Nonlinear System Identification and Analysis with Applications to Power Amplifier Modeling and ...

196 Pages·2004·1.67 MB·English

Nonlinear System Identi(cid:12)cation and Analysis with
Applications to Power Ampli(cid:12)er Modeling and Power
Ampli(cid:12)er Predistortion
A Thesis
Presented to
The Academic Faculty
by
Raviv Raich
In Partial Ful(cid:12)llment
of the Requirements for the Degree
Doctor of Philosophy
School of Electrical and Computer Engineering
Georgia Institute of Technology
March 2004
Nonlinear System Identi(cid:12)cation and Analysis with
Applications to Power Ampli(cid:12)er Modeling and Power
Ampli(cid:12)er Predistortion
Approved by:
Professor G. Tong Zhou, Advisor Professor J. Stevenson Kenney
Professor Erik I. Verriest Professor Xiaoming Huo
(School of Industrial and Systems Engi-
neering)
Professor Douglas B. Williams
Date Approved: 31 March 2004
To my family, my wife Genevieve,
and my daughter Yael
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ACKNOWLEDGEMENTS
I would like to express my gratitude to the people without whom this dissertation would
not have been completed.
First, I would like to thank my advisor, Dr. G. Tong Zhou who continuously supported
my work and ideas throughout my studies at Georgia Tech. Whenever I needed, Dr. Zhou
always made herself available and put a great deal of time and e(cid:11)ort into our joint work. I
am thankful for her close guidance and e(cid:11)orts to prepare me for an academic career.
Next, I would like to thank my thesis defense committee: Dr. Erik Verriest, Dr. Doug
Williams, Dr. J. Stevenson Kenney, and Dr. Xiaoming Huo for helping to improve my
dissertation with their thoughtful advice and suggestions.
Iwouldliketothankmygroupmembers: KrishnaNagarajan,MuhammadIkram,Yong-
sub Kim, Gail Rosen, Lei Ding, Hua Qian, Chunpeng Xiao, Ning Chen, Vince Emanuele,
Thao Tran, and Bob Baxley for useful technical discussions and fruitful collaborations.
I would also like to thank some of the students from the center for signal and image
processing (CSIP): Amer Abufadel, Volkan Cevher, Kevin Chan, Kerkil Choi, Farshid Del-
gosha,ErdemErtan,MajidFozunbal, JohnGlotzbach,PaulHong,JayJeon,ApurvaMody,
Jonathan Morris, Robert Morris, Greg Slabaugh, Martin Tobias, Nik Vasiloglou, and Ra-
jbabu Velmurugan for their willingness to help, insightful discussions, and mostly for their
friendship, which made my graduate experience much more enjoyable.
To my family, I owe a debt of gratitude for their love and support along the path of my
academic pursuits.
Last but not least, I would like to express my deepest gratitude to my wife Genevieve
for her continuous support and encouragement throughout my entire graduate career from
the time of application to the time of graduation. Without her help, kindness, patience,
and mostly her love, I would not have been able to complete this work.
iv
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
CHAPTER I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Characterization and Modeling of Power Ampli(cid:12)ers . . . . . . . . . . . . . 2
1.2.1 Memoryless Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Volterra Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Wiener Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Hammerstein Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 Wiener-Hammerstein Model . . . . . . . . . . . . . . . . . . . . . . 8
1.2.6 Parallel Hammerstein Model . . . . . . . . . . . . . . . . . . . . . . 9
1.3 PA Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Power Ampli(cid:12)er Linearization . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 PA Linearization Architecture . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Predistortion Models . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Communications Aspects of Power Ampli(cid:12)ers . . . . . . . . . . . . . . . . 22
1.6 Organization of this Dissertation . . . . . . . . . . . . . . . . . . . . . . . 24
CHAPTER II ON THE MODELING OF MEMORY NONLINEAR EF-
FECTS OF POWER AMPLIFIERS FOR COMMUNICATION APPLI-
CATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Baseband Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 The Quasi-memoryless Case . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 The (strictly-) Memoryless Case . . . . . . . . . . . . . . . . . . . . 29
2.3 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
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2.3.1 Diagonal Kernel System . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Hammerstein System . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Separable Kernel System . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.4 Wiener System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
CHAPTER III ON BASEBAND REPRESENTATION OF PASSBAND
NONLINEARITIES 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Passband and baseband representations . . . . . . . . . . . . . . . . . . . . 35
3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
CHAPTERIV AHAMMERSTEINPREDISTORTIONLINEARIZATION
DESIGNBASEDONTHEINDIRECTLEARNINGARCHITECTURE1 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Indirect Learning Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Identi(cid:12)cation of The Hammerstein Predistorter . . . . . . . . . . . . . . . 44
4.3.1 Narendra-Gallman algorithm. . . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Optimal two stage identi(cid:12)cation algorithm . . . . . . . . . . . . . . 46
4.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
CHAPTER V ORTHOGONAL POLYNOMIALS FOR POWER AMPLI-
FIER MODELING AND PREDISTORTER DESIGN1 . . . . . . . . . 51
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 The polynomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.1 The conventional polynomial model . . . . . . . . . . . . . . . . . . 53
5.2.2 Orthogonal polynomial bases . . . . . . . . . . . . . . . . . . . . . 57
5.2.3 Discussions on the orthogonal polynomial basis . . . . . . . . . . . 59
5.2.4 PA modeling example . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.5 Extension to the memory polynomial case . . . . . . . . . . . . . . 65
5.3 Orthogonal polynomial predistortion . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Predistortion via the indirect learning architecture . . . . . . . . . 67
vi
5.3.2 Simulations { predistortion linearization . . . . . . . . . . . . . . . 69
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.A Orthogonality Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
CHAPTERVI ORTHOGONALPOLYNOMIALSFORCOMPLEXGAUS-
SIAN PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.1 The Numerical Instability Problem in PA Modeling . . . . . . . . . 77
6.2.2 The Numerical Instability Problem in Predistortion Linearization . 78
6.3 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.1 Wiener G-Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.2 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Numerical Stability Improvement . . . . . . . . . . . . . . . . . . . . . . . 87
6.5.1 Deviation from the Gaussian Assumption. . . . . . . . . . . . . . . 89
6.5.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.A Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.B Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.C Derivation of the Orthogonal Polynomial Predistorter . . . . . . . . . . . . 95
CHAPTER VII STATISTICAL ANALYSIS OF A BANDPASS NONLIN-
EARITY WITH NONSTATIONARY INPUT . . . . . . . . . . . . . . . 97
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Digitally-Modulated Signals and Spectral Analysis for Polynomial Nonlin-
earity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.3.1 Digitally-Modulated Signals . . . . . . . . . . . . . . . . . . . . . . 104
7.3.2 Cyclostationary Spectral Analysis . . . . . . . . . . . . . . . . . . . 106
7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.A Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.B Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
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7.C Derivation of (7.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.D Derivation of (7.33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
CHAPTER VIII ANALYZING SPECTRAL REGROWTH OF QPSK AND
OQPSK SIGNALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
CHAPTER IX SPECTRAL ANALYSIS OF POLYNOMIAL NONLIN-
EARITY WITH APPLICATIONS TO RF POWER AMPLIFIERS . 126
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.2 Cyclostationary input and spectral analysis. . . . . . . . . . . . . . . . . . 128
9.3 Quasi-memoryless PA model . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.3.1 Closed form expression for spectral regrowth . . . . . . . . . . . . . 130
9.3.2 Case study: The e(cid:11)ect of AM/PM conversion on spectral regrowth 132
9.4 Memory polynomial PA model . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.4.1 Closed form expression . . . . . . . . . . . . . . . . . . . . . . . . . 137
9.4.2 Case study: Asymmetric spectral regrowth and memory e(cid:11)ects . . 138
9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.6 Proofs of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.6.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.6.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
CHAPTER X OPTIMIZATION OF SNDR FOR AMPLITUDE LIM-
ITED NONLINEARITIES1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.2 The SNDR Criterion and Its Optimization . . . . . . . . . . . . . . . . . . 146
10.2.1 SNDR De(cid:12)nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10.2.2 Optimization of the SNDR . . . . . . . . . . . . . . . . . . . . . . . 147
10.2.3 Examples { Optimal Threshold Selection and Application . . . . . 151
10.3 Relationship between SNDR and Capacity . . . . . . . . . . . . . . . . . . 153
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10.3.1 Lower Bound on Capacity . . . . . . . . . . . . . . . . . . . . . . . 155
10.3.2 Upper Bound on Capacity . . . . . . . . . . . . . . . . . . . . . . . 156
10.3.3 Example on Capacity and Bounds . . . . . . . . . . . . . . . . . . . 157
10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.AProof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.B Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
CHAPTER XI CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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LIST OF TABLES
Table 1.1 Spurious emission limits when transmitting [1] . . . . . . . . . . . . . . . 12
Table 1.2 Digital predistortion: reduced power consumption, less heat dissipation
and cooling costs, and higher reliability can be achieved. . . . . . . . . . 16
Table 3.1 Nonlinear systems and their representations . . . . . . . . . . . . . . . . . 38
Table 5.1 Orthogonal polynomial basis functions (x) for 1 k 7. . . . . . . . . 59
k
(cid:20) (cid:20)
Table 6.1 The (cid:12)rst (cid:12)ve Orthonormal polynomials for unit variance z~. . . . . . . . . 82
Table 6.2 The (cid:12)rst six Hermite-related orthonormal polynomials . . . . . . . . . . . 83
Table 9.1 Estimated polynomial PA model coe(cid:14)cients for three scenarios: (i) when
both AM/AM and AM/PM conversions are present; (ii) when only the
AM/AMconversionispresent((cid:8)(r) = 0); and(iii)whenonlytheAM/PM
conversion is present (A(r) = 11:75 was used). . . . . . . . . . . . . . . . 135
Table 9.2 MemorypolynomialPAcoe(cid:14)cientsextractedforarealPAwithmaximum
nonlinearity order 2K +1 = 7 and maximum lag Q = 2. . . . . . . . . . . 139
Table 10.1SER vs. SNR for unclipped OFDM and OFDM with optimum clipping
using threshold (cid:17)?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
x