd adjacent channel

are observed (Figure 4.20). However, in memory polynomial predistortion, 8dB linearization

in 1st adjacent channel and 2.3dB in 2nd are obtained (Figure 4.21). So, by taking memory

effects into account, up to 3 dB improvement is obtained in 1st adjacent channel when the

65

Figure 4.13: The output of the amplifier without predistortion with 35 kHz 8psk input

Figure 4.14: The output of the amplifier with memoryless predistortion with 35 kHz 8pskinput

input is 80 kHz π/2 bpsk signal.

In conclusion, the memory polynomial predistortion performance is better than the mem-

oryless predistortion. The strong AM/AM and AM/PM nonlinearity is compensated also

66

Figure 4.15: The output with memory polynomial predistortion, memory depth=2,PEP=54,66dBm with 35 kHz 8psk input

Figure 4.16: The output of the amplifier without predistortion with 80 kHz 8psk input

by memoryless predistortion but the weak nonlinearity coming from memory effects is just

linearized by predistortion with memory. Furthermore, according to the input given to the

amplifier, the optimal memory depth changes according to the signal statistics, bandwidth etc

67

Figure 4.17: The output of the amplifier with memoryless predistortion with 80 kHz 8pskinput

Figure 4.18: The output with memory polynomial predistortion, memory depth=6,PEP=54.02dBm with 80 kHz 8psk input

as expected.

68

Figure 4.19: The output of the amplifier without predistortion with 80 kHz π/2 bpsk input

Figure 4.20: The output of the amplifier with memoryless predistortion with 80 kHz π/2 bpskinput

4.4 Model Validity Range

4.4.1 Predistortion Performance With Different Data of Same Modulation

In predistortion process, a number of input samples are taken and it is used in predistortion

algorithm. If the input is wideband modulated signal like 8psk or π/2 bpsk, the input samples69

Figure 4.21: The output with memory polynomial predistortion, memory depth=4,PEP=54.12dBm with 80 kHz π/2 bpsk input

Table 4.1: 35kHz 8psk input predistortion with different sets

Data set L.3rd adj L.2nd adj L.1st adj Main ch U.1st adj U.2nd adj U.3rd adj8psk set1 -62,8 -53,6 -43,4 49,2 -44,4 -54,9 -61,48psk set2 -59,8 -51,6 -42 49,2 -42,1 -51,8 -60,78psk set3 -60,7 -54,3 -43,2 49,2 -42,6 -53 -60,88psk set4 -61,6 -54,1 -43 49,2 -43 -52,7 -61,2

are random. In order to be sure that the amplifier’s forward and inverse models are estimated

accurately, it is necessary to take all random points. This can be checked in such a way that the

predistorter is trained with a set of data and tested with other sets of data of same modulation

and bandwidth. If the linearization performances of those other sets are close to the main

set which is used for training, then it can be said that this predistortion function is estimated

accurately. This is tried in the predistortion test bench with 35 kHz 8psk and 35 kHz π/2

bpsk input data. 8psk set1 and p2bpsk set1 are the training data sets for 8psk and π/2 bpsk

modulated input signals respectively. The results are given in Table 4.1 and Table 4.2 and as

seen in the tables, the linearization performances are close. It was shown in Chapter 3, now

in hardware implementation it is verified.

This is an important feature of predistortion considering the adaptation. In hardware

implementation of the predistorter, when the data stream comes, there is no time to take

70

Table 4.2: 35kHz 8psk input predistortion with different sets

Data set L.3rd adj L.2nd adj L.1st adj Main ch U.1st adj U.2nd adj U.3rd adjp2bpsk set1 -66,3 -59,3 -46,2 50,2 -44,2 -57,3 -64,8p2bpsk set2 -65,1 -56,3 -44,5 50,2 -43,7 -56,1 -62,6p2bpsk set3 -66,3 -58,7 -46,7 50,2 -45 -57,4 -64,1p2bpsk set4 -65,4 -57 -47,7 50,2 -44,1 -56,3 -65,1

Table 4.3: 35kHz π/2 bpsk input predistorted with model trained by 35 kHz 8psk (units:dBm)

Predistorter L.3rd adj L.2nd adj L.1st adj Main ch U.1st adj U.2nd adj U.3rd adjNo predistortion -65,5 -53,5 -32,6 52 -31,8 -51,7 -65

Trained by π/2 bpsk -66,1 -58,8 -46 50,3 -45,8 -57,9 -63,4Trained by 8psk -66,1 -58,8 -38,1 50,5 -37,6 -55,3 -62,4

the output of it, adapt the predistorter and put the input into predistorter polynomial and send

the output of the predistorter to the amplifier. By this feature, a training input is sent to the

amplifier and predistorter polynomial is estimated. Of course this input must have the same

modulation, bandwidth, output power etc since the characteristics of the amplifier change

with respect to those. After estimating the predistorter function, there is no need to update it

until the amplifier characteristics change.

4.4.2 Predistortion Performance With Different Modulations

In this section, the predistorter function is estimated using 35 kHz and 80 kHz 8psk modulated

input, then this function is used for linearization of 35 kHz and 80 kHz π/2 bpsk modulated

input signals respectively. The results are given in Table 4.3 and Table 4.4.

As seen in tables, when 35 kHz π/2 bpsk test data is predistorted with 35 kHz π/2 bpsk

trained function, 13.4dB linearization in 1st adjacent channel and 5.3dB in 2nd adjacent chan-

nel are obtained. However, when 35 kHz π/2 bpsk test data is predistorted with 35 kHz 8psk

trained function, 5.5dB linearization in 1st adjacent channel and 3.6dB in 2nd adjacent chan-

nel are obtained. Therefore, although there exists linearization with predistorter trained with

Table 4.4: 80kHz π/2 bpsk input predistorted with model trained by 80 kHz 8psk (units:dBm)

Predistorter L.3rd adj L.2nd adj L.1st adj Main ch U.1st adj U.2nd adj U.3rd adjNo predistortion -66,1 -55,5 -36,8 51,3 -35,2 -54,5 -65,7

Trained by π/2 bpsk -63,3 -59,9 -47,2 50,3 -46,4 -58,3 -63,5Trained by 8psk -64 -59,6 -39,3 50,5 -39,2 -58 -62,4

71

different modulation input, it is much lower. Furthermore, when 80 kHz π/2 bpsk test data is

predistorted with 80 kHz π/2 bpsk trained function, the linearization 10.4dB in 1st adjacent

channel and 3.8dB in 2nd adjacent channel are measured. Whereas, when 80 kHz π/2 bpsk

test data is predistorted with 80 kHz 8psk trained function, the linearization 2.5dB in 1st adja-

cent channel and 3.5dB in 2nd channel are measured. As in 35 kHz case, the linearization with

predistorter function trained by data of different modulation provides very low linearization.

This is an expected result because the characteristics of the amplifier changes with the input

statistics. So, if the inverse model of the amplifier is estimated with 8psk modulated input,

its performance is high when the input for predistortion is 8psk modulated. If it is used for

linearization of another modulation type like π/2 bpsk, the performance decreases since the

amplifier model changes with this input type. This is also proved in Simulation chapter, so

what is obtained in the hardware implementation is consistent.

4.4.3 Predistortion Performance Change by Power Sweep

Predistortion depends on the peak envelope power (PEP) as mentioned in Power Alignment

section. Once the predistorter function is estimated at a PEP value, it is optimal at a PEP value.

The predistortion performance changes as PEP changes. The predistortion performance with

respect to power sweep is studied using two-tone input with 20 kHz spacing, 35 kHz 8psk

modulated input and 35 kHz π/2 bpsk modulated input.


20 kHz two-tone input is optimized at 54.45dBm PEP. The maximum linearization, 31.65

dBm, is observed at IMD3 at that PEP as it is seen in Figure 4.22. As either PEP increases or

decreases, the linearization performance decreases. However, when PEP is between 54.2dBm

and 54.7dBm, the predistorter still provides linearization.


An input signal having 8psk modulation and 35 kHz bandwidth is predistorted at 54.6 dBm

PEP value. As expected, maximum linearization, about 12.5dB in 1st adjacent channel, is

observed at that PEP as seen in Figure 4.23 and Figure 4.26. As PEP changes, the linearization

gets worse. Between 54.35 dBm to 54.85 dBm PEP, there exists still acceptable linearization

which is higher than 5dB for both in 1st and 2nd adjacent channels.

72

Figure 4.22: Linearization of 20 kHz two-tone input by power sweep

Figure 4.23: Linearization of 35 kHz 8psk input by power sweep (Lower side)

35 kHz π/2 bpsk modulated input is predistorted at 54.55 dBm PEP value. The maximum

linearization, about 14.7dB, in 1st adjacent channel and 7.2dB in 2nd adjacent channel, is

observed at that PEP. As PEP changes, the linearization decreases. Between 54.35 dBm to

54.85 dBm PEP, the predistorter function still provides linearization about higher than 5dB in

1st adjacent channel (Figure 4.25 and Figure 4.26).

As observed in three different input cases, the predistorter function is optimal at a PEP

73

Figure 4.24: Linearization of 35 kHz 8psk input by power sweep (Upper side)

Figure 4.25: Linearization of 35 kHz π/2 bpsk input by power sweep (Lower side)

value and the linearization gets worse as the peak power changes. However, although the

linearization amount decreases, the linearization still exists in acceptable region. This is im-

portant in practical cases because once the input data stream is sent to the amplifier, the pre-

distorter function is estimated and it is used until the input type changes. During this period,

the peak power of the amplifier can change a little due to some reasons like temperature, bias

change etc. So, it is critical to know that the predistortion function is still valid although the

74

Figure 4.26: Linearization of 35 kHz π/2 bpsk input by power sweep (Upper side)

peak power changes up to a point.

4.4.4 Predistortion Performance Change by Bandwidth Sweep

Predistortion performance of wideband signal of one bandwidth depends on the bandwidth

that the predistorter function is estimated. As the bandwidth gets away from the estimation

bandwidth value, the predistorter performance decreases. The predistortion performance with

respect to bandwidth sweep is studied using two-tone inputs with 1 kHz and 20 kHz spacing,

35 kHz and 80 kHz 8psk modulated inputs and 35 kHz and 80 kHz π/2 bpsk modulated input.


Bandwidth has a meaning when the input is wideband. For two-tone input, the frequency

spacing is a metric instead of bandwidth. So, the predistorter functions are estimated at 1 kHz

and 20 kHz spacing separately and they are tried with various frequency spacing two-tone

inputs.

For 1 kHz two-tone input trained predistortion, the maximum performance is obtained at

1 kHz spacing as expected. When 1 kHz two-tone input is predistorted, linearizations 25.3dB

in IMD3, 13.3dB in IMD5 and 13.3dB in IMD7 are obtained as seen in Figure 4.27. As

bandwidth increases, the linearization amount decreases. However, up to 30 kHz frequency

spacing, the predistorter function estimated with 1 kHz spacing two-tone input still provides

75

acceptable amount of linearization like above 5dB and up to 4 kHz above 15dB.

Figure 4.27: Linearization of 1 kHz two-tone input by bandwidth sweep

For 20 kHz two-tone trained predistortion, the predistortion performance is the best at 20

kHz frequency spacing. The linearization amounts of 20 kHz two-tone input are 30.1dB in

IMD3, 19,6dB in IMD5 and 9.6dB in IMD7. As spacing changes, the predistortion perfor-

mance decreases. However, the frequency spacing between 10 kHz to 35 kHz, the lineariza-

tion still exists which is above 5dB, besides, between 15 kHz to 30 kHz, the linearization is

above 15dB (Figure 4.28).


As wideband input, 8psk and π/2 bpsk modulated inputs of 35 kHz and 80 kHz bandwidth

are applied. The predistorter functions are estimated using these inputs and they are tried with

different bandwidths.

When 35 kHz 8psk input trained predistortion, the maximum performance at 35 kHz is

15.8dB linearization in 1st adjacent channel, 12.1dB in 2nd channel and 3.5dB in 3rd channel.

As bandwidth either increases or decreases, the linearization performance decreases. Up to 90

kHz bandwidth, the linearization is obtained in all adjacent channels (Figure 4.29 and Figure

4.30). Besides, when the bandwidth is between 25 kHz to 40 kHz, the linearization in 1st

adjacent channel amount is above 10dB.

76

Figure 4.28: Linearization of 20 kHz two-tone input by bandwidth sweep

Figure 4.29: Linearization of 35 kHz 8psk input by bandwidth sweep (Lower side)

When the predistorter function trained with 80 kHz 8psk input is used, maximum lin-

earization is obtained with 80 kHz input as expected with 11.6dB in 1st adjacent channel and

6.9dB in 2nd adjacent channel. Between the bandwidths 65 kHz to 100 kHz, the predistortion

function provides acceptable linearization. Spectral regrowth is seen in 3rd adjacent channel.

77

Figure 4.30: Linearization of 35 kHz 8psk input by bandwidth sweep (Upper side)

This is because the polynomial order is not enough to predistort after 2nd adjacent channel

(Figures 4.31 and 4.32). It can be also caused since 3rd adjacent channel power is close to

noise floor.

Figure 4.31: Linearization of 80 kHz 8psk input by bandwidth sweep (Lower side)

When the predistorter function is trained by using 35 kHz π/2 bpsk input, the maximum

78

Figure 4.32: Linearization of 80 kHz 8psk input by bandwidth sweep (Upper side)

linearization which is obtained at 35 kHz is 13.8dB in 1st adjacent channel and 6.1dB in 2nd

adjacent channel. The predistorter provides desirable linearization, above 10dB, of the inputs

between 20 kHz to 50 kHz bandwidth (Figure 4.33 and Figure 4.34).

Figure 4.33: Linearization of 35 kHz π/2 bpsk input by bandwidth sweep (Lower side)

When the predistorter function is trained by using 80 kHz π/2 bpsk input, the maximum

79

Figure 4.34: Linearization of 35 kHz π/2 bpsk input by bandwidth sweep (Upper side)

linearization which is obtained at 35 kHz is 13.8dB in 1st adjacent channel and 6.1dB in 2nd

adjacent channel. The predistorter provides linearization above 5dB of the inputs between 70

kHz to 115 kHz bandwidth (Figure 4.35 and Figure 4.36). The predistorter function trained

with 80 kHz π/2 bpsk input provides linearization of 8.5dB in 1st adjacent channel and 2.1dB

in 2nd adjacent channel. There is no linearization in 3rd adjacent channel like happened in 80

kHz 8psk input case since the polynomial order is not high enough and also because of its

level which is close to the noise floor.

In conclusion, the predistorter function has its best performance at which bandwidth it is

trained. However, it is still valid as bandwidth changes. The bandwidth adaptability depends

on the input specifications like modulation, amount of change in bandwidth etc.

80

Figure 4.35: Linearization of 80 kHz π/2 bpsk input by bandwidth sweep (Lower side)

Figure 4.36: Linearization of 80 kHz π/2 bpsk input by bandwidth sweep (Upper side)

81

CHAPTER 5

CONCLUSION

The new digital modulation techniques like OFDM, WCDMA, EDGE etc, which are needed

for wireless communication systems are invented because of the dramatically increase of the

users and limited spectrum. These signals have high peak to average ratios. This means, if

the amplifier works at back-off power in order to be linear, it works with very low efficiency.

Low efficiency causes power loss and heating problems. It is also not cost efficient since the

power amplifier needs to be worked at back-off power which means high cost transistor with

much more power. It is expected to get as much power as the amplifier can give.

If the power amplifier works with high efficiency, it works nonlinearly. Nonlinearity

causes spectral regrowth which means the increase of adjacent channel power. It decreases

the spectral efficiency. Besides, there are limitations of the RF communication systems to

have adjacent channel power up to a specific value in order not to prevent the others’ com-

munication in allocated adjacent channels. Furthermore, the nonlinearity causes deformation

in constellation which means the information is distorted in the amplifier before sending. It

cannot be sent correctly. So, the signal is expected to be sent linearly.

The efficiency and linearity is a trade-off for the power amplifiers. However, the lineariza-

tion techniques are investigated in order to provide the amplifier work close to saturation but

linearly. Working close to saturation prevents the low efficiency, so this problem is also solved.

Many linearization techniques are introduced in 2nd Chapter and because of its advantages;

memory polynomial digital predistortion is chosen to study.

Memory polynomial digital predistortion like other baseband predistortion methods, is

easy to apply. It doesn’t need deep knowledge of the amplifier and its functionality. It takes

the amplifier as a black box and models the characteristics of it. The baseband predistortion,

as the name implies, works in baseband. It needs a facility to convert the RF to baseband

and vice versa. It also needs a processor for the operations for modeling. In today’s re-

82

ceiver and transmitter, these infrastructures already exist. Therefore, there is no need to add

any other circuitry or any component to the system. It is also cost efficient. Among vari-

ous baseband predistortion techniques, the memory polynomial has one more advantage. It

provides the linearization of both static and dynamic nonlinearities since it takes the memory

effects of the amplifier into account. The memoryless predistortion techniques only linearize

the static AM/AM and AM/PM nonlinearities. However, the amplifier has dynamic nonlin-

earities caused by memory effect which depends on the input modulation, bandwidth, bias

network and matching circuit of the amplifier etc. The memory polynomial digital predis-

tortion linearizes these dynamic nonlinearities by considering the past input samples as well

as the current input sample. There are any other predistortion techniques which take mem-

ory effect into account like Volterra series; however, they are more complicated than memory

polynomial for application.

In the third chapter, the memory polynomial digital predistortion is investigated with

MATLAB. The amplifier model and predistortion model are memory polynomial model. In

this chapter, 1 kHz and 20 kHz spacing two-tone input; 40 kHz and 80 kHz 8psk input; and 40

kHz and 80 kHz π/2 bpsk input are used in investigations. The amplifier model is extracted

by using real amplifier measurement result. Parameter extraction procedure is explained in

hardware implementation chapter. In simulations part, memory depth and polynomial order

effects on predistortion performance are investigated.

In memory depth effect on predistortion performance investigation, for two-tone input

signals, it is seen that as spacing decreases, the linearization amount increases since it is

easier to linearize the input with less memory depth. Moreover, it is observed that the optimal

memory depth of the predistorter model increases as the frequency spacing of two-tone data

increases. It can be explained such that; as spacing increases, the frequency of envelope signal

increases. This increase causes the signal to spend more electrical time (phase) in transistor

and it can be modelled with higher memory depth. Like two-tone input case, it is observed that

as bandwidth of wideband modulated signal increases, the optimal memory depth increases.

This result is consistent with the fact that as the bandwidth increases, the memory effects of

the amplifier increase. Moreover, comparing the 8psk modulated input and π/2 bpsk input,

their optimal memory depths are different if the bandwidth remains the same. This shows that

the amplifier has different amount of memory effects, here different memory depths, as the

statistics of the input signal changes. The difference of the amplifier response to these signals

are caused by the difference in power distribution of them. For π/2 bpsk signal, the power

83

is more evenly distributed over bandwidth than 8psk signal. Thus, the amplifier shows lower

memory effect with this input. Furthermore, in all input cases, it is seen that if the memory

depth is increased more than the optimal value, the linearization decreases dramatically. This

is because the predistortion function is estimated with the previous samples that the amplifier

is not actually affected.

In the third chapter, polynomial order effect on predistortion performance is also studied.

The inputs which are used for memory depth effect investigation are used for polynomial ef-

fect investigation. For all types of input signals, it is observed that the optimal order increases

as the bandwidth increases. After these optimal polynomial orders, if the polynomial order in-

creases, the linearization amount remains the same. Moreover, it is observed that the statistics

of the input signal determines the optimal polynomial order. The uneven power distribution

over bandwidth and having random data makes the model extraction difficult. Therefore,

higher polynomial order is necessary for these kind of input signals.

Another issue mentioned in the third chapter is the validity range of the predistorter model.

Firstly, a predistorter function is estimated with a set of modulated signal, and then this pre-

distorter function is used for linearization of other sets of signal of same modulation and

bandwidth. It is observed that this predistorter function is valid and other sets of data can

be linearized as the training data itself. This is an important result because this shows that

the predistortion function is modeled well enough with this data of random samples so other

random sampled data of same modulation can be linearized. It is also critical in practical

case. Once the input set of one modulation and bandwidth is used for predistortion function

estimation, then other sets of this input type can be predistorted with this predistorter without

calculating every time the data comes. Furthermore, another trial is conducted in order to

investigate the performance change of the predistortion as the bandwidth of the same modu-

lation input changes. A predistorter function is estimated with a signal of one bandwidth and

modulation, and then this function is used for linearization of another signal of same mod-

ulation but different bandwidth. It is observed that the linearization amount decreases. This

result shows that the characteristics of the amplifier changes with the bandwidth of the input

signal although the signal has the same modulation. It can be explained with the fact that the

memory effect of the amplifier changes as the bandwidth of the signal changes. Therefore, the

predistorter model which is estimated with one bandwidth cannot linearize enough the signal

of another bandwidth. Finally, cross validation of the predistortion with different modulation

signals is investigated. It is seen that the linearization decreases dramatically comparing with

84

the linearization obtained with the predistorter which is estimated with the data itself. It is

because the amplifier shows different responses when the input statistics change.

In the fourth chapter, the hardware implementation of predistortion is introduced. The

inputs used in this chapter is two-tone signal, 8psk and π/2 bpsk modulated signals of different

bandwidths. In experimental procedure, firstly, the linearization performance comparison is

done between the memoryless predistortion and predistortion with memory. With all types

of inputs, it is observed that the memory polynomial predistortion has higher linearization

performance than the memoryless predistortion. The difference of predistortion performance

between the memoryless and memory predistortion is higher when the bandwidth of the signal

increases since the amplifier memory effect increases.

Another issue investigated in the fourth chapter is the model validity range. Firstly, pre-

distortion performance is observed with the predistorter function estimated by using one set

of data of a specific modulation type and bandwidth, and then this predistorter model is tried

with other sets of same specifications as conducted in the third chapter. It is seen that the

linearization performances are close even though other sets are used. Therefore, the results

found in simulation are verified in hardware implementation. Secondly, the cross validation

of model with different type of modulation is checked by keeping the bandwidth constant as

conducted in the third chapter. The linearization amount decreases dramatically comparing

the linearization by the predistorter which is estimated with the signal itself. This is an ex-

pected result since the amplifier shows different characteristics as the input statistics change.

Therefore, the predistorter function must be changed according to the characteristics of the

amplifier for linearization.

In the validity range of model section in the fourth chapter, the predistortion performance

change with respect to power sweep is mentioned. As seen in these power sweep experiments,

the predistortion performance is highest at the power level at which the predistorter function

is estimated and it gradually decreases at different PEP powers. This is because the gain,

AM/AM and AM/PM characteristics and memory effect of the amplifier change with respect

to PEP. Although at some range of power the predistorter is still provides linearization in

acceptable region, it must be estimated again as the PEP of the amplifier changes in order to

get highest linearization. Nonetheless, this power range of linearization is crucial because if

the input is sent through the system and it lasts long, the amplifier’s peak power can change

with respect to time. It is desired that the predistorter function is still valid, insensitive to

those small changes and doesn’t need to be estimated again and again.

85

Finally, in the validity range of model section in the fourth chapter, the predistortion

performance change with respect to bandwidth sweep is investigated. It is observed that

the predistorter has highest performance with the bandwidth that is used for estimating the

model. As the input of the bandwidth changes, the amplifier response changes. For example,

the memory effect of the amplifier is very sensitive to the input signal bandwidth. Therefore,

for each input bandwidth, new predistorter function is to be estimated in order to get highest

performance.

In conclusion, the linearization method, memory polynomial digital predistortion is pre-

sented in this thesis. Other baseband predistortion techniques are mentioned and memory

polynomial model is chosen to study since it has more advantages compared to them. Memo-

ryless predistortion and predistortion with memory performances are compared and increase

in linearization is shown by taking memory effects into account. Simulations and hardware

implementation are presented and satisfactory linearization amounts are obtained.

As future work, a closed loop hardware implementation can be done with an appropriate

signal processor instead of running the algorithm in MATLAB. There can be problems in this

analog quadrature modulation topology, appropriate line up component choice can be studied

in detail. The memory effect of the amplifier can be analyzed deeply and appropriate memory

depths can be found for the amplifier model. Then this memory depth is used in the model

of the predistorter instead of finding the optimal memory depth by trial. Moreover, with this

model, the memory effect is taken into account by considering each previous sample one by

one up to memory depth. If it is wanted to go to very former sample, it is necessary to take all

the previous samples up to that sample into account. This is impossible since it makes the ma-

trix really huge and impossible to solve. Therefore, a model as proposed in [29] can be tried

for predistortion which takes necessary former samples for memory effect compensation, not

all of them.

86

REFERENCES

[1] Joseph F. White, High Frequency Techniques, IEEE Press, 2004.

[2] Matthew M. Radmanesh, Radio Frequency and Microwave Electronics Illustrated,Prentice Hall, 2001.

[3] Durukal M., Power Amplifier Linearization By Predistortion, M.S. Thesis, Dept. of Elec-trical and Electronics Eng., Bilkent Univ., Ankara, Turkey, 2006.

[4] Wendy Van Moer, Yves Rolain and Alain Geens Measurement Based Nonlinear Model-ing of Spectral Regrowth, IEEE MTT-S Digest, 2000.

[5] Joel Vuolevi, T h o Rahkonen and Jani Manninen Measurement Technique for charac-terizing memory effects in rf power amplifiers, IEEE, 2000.

[6] Yan Ye, Taijun Liu, Tiefeng Xu and Fadhel M. Ghannouchi, Analysis and Decomposi-tion of the Nonlinearities in RF Power Amplifiers, IEEE, 2010.

[7] Roman Marsalek Contributions to the power amplifer linearization using digital base-band adaptive predistortion, Ph.D. dissertation, Universite de Marne La Vallee, 2003.

[8] Jeong Y.C., Microwave Circuits Design Laboratory Notes, Topic:Linearizing Principleson High Power Amplifier, School of Electronics and Information Eng., Chonbuk NationalUniv., 2004.

[9] Dennis R. Morgan, Zhengxiang Ma, Jaehyeong Kim, Michael G. Zierdt, and John Pasta-lan, A Generalized Memory Polynomial Model for Digital Predistortion of RF PowerAmplifiers, IEEE Transactions on Signal Processing, 2006

[10] Fadhel M. Ghannouchi and Oualid Hammi, Behavioral Modeling and Predistortion,IEEE Microwave Magazine, 2009.

[11] James K. Cavers, A Linearizing Predistorter With Fast Adaptation, IEEE, 1990.

[12] Oualid Hammi, Fadhel M. Ghannouchi, Slim Boumaiza and Bill Vassilakis, A Data-Based Nested LUT Model for RF Power Amplifiers Exhibiting Memory Effects, IEEEMicrowave and Wireless Components Letters, 2007.

[13] Anding Zhu, JosA c© C. Pedro and Thomas J. Brazil, Dynamic Deviation Reduction-Based Volterra Behavioral Modeling of RF Power Amplifiers, IEEE Transactions onMicrowave Theory and Techniques, 2006.

[14] Anding Zhu, JosA c© Carlos Pedro and Telmo Reis Cunha, Pruning the Volterra Seriesfor Behavioral Modeling of Power Amplifiers Using Physical Knowledge, IEEE Trans-actions on Microwave Theory and Techniques, 2007.

[15] J.Kim and J.Konstantinou, Digital Predistortion of wideband signals based on poweramplifier model with memory, Electronics Letters, 2001.

87

[16] Oualid Hammi, Fadhel M. Ghannouchi and Bill Vassilakis, A Compact Envelope-Memory Polynomial for RF Transmitters Modeling With Application to Baseband andRF-Digital Predistortion, IEEE Microwave and Wireless Components Letters, 2008.

[17] Taijun Liu, Slim Boumaiza and Fadhel M. Ghannouchi, Deembedding Static Nonlinear-ities and Accurately Identifying and Modeling Memory Effects in Wide-Band RF Trans-mitters, IEEE Transactions on Microwave Theory and Techniques, 2005.

[18] P.L. Gilabert, G. Montoro and E. Bertran, On the Wiener and Hammerstein Models forPower Amplifier Predistortion, IEEE, 2005.

[19] Taijun Liu, Slim Boumaiza and Fadhel M. Ghannouchi, Augmented Hammerstein Pre-distorter for Linearization of Broad-Band Wireless Transmitters, IEEE Transactions onMicrowave Theory and Techniques, 2006.

[20] Cuiping Yu, Yongsheng Feng, Yuanan Liu and Gang Xie, Memory Polynomial DigitalPredistortion for Power Amplifiers, IEEE, 2007.

[21] S.A.Maas, Nonlinear Microwave Circuits, Piscataway, IEEE, 1997.

[22] Martin Schetzen, Theory of pth-Order Inverses of Nonlinear Systems, IEEE Transactionson Circuits and Systems, 1976.

[23] Changsoo Eun and Edward J. Powers, A New Volterra Predistorter Based on the IndirectLearning Architecture, IEEE Transactions on Signal Processing

[24] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice- Hall, 1991.

[25] Oualid Hammi, Fadhel M. Ghannouchi and Bill Vassilakis, On The Sensitivity of RFTransmitters’ Memory Polynomial Model Identification to Delay Alignment Resolution,IEEE Microwave and Wireless Components Letters, 2008.

[26] Jeckeln et al., Adaptive Digital Predistortion for PAs with Real Time Modeling of Mem-oryless Complex Gains, US Patent No. 6,072,364, Jun. 6, 2002.

[27] Oualid Hammi and Fadhel M. Ghannouchi, Power Alignment of Digital Predistorters forPower Amplifiers Linearity Optimization, IEEE Transactions on Broadcasting, 2009.

[28] Magnus Isaksson, David Wisell and Daniel RA¶nnow, A Comparative Analysis of Be-havioral Models for RF Power Amplifiers, IEEE Microwave and Wireless ComponentsLetters, 2006.

[29] A.H. Yuzer, Modeling of Asymmetric Intermodulation Distortion and Memory Effects ofPower amplifiers, Ph.D. dissertation, Dept. of Electrical and Electronics Eng., METU.,Ankara, Turkey 2011.

88

Documents

LINEARIZATION OF RF POWER AMPLIFIERS BY USING MEMORY