Linköping Studies in Science and Technology. Licentiate Thesis. No. 1605 Estimation of Inverse Models Applied to Power Ampliﬁer Predistortion Ylva Jung LERTEKNIK REG AU T O MA RO TI C C O N T L LINKÖPING Division of Automatic Control Department of Electrical Engineering Linköping University, SE-581 83 Linköping, Sweden http://www.control.isy.liu.se ylvju@isy.liu.se Linköping 2013 This is a Swedish Licentiate’s Thesis. Swedish postgraduate education leads to a Doctor’s degree and/or a Licentiate’s degree. A Doctor’s Degree comprises 240 ECTS credits (4 years of full-time studies). A Licentiate’s degree comprises 120 ECTS credits, of which at least 60 ECTS credits constitute a Licentiate’s thesis. Linköping Studies in Science and Technology. Licentiate Thesis. No. 1605 Estimation of Inverse Models Applied to Power Ampliﬁer Predistortion Ylva Jung ylvju@isy.liu.se www.control.isy.liu.se Department of Electrical Engineering Linköping University SE-581 83 Linköping Sweden ISBN 978-91-7519-571-1 ISSN 0280-7971 LIU-TEK-LIC-2013:39 Copyright © 2013 Ylva Jung Printed by LiU-Tryck, Linköping, Sweden 2013 Till Daniel Abstract Mathematical models are commonly used in technical applications to describe the behavior of a system. These models can be estimated from data, which is known as system identiﬁcation. Usually the models are used to calculate the output for a given input, but in this thesis, the estimation of inverse models is investigated. That is, we want to ﬁnd a model that can be used to calculate the input for a given output. In this setup, the goal is to minimize the diﬀerence between the input and the output from the cascaded systems (system and inverse). A good model would be one that reconstructs the original input when used in series with the original system. Diﬀerent methods for estimating a system inverse exist. The inverse model can be based on a forward model, or it can be estimated directly by reversing the use of input and output in the identiﬁcation procedure. The models obtained using the diﬀerent approaches capture diﬀerent aspects of the system, and the choice of method can have a large impact. Here, it is shown in a small linear example that a direct estimation of the inverse can be advantageous, when the inverse is supposed to be used in cascade with the system to reconstruct the input. Inverse systems turn up in many diﬀerent applications, such as sensor calibration and power ampliﬁer (pa) predistortion. pas used in communication devices can be nonlinear, and this causes interference in adjacent transmitting channels, which will be noise to anyone that transmits in these channels. Therefore, linearization of the ampliﬁer is needed, and a preﬁlter is used, called a predistorter. In this thesis, the predistortion problem has been investigated for a type of pa, called outphasing power ampliﬁer, where the input signal is decomposed into two branches that are ampliﬁed separately by highly eﬃcient nonlinear ampliﬁers, and then recombined. If the decomposition and summation of the two parts are not perfect, nonlinear terms will be introduced in the output, and predistortion is needed. Here, a predistorter has been constructed based on a model of the pa. In a ﬁrst method, the structure of the outphasing ampliﬁer has been used to model the distortion, and from this model, a predistorter can be estimated. However, this involves solving two nonconvex optimization problems, and the risk of obtaining a suboptimal solution. Exploring the structure of the pa, the problem can be reformulated such that the pa modeling basically can be done by solving two least-squares (ls) problems, which are convex. In a second step, an analytical description of an ideal predistorter can be used to obtain a predistorter estimate. Another approach is to compute the predistorter without a pa model by estimating the inverse directly. The methods have been evaluated in simulations and in measurements, and it is shown that the predistortion improves the linearity of the overall power ampliﬁer system. v Populärvetenskaplig sammanfattning Matematiska beskrivningar, här kallade modeller, används i många tekniska tilllämpningar. Ett exempel är utveckling av bilar, där man med simuleringar kan utvärdera olika designval på ett kostnadseﬀektivt sätt. Ett annat är ﬂygtillämpningar där riktiga tester på ﬂygplanet skulle kunna leda till fara för piloten. Dessa modeller kan skattas från uppmätt data från systemet, och detta förfarande kallas systemidentiﬁering. Ett system är den avgränsade del av världen som vi är intresserade av, i exemplen ovan bilen och ﬂygplanet. I systemidentiﬁering är målet att ﬁnna en modell som så bra som möjligt beskriver utsignalen, baserat på tidigare in- och utsignaler som har kunnat mätas. I denna avhandling undersöks hur inversa modeller kan skattas. Här ska inversen användas i kombination med det ursprungliga systemet, med målet att utsignalen från de seriekopplade systemen (det ursprungliga och dess invers) ska vara densamma som insignalen. Skattning av inversa system kan göras på ﬂera sätt. Inversen kan baseras på en modell av systemet som sedan inverteras, eller skattas direkt genom att insignalen och utsignalen byter plats i systemidentiﬁeringsproblemet. Hur inversen skattas påverkar modellen genom att olika egenskaper hos systemet fångas, och detta kan därför ha en stor inverkan på slutresultatet. I ett litet förenklat exempel visas att det kan löna sig att skatta inversen direkt när den ska användas i serie med systemet för att återskapa insignalen. Linjärisering av eﬀektförstärkare är ett exempel där inversa system används. Eﬀektförstärkare används i många tillämpningar, bland annat mobiltelefoni, och dess uppgift är att förstärka en signal vilket är ett steg i överföringen av information. I exemplet med mobiltelefoner kan det exempelvis vara en persons röst som är signalen, vilken ska överföras från telefonen via luften och vidare till mottagaren. Eﬀektförstärkare kan vara olinjära, vilket medför att de sprider eﬀekt till närliggande frekvensband. För den som ska sända i dessa frekvensband uppfattas detta som brus, och det ﬁnns gränser för hur mycket spridning som får ske. För att uppfylla dessa krav på spridning krävs alltså ofta linjärisering. Genom att modellera förstärkarens olinjäriteter och invertera dem kan man få ett system som inte sprider eﬀekt i frekvensbandet. I detta sammanhang säger man att en förkompensering, kallad fördistortion, används. I denna avhandling tillämpas fördistortion på en typ av eﬀektförstärkare, som kallas outphasing-förstärkare. Detta är en olinjär eﬀektförstärkarstruktur som delar upp signalen i två delar, där delarna förstärks av eﬀektsnåla förstärkare för att sedan adderas ihop. Om denna uppdelning och summation inte är perfekta uppstår olinjäriteter, och fördistortion krävs. Fördelen med olinjära eﬀektförstärkare är att dessa kan göras mer eﬀektsnåla, vilket direkt speglas i exempelvis batteritiden för en mobiltelefon. Här presenteras ﬂera olika metoder för att ta fram en fördistortion. Metoderna har utvärderats på fysiska förstärkare i mätningar, vilka visar att en förbättring kan uppnås vid användning av fördistortion. vii Acknowledgments I want to start by expressing my deepest gratitude to my supervisor Dr. Martin Enqvist. Your never-ending knowledge, patience and encouragement is rather remarkable. The (almost) inﬁnite amount of comments and questioning on every detail is very much appreciated. Thank you so much for the help and the time. I also want to thank Prof. Lennart Ljung for letting me join this great group. The way you and your successor Prof. Svante Gunnarsson lead the group in large and also handle smaller matters is impressing. All administrative help from Ninna Stensgård and her predecessor Åsa Karmelind is also appreciated. Without Dr. Jonas Fritzin and Prof. Atila Alvandpour, I would not have gotten into this ﬁeld of research, I appreciate the nice cooperation and especially Jonas for answering all my questions about the hardware. Many thanks to Lic. Patrik Axelsson, Lic. Daniel Eriksson, M.Sc. Manon Kok, Lic. Roger Larsson, Dr. Christian Lyzell and M.Sc. Maryam Sadeghi Reineh for proofreading parts of this thesis, your comments have been very valuable to clarify and improve the thesis. I am also grateful to the LATEX gurus Dr. Gustaf Hendeby and Dr. Henrik Tidefelt for making the great template so I don’t have to do much more than writing. And whenever I do need help, Lic. (soon to be Dr.) Daniel Petersson always comes to the rescue. Thank you so much! I am very happy to have happened to end up in the group of Automatic Control. The ﬁka-breaks are always nice, whether the discussions concern work matters or something completely diﬀerent. I hope there will be more great beer drunk and barbecues had, and that the girl-power group will continue and make us even more ﬁt and powery. Special thanks to my former room mate Patrik Axelsson for taking care of me when I was new and lost and helping me with whatever, and to my present room mate Maryam Sadeghi Reineh for dragging me along to ﬁka ;-) Thanks to life (though not appreciated at the time) for making me see what is important and that this is not necessarily what I had in mind. This work has been supported by the Excellence Center at Linköping-Lund in Information Technology (ELLIIT), the Center for Industrial Information Technology at Linköping University (CENIIT) and the Swedish Research Council (VR) Linneaus Center CADICS, which is gratefully acknowledged. I am also very grateful that I have family and friends outside of work. Friends in Linköping, those who left, those from back home and from over the world, I’m so glad that you are still in my life. I just wish there were more time for us and shorter distances! And most of all, to Daniel, the one who puts up with me the most and knows how to get me up in the morning. I’m so glad you’re mine! Thank you for your patience, love and encouragement :) Linköping, August 2013 ix Contents Notation xv 1 Introduction 1.1 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1 1 3 4 System Inversion 2 Introduction to Model Estimation 2.1 System Identiﬁcation . . . . . . . . . 2.2 Transfer Function Models . . . . . . . 2.3 Prediction Error Method . . . . . . . 2.4 Linear Regression . . . . . . . . . . . 2.5 Least-squares Method . . . . . . . . . 2.6 The System Identiﬁcation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10 11 12 13 14 15 3 Introduction to System Inversion 3.1 Inversion by Feedback . . . . . . . . . . . . . . . . 3.1.1 Feedback and Feedforward Control . . . . . 3.1.2 Iterative Learning Control . . . . . . . . . . 3.1.3 Exact Linearization . . . . . . . . . . . . . . 3.2 Analytic Inversion . . . . . . . . . . . . . . . . . . . 3.2.1 Problems Occurring with System Inversion 3.2.2 Postinverse and Preinverse . . . . . . . . . . 3.2.3 Volterra Series . . . . . . . . . . . . . . . . . 3.3 Inversion by System Simulation . . . . . . . . . . . 3.3.1 Separation of a Nonlinear System . . . . . . 3.3.2 Hirschorn’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 18 19 20 21 22 22 23 24 26 26 27 4 Estimation of Inverse Models 4.1 System Inverse Estimation . . . . . . . . . . . . . . . . . . . . . . . 33 34 xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Contents 4.2 Inverse Identiﬁcation of LTI Systems . . . . . . . . . . . . . . . . . 4.3 An Illustrative Linear Dynamic Example . . . . . . . . . . . . . . . 4.4 Inverse Identiﬁcation of Nonlinear Systems . . . . . . . . . . . . . II 35 37 39 Power Ampliﬁer Predistortion 5 Power Ampliﬁers 5.1 Power Ampliﬁer Fundamentals . . . . 5.1.1 Basic Transmitter Functionality 5.2 Power Ampliﬁer Characterization . . . 5.2.1 Gain . . . . . . . . . . . . . . . 5.2.2 Eﬃciency . . . . . . . . . . . . . 5.2.3 Linearity . . . . . . . . . . . . . 5.3 Classiﬁcation of Power Ampliﬁers . . . 5.3.1 Transistors . . . . . . . . . . . . 5.3.2 Linear Ampliﬁers . . . . . . . . 5.3.3 Switched Ampliﬁers . . . . . . 5.3.4 Other Classes . . . . . . . . . . 5.4 Outphasing Concept . . . . . . . . . . 5.5 Linearization of Power Ampliﬁers . . . 5.5.1 Volterra series . . . . . . . . . . 5.5.2 Block-oriented Models . . . . . 5.5.3 Outphasing Power Ampliﬁers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 47 48 50 51 51 52 55 55 56 57 58 58 61 62 62 63 6 Modeling Outphasing Power Ampliﬁers 6.1 An Alternative Outphasing Decomposition 6.2 Nonconvex PA Model Estimator . . . . . . . 6.3 Least-squares PA Model Estimator . . . . . . 6.4 PA Model Validation . . . . . . . . . . . . . 6.5 Convex vs Nonconvex Formulations . . . . . 6.6 Noise Inﬂuence . . . . . . . . . . . . . . . . 6.7 Memory Eﬀects and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 65 67 69 71 82 82 84 7 Predistortion 7.1 A DPD Description . . . . . . . . . . . . . . . . . . . . . 7.2 The Ideal DPD . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Nonconvex DPD Estimator . . . . . . . . . . . . . . . . . 7.4 Analytical DPD Estimator . . . . . . . . . . . . . . . . . 7.5 Inverse Least-Squares DPD Estimator . . . . . . . . . . . 7.6 Simulated Evaluation of Analytical and LS Predistorter 7.7 Recursive Least-Squares and Least Mean Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 85 87 88 89 90 94 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Predistortion Measurement Results 101 8.1 Signals Used for Evaluation . . . . . . . . . . . . . . . . . . . . . . 101 8.2 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.3 Evaluation of Nonconvex Method . . . . . . . . . . . . . . . . . . . 104 xiii Contents 8.3.1 Measured Performance of EDGE Signal . . . . . . . . . . 8.3.2 Measured Performance of WCDMA Signal . . . . . . . . . 8.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Evaluation of Least Squares PA and Analytical Inversion Method 8.4.1 Measured Performance of WCDMA Signal . . . . . . . . . 8.4.2 Measured Performance of LTE Signal . . . . . . . . . . . . 8.4.3 Evaluation of Polynomial Degree . . . . . . . . . . . . . . 8.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 105 107 109 110 111 115 115 9 Concluding Remarks 117 9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.2 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A Power Ampliﬁer Implementation 119 A.1 +10.3 dBm Class-D Outphasing RF Ampliﬁer in 90 nm CMOS . . 119 A.2 +30 dBm Class-D Outphasing RF Ampliﬁer in 65 nm CMOS . . . . 121 Bibliography 125 Notation Outphasing Amplifiers Notation Meaning Δψ (s1 , s2 ) arg(s1 ) − arg(s2 ), angle diﬀerence between outphasing signals, deﬁned on page 65 same as Δψ (s1 , s2 ) angle diﬀerence between predistorted outphasing input signals angle diﬀerence between predistorted outphasing output signals angle diﬀerence between s˜k and sk , deﬁned in (6.6)(6.7), page 67, and Figure 6.1, page 66 phase distortion in the ampliﬁer branch k, deﬁned in (6.9) gain factors of each branch in pa, should ideally be g1 = g2 = g0 phase predistorter functions in the ampliﬁer branch k, deﬁned in (7.1) outphasing input signals, decomposed in standard way (5.11) predistorted outphasing input signal in branch k, decomposed with identical gain factors using (5.11) outphasing input signal in branch k, decomposed with nonidentical gain factors using (6.3) outphasing output signal in branch k, decomposed with nonidentical gain factors using (6.3) predistorted outphasing output signal in branch k, decomposed with nonidentical gain factors using (6.3) an estimate of the value of x Δψ Δψ (s1,P , s2,P ) Δψ (y1,P , y2,P ) ξk fk g1 , g2 hk sk sk,P s˜k yk yk,P x̂ xv xvi Notation Power Amplifier Glossary Notation Deﬁnition aclr, acpr adjacent channel leakage (power) ratio, a linearity measure that describes the amount of power spread to neighboring channels, page 52. am-am, am-pm amplitude modulation to amplitude modulation or phase modulation, respectively, a plot mapping the output amplitude (or phase distortion) to the input amplitude to determine the distortion induced by the circuit, for example a power ampliﬁer, page 52. the circuit that handles the addition of signals in, for example, Figure 5.13, page 59. decibel to carrier, the power ratio of a signal to a carrier signal, expressed in decibels. power level expressed in dB referenced to one milliwatt, so that zero dBm equals one mW and one dBm is one decibel greater (about 1.259 mW). drain eﬃciency and power added eﬃciency are eﬃciency measures for power ampliﬁers, page 51. direct and indirect learning architectures are two approaches to estimate a power ampliﬁer predistorter, see Method B and Method C on page 34. digital predistortion, a linearization technique for power ampliﬁers that modiﬁes the input to counteract power ampliﬁer distortion from nonlinearities and dynamics, page 61. dynamic range, deﬁning the ratio of the maximum and minimum output amplitudes an ampliﬁer can achieve, page 60. a signal separation into an imaginary part (quadrature, q) vs real part (in-phase, i), page 48. local oscillator, a circuit that produces a continuous sine wave. Usually drives a mixer in a transmitter/receiver, page 48. translates the signal up or down to another frequency, page 48 and Figure 5.2. combiner dBc dBm de, pae dla, ila dpd dr iq lo mixer outphasing, linc pa rf scs an outphasing ampliﬁer, also called linear ampliﬁcation with nonlinear components, is a nonlinear ampliﬁer structure. power ampliﬁer, used to increase the power of a signal, so that the output is a magniﬁed replica of the input. radio frequency, ranging between 3 kHz and 300 GHz. signal component separator, (here) decomposes the signal into outphasing signals according to (5.11). xvii Notation Abbreviations A-O Abbreviation ac aclr acpr am am-am am-pm bjt cmos dac db dc de dla dpd dr edge evm fpga fet fir fm gsm gprs iir ila ilc iq linc lms lo ls lte lti lut mimo mosfet nmos Meaning Alternating current Adjacent channel leakage ratio Adjacent channel power ratio Amplitude modulation Amplitude modulation to amplitude modulation Amplitude modulation to phase modulation Bipolar junction transistor Complementary metal-oxide-semiconductor Digital-to-analog converter Digital baseband Direct current Drain eﬃciency Direct learning architecture Digital predistortion or predistorter Dynamic range Enhanced data rates for gsm evolution Error vector magnitude Field programmable gate array Field-eﬀect transistor Finite impulse response Frequency modulation Global system for mobile communications General packet radio service Inﬁnite impulse response Indirect learning architecture Iterative learning control in-phase component (i, real part) vs quadrature component (q, imaginary part) Linear ampliﬁcation with nonlinear components Least mean squares Local oscillator Least squares Long term evolution Linear time invariant Look-up table Multiple-input multiple-output Metal-oxide-semiconductor ﬁeld-eﬀect transistor N-channel metal-oxide-semiconductor xviii Notation Abbreviations P-Z Abbreviation pa pae papr pd pem pm pmos pvt pwm rbw rf rls rms rx scs siso sls tx wcdma Meaning Power ampliﬁer Power added eﬃciency Peak-to-average power ratio Predistortion or predistorter Prediction-error (identiﬁcation) method Phase modulation P-channel metal-oxide-semiconductor Process, voltage and temperature Pulse-width modulated Resolution bandwidth Radio frequency Recursive least squares Root mean square Receiver Signal component separator Single-input single-output Separable least-squares Transmitter Wideband code-division multiple access 1 Introduction Inverse systems and models thereof show up in numerous applications. This entails a need for estimation of models of inverse systems. The concept of building models based on measured data is called system identiﬁcation, and many theoretical results exist concerning the properties of the estimated models. However, less is known when the goal is to estimate the inverse. Should it be based on a forward model, or should the inverse be estimated directly? Inverse models produced in diﬀerent ways will capture diﬀerent properties of the system and more insights are needed. In this chapter, a short research motivation will be given, followed by an outline of the thesis. Then follows an overview of the contributions of the thesis, and some clariﬁcations of the author’s role in the work. 1.1 Research Motivation Power ampliﬁers (pas) are used in many applications, such as communication devices (mobile phones) and loudspeakers. In a hand-held device such as a mobile phone, the power eﬃciency is an important property as it will reﬂect directly on the battery time. In order to match the increasing demand for lower power consumption, nonlinear power ampliﬁers have been developed. These nonlinear pas can be made more power eﬃcient than linear ones, but introduce other problems. A nonlinear device will not only transmit power in the frequency band where the input signal is, but also risks spreading power to neighboring transmitting channels. For anyone transmitting in these frequency bands, this will be perceived as noise. Therefore, there are standards describing the amount of power that is allowed to be spread to adjacent frequencies. So, for the power ampliﬁer to be useful, linearization is needed, limiting the interference in the neighboring channels. Since the distorted output of the power ampliﬁer is an ampliﬁed version 1 2 1 u S −1 S (a) yu u S y S −1 Introduction yu (b) Figure 1.1: An inverse S −1 of the system S is used to undo the eﬀects of the system S such that yu = u. In (a), a preinverse is used, where the inverse S −1 is applied before the system S, and in (b), a postinverse is applied, where the order of the system and the inverse is reversed. of the input, it is preferable to work with the input. Thus, the goal is to ﬁnd a preﬁlter that inverts the nonlinearities, called a predistorter. Small loudspeakers, in mobile phones for example, can also show a nonlinear behavior due to limitations in the movement of the cone. This will distort the sound and make listening to music less agreeable. The idea behind a compensation of the nonlinearities is similar to that of the power ampliﬁer predistortion, since only the input is available for modiﬁcation. Once the signal has been converted to sound, it cannot be altered. In the power ampliﬁer and loudspeaker applications, the goal is to ﬁnd a preﬁlter that inverts the nonlinearities introduced by the power ampliﬁer or loudspeaker, but the same type of inversion problems can be found also in other areas. In sensor applications it is rather a postdistortion that is needed. If the sensor itself has dynamics or a nonlinear behavior, the sensor output is not the true signal but will also contain some sensor contamination. This has to be handled at the sensor output since this is where the user can get access to the signal. The need for calibration is also relevant in other applications, such as analogto-digital converters (adc s). In an adc, an analog (continuous) input signal is converted to a digital output, which is limited to a number of discrete values. A small error in the analog input risks causing a larger error in the output, since the discrete signal is limited to certain values. Inversion of systems also appear in other areas, not directly connected to preor postinversion. One application where models of both the system S and its inverse S −1 are used is robotics. The forward kinematics, describing how to compute the robot tool pose as a function of the joint variables, is used for control as well as the inverse kinematics, how to compute joint conﬁguration from a given tool pose. In all of the above applications, the question is how to ﬁnd an inverse S −1 to the system S. The application will determine if it is a preinverse or a postinverse that is desired. In Figure 1.1, the two diﬀerent approaches are illustrated. If an inverse cannot be found analytically, it can be estimated. This opens up for questions regarding this inverse estimation. Diﬀerent methods can be applied. Either it can be based on an inverted model of the system itself, or the method can try to estimate the inverse directly. That the choice of estimation method matters is motivated by Example 1.1. 1.2 3 Outline 200 100 0 −100 −200 20 21 22 23 25 24 Time [s] 26 27 28 Figure 1.2: The input u (black solid line), and the reconstructed input yu using an inverted estimated forward model (black dashed line) and the inverse model estimated directly (gray solid line). The estimation of the inverse (gray) cannot perfectly reconstruct the input (black solid), but is clearly better than the inverted forward model (dashed). Example 1.1: Introductory example Consider a linear time-invariant (lti) system. The goal is to reconstruct the input by modifying the measured output. When the structure of the inverse is set, in this case to a ﬁnite impulse response (fir) system, what is the best way to estimate it? Should the inverse be estimated directly or should an inverted model of the system itself be used? These two approaches have been applied to noise-free data, and the results are presented in Figure 1.2. We see here that the two models, both descriptions of the system inverse, capture very diﬀerent aspects of the system, and that the method chosen can have a large impact. This example is described in more details in Section 4.3. 1.2 Outline The thesis is divided into two parts. The ﬁrst introduces system inversion and the estimation of inverse models. The second part concerns using estimated inverse models for power ampliﬁer predistortion. Part I – System inversion gives a background to the problem of estimating inverse models. A short introduction to model estimation is provided in Chapter 2 and a background to system inversion in Chapter 3. In Chapter 4, some ideas concerning the estimation of inverse models are presented, and three basic approaches are explained. In particular, some conclusions concerning linear, time-invariant systems are presented. 4 1 Introduction In Part II – Power ampliﬁer predistortion, the estimation of inverse models is applied to outphasing power ampliﬁers. Here, the goal is to ﬁnd an inverse such that the output of the power ampliﬁer is an ampliﬁed replica of the input, counteracting the distortion caused by the ampliﬁer. An introduction to power ampliﬁer functionality and characterization is given in Chapter 5 as well as an overview of earlier predistortion methods. This chapter also contains a description of the outphasing power ampliﬁer, which is a nonlinear ampliﬁer structure that needs predistortion, and for which the predistorter methods in this thesis were produced. Modeling approaches for the power ampliﬁer are presented in Chapter 6 and methods for ﬁnding a predistorter in Chapter 7. The predistortion methods are evaluated on real power ampliﬁers in Chapter 8. The thesis is concluded by Chapter 9 where some conclusions and a discussion on ideas for future research are presented. Some additional information about the power ampliﬁers used is given in the appendix. 1.3 Contributions The contributions in this thesis are in two areas, power ampliﬁer predistortion and the more general ﬁeld of estimating models of inverse systems. The power ampliﬁer predistortion was ﬁrst presented in Jonas Fritzin, Ylva Jung, Per N. Landin, Peter Händel, Martin Enqvist, and Atila Alvandpour. Phase predistortion of a Class-D outphasing RF ampliﬁer in 90nm CMOS. IEEE Transactions on Circuits and Systems-II: Express Briefs, 58(10):642–646, October 2011a. where a novel model structure for the outphasing power ampliﬁers was used. A predistorter that changes only the phases of the outphasing signals was shown to successfully reduce the distortion introduced by the power ampliﬁer. The proposed model and predistorter structures were produced in close collaboration between the paper’s ﬁrst three authors. The theoretical motivation of the predistorter model has been developed by the author of this thesis. The nonconvex predistortion method presented in the above publication was then developed into a method that makes use of the structure of the outphasing power ampliﬁer. It basically consists of solving least-squares problems, which are convex, and performing an analytical inversion, and it is suitable for online implementation. This is presented in Ylva Jung, Jonas Fritzin, Martin Enqvist, and Atila Alvandpour. Leastsquares phase predistortion of a +30dbm Class-D outphasing RF PA in 65nm CMOS. IEEE Transactions on Circuits and Systems-I: Regular papers, 60(7):1915–1928, July 2013. The derivation of this least-squares predistortion method has mainly been done by the author of this thesis, whereas the paper’s second author has been responsible for the power ampliﬁer and hardware issues. In addition to the reformulation of the nonconvex problem, the paper provides a theoretical description of 1.3 Contributions 5 an ideal outphasing predistorter, that is, one that does not change neither the amplitude nor the phase of the output. This involves a mathematical description of the branch decomposition and the impact of unbalanced ampliﬁcation in the two branches. Inverse models can either be estimated directly, or based on a model of the (forward) system. Some insights into diﬀerent approaches to estimate models of inverse systems are discussed in Chapter 4. These results concerning the estimation of inverse models have been accepted for presentation at the 52nd IEEE Conference on Decision and Control (CDC): Ylva Jung and Martin Enqvist. Estimating models of inverse systems. In 52nd IEEE Conference on Decision and Control (CDC), Florence, Italy, To appear, December 2013. The paper also contains the postinverse application of Hirschorn’s method presented in Section 3.3.2. The contents of Appendix A are included here for the sake of completeness and are not part of the contributions of this thesis. The power ampliﬁers and the characterization thereof were done at the Division of Electronic Devices, Department of Electrical Engineering at Linköping University, Linköping, Sweden, by Jonas Fritzin, Christer Svensson and Atila Alvandpour. Part I System Inversion 2 Introduction to Model Estimation In many cases it is costly, tedious or dangerous to perform real experiments, but we still want to extract information somehow. The limited part of the world that we are interested in is called a system. This system can be pretty much anything. It can for example be interesting for a car manufacturer to know how the car will react to a change in the accelerator. Or in a paper mill, how the moist content of the wood will aﬀect the quality of the paper. For a diabetic it is essential to know how the blood sugar level depends on food intake and exercise. A pilot needs to know how an airplane reacts to the control of diﬀerent rudders, and in economics it is necessary to know how a change in the interest rate will inﬂuence the customers’ willingness to borrow or save money. What we see as a system depends on the application. In the car analogy, the system can be only the engine, or the whole car. In the blood sugar level we can either be interested only in how food intake eﬀects the glucose levels, or how exercise contributes. In many of these applications one does not want to perform experiments directly, but instead start the evaluation using simulations. This leads to a need for models of the systems. One way is to use physical modeling where the models are based on what we know of the system by using the knowledge of, for example, the forces, moments, ﬂows, etc. In the engine example, it is possible to calculate the output and the connection between the accelerator and the engine torque. This method is sometimes called white box modeling. Another modeling approach is to gather data from the system and construct a model based on this information. This approach is called system identiﬁcation and will be presented in this chapter. 9 10 2 Introduction to Model Estimation v u S y Figure 2.1: A system S with input u, output y, and disturbance v. For the blood glucose example, the system S is the patient, or rather a part of the body’s metabolism system, the input u could represent food intake, the output y is the measured blood glucose level and the disturbance v is for example an infection that eﬀects the body’s insulin sensitivity. 2.1 System Identiﬁcation System identiﬁcation deals with the problem of identifying properties of a system. More speciﬁcally, it treats the problem of using measured data to extract a mathematical model of a system we are interested in. The introduction and notation presented here is based on Ljung [1999], but other standard references include Pintelon and Schoukens [2012] and Söderström and Stoica [1989]. Since we are dealing with sampled data, t will be used to denote the time index. Also, for notational convenience, the sample time Ts will be assumed to be one, so that Δ Δ y(tTs ) = y(t) and y((t + 1)Ts ) = y(t + 1) is the measurement after y(t), but this can of course easily be adapted to other choices of Ts . The observable signals that we are interested in are called outputs, denoted y(t), and in the examples above this can be the car speed/engine velocity, or the glucose level in the blood. The system can also be aﬀected by diﬀerent sources that we are in control of – the accelerator or the food intake – called inputs, u(t). Other external sources of stimuli that we cannot control or manipulate are called disturbances, v(t), – such as a steep uphill aﬀecting the car or a fever or infection which eﬀect the insulin sensitivity. Some disturbances are measurable and for others the eﬀects can be noted, but the signal itself cannot be measured. The diﬀerent concepts are presented in Figure 2.1. A system has a number of properties connected to it. A system is linear if its output response to a linear combination of inputs is the same linear combination of the output responses of the individual inputs. That is f (αx + βy) = f (αx) + f (βy) = αf (x) + βf (y), with x and y independent variables and α and β real-valued scalars. The ﬁrst equality makes use of the additivity (also called the superposition property), and the second the homogeneity property. A system that is not linear is called nonlinear. Since this includes “everything else”, it is hard to do a classiﬁcation and come to general conclusions. Most results in system identiﬁcation are therefore developed for linear systems, or some limited subset of nonlinear systems. The system is time invariant if its response to a certain input signal does not depend on absolute time. A system is said to be dynamical if it has some memory or history, i.e., the output does not only depend on the current input but also previous inputs and outputs. If it depends only on the current input, it is static. 2.2 11 Transfer Function Models In system identiﬁcation, the goal is to use the known input data u and the measured output data y to construct a model of the system S. Here, only singleinput single-output (siso) systems are considered, but the ideas can most of the time be adapted to multiple-input multiple-output (mimo) systems. It is usually neither possible nor desirable to ﬁnd a model that describes the whole system and all its properties, but rather one wants to construct a model which captures and can describe some interesting subset thereof, needed for the application. It is up to the user to deﬁne such criteria as to what needs to be captured by the model. 2.2 Transfer Function Models One way to present a linear time invariant (lti) system is via the transfer function model y(t) = G(q, θ)u(t) + H(q, θ)e(t) (2.1) where q is the shift operator, such that qu(t) = u(t + 1) and q −1 u(t) = u(t − 1), and e(t) is a white noise sequence. G(q, θ) and H(q, θ) are rational functions of q and the coeﬃcients in θ, where θ consists of the unknown parameters that describe the system. Depending on the choice of polynomials in G(q, θ) and H(q, θ), diﬀerent structures can be obtained. The most general structure is A(q)y(t) = C(q) B(q) u(t) + e(t) F(q) D(q) (2.2) where the polynomials are described by X(q) = 1 + x1 q −1 + · · · + xnx q −nx for X = A, C, D, F, and nx is the order of the polynomial and a possible delay nk in B(q), B(q) = bnk q −nk + · · · + xnk +nb −1 q −(nk +nb −1) , such that there can be a delay between input and output. This structure is often too general, and one or several of the polynomials will be set to unity. Depending on the polynomials used, diﬀerent commonly used structures will be obtained. When the noise is assumed to enter directly at the output, such as white measurement noise, or when we are not interested in modeling the noise, the structure is called an output error (oe) model, which can be written y(t) = B(q) u(t) + e(t), F(q) i.e., the polynomials A(q), C(q) and D(q) have all been set to unity. Many such structures exist (see Ljung [1999] for more examples) and are called black-box models, since the model structure reﬂects no physical insight but acts like a black box on the input, and delivers an output. One strength of these structures is that 12 2 Introduction to Model Estimation they are ﬂexible and, depending on the choice of G(q, θ) and H(q, θ), they can cover many diﬀerent cases. A model which does not belong to the black-box model structure, and is not completely obtained from physical knowledge of the system is called a gray-box model. This can for example be a physical structure with unknown parameters, such as an unknown resistance in an elsewise known circuit. It can also be a some properties of the data that can be explored in the choice of model structure. The latter is done in the power ampliﬁer modeling in Chapter 6. 2.3 Prediction Error Method In order to say something about the system, we need a model that can predict what will happen next. At the present time instant t, we have collected data from previous time instants t − 1, t − 2, . . . , and this can be used to predict the output. The one-step-ahead predictor of (2.2) is D(q)A(q) D(q)B(q) u(t) + 1 − y(t), (2.3) ŷ(t) = C(q)F(q) C(q) and depends only on previous output data. The unknown parameters in the polynomials A(q), B(q), C(q), D(q) and F(q) are gathered in the parameter vector θ, θ = [a1 . . . ana bnk . . . bnk +nb −1 c1 . . . cnc d1 . . . dnd f 1 . . . f nf ]T . The predictor ŷ(t) is often written ŷ(t|θ) to point out the dependence on the parameters in θ. By deﬁning the prediction error ε(t) = y(t) − ŷ(t|θ), (2.4) a straightforward modeling approach is to try to ﬁnd the parameter vector θ̂, that minimizes this diﬀerence, θ̂ = arg min V (θ), V (θ) = 1 N θ N l(ε(t)) (2.5a) (2.5b) t=1 where l( · ) is a scalar valued, usually positive, function. Finding the parameters by this minimization is called a prediction-error (identiﬁcation) method (pem). This idea is illustrated in Figure 2.2. Except for special choices of the model structures G(q, θ) and H(q, θ) and the function l(ε) in (2.5b), there is no analytical way of ﬁnding the minimum of the minimization problem (2.5a). Numerical solutions have to be relied upon, which means that a local optimum might be found instead of the global one if the cost function is nonconvex, with more than one minimum. For results on the convergence of the parameters and other properties of the estimate, such as consistency and variance, see Ljung [1999]. 2.4 13 Linear Regression v(t) u(t) System Model y(t) _ ε(t) ŷ(t|θ) Figure 2.2: An illustration of the idea behind system identiﬁcation. 2.4 Linear Regression Another common way to describe the relationship between input and output of an lti system is through a linear diﬀerence equation where the present output, y(t), depends on previous inputs, u(t − nk ), . . . , u(t − nk − nb + 1), and outputs, y(t − 1), . . . , y(t − na ) , as well as the noise and disturbance contributions. This can for example be done for (2.2) when C(q), D(q) and F(q) are set to unity, so that G(q, θ) and H(q, θ) in (2.1) correspond to G(q, θ) = B(q) , A(q) H(q, θ) = 1 A(q) with A(q) = 1 + a1 q−1 + · · · + ana q −na B(q) = bnk q −nk + · · · + bnk +nb −1 q −(nk +nb −1) . The linear diﬀerence equation is then y(t)+ a1 y(t −1)+· · ·+ ana y(t − na ) = bnk u(t − nk )+· · ·+ bnk +nb −1 u(t − nk − nb +1)+ e(t), and we can write A(q)y(t) = B(q)u(t) + e(t). (2.6) This particular structure is called auto-regressive with external input (arx). Another special case is when the output only depends on past inputs, such that na = 0 in (2.6). This is called a ﬁnite impulse response (fir) structure. The predictor for an arx model is ŷ(t|θ) = −a1 y(t − 1) − · · · − ana y(t − na )+ bnk u(t − nk ) + · · · + bnk +nb −1 u(t − nk − nb + 1). By gathering all the known elements into one vector, the regression vector, φ(t) = [−y(t − 1), . . . , −y(t − na ) u(t − nk ), . . . , u(t − nk − nb + 1)]T and the unknown elements into the parameter vector, θ = [a1 . . . ana bnk . . . bnk +nb −1 ]T , (2.7) 14 2 Introduction to Model Estimation the predictor (2.7) can be written as a linear regression ŷ(t|θ) = φ T (t)θ, (2.8) that is, the unknown parameters in θ enter the predictor linearly. 2.5 Least-squares Method With the function l( · ) in (2.5b) chosen as a quadratic function, l(ε) = 1 2 ε , 2 and the predictor described by a linear regression, as in (2.8), we get V (θ) = N 2 1 y(t) − φ T (t)θ , 2N (2.9) t=1 called the least-squares (ls) criterion. A good thing about this criterion is that it is quadratic in θ, which means that the problem is convex and the minimum can be calculated analytically. The minimum is obtained for θ̂ LS ⎡ ⎤−1 N N ⎢⎢ 1 ⎥⎥ 1 T ⎢ = ⎢⎢⎣ φ(t)φ (t)⎥⎥⎥⎦ φ(t)y(t), N N t=1 (2.10) t=1 called the least-squares estimator. See Draper and Smith [1998] for a more thorough description of the ls method and its properties. Apart from the guaranteed convergence to the global optimum, a beneﬁt with ls solutions is that there exist many eﬃcient numerical methods to solve them. The recursive least-squares (rls) method can be used to solve the numerical optimization recursively [Björck, 1996]. Another option is the least mean square (lms) method, which can make use of the linear regression structure of the optimization problem, developed in (2.8). Separable Least-squares For some model structures, the parameter vector can be divided into two parts, θ = [ρ T η T ]T , so that one part enters the predictor linearly and the other nonlinearly, i.e., ŷ(t|θ) = ŷ(t|ρ, η) = φ T (t, η)ρ. Hence, for a ﬁxed η, the predictor is a linear function of the parameters in ρ. The identiﬁcation criterion is then V (θ) = V (ρ, η) = N 2 1 y(t) − φ T (t, η)ρ 2N t=1 2.6 The System Identiﬁcation Procedure 15 and this is an ls criterion for any given η. Often, the minimization is done ﬁrst for the linear ρ and then the nonlinear η is solved for. The nonlinear minimization problem now has a reduced dimension, where the reduction depends on the dimensions of the linear and nonlinear parameters. This method is called separable least-squares (sls) as the ls part has been separated out, leaving a nonlinear problem of a lower dimension, see Ljung [1999, p. 335-336]. 2.6 The System Identiﬁcation Procedure The process of constructing a model from data consists of a number of steps, which often have to be performed a number of times before a suitable model can be obtained. 1. A data set is needed, usually containing input and output data. The data should be “good enough”, so that it excites the desired properties of the system. This is called persistency of excitation. 2. Diﬀerent model structures should be examined, to evaluate which structure best captures the properties of the data. These structures should fulﬁll certain demands, such that two sets of parameters do not lead to the same model. This property is called identiﬁability. 3. A measurement of “goodness”, such as the criterion (2.5), has to be selected to decide which models best describe the data. 4. The model estimation step is where the parameters in θ are determined. In the ls method, this would consist of inserting the data into (2.10), and in the pem case, the minimization of (2.5) for a certain choice of predictor structure ŷ(t|θ) in (2.3). 5. Model validation. In this step, diﬀerent models should be evaluated to determine if the models obtained are good enough. The evaluation should be done on a new set of data, validation data, to ensure that the model is useful not only for the data for which it was estimated. Two important components of the model validation are the comparisons between measured data and model output as well as the residual analysis, where the statistics of the unmodeled properties of the data are evaluated. Some of these steps contain a large user inﬂuence, whereas others might be set or rather straightforward. The choice of model structure and model order, such as na and nb in (2.7), is often hard and needs to be repeated a number of times before a suitable model can be found. 3 Introduction to System Inversion Inverse systems are used in many applications, more or less visibly. One application example of this is power ampliﬁers in communication devices, which are often nonlinear, causing interference in adjacent transmitting channels [Fritzin et al., 2011a]. This interference will be noise to anyone that transmits in these channels, and there are measures describing the amount of power that is allowed to be spread to adjacent frequencies. So to be useful, linearization of the ampliﬁer is needed, limiting the interference in the neighboring channels. However, one does not want to work with the ampliﬁed signal, but rather with the input signal to the system, that is, before the signal is ampliﬁed. A preﬁlter that inverts the nonlinearities, called a predistorter, is thus preferable. In sensor applications it is rather a postdistortion that is needed. If the sensor itself has dynamics or a nonlinear behavior, the sensor output is not the true signal but will also contain some sensor contamination. This would have to be handled at the sensor output, since this is where the user can get access to the signal. In the area of robotics, there is a need for control such that the robot achieves the demands on precision. Smaller and lighter robots reduces the need for large motors, and also the cost and wear of the robot. However, this also introduces new problems such as larger oscillations and increases the demands on the control performance. In robotic control applications, a common strategy is to use feedback to control the joint positions. The last part of the robot, however, connecting the tool to the robot, is often controlled using open-loop control. Models of both the forward and the inverse kinematics are used for control. In the above applications, ﬁnding the inverse of the system is a crucial point; how should the input to or the output from the system be modiﬁed to obtain the desired dynamics from input to output? Each application entails its own restrictions and special conditions to attend to, and in this chapter, some aspects of 17 18 3 Introduction to System Inversion system inversion are discussed. For a nonlinear system, the inversion is nontrivial, and diﬀerent approaches can be used. A selection of methods is presented here. In this thesis, it is assumed that an inverse exists, that is, there is a one-to-one relation between input and output. This property is called bijectivity. Furthermore, we assume that the system and the inverse can both be written analytically, see Example 3.1 for a case when this is not the valid. Both the system and the inverse are assumed to be stable and causal (see for example Rugh [1996]). In this chapter, the main focus is on inversion, and a model of the system is supposed to be known, either by physical modeling or by system identiﬁcation. Diﬀerent approaches to estimate inverse models will be presented in Chapter 4. Example 3.1: Nonexisting Analytical Inverse Consider the system y(t) = e x(t) + sin(x(t)) for |x(t)| < 0.5π. The function e x + sin(x) is monotonic on [− π2 π2 ], and thus also invertible. However, no analytic expression of the inverse exists, and a numerical inverse will have to be used. Here, the methods are described in either continuous or discrete time. Diﬀerent frameworks are usually most easily described in one domain or the other, hence the mixed use in this chapter. Also, the systems are often continuous whereas the controllers are implemented in discrete time. The explicit dependency on time will sometimes be left out for notational convenience. 3.1 Inversion by Feedback The behavior of a system can be modiﬁed in a multitude of ways, often with the goal of making the output follow a desired trajectory, called reference signal, r. In the automatic control society the main choices are feedback and feedforward control. For the linear case many diﬀerent control strategies exist, the perhaps most common of which is the pid, consisting of a proportional (P), an integral (I) and a derivative (D) part. The P, I and D parameters of the controller can be trimmed to obtain a desired behavior of the controlled system [Åström and Hägglund, 2005]. In this section, a few feedback strategies will be introduced. An iterative control approach that can be used for linear and nonlinear systems is the iterative learning control (ilc). ilc works on systems with a repetitive input signal, such as a robot that performs the same task over and over again. It makes use of the output from the last repetition and tries to improve this so that the output better follows the reference signal. Another feedback solution for nonlinear systems is the exact input-output linearization, that makes use of a known model of the system to obtain overall linear dynamics, determined by the user. Though the classical view of feedback control is not that of system inversion, this is indeed one interpretation; the feedback system produces the input that 3.1 19 Inversion by Feedback r + e F u G y −1 Figure 3.1: A feedback controller F applied to the system G. leads to a desired output. This is also the goal of an inverse system, to produce an input by use of an output. We will start this chapter by covering a few control strategies. 3.1.1 Feedback and Feedforward Control Feedback control refers to a measured output of a system that is used to determine the input to said system. A standard solution is to look at the diﬀerence between the reference r(t) and the output y(t), called control error e(t) = r(t) − y(t). This signal can be used for control of the system. For example, if the control error is negative, a conclusion can be that the input u is too small, and should be increased, and vice versa. Many control strategies based on this idea have been constructed and are commonly used in industry. The idea is presented in Figure 3.1 where a feedback controller F is applied to the system G. On the other hand, if we know something about how the system will transform the input, we might want to use this to counteract later eﬀects. This is the concept of feedforward control, where the reference signal is altered and sent to the system, or fed forward. Often, feedforward and feedback control are used together to get the advantages of both approaches. Figure 3.2 shows a block diagram where the feedback loop in Figure 3.1 has been expanded to include a feedforward loop with the feedforward controller Ff . A common requirement is that the output should have a softer behavior than the reference, and this can be achieved via the ﬁlter Gm , which denotes the desired dynamics. The ideal choice of the feedforward controller is Ff = Gm /G. If feedforward control is used alone, with no feedback loop, it is often called open-loop control. Feedback control can handle phenomena like disturbances and model uncertainties, since it is based on the true output and not only the input and a model of the system. It can also handle unstable systems, which is not possible for a pure feedforward (open-loop) control, but a bad feedback loop may cause instability. Feedforward control has the advantage of not needing any measurements but the drawback is that ideal feedforward control (using Ff = Gm /G) requires perfect knowledge of the system, and that both G and Gm /G are stable. Also, there is no possibility to compensate for disturbances. However, if the disturbances are perfectly known or measurable, feedforward control from the disturbances can be applied and the disturbance compensated for. These are of course limiting assumptions. A beneﬁt with feedforward control is that two cascaded stable systems will always be stable, and a bad controller can therefore not destabilize the system. 20 3 Introduction to System Inversion Ff r Gm yr + uf F us + u G y −1 Figure 3.2: Feedforward controller Ff and feedback controller F applied to a system G. Gm is used here to describe the desired dynamics between reference and output. 3.1.2 Iterative Learning Control As discussed in the introduction, iterative learning control (ilc) can be seen as an iterative inversion method [Markusson, 2001]; the goal is to ﬁnd the input that leads to the desired output. In this section, the basic concepts of ilc will be described, but for a more thorough analysis see for example Wallén [2011], Moore [1993] and the references therein. The ilc concept comes from the industrial robot application, where the same task or motion is performed repeatedly. The idea is to use the knowledge of how the controller performed in the last repetition and improve the performance in each iteration. The system S in this setting is described by the input u, the output y, and the reference r over a ﬁnite time interval. The task is assumed to be repeated, so that the reference r and the starting point are the same for each iteration. The time index is t, where t ∈ [0, N − 1] for each repetition, and each repetition is of length N . A basic ﬁrst order ilc algorithm is described by uk+1 (t) = Q(q) (uk (t) + L(q)ek (t)) (3.1) where ek (t) = r(t) − yk (t) and k is the iteration index, and indicates how many times the task has been repeated. Here, q is the shift operator such that q−1 u(t) = u(t − 1) and Q(q) and L(q) denote linear or nonlinear operators, chosen by the user. It is important that this choice leads to convergence and an input where the output achieves the desired performance. Also, the learning should be fast enough. There are structured ways to determine Q(q) and L(q), which can be based on a model of the system. The concepts of stability and convergence of ilc systems are treated in, for example, Wallén [2011]. It can be shown that ilc is robust to model errors, such that for a linear system, a relative model error of 100% can be tolerated [Markusson, 2001]. Even a rather simple model can therefore perform well. Iterative methods are used in many applications, also outside the control community. The common factor is that the information found in the output y is used to improve the input, but the algorithm is not necessarily similar to (3.1). One 3.1 21 Inversion by Feedback application where iterative solutions are often used is analog-to-digital converter (adc) correction, such as in Soudan and Vogel [2012]. 3.1.3 Exact Linearization In exact linearization (also called input-output linearization) [Sastry, 1999], the output from a nonlinear system S, ẋ = f (x) + g(x)u y = h(x), (3.2) which is aﬃne in u, is diﬀerentiated enough times to obtain a relation between the diﬀerentiated output y (n) and the input, u. Diﬀerentiating y with respect to time, we obtain ∂h ∂h f (x) + g(x)u ∂x ∂x = Lf h(x) + Lg h(x)u, ẏ = where Lf h(x) and Lg h(x) are the Lie derivatives of h with respect to f and g, respectively. If Lg h(x) 0, a relation between the diﬀerentiated output ẏ and the input u has been obtained and an input, u= 1 (−Lf h(x) + r) Lg h(x) can be calculated that leads to a linear relation between output and reference, ẏ = r. If Lg h(x) = 0, a second diﬀerentiation can be done, ∂Lf h f (x) + ∂Lf h g(x)u ∂x ∂x = L2f h(x) + Lg Lf h(x)u, ÿ = from which a control law can be calculated if Lg Lf h(x) 0. In this manner, one can continue until there is a direct relation between y (γ) and r through the control law u= 1 γ−1 Lg Lf h(x) γ (−Lf h(x) + r) = α(x) + β(x)r. (3.3) Here, γ is the smallest integer for which Lg Lif ≡ 0 for i = 0, 1, . . . , γ − 2 and γ−1 Lg Lf h(x) 0 and it is called the relative degree of the system. The system (3.2) with control input (3.3) now describes a system with linear dynamics. Thus, linear theory can be used to obtain the desired dynamics, Gm , chosen by the user, and the linear feedback loop can be combined with the nonlinear one. The overall system from r to y (the nonlinear system with the nonlinear 22 3 Introduction to System Inversion and linear feedback) will thus be linear, and the dynamics will be described by the transfer function Gm . Exact linearization requires knowledge of all the states, and is therefore often used in combination with a nonlinear observer. This can lead to a complicated feedback loop. Here, it is assumed that any zero dynamics present are stable. The above system and the derivation of the feedback loop is described in continuous time. A discrete-time description can also be done, as presented in Califano et al. [1998]. 3.2 Analytic Inversion In the above feedback loops, only the system itself, or a model thereof, is used to produce an inverse. No explicit inversion is done. Another approach is to perform an analytic inversion of the system, which can be applied at the input to, or the output from, the system, see Figure 1.1. The output from this cascaded system should have the desired dynamics. If the goal is to make the output exactly the same as the reference, a “true” inverse has to be found. But even for other cases, the inversion can be seen as a case where the unwanted nonlinear and linear dynamics have been inverted. For example, in the exact linearization case, the nonlinear and dynamical behavior of the system are inverted, and in the end a system with some user-deﬁned linear dynamics is obtained. This approach has already been used in the feedforward controller Ff = Gm /G, where the system G is inverted. Finding a system inverse can be done in multiple ways. One method for ﬁnding an inverse to dynamic systems uses Volterra series, which is a nonlinear extension of the impulse response concept from the linear case. This leads to an analytical inverse. Other systems that might be analytically invertible are blockoriented systems, which consist of a static nonlinearity and a linear dynamic system. A brief overview of Volterra series will be presented here together with a short discussion on the use of preinverse and postinverse and problems that occur with inversion. 3.2.1 Problems Occurring with System Inversion For a stable and minimum-phase lti system G, it is rather straightforward to ﬁnd an inverse G −1 . However, if these conditions are not fulﬁlled, we quickly run into problems, even for linear systems. Any nonminimum-phase zeros of the original system will become unstable poles of the inverse system. However, if the system is nonminimum phase, the inverse can be used if noncausal ﬁltering is allowed. If a delay can be allowed, time-reversed input and output sequences can be used together with a matching, stable inverse [Markusson, 2001]. Another trouble with inverse systems concerns whether the system is proper or not. A proper transfer function is one where the order of the denominator is greater than or equal to that of the numerator. A strictly proper transfer function is one where the order of the denominator is greater than that of the numerator. 3.2 23 Analytic Inversion The ampliﬁcation of a proper system always approaches a value as the frequency goes to inﬁnity. If the transfer function is strictly proper, the ampliﬁcation will approach zero at high frequencies. For a transfer function that is not proper, however, the ampliﬁcation will approach inﬁnity when the frequency approaches inﬁnity. That is, high frequency contents will be ampliﬁed. This means that the inverse of a strictly proper system will be improper. Here, the goal is not to cover all problems with the inversion of systems, but to give some insights to the problems that can occur. 3.2.2 Postinverse and Preinverse As is commonly known, the ordering of a linear system does not matter, i.e., the output from A ∗ B equals the output from B ∗ A when A and B are linear dynamical systems. This property is called commutativity. However, this does not apply to nonlinear systems, as shown in Example 3.2. Example 3.2: Noncommutativity of Nonlinear Systems Consider the two functions f 1 (x) = 2x and f 2 (x) = x2 . If the order of the systems is f 1 , f 2 , the output is y12 = 4u 2 and with the reversed order, the output is y21 = 2u 2 y12 . Thus, for nonlinear systems, the output depends on the order of the systems. For some nonlinear systems, this is not true and the systems can change order without changing the output. One example where two nonlinear systems commute, is where one of the systems is the inverse of the other as in Example 3.3 for a Hammerstein-Wiener system. When an exact inverse exists, the preinverse and the postinverse are the same. However, it is often not possible to determine the exact inverse, and an approximate inverse has to be used. This approximate function does not necessarily commute with the system. Another example of nonlinear systems that commute are the Volterra series and the p-th order Volterra inverse that will be described in the next section. But, in general, the commutative property does not apply to nonlinear systems. See Mämmelä [2006] for an extended discussion on commutativity in linear and nonlinear systems. Example 3.3: Analytical Inversion Consider the Hammerstein system with the static nonlinearity f H (x) = x3 which is invertible for all x, followed by the minimum-phase linear dynamic system s+1 GH (s) = , s+2 24 3 Introduction to System Inversion u ( · )3 z s+1 s+2 y (a) √ 3 s+2 s+1 · (b) Figure 3.3: (a) A Hammerstein system with invertible static nonlinearity followed by a linear, stable minimum-phase dynamical system. For such a system, an analytical inverse exists, as shown in (b). as shown in Figure 3.3a. For this system, an analytical inverse exists, namely the Wiener system √ s+2 , f W (x) = 3 x, GW (s) = s+1 see Figure 3.3b. This inverse is also an example of where a nonlinear system and its inverse are commutative, that is, the two systems can be placed in whichever order. This Wiener system can thus be used as a preinverse or a postinverse. Diﬀerent approximate modeling approaches, which will be further considered in Chapter 4, lead to either a preinverse or a postinverse that are not necessarily equal. Which one that is requested is connected to the application, such that for power ampliﬁer linearization a preinverse is desired, and for sensor calibration a postinverse. However, for power ampliﬁer predistortion, the commutativity property is often considered approximately valid, and the pre- and postinverses are used interchangeably without further consideration [Abd-Elrady et al., 2008, Paaso and Mämmelä, 2008]. 3.2.3 Volterra Series In the linear systems theory, a common way to describe the output, y(t), of the system aﬀected by the input u(t), is by the impulse response g( · ), ∞ g(τ)u(t − τ)dτ, y(t) = (3.4) −∞ usually with the added constraints that the system is causal and the input zero for t < 0, so that the integral is limited to [0, t]. It can also be described by the corresponding Laplace relation Y (s) = G(s)U (s) (3.5) where Y (s) and U(s) are the Laplace transformed versions of y(t) and u(t), respectively, and G(s) is the transfer function. This is not possible for nonlinear systems. 3.2 25 Analytic Inversion However, if the nonlinear system is time invariant with certain restrictions, an input-output relation can be determined. These conditions include convergence of the inﬁnite sums and integrals that occur [Sastry, 1999], but will not be further considered here. The input-output relation can be described by ∞ ∞ ∞ h1 (τ1 )u(t − τ1 )dτ1 + y(t) = −∞ ∞ −∞ −∞ ∞ hn (τ1 , . . . , τn )u(t − τ1 ) . . . u(t − τn )dτ1 . . . dτn + . . . ... + −∞ h2 (τ1 , τ2 )u(t − τ1 )u(t − τ2 )dτ1 dτ2 + . . . (3.6) −∞ where hn (τ1 , . . . , τn ) = 0 for any τj < 0, j = 1, 2, . . . , n. The relation (3.6) is called a Volterra series (sometimes Volterra-Wiener series) and the functions hn (τ1 , . . . , τn ) are called the Volterra kernels of the system. The expression (3.6) can also be written as y(t) = H1 [u(t)] + H2 [u(t)] + · · · + Hn [u(t)] + . . . (3.7) where ∞ ∞ ... Hn [u(t)] = −∞ hn (τ1 , . . . , τn )u(t − τ1 )u(t − τ2 ) . . . u(t − τn )dτ1 . . . dτn (3.8) −∞ is called an n-th order Volterra operator. When considering an lti single input-single output (siso) system, the Volterra series reduces to the standard form, and the kernel h1 ( · ) in (3.6) corresponds to g( · ) in (3.4). See for example Schetzen [1980] for a more thorough description of Volterra series. The counterpart of the transfer function is based on the multivariable Fourier transform, ∞ Hp (j ω1 , . . . , j ωp ) = ∞ ... −∞ hp (τ1 , . . . , τp )e−j(ω1 τ1 +···+ωp τp ) dτ1 . . . dτp (3.9) −∞ called the p-th order kernel transform. The inverse relation is 1 hp (τ1 , . . . , τp ) = (2π)p ∞ ∞ Hp (j ω1 , . . . , j ωp )e j(ω1 τ1 +···+ωp τp ) dω1 . . . dωp . ... −∞ −∞ (3.10) In analogy to the linear case, these functions are sometimes referred to as higher order transfer functions. The discrete counterpart of the Volterra operators (3.8) is [Tummla et al., 1997] Hn [u(t)] = ∞ i1 =−∞ ... ∞ in =−∞ (n) hi1 ,i2 ,...,in u(n − i1 ) . . . u(n − in ). (3.11) 26 3 Introduction to System Inversion This version is often used in data-based modeling, where the models are based on sampled data. p -th Order Volterra Inverse A p-th order inverse, H−1 (p) , is deﬁned as a system that, when connected in series with the nonlinear system H results in a system, Q, in which the ﬁrst-order Volterra kernel is a unit pulse and the other Volterra kernels are zero, qk = 0, k = 2, . . . , p. The Volterra kernels for k > p might however be nonzero but are generally considered to be negligible [Zhu et al., 2008]. The inverse, H−1 (p) , can be determined by using the Volterra series (assumed known) of the system, and the desired output. This is done in a sequential way by ﬁrst ﬁnding the ﬁrst order Volterra operator, H−1 (1) , and then solving for the higher order Volterra operators H−1 (n) , n = 2, . . . , p which then only depend on the system H and lower order operators of the inverse, see Schetzen [1980, Chapter 7] for a thorough discussion. The ordering of the system H and the inverse H−1 (p) will aﬀect the output, but it can be shown [Schetzen, 1980] that the ﬁrst p Volterra operators of the connected systems are the same. The order of the system H and the inverse H−1 (p) can thus be interchanged and the postinverse H−1 (p) can also be used as a preinverse, if only nonlinearities up to order p are of interest. 3.3 Inversion by System Simulation Some approaches to avoid the explicit inversion of a system are based on a simulation of the true system, without including any feedback from the actual system. The exact linearization described in Section 3.1.3 can be modiﬁed such that it uses a simulated output in the feedback loop. Another approach is to decompose the original system to avoid the explicit inversion of the nonlinear system. The idea with these inversion methods is that the inverse can be used as a preinverse or a postinverse, thus avoiding a feedback loop. 3.3.1 Separation of a Nonlinear System A way to avoid the explicit inversion of a nonlinear system is presented in Markusson [2001, p. 51]. There, the nonlinear system S is separated into a linear part, L, and a nonlinear part, N , where operator notation is used. The inverse of S = L + (S − L) = L + N = L(I + L−1 N ), can then be written S −1 = (I + L−1 N )−1 L−1 . We have thus obtained a postinverse S −1 such that S −1 S = (I + L−1 N )−1 L−1 (L + N ) = (I + L−1 N )−1 (I + L−1 N ) = I . This can also be used as a preinverse, since S S −1 = (L + N )(I + L−1 N )−1 L−1 = L(I + L−1 N )(I + L−1 N )−1 L−1 = I . 3.3 27 Inversion by System Simulation y(t) + L−1 (q−1 ) w(t) N (w, t) Figure 3.4: An inversion method that only uses the inverse of the linear part L of a nonlinear system S = L + N . The inverse (I + L−1 N )−1 L−1 can be obtained in a feedback loop with the nonlinear part N in the feedback and the linear inverse L−1 in the forward path (compare to the sensitivity function for lti systems), see Figure 3.4. It follows that the nonlinear part N does not have to be explicitly inverted, and that only the linear part L is to be inverted. The output from the inverted system is denoted w(t) to separate it from the true input u(t), since diﬀerent initial conditions of the true system and the model will produce an output that is not exactly equal to the input. Unknown initial states are discussed in Markusson [2001, p. 45], in a maximum likelihood (ml) setting. 3.3.2 Hirschorn’s Method Another approach to invert nonlinear systems is Hirschorn’s method, where exact linearization is used in order to construct a linear system [Hirschorn, 1979]. Given that the model is good enough, it should be possible to use the model not only in the feedback, as it is used in the construction of u in (3.3), but also as a simulation model. Preinversion If instead of the measured output from the system, the output from the simulated model is fed back to the controller, see Figure 3.5, the overall system (from reference r to output ys ) will by construction be linear with the dynamics Gm . Also, the input calculated for this (simulated) system leads to the desired dynamics, and the same input signal can be used also for the true system. The system from r to u will be denoted S † . A pure open loop controller is thus obtained, as in Hirschorn [1979], see Figure 3.6, and this is called Hirschorn’s method. The simulated feedback can also be interpreted as an observer with no measurement inputs. Postinversion Let the nonlinear system be denoted S and the precompensation be denoted S † , since it is not really an inverse of S, but rather creates a system that, in series with S will be linear. The dynamics of the overall linear system is Gm . The method described above can be seen as an inversion of the nonlinearities of the system – the output from the overall system will be linear with dynamics 28 3 Introduction to System Inversion ylin S r Ŝ u ys S† Controller Figure 3.5: A block diagram of Hirschorn’s method, where the system S is replaced by a model Ŝ in the exact linearization feedback loop. The input signal calculated in this way is then also applied to the real system S. The simulation system and feedback loop that leads to an overall linear behavior between r and ys is denoted S † . The input to S † is the reference r and the output is the control signal u. r S† u S ylin Gm Figure 3.6: The predistortion block S † obtained using Hirschorn’s method in series with the real system leads to an overall linear behavior between r and ylin . 3.3 29 Inversion by System Simulation u S† r̃ y S Figure 3.7: The (possibly ﬁctitious) reference signal r̃ can be seen as input to the block S † , creating the input u to the nonlinear system S. 1 Gm y r̄ S† ū ylin Gm Figure 3.8: Hirschorn’s method applied as postdistortion, when the output can be assumed to be created according to Figure 3.7. The block S † cannot simply be applied at the output y, but has to be manipulated to obtain a linear behavior between u and ylin . Gm chosen by the user. This is based on the assumption that the model is accurate enough, of course. This is a setup where preinverse and postinverse are not interchangeable; Hirschorn’s method tells us only how to determine the input to the nonlinear system such that the reference-to-output has the linear dynamics Gm , not how to manipulate the output to make it a linear response to the input. If it is this postinverse that is wanted, a diﬀerent setup is needed. It is known that S † in cascade with S leads to a linear system Gm , so that y = Gm r (3.12) with r the reference, cf. Figure 3.6. The goal is to obtain a linear response to u by using a postinverse on the output y. Assume that u was actually created by a preﬁlter, S † , with u as output and the ﬁctitious signal r̃ as input, as in Figure 3.7. An estimate of this signal can then be obtained by r̄ = 1 y, Gm (3.13) where, if no transients or noise are present, r̄ = r̃. An estimate of the input u, called ū, can be obtained by ﬁltering r̄ by S † . Now, to obtain the desired dynamics, ū must be ﬁltered by the linear function Gm , see Figure 3.8. The cascade of these three blocks (1/Gm , S † and Gm ), thus make up a postdistorter that leads to a linear response between u (not available for manipulation) and ylin in Figure 3.9. This method is illustrated in Example 3.4. u S y 1 Gm r̄ S† ū Gm ylin Figure 3.9: Hirschorn’s method used as postdistortion. The postinverse consists of the three blocks 1/Gm , S † and Gm . Used in this way, the overall behavior between u and ylin will be linear. 30 3 Introduction to System Inversion 1 0 −1 100 104 108 112 116 120 time [s] Figure 3.10: The output from the nonlinear system (3.14) in gray and the desired dynamics from Gm (3.15) in black. Example 3.4: Hirschorn’s Postinverse Consider the nonlinear system ẋ1 = −x13 + x2 + w1 ẋ2 = −x2 + u + w2 y = x1 (3.14) with process noise wi ∈ N (0, 0.05) and a multisine input. The nonlinear feedback u = −3x15 + 3x12 x2 + x2 + ũ leads to a linear system ÿ = ũ. Now, linear theory can be applied and pole placement has been used to get an overall system response from reference r to output y corresponding to the one from Gm (s) = s2 1 . + 5s + 6 (3.15) The output from the nonlinear system (3.14) is plotted in Figure 3.10 together with the output from the desired dynamics Gm . A preinverse S † has been constructed as in Figure 3.5. S † has been used as a preinverse, as well as a postinverse for evaluation purposes. The results are shown in Figure 3.11. Here, it is clear that the desired preinverse and postinverse are not the same, and that S † cannot straight away be used as a postinverse. If instead, the output y is ﬁltered by the cascaded systems 1/Gm , S † and Gm , as in Figure 3.8, the result improves considerably, as shown in Figure 3.11. The remaining errors are primarily caused by the noise. For noise-free data, the preinverse performs perfectly whereas the postinverse has some minor errors. 3.3 31 Inversion by System Simulation 0 −0.5 100 104 108 112 116 120 time [s] Figure 3.11: A Hirschorn postinverse applied to the system (3.14). The output from Gm (3.15) is plotted in solid black and the output when S † was used as a preinverse in dashed black. The output from the system in series with the inverse S † is plotted in dashed gray when S † is used as a postinverse with no extra ﬁltering. When the postdistortion is constructed according to Figure 3.8, the result improves considerably. Here, the postinverse consists of three blocks, 1/Gm , S † and Gm and the postdistorted output is plotted in solid gray. Note the scale diﬀerence from Figure 3.10. 4 Estimation of Inverse Models An inverse model is here estimated with the purpose of using it in cascade with the system itself, as an inverter, and a good inverse model in this setting would be one that, when used in series with the original system, reconstructs the original input, see Figure 4.1. In estimation, one should usually estimate the system in the setting it should be used, concerning for example the choice of input and the experimental conditions [Ljung, 1999, Gevers and Ljung, 1986, Pintelon and Schoukens, 2012]. It is important to choose the input signal to capture the signiﬁcant characteristics of the system. Usually, it ought to resemble the conditions in which it is intended to be used, but what does that correspond to in this case? Will it be the input spectrum that decides the weighting, or is it rather the output spectrum that should be weighted to reﬂect the relative importance of the model ﬁt, since the output is in this case seen as the input to the inverse system to be estimated? Another important topic in system identiﬁcation is the choice of loss function, V in (2.5b). u S −1 S yu (a) u S y S −1 yu (b) Figure 4.1: The intended use of the estimated inverses. Figure (a) shows predistortion, where the inverse S −1 is applied before the system S, and (b) shows postdistortion, where the order is reversed. 33 34 4 Estimation of Inverse Models Table 4.1: Inputs and outputs to the identiﬁcation procedure, using the different methods. Input Output Requires Model Method A u y forward Ŝ Method B u u inverse Method C y u inverse It should reﬂect the goal of the identiﬁcation, and, depending on how it is chosen, diﬀerent properties of the estimated model will be emphasized. In this setup, the goal is to make use of these degrees of freedom and the ﬂexibility of the model to obtain an accurate input estimate. In this chapter, some aspects of system inverse estimation are discussed. The contents are also presented in Jung and Enqvist [2013]. 4.1 System Inverse Estimation In system identiﬁcation, the goal is to achieve as good a model as possible to explain the behavior of y by a prediction or simulation ŷ(t|θ), which depends on the estimated model parameters θ and the input u. This is done using measured data, usually input data u(t) and output data y(t), (2.3), see Chapter 2. Here, a model describing the system itself will be referred to as a forward model and a model describing the inverse will be called an inverse model. The inverse model is estimated with the purpose of using it in series with the system itself, as an inverter, see Figure 4.1. In this setup, the goal is to minimize the diﬀerence between the input u and the output from the cascaded systems, yu . A good model in this setting would be one that, when used in series with the original system, regains the original input, so that yu = u. There are three main approaches to the estimation of an inverse of a system S, described in more detail below. Method A In a ﬁrst step, the forward model Ŝ is estimated in the standard way, with input data u and output data y. Step two is to invert the resulting model to obtain an approximate inverse Ŝ −1 . Method B In a pre-step, the forward model Ŝ is estimated in the standard way, with input data u and output data y. This model is used in series with an inverse model, Ŝ −1 , and the inverse model parameters are estimated in this setting, by trying to minimize the diﬀerence between the input u and the simulated, distorted output yu . Method C The identiﬁcation is done in one step, by identifying the inverse directly, using input data y and output data u. The inputs and outputs to the diﬀerent approaches are summarized in Table 4.1. The identiﬁcation in the ﬁrst approach, Method A, is the standard one, as described in, for example, Ljung [1999] and Pintelon and Schoukens [2012], and 4.2 Inverse Identiﬁcation of LTI Systems 35 the inversion is discussed in Åström and Hägglund [2005] in the feedforward control application. The use of feedforward control based on an inverse model of the system in the presence of plant uncertainty is discussed in Devasia [2002]. A good thing with Method A is that the identiﬁcation uses standard methods, but on the other hand, an inversion is required, and the weighting of the model ﬁt is not necessarily optimal for the use intended here. The second approach, Method B, is often used in power ampliﬁer predistortion [Fritzin et al., 2011a, Abd-Elrady et al., 2008, Paaso and Mämmelä, 2008]. In this application, it is also called direct learning architecture (dla). The quality of the inverse and the forward models are closely coupled, and two choices are available. Since it is often preferable to obtain a rather simple inverse model (for example in the predistorter case), this restriction can also be applied to the forward model, so that the same model structure is used for the forward and the inverse models. Another approach is to use a more complex forward model, making sure that as much as possible of the system behavior is captured, and then let the inverse model be less complex. The choice in the end comes down to the implementation – if the forward model has to be implemented, also this model needs to have a limited complexity. A good thing with this approach is that the estimation of the inverse is done with no noise present, but it also requires two, possibly nonconvex, minimizations with the risk of obtaining local minima. The quality of the inverse also clearly depends on the quality of the forward model. The third approach, Method C, is also called indirect learning architecture (ila) in power ampliﬁer predistortion applications. It has been evaluated in pa predistortion applications in Abd-Elrady et al. [2008] and Paaso and Mämmelä [2008]. For this approach to be applicable in predistortion, it is assumed that the predistorter and the postdistorter are interchangeable (commutativity), see also Section 3.2.2. An advantage with this method is that the inverse is estimated in the setting in which it is going to be used, and that the weighting is possibly better than for Method A. A drawback is that the measured output is used as input, which risks causing a biased estimate [Amin et al., 2012]. It can be an easier approach, since the estimation is done in one step. Furthermore, there is no need to construct a model for the forward system that will later be discarded. In power ampliﬁcation predistortion applications, Method C (ila) is more commonly used than Method B (dla), as investigated in Paaso and Mämmelä [2008]. In Paaso and Mämmelä [2008], comparisons performed indicate that the dla performs better in the simulation setup used, whereas in Abd-Elrady et al. [2008] the ila seems to perform slightly better. 4.2 Inverse Identiﬁcation of LTI Systems To simplify the discussion, we will start by looking at lti dynamical systems. The model estimation is done in open loop and assuming the output was created according to y(t) = G0 (q)u(t) + H0 (q)e0 (t) (4.1) 36 4 Estimation of Inverse Models where G0 is the true system, H0 is the true noise dynamics and e0 is a white noise sequence. In system identiﬁcation, the goal is often to ﬁnd the minimizing argument of a function of the prediction error ε(t, θ) θ̂ = arg min θ N N 1 1 ε(t, θ)2 = arg min [y(t) − ŷ(t|θ)]2 , N N θ t=1 (4.2) t=1 where y(t) is the measured output and ŷ(t|θ) is the predicted output given the model parameters θ. Here, we use a ﬁxed noise model H∗ ≡ 1 such that the prediction is described by ŷ(t|θ) = G(q, θ)u(t). Looking at the identiﬁcation from a frequency domain point of view, the minimization criterion in (4.2) can asymptotically be written as [Ljung, 1999, (8.71) p. 266] π 2 θ̂ = arg min G0 (e iω ) − G(e iω , θ) Φu (ω)dω θ (4.3) −π where G(e iω , θ) is the model and Φu (ω) is the spectrum of the input signal. The estimation will thus be done in a way to emphasize the model ﬁt in frequency bands where the transfer function and the input spectrum are large enough to have a signiﬁcant impact on the total criterion. The minimization is done with respect to the product of model ﬁt (|G0 − G|2 ) and input spectrum. If the input is white noise (ﬂat spectrum), it is thus more important to obtain a good model ﬁt at frequencies with a large transfer function magnitude. If instead the goal is to estimate the inverse model to be used as described in Section 4.1, the minimization criterion in the time domain can be written θ̂ = arg min θ 2 N 1 1 y(t) u(t) − N G(q, θ) (4.4) t=1 and the frequency domain equivalent to (4.4), when y is noise-free, is θ̂ = arg min Vinv (θ). θ (4.5) 4.3 37 An Illustrative Linear Dynamic Example The loss function is 2 π 1 1 − Φ (ω)dω Vinv (θ) = G0 (e iω ) G(e iω , θ) y −π π = −π π = −π π = 2 1 1 iω 2 − |G0 (e )| Φu (ω)dω iω iω G0 (e ) G(e , θ) 2 G0 (e iω ) 1 − Φu (ω)dω G(e iω , θ) 2 G(e iω , θ) − G0 (e iω ) −π Φu (ω) dω |G(e iω , θ)|2 (4.6) (4.7) using Φy = |G0 (e iω )|2 Φu if no noise is present. The loss function in (4.7) is similar to the weighting for the input error case where H = G so that y(t) = Gu + Ge = G(u + e), that is, the error enters the system at the same place as the input [Åström and Eykhoﬀ, 1971]. Comparing the minimization criterion for the forward estimation in (4.3) to the one for the inverse estimation in (4.6), the weighting is clearly diﬀerent. In the forward case, a relative model error at a frequency where the system ampliﬁcation is small, will aﬀect the criterion much less than a model error at a frequency where the system ampliﬁcation is large. In the inverse estimation case, a relative model error will have the same eﬀect on the criterion for two frequencies with the same input spectral density, and does not depend on the system ampliﬁcation at that frequency. The weighting, and thus the model ﬁt, between the diﬀerent frequencies will be shifted to better reﬂect the importance of a good ﬁt also at frequencies with a small transfer function magniﬁcation. The time domain criterion (4.4) thus leads to the frequency domain description (4.6), and the weighting is automatically done to match the use of the inverse model estimate. Here, only the case when the system and its inverse are both stable and causal will be investigated. See Section 3.2.1 for a brief discussion on the problems involved in system inversion. 4.3 An Illustrative Linear Dynamic Example Let us look at a small example. The goal is to obtain a system inverse to be used in series with the original system in order to retrieve the input, see Figure 4.1a. The input u and the noise-free output y are measured. The system has two resonance frequencies, at ω = 1 rad/s and ω = 10 rad/s. The magnitudes of the two resonance peaks are very diﬀerent, with the ﬁrst one a hundred times larger than the second one. The true system, G0 is described by G0 (s) = 10 s 4 + 1.1s3 + 101.1s 2 + 11s + 100 (4.8) 38 4 Estimation of Inverse Models Magnitude 100 10−2 10−4 10−6 10−1 100 101 Frequency [rad/s] 102 Figure 4.2: The Bode magnitude plot of G0 in (4.8) in the solid line. The stars mark the amplitude of the multisine input (u in (4.9)) components at each frequency. and the Bode magnitude diagram is shown in Figure 4.2. The input consists of three sinusoids around each of the two resonance peaks such that the input power is concentrated in two bands, centered around the resonance frequencies, i.e., u= 6 k=1 ak sin(ωk t + φ k ) (4.9) with ak = 1 for k = 1, 2, 3, ak = 10 for k = 4, 5, 6, ωk = 0.9, 1, 1.1, 9, 10, 11 and φ k ∼ U [−π π]. The input amplitude and the frequency points are illustrated by the stars in Figure 4.2. The sampling time is Ts = 0.02 s and N = 10 000 simulated measurements have been collected. With the goal of using an fir model as a preﬁlter to recover the input u, two models have been estimated, using Method A and Method C in Section 4.1. An fir model depends only on previous input signals, as described on page 13. As the system is linear, the ordering of the two systems does not matter, and the preinverse and postinverse are interchangeable. First, a forward model has been estimated as an output error (oe) model using System Identiﬁcation Toolbox in Matlab [Ljung, 2003], with [nb nf nk] = [1 3 0]. This model has then been inverted resulting in an fir model with 4 terms, according to Method A. The approximative inverse using Method C is an fir model with 4 terms, i.e. [nb nf nk] = [4 0 0], and will have a very diﬀerent weighting. Hence, the two inverses will catch diﬀerent behaviors of the system. The system G0 , (4.8), is a fourth order system whereas the model is third order. Thus, 4.4 Inverse Identiﬁcation of Nonlinear Systems 39 the model cannot perfectly model the system but should be able to capture one resonance peak and the overall behavior of the system. As can be seen in the Bode magnitude plot in Figure 4.3, the Method A model has a much better ﬁt around ω = 1 rad/s and almost perfectly models the resonance peak, but completely misses the second resonance peak at ω = 10 rad/s. The inverse estimate, the Method C model, on the other hand, does not manage to catch either of the resonance peaks in a satisfactory way but catches both of the resonance frequencies. That is, the ampliﬁcation at ω = 1 and 10 rad/s is well captured, but not the resonance peaks around. Estimating the forward model in the standard way will clearly focus on the frequencies where the product of model ﬁt (|G0 − G|2 , connected to the transfer function ampliﬁcation) and input spectrum is large. When this system approximation is then inverted, according to Method A, the errors around ω = 10 rad/s will become prominent. The results in the time and frequency domains are presented in Figures 4.4 and 4.5. In the time domain plot in Figure 4.4, it is clear that the Method C model better reconstructs the input than the Method A model. In Figure 4.5, the periodograms of the reconstructed inputs are shown, zoomed in around the input frequencies. At the lower frequency around ω = 1 rad/s, the Method A model captures the input almost perfectly, but around ω = 10 rad/s, the reverse is true and the Method C model performs better. As shown in this small example, there are clearly occasions when it is advantageous to estimate an approximate inverse directly as opposed to estimating the forward model and then inverting it. 4.4 Inverse Identiﬁcation of Nonlinear Systems It is hard to say anything about the estimation of inverse systems for a general nonlinear system. Depending on the type of nonlinearity and how it enters, the eﬀects will be diﬀerent. So the choice of how to estimate the inverse is closely connected to the system itself. For a linear system, a binary input signal is enough to extract all information in an identiﬁcation experiment. One example of where this can be used is in identiﬁcation of Hammerstein systems, which are block-oriented systems where a static nonlinearity is followed by a linear dynamical system. The linear part of the Hammerstein system can be modeled perfectly by using a binary input signal. In a second experiment, where the input is no longer binary, the model of the linear system can be used to simplify the estimation of the nonlinearity. The estimates of the nonlinearity and the linear dynamics can be inverted and a Wiener system is obtained, according to Method A, assuming a stable, minimum-phase dynamical system and an invertible nonlinearity. This is similar to Example 3.3 on page 23 where an exact inverse could be found (but there the system was assumed known). One way of ﬁnding an inverse to a more general nonlinear system is by using Hirschorn’s method, described in Section 3.3.2. A question is how to estimate this system inverse. In the case where the structure of the nonlinear system is 40 4 Estimation of Inverse Models Magnitude 106 104 102 100 10−1 100 101 Frequency [rad/s] 102 Figure 4.3: The Bode magnitude response of G0−1 (black solid line), the inverted forward model, Method A, (black dashed line) and the inverse model estimate using Method C (gray solid line). The inverted forward model perfectly catches the resonance peak at ω = 1 rad/s, whereas the direct estimation of the inverse does not model either of the resonance peaks in a satisfactory way. The Method C model instead has an accurate modeling of both peak frequencies, that is, it manages to accurately model the ampliﬁcation at ω = 1 and 10 rad/s, but not the resonance peaks. 4.4 41 Inverse Identiﬁcation of Nonlinear Systems 200 100 0 −100 −200 20 21 22 23 25 24 Time [s] 26 27 28 Figure 4.4: The input u (black solid line), and the reconstructed input yu using the inverted forward model (black dashed line) and the inverse model estimate using Method C (gray solid line). The estimation of the inverse cannot perfectly reconstruct the input, but is clearly better than the inverted forward model. 42 4 Estimation of Inverse Models 104 103 10−0.1 100 100.1 100.9 101 Frequency [rad/s] 101.1 107 105 103 Figure 4.5: Periodogram of the input u (black solid line), and the reconstructed input yu using the inverted forward model (black dashed line) and the inverse model estimate using Method C (gray solid line) around ω = 1 rad/s (top) and ω = 10 rad/s (bottom). It is clear also in the frequency domain that the forward model better captures the behavior around ω = 1 rad/s than the inverse estimation, but the reverse is true around ω = 10 rad/s. 4.4 43 Inverse Identiﬁcation of Nonlinear Systems u S y 1 Gm r̄ S† ū Figure 4.6: Estimation of Hirschorn inverse model. By ﬁltering the output y through the inverse dynamics of the desired dynamics Gm , an estimate of the reference signal r̄ can be obtained. known, but where there are unknown parameters that need to be estimated, the identiﬁcation can be done in several ways, just as described in Section 4.1. Method A would correspond to measuring the input u and the output y, and identifying the unknown parameter values in the standard (forward) way. This estimated model could then be used to provide the inverse, since if a model of the forward system is available, a model of the inverse system is as well. Since the exact linearization framework provides us with an inverse once the forward model is known, Method B does not really have an equivalence in this case – once the forward model is known, the exact inverse to match it is also known. In the general case, this forward model could be used to estimate an approximate inverse. Method C would correspond to estimating the inverse S † directly. The order of the inverse and the output are reversed in Method C, so that y is used as input and u as output. In Hirschorn’s method, the inverse takes the reference r as input and the output is the control signal u. So, in order to ﬁnd the inverse of S † , we would need u as output and the reference r as input. But, as the data was collected in open loop with no pre- or postdistorter, the signal r is not available, only u and y. Now, as in Section 3.3.2, assume that the system was actually preceded by a system S † , fed by a ﬁctitious reference signal r̃, and that the overall behavior from r̃ to y is in fact linear with dynamics described by Gm . If this is true, then the signal r̄ would be obtained by ﬁltering y with 1/Gm , and the system S † can be identiﬁed using r̄ as input and u as output, see Figure 4.6. So, this equals ﬁnding the inverse by using (a ﬁltered version of) the output y as the input and u as output as in Method C. A beneﬁt with Hirschorn’s method is that it provides a parameterized inverse, so that the structure of this inverse system is already known. Part II Power Ampliﬁer Predistortion 5 Power Ampliﬁers An electronic ampliﬁer, or power ampliﬁer (pa) is used to increase the power of a signal, so that the output is a magniﬁed replica of the input. There are many diﬀerent constructions of ampliﬁers, and they can be characterized by diﬀerent measures such as gain, eﬃciency and linearity. Ampliﬁers are commonly used in many applications, such as audio applications and telecommunications, both in base stations and hand-held devices. This chapter provides a basis to understand the ampliﬁer related problems described in later chapters. It is by no means a complete description of pas, but should be enough to understand this thesis. It also introduces the concepts of predistortion and linearization as well as the outphasing pa. 5.1 Power Ampliﬁer Fundamentals Today, wireless communication is used everywhere to transfer information. An important part of the technology is the possibility to transmit and receive the information, and the devices used are called transmitter (tx) and receiver (rx). The transmitter converts the information to an electrical signal suitable for the transmission in the given medium (in this case air, but in standard communication this can be a wire, ﬁber-optics, etc.). At the other end of the transmitting medium, a device is needed to receive the message and convert it into the original form – the receiver. This process of sending and propagating an information signal over a medium is called a transmission. It is often desired that the equipment should be able to both send and receive information (a phone for example, where one can speak and listen), that is, a device that contains both a transmitter and a receiver. Such a circuit is called a transceiver. The physical circuit is connected to a chip. By combining the receiver and transmitter into a transceiver, the circuits can 47 48 5 Power Ampliﬁers i dac Digital Baseband (db) 0◦ lo xBB x 90◦ pa Matching Network q dac Figure 5.1: Block diagram of a direct-conversion transmitter. The baseband signal (xBB ) is upconverted to radio frequencies by the modulator and passes through a pa before being sent to the antenna. be used for multiple purposes, reducing the number of components (and thus the cost) as well as the size of the chip, leading to more functionality per area. Such shareable components are antennas, oscillators, ampliﬁers, tuned networks and ﬁlters, frequency synthesizers and power supplies [Frenzel, 2003]. 5.1.1 Basic Transmitter Functionality A standard transmitter includes a digital baseband (db), digital-to-analog converters (dac s), mixers (x) (further explained in Example 5.1), two local oscillators (los) that are 90◦ out of phase, a combiner, a power ampliﬁer and a matching network before the antenna. The signal of interest, xBB , is split into an in-phase channel, I , and a quadrature channel, Q, xBB (t) = I (t) + j Q(t) (5.1) by the db, corresponding to the real (I ) and imaginary (Q) parts of the signal, to generate two independent signals. Complex signals are commonly used in diﬀerent modulation techniques in communications applications, see for example Frenzel [2003]. The I and Q signals are upconverted to the radio frequency (rf, ranging between 3 kHz and 300 GHz) carrier frequency, ωc , and recombined, see Figure 5.1. The upconversion is done by a quadrature modulator, usually implemented by two mixers and two lo signals with a phase diﬀerence of 90◦ . The power of the recombined output signal, x(t) = r(t) cos(ωc t + α(t)) where (5.2) r(t) = I 2 (t) + Q 2 (t) (5.3) and α(t) = arctan(Q(t)/I (t)) (5.4) is often too low for transmission, and it has to pass through a power ampliﬁer before being sent to the antenna. 5.1 49 Power Ampliﬁer Fundamentals Envelope Modulation signal Modulated signal Carrier signal Figure 5.2: Amplitude modulation. The information in the modulation signal is upconverted in the mixer to the carrier frequency (frequency of the carrier signal) and the shape (envelope) of the modulated signal contains the original information in the modulation signal. Example 5.1: Amplitude modulation Modulation is the process of varying the properties of a high-frequency signal, the carrier signal (usually a sine wave) with a modulation signal that contains the information to be transmitted. The modulation can be performed using a mixer, a component that multiplies the two (possibly shifted) inputs. When amplitude modulation (am) is used, the information can be found in the amplitude of the modulation signal. The imaginary line that connects the peaks of the modulated signal is the information signal, and is called the envelope. Other common analog modulation techniques include phase modulation (pm) and frequency modulation (fm). Here, the envelope of the signal is kept constant but the phase shift or the frequency, respectively, of the carrier frequency is varied. These modulation techniques can also be combined into more complex modulation techniques. For the example in Figure 5.2, the modulation (information) signal is a sine wave. The carrier is a sine wave of much higher frequency, and the modulated output is a high frequency signal where the shape of the envelope contains the information in the modulation signal. The amplitude modulation in Example 5.1 is an analog modulation scheme that can be used for continuous signals. If the baseband signal is digital, a digital modulation is needed, which will be introduced in Example 5.2. Example 5.2: Digital modulation One digital modulation scheme is phase-shift keying (psk) that changes, modulates, the phase of the carrier signal. A digital modulation uses a ﬁnite number of distinct signals to represent digital data. In psk, the phase is unique for each signal section, or symbol, that is transmitted. The demodulator, at the receiver end, should interpret the signal and map it back to the original symbol. This 50 5 Power Ampliﬁers q 1 i 01 11 i 00 10 0◦ lo 90◦ 10 q 0 (a) (b) Figure 5.3: (a) Constellation diagram for quadrature phase-shift keying, a digital modulation scheme. The four symbols represent the bits 00, 01, 11 and 10. (b) shows an example where the symbol 10 is to be transmitted. The i part is 1 and the q part is 0. The bits are modulated by a carrier signal, a sinusoidal with a 90◦ phase shift between the i and q parts, and the signals are added. Typically, the zero is coded as −1. The phase of the output is unique and can be mapped back to the i and q parts, as seen in Figure 5.4. requires the receiver to be able to compare the phase of the received signal to a reference signal. Such a system is termed coherent. One type of digital psk modulation is quadrature phase-shift keying (qpsk) which uses four phases, and can encode two data bits per symbol. In a constellation diagram, the qpsk scheme has four points spread out around a circle, as seen in Figure 5.3a. We will here look at an example where the symbol to be transmitted is 10. The iq decomposition is done such that the odd-numbered bit (1) is the i component and the even-numbered bit (0) is the q component, as seen in Figure 5.3b. The bits are modulated by the carrier signal, a sinusoidal with a 90◦ phase shift between the i and q branches, and the signals are added. The resulting signal is unique, as seen in the bottom row of Figure 5.4, and can be mapped back to the i and q components. 5.2 Power Ampliﬁer Characterization The choice of pa is a trade-oﬀ between diﬀerent properties such as output power, eﬃciency and linearity, and will depend on the application. If power eﬃciency is an important property, such as in handheld devices where it will reﬂect directly on the battery time, a lower linearity might be accepted, whereas an audio ampliﬁer, always connected to the power net, might focus more on the linearity and gain than on the eﬃciency. Any number of pas can be cascaded in order to combine the beneﬁts of each step. 5.2 51 Power Ampliﬁer Characterization 1 0 0 1 I 1 0 1 0 Q Signal 11 0 Tsym 00 01 10 Data 2Tsym 3Tsym 4Tsym Time Figure 5.4: The modulated signals in the iq modulation, where the two carrier waves are sinusoidal with a 90◦ phase shift. The odd-numbered bits encode the in-phase (i) component and the even-numbered bits encode the quadrature (q) component. The total signal is shown at the bottom, together with the mapping. The digital data transmitted by this signal is 1 1 0 0 0 1 1 0. Tsym is the symbol duration. 5.2.1 Gain An ampliﬁer is of course supposed to amplify the input signal, and this property is described by the gain. The gain of an ampliﬁer expresses the relationship between the input and the output [Frenzel, 2003], and is usually described by the voltage gain, AV , V (5.5) AV = out , Vin where Vin and Vout are the input and output voltages, respectively. It can also be expressed by the power gain, AP , AP = Pout , Pin where Pin and Pout are the input and output powers, respectively, see Figure 5.5. The gain is usually expressed in decibels (dB), so that the power gain is Pout AP = 10 log10 . (5.6) Pin 5.2.2 Efﬁciency Another important property of a pa is the eﬃciency, which describes the amount of power needed to perform the ampliﬁcation. A part of the input power will be dissipated in the circuit and can be counted as losses. The eﬃciency of a pa will 52 5 Power Ampliﬁers Ampliﬁer Pin Pout pa Input signal Output signal Figure 5.5: Ampliﬁer with input and output. The power gain is AP = Pout Pin . directly aﬀect the battery time for a cell phone for example, and a high eﬃciency is desired. The output eﬃciency, η, of a pa is deﬁned as the ratio between the output power at the fundamental frequency, Pout , and the dc supply power of the last ampliﬁer stage, PDC , [Cripps, 2006] η= Pout , PDC (5.7) and is often denoted drain eﬃciency (de). Another eﬃciency measure is the power added eﬃciency (pae), pae = Pout − Pin , PDC (5.8) where PDC now represents the total power consumption of all ampliﬁer stages constituting the whole pa [Razavi, 1998]. 5.2.3 Linearity By assigning transmissions diﬀerent frequency bands, many transmissions can be done at the same time. For this setup to work, each of these transmissions must send only in the allotted slot, or channel. A radio transmission is allocated a frequency band with a certain bandwidth, ωb , around a center frequency, f c , where power may be transmitted. Any power falling outside the boundaries will cause disturbances in the neighboring channels. Broadening of the spectrum can be caused by, for example, nonlinearities in the pa. So to be practically useful in radio communications, pas need to be linear. This means that the signal should be ampliﬁed in such a way that the output is an exact replica of the input but with a larger amplitude, and not be transferred to other frequencies. This is not possible in practice, and the level of linearity, or rather nonlinearity, is quantiﬁed by measures such as spectral mask, adjacent channel power ratio (acpr) and error vector magnitude (evm). Spectral mask A spectral mask is a nonlinearity measure describing the amount of power that is allowed to be spread to adjacent frequencies. It is usually speciﬁed in decibel to carrier (dBc, the power ratio of a signal to a carrier signal, expressed in decibels) or in power levels given in dBm (power expressed in dB with one milliwatt as reference) in a speciﬁed bandwidth at deﬁned frequency 5.2 Power Ampliﬁer Characterization 53 Table 5.1: Spectral mask limitations for an edge signal Oﬀset [kHz] 100 200 250 400 600 1000 Limit [dBc] 0 -30 -33 -54 -60 -60 Figure 5.6: Spectrum at 1.95 GHz for (a) measured output without dpd, (b) measured output with predistortion (linearization) and (c) the input signal for a wcdma signal. The measured aclr are printed in gray for the original output signal (without predistortion) and in black for the predistorted output. The gray shadows represent the passband in which the integration takes place. oﬀsets [Fritzin, 2011]. See Table 5.1 for an example of the spectral mask limits for an edge signal. Adjacent Channel Power Ratio The acpr is a measure that, like the spectral mask, describes the amount of power spread to neighboring channels. It is deﬁned as the power in a passband away from the main signal divided by the power in a passband within the main signal [Anritsu, 2013]. The power at frequencies that are not in the main signal is the power transmitted in neighboring channels, i.e., the distortion caused by nonlinearities. Another measure is the alternate channel power ratio, which is deﬁned as the ratio between the power in a passband two channels away from the main signal, over the power within the main signal. The bandwidths and limits are connected to the standard used (for example wcdma and lte). For a wcdma signal, the acpr can be calculated by integrating the spectrum over a bandwidth of ωb = 3.84 MHz at ±5 MHz distance from the 54 5 Q Power Ampliﬁers Magnitude error Error vector Measured signal Ideal (reference) signal Phase error I Figure 5.7: Error vector magnitude (evm) and related quantities. center frequency, as f c +l ·5+1.92 acpr = wcdmaspectrum df f c +l · 5−1.92 f c +1.92 . (5.9) wcdmaspectrum df f c −1.92 Here, f c is the center frequency in the main signal and l = ±1 for the adjacent and l = ±2 for the alternate channel power ratio. acpr is also named adjacent power leakage ratio (aclr). An example of the aclr can be seen in Figure 5.6. Error Vector Magnitude The error vector magnitude (evm) is a description of the quality of a signal with both magnitude and phase, such as the iq signals as described in Section 5.1. The error vector is deﬁned as the diﬀerence between the ideal signal and the measured signal [Agilent, 2013], see Figure 5.7. Gain Compression, AM - AM and AM - PM At some point, a change in input amplitude does not result in a corresponding change in output amplitude, as illustrated in Figure 5.8. This phenomenon is called gain compression. This leads to nonlinearities in the output, since diﬀerent amplitudes of the input will be ampliﬁed in diﬀerent ways. Other nonlinearity measures describing the amplitude and phase distortion are the amplitude modulation to amplitude modulation (am-am) and the amplitude modulation to phase modulation (am-pm). The am-am maps the input amplitude to the output amplitude (similar to the gain compression graph in Figure 5.8) and deviations from the straight line will result in output distortion. The am-pm maps the input amplitude to the output phase, where an increasing input amplitude results in an additional output phase shift [Cripps, 2006]. 5.3 55 Classiﬁcation of Power Ampliﬁers Pout Pin Figure 5.8: Gain compression due to saturation in an ampliﬁer transistor. The dashed line represents the ideal operation of the ampliﬁer, while the solid line is the true output of the pa and a consequence of gain compression. 5.3 Classiﬁcation of Power Ampliﬁers There are many diﬀerent types of ampliﬁers, but they can be divided into two basic types; linear and switched ampliﬁers, see for example Frenzel [2003] and Jaeger and Blalock [2008] for a more thorough description of the diﬀerent pa classes and the circuitry to implement them. Classical pas usually assume both the input and the output to be sinusoidal, which limits the eﬃciency. If this assumption is disregarded, higher eﬃciency can be achieved [Razavi, 1998]. Here, the diﬀerent classes are described. 5.3.1 Transistors An important part of power ampliﬁer implementation are the transistors, and we will start with a short overview of transistor functionality. A transistor is a device that uses a small signal to control a much larger signal. The two basic types of transistors are bipolar junction transistors (bjt s) and ﬁeld-eﬀect transistors (fets). The structure of the commonly used fets using semiconducting material has led to the name metal-oxide-semiconductor ﬁeld-eﬀect transistor (mosfet). Depending on how the silicon is doped, the fets can be either of p-type (pmos) or n-type (nmos), and thus have diﬀerent conduction capabilities with respect to the applied voltages at the transistor terminals. Doping is the process of intentionally introducing impurities into an extremely pure semiconductor for the purpose of modulating its electrical properties. Complementary metal-oxidesemiconductor (cmos) is a technology that typically uses complementary and symmetrical pairs of p-type and n-type mosfets for logic functions. The fets have three terminals, labeled gate (G), source (S) and drain (D), and a voltage at the gate controls the current between source and drain, see Figure 5.9. See for example Jaeger and Blalock [2008] for more insights into the workings and construction of transistors. For an nmos transistor, a high voltage at the gate leads to a large current between source and drain, and for a small gate voltage, there is no current. For a pmos transistor the relations are reversed, and a small gate voltage leads to a large current between source and drain, and a large gate 56 5 D Power Ampliﬁers S G G S D Figure 5.9: The symbols of nmos (left) and pmos (right) and the associated ports. The ports are labeled gate (G), source (S) and drain (D). A VDD Input LRFC CDC block B Vout , iout Vin RL C Figure 5.10: Generic Class A/B/C power ampliﬁer. The biasing of the transistor determines the conduction angle of the pa, as illustrated in the ampliﬁcation of a sinewave input (left). The conduction angles are (from top to bottom) 360◦ for the Class A, 180◦ for the Class B and 90◦ for the Class C here. voltage opens the circuit and no current ﬂows. Common uses for transistors are as ampliﬁers and switches, depending on the circuitry surrounding them. 5.3.2 Linear Ampliﬁers Linear ampliﬁers provide an ampliﬁed replica of the input. The drawback is that linear ampliﬁers often require a high power level and provide a rather low eﬃciency, as they operate far from their maximum output power where the linearity is limited. Class A Ampliﬁers A Class A ampliﬁer operates linearly over the whole input and output range. It is said to conduct for 360◦ of an input sine wave, that is, it will amplify for the whole of the input cycle, see Figure 5.10. Since the device is always conducting, a lot of power will be dissipated and the maximum achievable output eﬃciency is low, only 50%. 5.3 Classiﬁcation of Power Ampliﬁers 57 Class B Ampliﬁers In a Class B ampliﬁer, the device is biased so that it only conducts for half of the input cycle, i.e., it has a conduction angle of 180◦ , see Figure 5.10. In this region the ampliﬁer is linear, and at the rest of the input it is turned oﬀ, and the eﬃciency reaches η = π/4 ≈ 78.5%, with η deﬁned in (5.7). Class B ampliﬁers are often connected in a push-pull circuit, so that two ampliﬁers are connected, each of them conducting for half of the cycle, and together they conduct for the whole 360◦ . The eﬃciency is still the same, and in theory this will be a completely linear ampliﬁer. In practice, however, if the biasing of the two ampliﬁers is not perfect, this will cause cross-over distortion at the time of switching between the two ampliﬁers [Jaeger and Blalock, 2008]. Class AB Ampliﬁers The Class AB ampliﬁer uses the same idea as the Class B conﬁguration with two ampliﬁers, but the ampliﬁers are slightly overlapping such that the cross-over distortion is minimized. Each ampliﬁer thus has a larger conduction angle than the 180◦ of a Class B ampliﬁer, but less than the full 360◦ of a Class A ampliﬁer. This reduction of cross-over distortion is at the expense of eﬃciency. Class C Ampliﬁers Class C ampliﬁers have a conduction angle smaller than 180◦ , typically between 90◦ and 150◦ , see Figure 5.10. This causes a very distorted output consisting of short pulses, and the ampliﬁer usually has some form of resonant circuit connected to recover the original sine wave. 5.3.3 Switched Ampliﬁers The low eﬃciency of linear ampliﬁers is caused by the high power dissipation due to constant conduction. Switched ampliﬁers consist of transistors that are either on (conducting) or oﬀ (nonconducting). In the oﬀ state (cutoﬀ state), no current ﬂows so there is (almost) no dissipation. When the transistor is conducting, the resistance across it is very low, and so is the power dissipation. The output of a switched ampliﬁer is a square wave, which is passed through a ﬁlter to obtain a sinusoidal signal. Class D Ampliﬁers A Class D ampliﬁer consists of two transistors that alternately are on and oﬀ. The output is a pulse-width modulated (pwm) signal, which can be ﬁltered to obtain the fundamental sine wave, see Figure 5.11. With ideal switches and ideal series resonant network (C1 and L1 ) stopping all frequencies but the fundamental tone, the theoretical maximum eﬃciency is 100%. 58 5 VDD Power Ampliﬁers VDD Vin VDS C1 0 L1 Vin Vout , iout VDS RL Vout Figure 5.11: Class D power ampliﬁer. Class E Ampliﬁers In a Class E ampliﬁer, only one transistor is used (compared to the two for Class D). By choosing a suitable load matching network, the drain current and voltage can be shaped to not overlap each other, making the theoretical eﬃciency 100%. 5.3.4 Other Classes There exist many other classes including Class F (a variation of the Class E ampliﬁer) and Class S (a variation of switching ampliﬁer using pulse-width modulation), see for example Frenzel [2003]. 5.4 Outphasing Concept An outphasing ampliﬁer is based on the idea that a nonconstant envelope signal, with amplitude and phase information, can be decomposed into two constant envelope signals with phase information only. The two signals can then be ampliﬁed separately by two nonlinear and highly eﬃcient ampliﬁers and recombined, as presented in Cox [1974] and Chireix [1935]. The output signal will be amplitude and phase modulated, just like the input signal. Another name for the outphasing concept is linear ampliﬁcation with nonlinear components (linc). The outphasing concept is illustrated in Figure 5.12. Here, a nonconstant envelope-modulated signal s(t) = r(t)e jα(t) = rmax cos(ϕ(t))e jα(t) , 0 ≤ r(t) ≤ rmax (5.10) where rmax is a real-valued constant, and α and ϕ are angles, is used to create two constant-envelope signals, s1 (t) and s2 (t). This is done in the signal component 5.4 59 Outphasing Concept Q 2s(t) s1 (t) e(t) ϕ −e(t) s(t) α s2 (t) I rmax Figure 5.12: Outphasing concept and signal decomposition. s1 (t) = s(t) + e(t) A1 , g1 y1 (t) s(t) + scs s2 (t) = s(t) − e(t) A2 , g2 y(t) y2 (t) g1 = g2 = g0 Figure 5.13: Illustration of ideal power combining (the plus sign) of the two constant-envelope signals. The signals are ampliﬁed separately by two nonlinear ampliﬁers, A1 and A2 , and recombined to an ampliﬁed replica of the input s(t). separator (scs) in Figure 5.13 as s1 (t) = s(t) + e(t) = rmax e jα(t) e jϕ(t) s2 (t) = s(t) − e(t) = rmax e jα(t) e −jϕ(t) 2 rmax − 1. e(t) = j s(t) 2 r (t) (5.11) The outphasing signals s1 (t) and s2 (t) contain the original signal, s(t), and a quadrature signal, e(t), and are suitable for ampliﬁcation by switched ampliﬁers like Class D/E. By separately amplifying the two constant-envelope signals and combining the outputs of the two individual ampliﬁers as in Figure 5.13, the output signal is an ampliﬁed replica of the input signal. In theory, the two quadrature signals will cancel each other perfectly in the combiner, but in practice, implementation imperfections and asymmetries will cause distortion. Letting g1 and g2 denote two positive real-valued gain factors, in each branch s1 (t) and s2 (t), and δ denote a phase mismatch in the path for s1 (t), 60 5 Power Ampliﬁers Figure 5.14: The bandwidth of the quadrature signal e(t), and thus the outphasing signals s1 (t) = s(t) + e(t) and s2 (t) = s(t) − e(t), is much larger than that of the original signal s(t). Any remainders of the quadrature signal caused by pa imperfections will thus lead to degraded aclr and reduced margins to the spectral mask. From Fritzin [2011]. it is clear from y(t) = g1 e jδ s1 (t) + g2 s2 (t) = [g1 e jδ + g2 ]s(t) + [g1 e jδ − g2 ]e(t), (5.12) that besides the ampliﬁed signal, a part of the quadrature signal remains. As the bandwidth of the quadrature signal, e(t), is larger than the original signal, s(t), see Figure 5.14, this would lead to a degraded aclr and reduced margins to the spectral mask [Birafane and Kouki, 2005, Birafane et al., 2010, Romanò et al., 2006]. The phase and gain mismatches between s1 (t) and s2 (t) must be minimized in order not to allow a residual quadrature component to distort the spectrum or limit the dynamic range (dr), max(|y(t)|) |g1 + g2 | = 20 log10 , (5.13) cDR = 20 log10 min(|y(t)|) |g1 − g2 | of the pa [Birafane and Kouki, 2005]. The dr deﬁnes the ratio of the maximum and minimum output amplitudes the pa can achieve. However, all phases and amplitudes within the dr can be reached by changing the phases of the outphasing signals s1 (t) and s2 (t). Since an outphasing ampliﬁer only uses two states (on or oﬀ), it will not experience problems like the conventional pas such as gain compression (see Section 5.2.3), where the peak amplitudes are clipped. Instead, the smallest amplitudes will not be properly ampliﬁed in outphasing pas, since any mismatch of the ampliﬁer gains will make it impossible for s1 (t) and s2 (t) to cancel each other, compare Figures 5.12 and 5.15. Thus, the dr in an outphasing pa limits the spectral performance when amplifying modulated signals. 5.5 61 Linearization of Power Ampliﬁers Q Q s1 + s2 I s1 + s2 I Figure 5.15: The outphasing concept when the gain factors g1 and g2 are not identical. In the left ﬁgure, the outphasing signals are parallel and the resulting output is the maximal one. In the right ﬁgure, the nonidentical gain factors cannot cancel each other, and some remains are left. The dynamic range (the ratio between the maximal and minimal amplitudes, see (5.13)) of the power ampliﬁer will determine the limit of small amplitude clipping. As the output of a Class D stage can be considered as an ideal voltage source whose output voltage is independent of the load [Yao and Long, 2006], i.e., the output is connected to either VDD or GND, the constant gain approximations g1 and g2 are appropriate and make Class D ampliﬁers suitable for nonisolating combiners like transformers [Xu et al., 2010]. The implementation of the combiner (the plus sign in Figure 5.13) can be done in a multitude of ways, see for example Fritzin [2011] and the references therein. 5.5 Linearization of Power Ampliﬁers The increased use of nonlinear ampliﬁers in an attempt to improve eﬃciency also requires new linearization methods. As described in Chapter 3, there are diﬀerent approaches to do linearization. Since it is desirable to work with the original signal, and not with the ampliﬁed output of the pa, a preﬁlter is desired, also called a predistorter [Kenington, 2000]. Originally, these predistorters consisted of small analog circuits, but now they are often implemented in a look-up table (lut) or a digital signal processor (dsp). Such an implementation is called a digital predistorter (dpd). The idea behind predistortion is presented in Figure 5.16. The predistortion can be divided into two parts, the construction of the predistorter functions and the implementation of the obtained dpd. The implementation of predistortion methods entails further considerations, and as concluded in Guan and Zhu [2010], “diﬀerent methodologies or implementation structures will lead to very diﬀerent results in terms of complexity and cost from the viewpoint of hardware implementation”. An implementation using a look-up table will grow quickly with the resolution of the dpd, and thus needs a large chip area, but avoids the necessity of calculations needed in a polynomial implementation (leading to a larger power consumption). The implementation issues have not been considered in this thesis. 62 5 Predistorter System Power Ampliﬁers Linear system Figure 5.16: The main idea behind predistortion is to compensate for future nonlinearities and dynamics so that the overall system is linear. 5.5.1 Volterra series The theory of p-th order Volterra inverses, introduced in Section 3.2.3, allows for the simpler postinverse (see Section 4.1) to be calculated and then used as the desired preinverse. This is used in the predistortion, or linearization, of for example rf power ampliﬁers. See also Section 3.2.2 for a discussion on preinverse versus postinverse. Since Volterra series consist of an inﬁnite sum of integrals, the use of general Volterra theory is rather limited. To reduce the complexity a pruned, or truncated, version of the Volterra series is often used, where the memory length and/or the order of nonlinearity is limited. This heavily reduces the complexity of the sum, but the computational growth is still exponential/polynomial in memory length/order of nonlinearity, limiting the practical use of Volterra series. Using pruned Volterra series as a means for modeling and predistortion of high-power ampliﬁers is presented in Tummla et al. [1997] and is shown to work for simulated data with memory length of 1 and nonlinearity order of 7. In Zhu et al. [2008], pruning techniques have been applied to drastically reduce the number of terms in the (discrete time) Volterra series and the method was applied to experimental data. Here, a memory length of 2 and an order of nonlinearity of 11 was used. Volterra based predistorters have also been implemented in ﬁeld programmable gate array (fpga), shown in Guan and Zhu [2010]. An fpga is a circuit that can be conﬁgured by the user and are used to implement complex digital computations. 5.5.2 Block-oriented Models Since general nonlinear systems are very diﬃcult to model, a common assumption is that the dynamics are linear, and that the nonlinearity is static, which gives a block-oriented model. This will be the case when there is, for example, a nonlinear actuator (due to saturation) in a control application. A Hammerstein system consists of a static nonlinear system followed by a linear dynamic system and in a Wiener system, the static nonlinearity is at the output of the linear dynamics, see also Example 3.3. One way to broaden the use of the Hammerstein system is to use a more general parallel Hammerstein system, where multiple Hammerstein systems are branched. This structure is often 5.5 Linearization of Power Ampliﬁers 63 used in modeling of power ampliﬁers, where a basic assumption is that the main part of the signal is ampliﬁed in a nonlinear way through the pa, and distortions are added to the output. The number of branches in the parallel Hammerstein structure determines the complexity of the model. In Gilabert et al. [2006], a Wiener model of the pa has been used in combination with a Hammerstein structure predistorter, with memoryless nonlinearities followed by linear blocks using ﬁnite impulse response (fir) and inﬁnite impulse response (iir) ﬁlters. The implementation of a Hammerstein predistorter in fpga technique is discussed in Xu et al. [2009] using a wcdma input signal. 5.5.3 Outphasing Power Ampliﬁers In outphasing pas, there is no linearity between the individual outphasing signals, and any gain or phase mismatch between the two signal paths will cause spectral distortion, see for example Birafane and Kouki [2005] and Romanò et al. [2006]. Typical requirements are approximately 0.1−0.5 dB in gain matching and 0.2 ◦ − 0.4 ◦ in phase matching, which is very hard to achieve [Zhang et al., 2001]. The gain mismatch could be eliminated by adjusting the voltage supplies in the output stage [Moloudi et al., 2008], but this would require an extra, adjustable voltage source on the chip, which is undesirable. For the outphasing ampliﬁer, all amplitudes (within the dynamic range) and phases can be achieved by tuning the outphasing signals s1 (t) and s2 (t), see Figures 5.12 and 5.13. This can be used in the predistortion, so that the two signals are adjusted in a way to compensate for gain errors and possibly other unwanted eﬀects in the pa. Earlier predistortion methods for outphasing pas compensate for the gain and phase mismatches in the signal branches. In Myoung et al. [2008], a mismatch detection algorithm has been evaluated using four test signals. These two-tone signals are used to calculate the amplitude and phase mismatches of the ampliﬁer using a closed-form expression, later used for predistortion. Chen et al. [2011] presents a signal component separator (scs) implementation with a built-in compensation for branch mismatches in phase and amplitude. The scs performs the decomposition of the original signal s(t) into the outphasing signals s1 (t) and s2 (t), (5.11). By taking gain and phase mismatches into account, the scs has a built-in predistorter. Helaoui et al. [2008] discuss the impact of the combiner on the outphasing pa performance. The choice of combiner is a trade-oﬀ between linearity and power consumption. Nonlinearities can be introduced by a nonisolated combiner such that the output distortion depends on the input power. These nonlinearities were successfully reduced by the use of a predistorter. The solutions in Myoung et al. [2008] and Chen et al. [2011] consider the gain mismatch between the two branches and compute the ideal phase compensation when the outputs are approximated as two signals with constant amplitudes. This is possible when there is no interaction between the ampliﬁer stages. In this thesis, the outputs are still considered as two constant amplitude signals generating amplitude and phase distortion. Furthermore, an amplitude dependent phase distortion, occurring due to the interaction and signal combining of 64 5 Power Ampliﬁers the ampliﬁers’ outputs, is also considered. Parts of the results in Chapters 6-8 can also be found in Fritzin et al. [2011a] and Jung et al. [2013]. The nonconvex algorithm, presented in Fritzin et al. [2011a], has in Landin et al. [2012] been developed to include a method for ﬁnding good initial values to the nonlinear optimization. However, the basic problem of nonconvexity has not been solved there and local minima still risk posing problems in the optimization. In Jung et al. [2013], the nonconvex formulation has been reformulated into a convex method. In this method, the pa model is estimated in a least-squares setting and an analytical calculation of the predistorter is used. Furthermore, a theoretical characterization of an outphasing pa is presented and form a basis for an ideal dpd. This characterization has also been used to obtain an estimate thereof. 6 Modeling Outphasing Power Ampliﬁers In this chapter, one way of modeling of the outphasing power ampliﬁer using knowledge of the physical structure of outphasing ampliﬁers is presented. It consists of a new decomposition of the outphasing signals making use of the knowledge of the uneven ampliﬁcation in the two branches, as well as a way to incorporate the possible nonlinearities in the branches. Despite the fact that the pa is analog and the baseband model is in discrete time, the notation t is used to indicate the dependency on time. Based on the context, t may thus be a continuous or discrete quantity and denote the time or the time indexation. For notational convenience, the explicit dependency on time will be omitted in parts of this chapter and the following one. 6.1 An Alternative Outphasing Decomposition As mentioned in Chapter 5, the pa output signal y(t) is a distorted version of the input signal. The nonlinearities are due to (i) the nonidentical gain factors g1 and g2 , and (ii) nonlinear distortion in the ampliﬁer branches. First, a novel decomposition will be described, accounting for the nonidentical gain factors g1 and g2 , followed by a description of how these can be used in the modeling of the outphasing power ampliﬁer. Since it is desired that the predistorter should invert all eﬀects of the pa except for the gain, the signals can be assumed to be normalized such that (6.1) max |s(t)| = max |y(t)| = 1. t t As described in Figure 5.12, the amplitude information of the original input signal s(t) can be found in the angle between s1 (t) and s2 (t). Let Δψ (s1 , s2 ) = arg(s1 ) − arg(s2 ) 65 (6.2) 66 6 Modeling Outphasing Power Ampliﬁers Q 2s(t) s1 (t) 2s˜1 (t) s˜2 (t) b1 ξ1 s(t) b2 2s˜2 (t) s2 (t) ξ2 b2 s(t) s˜1 (t) I (a) b1 (b) Figure 6.1: (a) Decomposition of the input signal s(t) into s1 (t) and s2 (t) when g1 = g2 = g0 = 0.5 and into s˜1 (t) and s˜2 (t) when decomposed as in (6.3) with nonidentical gain factors g1 and g2 . (b) Trigonometric view of the decomposition of s(t) using nonidentical gain factors. Note that |s˜k | = gk , k = 1, 2. denote the phase diﬀerence of the outphasing signals s1 (t) and s2 (t). Since the amplitude of the nondecomposed signal in the outphasing system is determined by Δψ (s1 , s2 ), this diﬀerence can be used instead of the actual amplitude in many cases. For notational convenience, Δψ will be used instead of Δψ (s1 , s2 ), unless speciﬁed otherwise. Here, all phases are assumed unwrapped. To describe the distortions caused by the imperfect gain factors, consider again the decomposition of s(t) into s1 (t) and s2 (t) in (5.11). This is only valid when g1 = g2 but we can use an alternative decomposition of s(t) into s˜1 (t) and s˜2 (t) such that s˜1 (t) + s˜2 (t) = s(t), |s˜k | = gk , k = 1, 2, (6.3a) and arg(s˜1 ) ≥ arg(s˜2 ). (6.3b) (6.3c) Assuming knowledge of g1 and g2 = 1 − g1 and given s(t), the signals s˜1 (t) and s˜2 (t) can be computed from (6.3). Let b1 = arg(s˜1 ) − arg(s) and b2 = arg(s) − arg(s˜2 ) denote the angles between the decomposed signals and s(t) as shown in Figure 6.1a. Figure 6.1b shows that the decomposition can be viewed as a trigonometric problem and application of the law of cosines gives and g22 = g12 + |s|2 − 2g1 |s| cos(b1 ) (6.4) g12 = g22 + |s|2 − 2g2 |s| cos(b2 ). (6.5) 6.2 67 Nonconvex PA Model Estimator The angles b1 and b2 that deﬁne s˜1 (t) and s˜2 (t) can be computed from these expressions and can be viewed as functions of Δψ since |s| = rmax cos(Δψ /2). This means that the angles Δ 1 Δ 2 ψ (6.6) Δ 1 Δ − b2 2 ψ (6.7) ξ1 (Δψ ) = arg(s˜1 ) − arg(s1 ) = b1 − and ξ2 (Δψ ) = arg(s˜2 ) − arg(s2 ) = can also be viewed as functions of Δψ . When the goal is to model the phase distortions in the two branches, this alternative way of deﬁning the decomposition reﬂects the physical behavior better than the standard outphasing decomposition in (5.11). The output y(t) can be decomposed in the same way to y1 (t) and y2 (t), taking the gain factors g1 and g2 into account. 6.2 Nonconvex PA Model Estimator A ﬁrst step on the way to model the outphasing pa is to observe that although the two branches are identical in theory, once implemented in hardware this will not be the case. Since the signals s1 (t) and s2 (t) are ampliﬁed by two diﬀerent ampliﬁers, there might be a small ampliﬁcation diﬀerence resulting in a gain oﬀset between these signals, as well as a time delay stemming from the fact that s1 (t) and s2 (t) take diﬀerent paths to the power combiner. With this insight, a ﬁrst model structure with a gain mismatch between g1 and g2 and a phase shift δ in one branch is proposed. This leads to a model structure described by y(t) = g1 e jδ s1 (t) + g2 s2 (t), (6.8) where g1 , g2 and δ are real-valued constants. When adding more complex behavior to the model structure, the structure of the physical pa must still be kept in mind. The separation of the two branches is still valid, but each branch can be aﬀected by other factors than the gain difference and possible phase shift. As the amplitudes of the outphasing signals are ﬁxed, a phase dependent distortion in each branch is proposed. To model an amplitude dependent phase shift while keeping in mind the constant amplitude of the signals s1 (t) and s2 (t), a model structure with an exponential function can be used. An amplitude-dependent phase distortion in yk (t), k = 1, 2 (the two ampliﬁer branches) can be written as yk (t) = gk e j f k (Δψ ) sk (t), y(t) = y1 (t) + y2 (t), k = 1, 2, (6.9a) (6.9b) as in Figure 6.2. Here, f 1 and f 2 are two real-valued functions describing the phase distortion (6.10) arg(yk ) − arg(sk ) = f k (Δψ ), k = 1, 2, 68 6 s1 (t) Modeling Outphasing Power Ampliﬁers y1 (t) g1 f1 s(t) + scs s2 (t) g2 f2 y(t) y2 (t) Figure 6.2: A schematic picture of the ampliﬁer branches setup. Note that the functions f k , k = 1, 2, are not functions of the input to the block only but are used to show the general functionality of the pa with the separation of the two branches. in each signal path. Furthermore, g1 and g2 are the gain factors in each ampliﬁer branch. Hence, an ideal pa would have f 1 = f 2 = 0 and g1 = g2 = g0 and any deviations from these values will cause nonlinearities in the output signal and spectral distortion as previously concluded. The functions f 1 and f 2 describing the phase distortion in the separate branches can be described by arbitrary basis functions. Here, polynomials fˆk = p(ηk , Δψ ) = n ηk,i Δiψ , k = 1, 2, (6.11) i=0 where ηk = ηk,0 ηk,1 . . . ηk,n T , have been used as parameterized versions of the functions f k , motivated by the Stone-Weierstrass theorem, see Rudin [1976, Theorem 7.26]. The model parameters in the given model structure are estimated by minimizing a quadratic cost function [Ljung, 1999] as in θ̂ = arg min V (θ), (6.12) θ V (θ) = N y(t) − ŷ(t, θ)2 (6.13) t=1 with ŷ(t, θ) = g1 e j p(η1 ,Δψ (s1 ,s2 )) s1 (t) + g2 e j p(η2 ,Δψ (s1 ,s2 )) s2 (t) (6.14) where θ = [g1 g2 η1T η2T ]T ∈ R2n+4 , y(t) is the measured output data and ŷ(t, θ) is the modeled output. The model (6.14) can be compared to the structure (6.9), where y(t) = g1 e j f 1 (Δψ ) s1 (t) + g2 e j f 2 (Δψ ) s2 (t). This structure leads to a nonlinear and nonconvex optimization problem, so the minimization algorithm might ﬁnd a local optimum instead of a global. In order to obtain a good minimum in a nonconvex optimization problem, it is essential to have good initial 6.3 Least-squares PA Model Estimator 69 values, and one way to obtain these is presented in Landin et al. [2012]. For further discussions on convexity and nonconvexity, see Section 6.5. Here, a model of the pa was estimated by minimizing a quadratic cost function measuring the diﬀerence between the measured and modeled output signal. This estimation problem involves solving a nonconvex optimization problem. However, using the knowledge of the structure of the outphasing ampliﬁer, there is an alternative way which essentially only involves solving standard leastsquares problems, presented in the next section. 6.3 Least-squares PA Model Estimator The output distortions originate both from imperfect gain factors and nonlinearities in the ampliﬁers. Once the gain factor impact has been accounted for, the ampliﬁer nonlinearities can be modeled. This means that the modeling optimization problem can also be rewritten as a separable least squares (sls) problem, also presented in Jung et al. [2013]. A separable least squares problem is when one set of parameter enters the model linearly and one set nonlinearly. Given the nonlinear parameters, the linear part can be solved for eﬃciently, leaving a nonlinear problem of a lower dimension [Ljung, 1999]. See also Section 2.5 for a short introduction to sls problems. Often, the minimization is done ﬁrst for the linear part and then the nonlinear parameters are solved for and this nonlinear minimization problem now has a reduced dimension. Here, the idea is to use knowledge of the gain factors to make a nonlinear transformation of the data using the decomposition (6.3). Once this decomposition is done, the minimization can be rewritten as a least-squares (ls) problem in the phase distortion in the two branches. This is not the usual sls method since it involves a nonlinear transformation of the data, but the basic idea of separating out the nonlinear parameters to obtain a ls problem still applies. We will here explore two ways of estimating the gain factors g1 and g2 . One is based on the dynamic range of the pa and the other is based on a parameter gridding of possible values of g1 and g2 . Assuming the gain factors to be known, we know what the phases of the outputs from the two outphasing branches must be in order for the two signals to sum up to the measured output y(t). It is now possible to decompose the output y(t) into y1 (t) and y2 (t), using the decomposition in Section 6.1. What is left to determine is the phase distortion in the branches, described by the functions f k . Since the gain factor inﬂuence is handled by the alternative decomposition of y(t), the phase distortion is now described by the diﬀerence between the phase of the input sk (t) and the output yk (t), k = 1, 2 and this can be formulated as a least-squares problem. Consider ﬁrst the two gain factors g1 and g2 = 1 − g1 , where the relation between them comes from the normalization (6.1). Let g1 = g0 ± Δ g , g2 = g0 ∓ Δ g , (6.15) 70 6 Modeling Outphasing Power Ampliﬁers where Δg ≥ 0 represents the gain imbalance between the ampliﬁer stages and g0 = 0.5. Inserting (6.15) into (5.13) gives g (6.16) cDR = 20 log10 0 . Δg Hence, the imbalance term Δg can be computed as Δg = g0 · 10−cDR /20 , (6.17) making it possible to ﬁnd approximations of g1 and g2 from the dynamic range of the output signal. The value of cDR can be estimated from measurements as the ratio between the maximum and minimum output amplitudes. The estimate is noise sensitive, but this can be handled by averaging multiple realizations. These approximations are valid for input signals with large peak to minimum power ratios, like wcdma and lte, where the pa generates an output signal including its peak and minimum output amplitudes, i.e., its full dynamic range. If this is not fulﬁlled or the noise inﬂuence is too large, an alternative approach is to evaluate a range of values of g1 and g2 = 1 − g1 and then solve the pa modeling problem for each pair of gain factors, as in the usual sls approach. Once the gain factors have been determined, s(t) can be decomposed into s˜1 (t) and s˜2 (t), and y(t) into y1 (t) and y2 (t) using (6.3) to (6.5). Furthermore, the standard outphasing decomposition of s(t) into s1 (t) and s2 (t) as in (5.11) will be used in the sequel. Since the gain factor mismatch has been accounted for, it is now possible to determine the impact of the nonlinearities on the two branches. The phase distortion in each signal path caused by the ampliﬁers can thus be modeled from measurements of s(t) and y(t). Here, polynomials p(ηk , Δψ ) = n ηk,i Δiψ , k = 1, 2, i=0 have been used as parameterized versions of the functions f k , as in (6.11). Estimates η̂k,i of the model parameters ηk,i have been computed by minimizing a quadratic cost function, i.e., η̂k = arg min Vk (ηk ), k = 1, 2, (6.18) ηk where Vk (ηk ) = N 2 arg (yk (t)) − arg (s˜k (t)) − p ηk , Δψ (s1 (t), s2 (t)) , (6.19) t=1 and ηk = ηk,0 ηk,1 . . . ηk,n T . The cost function (6.19) can be motivated by the fact that the true functions f k satisfy (6.10) when the ampliﬁer is described by (6.9). Minimization of V1 and V2 6.4 71 PA Model Validation are standard least-squares problems, which guarantees that the global minimum will be found [Ljung, 1999]. Once the ls problem is solved for each setup of g1 and g2 , the problem of ﬁnding the best setup is now reduced to a one dimensional (possibly nonconvex) optimization problem over g1 (g2 = 1 − g1 ), which is much easier to solve than the original, multidimensional problem. A problem this small can be solved at a small computational cost. The parameter estimates η̂k deﬁne function estimates fˆk (z) = p(η̂k , z), k = 1, 2, (6.20) that, together with the gain factor estimates ĝ1 and ĝ2 describe the power ampliﬁer behavior. The diﬀerent steps are also described in Part A – Estimation of pa model in Algorithm 1, page 91. The alternative decomposition described in Section 6.1 depends on the gain factors g1 and g2 via a nonlinear relation, but with these given, the problem is reduced to a ls-problem in the phase as in (6.19). If the gain factor estimation is done using the dr as in (6.15) and (6.17), will result in two ls-problems g this −g to solve, and gridding of g1 will result in maxp min + 1 ls problems. The values M gmin and gmax bounds the values of g1 and g2 that one wants to evaluate and p M is the precision, so that g1 ∈ [gmin , gmin + p M , . . . , gmax ] and g2 = 1 − g1 . Compare to Algorithm 1, page 91, for notation. This is not the standard sls method, since a nonlinear transformation of the data is done before solving the ls problem, but the separation of the linear and nonlinear parameters applies. This separation reduces the optimization to a number of ls problems and a nonlinear optimization in only one dimension, g1 (g2 = 1 − g1 due to the normalization (6.1)). This is clearly a reduction from the nonlinear optimization in 2n + 4 dimensions of the original problem. 6.4 PA Model Validation As an evaluation of the diﬀerent approaches presented above, the models have been compared. The Figures 6.3-6.6 present the amplitude and phase of the measured output and the model output. The amplitude error |y − ŷ| and the phase error arg(y) − arg(ŷ) are also included. The ﬁrst simple model in (6.8), using only the gain factors g1 and g2 and a phase shift δ, is presented in Figure 6.3. The more complex model structure (6.14) is presented in Figures 6.4, 6.5 and 6.6, using the diﬀerent modeling methods. The model obtained by the nonconvex approach as in (6.12)-(6.14) is presented in Figure 6.4. The ls method using (6.18)-(6.19) and the dynamic range to obtain the gain factors is presented in Figure 6.5. In Figure 6.6, the ls method using gridding of g1 over a range of values and then determining the best ﬁt is presented. The more complex models perform very well, and rather similarly. This is easier to see in Figure 6.7, where the errors for the diﬀerent modeling methods are plotted together. Though the models all perform well, there are still errors. These errors are largest where the input amplitude is small, such as around time 72 6 Modeling Outphasing Power Ampliﬁers Table 6.1: pa Model Validation Method Delay only, model structure (6.8) Nonconvex ls, grid ls, dr g1 0.4911 0.4986 0.50 0.4994 g2 0.5089 0.5014 0.50 0.5006 |y − ŷ|22 , (6.13) 62.99 0.9985 1.119 0.9781 152 μs and 156.5 μs. The result of the dr ls model is also presented in an iq plot in Figure 6.8 where the signals are plotted in the complex plane. Also in this plot, the model shows a very good behavior, with a slightly worse performance for small amplitudes. The gain factor estimates are presented in Table 6.1 together with the cost function (6.13) for the diﬀerent methods. As seen in the rightmost column where |y − ŷ|22 , (6.13), is presented, the added model complexity with nonlinearities makes a large improvement in the model ﬁt. The ls method using dr and the nonconvex method achieve rather similar results with the gridding ls method slightly behind. The results of the nonconvex method depends on the number of iterations used in the optimization. Except for the ﬁrst simple model, the other methods perform very similarly with a very good ﬁt to validation data. This clearly shows that the nonlinear extension to the model has a signiﬁcant impact on the model properties. This also means that the choice of method comes down to other considerations than the ﬁt. The lack of guarantees of convergence to a global minimum of nonconvex optimization methods is a reason to avoid the method described in Section 6.2. If the ls method is chosen, this also entails the choice of gridding or using the dynamic range. Gridding is more robust against noise, since the dr estimation is done using only two measurements (the one with minimal and the one with maximal amplitude), so noise at either of these data points will have a large impact. A drawback with gridding is the risk of missing the “true” value, if the precision p M (diﬀerence in g1 and g2 ) is chosen too large. A decreasing p M , on the other hand, will increase the number of ls problems that need to be solved. Beneﬁts and drawbacks for the dynamic range method are the opposite. 6.4 73 PA Model Validation Amplitude 1 0.5 0 Phase [rad] 4 2 0 −2 −4 146 148 150 152 Time [µs] 154 156 Figure 6.3: Model validation of the model produced using the first structure (6.8), with gain factors g1 and g2 and a phase shift δ only. The upper plot shows the amplitude of the measured signal (solid pink), the model output (dashed blue) and the error (black). The lower plot shows the phase. 74 6 Modeling Outphasing Power Amplifiers Amplitude 1 0.5 0 Phase [rad] 4 2 0 −2 −4 146 148 150 152 Time [µs] 154 156 Figure 6.4: Model validation of the model produced using the original, nonconvex, optimization in (6.12)-(6.14). The upper plot shows the amplitude of the measured signal (solid pink), the model output (dashed blue) and the error (black). The lower plot shows the phase. 6.4 75 PA Model Validation Amplitude 1 0.5 0 Phase [rad] 4 2 0 −2 −4 146 148 150 152 Time [µs] 154 156 Figure 6.5: Model validation of the model produced using the convex method in (6.18)-(6.19) and the dynamic range has been used to determine the gain factors as in (6.17) and (6.15). The upper plot shows the amplitude of the measured signal (solid pink), the model output (dashed blue) and the error (black line). The lower plot shows the phase. 76 6 Modeling Outphasing Power Amplifiers Amplitude 1 0.5 0 Phase [rad] 4 2 0 −2 −4 146 148 150 152 Time [µs] 154 156 Figure 6.6: Model validation of the model produced using the convex method in (6.18)-(6.19) and g1 has been gridded in [gmin , gmax ] = [0.4, 0.6] with precision pM = 0.005. The upper plot shows the amplitude of the measured signal (solid pink), the model output (dashed blue) and the error (black line). The lower plot shows the phase. 6.4 77 PA Model Validation · 10−2 Amplitude 6 4 2 0 Phase [rad] 0.5 0 −0.5 −1 −1.5 151 152 153 154 Time [µs] 155 156 157 Figure 6.7: A summary of the model errors of the different models. The upper plot shows the amplitude error |y − ŷ| and the lower plot shows the phase errors arg(y) − arg(ŷ). The simple model (6.8) is plotted in black, the ls methods using dr in solid pink and gridding in dashed blue. The model obtained by the nonconvex method is plotted in a green dashed line. The three models describing a nonlinear behavior perform very well and in a very similar way, as seen in the figure where the lines are almost on top of each other. 78 6 Modeling Outphasing Power Amplifiers Imaginary part of signal (q) 0.8 0.4 0 -0.4 -0.8 -0.8 0 -0.4 0.4 Real part of signal (i) 0.8 Figure 6.8: iq plot (imaginary part, Q, vs real part, I) of the measured signal (solid pink) and the model output (dashed blue) and the error y − ŷ (black). The model was estimated by the ls method using dr to estimate g1 and g2 . The zoom-in in the upper right corner is a ten times amplification of the error signal. 6.4 PA Model Validation 79 The estimated phase distortion functions, fˆ1 and fˆ2 , from the models can be plotted as functions of Δψ and the results for a wcdma signal for the diﬀerent methods are rather similar. The function f˜ˆ describes the phase change between the two outphasing signals at the output, and thus the amplitude change of the output. The phase distortion functions f˜ˆ are presented in Figure 6.9 as deviations from the ideal phase distortion, which should be as close to zero as possible. The ideal phase distortion includes the compensation for nonequal gain factors. By this, it is clear that at amplitudes close to zero (Δψ close to π), a zero distortion will not be possible for nonequal gain factors. In Figure 6.10, the diﬀerent functions f k , k = 1, 2, are shown for the diﬀerent methods. The methods achieve rather similar results, but at the expense of the number of computations in the nonconvex approach, where 25 000 function evaluations have been performed to achieve the optimum. Even though the methods result in similar validation results, the largest differences are found close to the edges of the interval. In the wcdma signal, 99.1% of the measured data points have 0.8 ≤ Δψ ≤ 3.0, so the focus of the ﬁt is where the most data points are. Compared to Figure 5.12 and (6.2), it is clear that the data points with a very large Δψ (close to π) have a very small amplitude, and errors in the phase distortion modeling might not aﬀect as much as the data points with a small Δψ (large amplitude). It can thus be concluded that it could be more important to obtain a good model for small values of Δψ than for large values (something that could be achieved by weighting functions). It can also be noted that, if the amplitude of the input had been used instead of the angle Δψ = arg(s1 ) − arg(s2 ), more weight would have been put at the largest amplitudes. This is not done now since a large input amplitude equals a small Δψ and vice versa. In polynomial ﬁtting, the agreement with the function f is often bad at the outer parts of the interval to be approximated. If one can choose the points at which the polynomial is to be ﬁtted, Chebyshev points should be chosen, with more points at the outskirts of the interval [Dahlquist and Björck, 2008, p. 377379]. Here, we are ﬁtting a polynomial using the method of least squares, but the same reasoning holds. To obtain a smaller error at the peak power, more data points could have been collected there. Instead, the least-squares ﬁtting focuses on ﬁtting the overall performance, and hence more eﬀort is made to obtain a small error in the parts where there is a larger point density. For the signals used in this thesis, this area of larger point density is in the center of the interval, where an improvement will be clearly seen in for example Figure 8.9. We will return to this subject in Chapter 8 when evaluating the predistortion results. 80 6 Modeling Outphasing Power Amplifiers Output phase distortion [rad] 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 ∆ψ (s1 , s2 ) 2 2.5 3 Figure 6.9: Simulated output phase distortion of the models from the nonconvex method (dotted green) and the ls methods using dr (dashed blue) and gridding (pink) (the two model outputs are almost completely on top of each other). The lines describe the modeled phase difference as a function of the input signal amplitudes, that is, taking the different gain factors into account. The three methods evaluated estimate the phase shift almost equally for the middle range where most of the data points are (99.1% have 0.8 ≤ ∆ψ ≤ 3.0), but the differences are visible at the edges. 81 PA Model Validation Output outphasing signals phase distortion [rad] 6.4 0.2 0 −0.2 −0.4 −0.6 0 0.5 1 1.5 ∆ψ (s1 , s2 ) 2 2.5 3 Figure 6.10: Simulated outphasing output phase distortion of the models from the nonconvex method (green) and the ls methods using dr (blue) and gridding (pink). The lines describe the modeled phase in each branch as a function of the input signal amplitudes. Branch one is plotted in solid lines and branch two in dashed lines. 82 6.5 6 Modeling Outphasing Power Ampliﬁers Convex vs Nonconvex Formulations The minimization of the cost function (6.12)-(6.14) is a nonconvex optimization problem in 2n + 4 dimensions with possible presence of local minima. Nonconvex optimization problems can either be solved by a local optimization method or a global one. A local optimization method minimizes the cost function over points close to the current point, and guarantees convergence to a local minimum only. Global methods ﬁnd the global minimum, at the expense of eﬃciency [Boyd and Vandenberghe, 2004]. Hence, even under ideal conditions (noise-free data, true pa described exactly by one model with the proposed structure), there is no guarantee that the nonconvex approach will produce an optimal model of the pa in ﬁnite time. The least-squares approach in (6.18)-(6.19) does exactly this and results in a closed-form expression for the parameter estimate. This is a major advantage since it removes the need for error-prone sub-optimality tests and possible time-consuming restarts of the search algorithm. Additionally, the computation time for the iterative, nonconvex, and potentially sub-optimal solution is signiﬁcantly longer compared to the least-squares method. A two dimensional projection of the cost functions to be minimized, (6.13) in the nonlinear formulation and (6.19) in the ls reformulation, can be seen in Figure 6.11. All parameters but two have been ﬁxed at the optimum, and the linear term in each ampliﬁer branch (ηk,1 in (6.11)) has been varied. Clearly, there is a risk of ﬁnding a local minimum in the nonconvex formulation illustrated in (a) whereas there is only one (global) optimum in the least-squares formulation in (b). The local minima in themselves might not be a problem if they are good enough to produce a well performing dpd, but there are no guarantees that this is the case. Typically, a number of diﬀerent initial points need to be tested in order to get a reasonable performance. 6.6 Noise Inﬂuence Noise is always present in measurements, and the noise will eﬀect the models. The algorithms presented in this chapter are sensitive to noise especially in two steps; the normalization g1 + g2 = 1 in (6.1) and the calculation of cDR in (5.13). Both these calculations are based on very few measurements, one for the normalization (the largest amplitude) and two for the dr calculation (the smallest and the largest amplitudes), so noise at these instances might have a large inﬂuence on the estimation, and thus the performance of the predistorter. The measurements used for the modeling and model validation in this chapter were recorded using the same measurement setup and power ampliﬁer that will be used in Section 8.4. To avoid the inﬂuence of measurement noise, the same input was applied a number of times, K, and the output was measured, whereupon the average over the diﬀerent realizations was calculated. In measurements used for the pa model estimation described here, K = 10. No automatic synchronization between input and measured output is done, so a manual syn- 83 Noise Influence Linear Term in Amplifier Path 2 6.6 8 4 0 −4 −4 0 4 8 Linear Term in Amplifier Path 1 Linear Term in Amplifier Path 2 (a) 8 4 0 −4 −4 0 4 8 Linear Term in Amplifier Path 1 (b) Figure 6.11: Two dimensional projections of the cost functions of (a) the original nonconvex optimization problem (6.12)-(6.14) and (b) the leastsquares reformulation, (6.18)-(6.19) using the dynamic range for the estimation of g1 and g2 . All but two parameters in each amplifier branch have been fixed at the optimal value, and the linear terms (ηk,1 in (6.11)) are varied. In (a), the visible local minima are marked with 5 and the minimum obtained clearly depends on the initial point of the local optimization. In the leastsquares formulation illustrated in (b), there is only one minimum (the global one) and convergence is guaranteed. The + marks the global minimum. 84 6 Modeling Outphasing Power Ampliﬁers chronization has to be performed. This also means that the sample times of the output diﬀer between diﬀerent measurement sets and that the synchronization between input and output is not the same for diﬀerent data sets. When looking at the diﬀerent data sets, the most dominant noise eﬀect seems to stem from this time mismatch, which is evenly distributed around the mean value. The noise levels in general are very low. 6.7 Memory Effects and Dynamics A more complex model structure has also been investigated by adding memory, that is to say that the output depends not only on the current input but also on the previous inputs, as in the model structure p mem (α, β̄nm (s)) = nm n αmj β(s(t − m))j , (6.21) m=0 j=0 with a memory depth nm , where nm . β̄nm (s) = β s(t − m) (6.22) m=0 This approach did not lead to a better ﬁt in the model validation, nor did it give any signiﬁcant improvement in predistortion. If dynamics are present in the pa, it is not unreasonable to assume that they would appear in the combiner, since the ampliﬁer components in each branch can be assumed to contribute with little dynamics. This would mean that we have a parallel Hammerstein system with two parallel nonlinear, static branches (the ampliﬁers) followed by a dynamic system (the combiner). To investigate how such dynamics would eﬀect the method described above, a dynamical system has been simulated at the output of a static model. The model was estimated using the ls method with dr. The dynamical system was a ﬁrst order system with different values of the time constant in the range [0.2Ts 5Ts ], where Ts is the sample time. The same identiﬁcation method was then applied to this data. In this case, the decomposition of the output using an estimate of g1 and g2 (obtained by dynamic range or gridding), is no longer a good approximation of the system, and the method will not perform in a satisfactory way. Thus, further investigation of how to include dynamics is needed. 7 Predistortion Power ampliﬁers in communication devices are often nonlinear and/or dynamic, which causes interference in adjacent transmitting channels. To reduce this interference, linearization is needed. This is preferably done at the input, so that a preﬁlter inverts the nonlinearities/dynamics. This preﬁlter is called a predistorter (pd). Originally, these predistorters consisted of small analog circuits, but now they are often implemented in a look-up table (lut) or a digital signal processor (dsp). Such an implementation is called a digital predistorter (dpd). For the outphasing ampliﬁers evaluated in this thesis, the gain mismatch could be eliminated by adjusting the voltage supplies in the output stage, but this would require an extra adjustable voltage source on the chip, which is undesirable. Instead, the goal is to ﬁnd a predistorter that uses only the phases of the two outphasing signals. By adjusting the outphasing signals, it is possible to achieve all amplitudes (within the dynamic range) and phases, and this idea will be explored in the construction of a predistorter. In this chapter, a description of an ideal dpd will be presented and diﬀerent methods to obtain it will be described. As a ﬁrst step, the evaluation of the predistorters will be based on a model of the pa (described in Chapter 6), on simulated data only. In Chapter 8, the predistorters will be evaluated on real measurement data. 7.1 A DPD Description With the description of the power ampliﬁer in (6.9)-(6.10), it is clear that an ideal pa would have f 1 = f 2 = 0 and g1 = g2 = g0 = 0.5 and any deviations from these values will cause nonlinearities in the output signal and spectral distortion. In order to compensate for these eﬀects, a dpd can be used to modify the input outphasing signals to the two ampliﬁer branches, i.e., s1 (t) and s2 (t). 85 86 7 Predistortion dpd s1 (t) s1,P (t) h1 g1 f1 y1 (t) s(t) + scs s2 (t) h2 s2,P (t) g2 f2 y(t) y2 (t) Figure 7.1: A schematic picture of the ampliﬁers with predistorters. Note that the functions f k and hk , k = 1, 2, are not functions of the input to the block only, but are used to show the general functionality of the pa and the dpd with the separation of the two branches. Since the outputs of the Class D stages (the ampliﬁers in each branch) have constant envelopes, the dpd may only change the phase characteristics of the two input outphasing signals. With this in mind, a dpd that produces the predistorted signals sk,P (t) = e j hk (Δψ ) sk (t), k = 1, 2, (7.1) to the two ampliﬁer branches is proposed. Here, h1 and h2 are two real-valued functions that depend on the phase diﬀerence between the two signal paths. By modifying the signals in each branch using the dpd in (7.1), shown in Fig. 7.1, the predistorted pa output yP (t) can be written yP = g1 e j f 1 (Δψ (s1,P ,s2,P )) s1,P + g2 e j f 2 (Δψ (s1,P ,s2,P )) s2,P . Δ =y1,P (7.2) Δ =y2,P The output is thus a sum of the two predistorted branches. In each branch k = 1, 2, the phase of the input is changed to counteract the eﬀects of the nonequal gain factors and the pa nonlinearities. Each branch is predistorted separately and sent to the outphasing pa. We will start by describing the eﬀects of the predistorter on the output. The phase diﬀerence between the two paths after the predistorters is described by Δψ (s1,P , s2,P ) = arg(s1,P ) − arg(s2,P ) = [arg(s1 ) + h1 (Δψ )] − [arg(s2 ) + h2 (Δψ )] Δ = Δψ + h1 (Δψ ) − h2 (Δψ ) = h̃(Δψ ), (7.3) 7.2 87 The Ideal DPD and the phase diﬀerence between the two paths at the (predistorted) outputs by Δψ (y1,P , y2,P ) = arg(y1,P ) − arg(y2,P ) = arg(s1,P ) + f 1 (Δψ (s1,P , s2,P )) − arg(s2,P ) + f 2 (Δψ (s1,P , s2,P )) = arg(s1 ) + h1 (Δψ ) + f 1 (h̃(Δψ )) − arg(s2 ) + h2 (Δψ ) + f 2 (h̃(Δψ )) = Δψ + h1 (Δψ ) − h2 (Δψ ) + f 1 (h̃(Δψ )) − f 2 (h̃(Δψ )) = h̃(Δψ ) + f 1 (h̃(Δψ )) − f 2 (h̃(Δψ )) Δ = f˜(h̃(Δψ )). (7.4) These phase diﬀerences correspond to the amplitude of the signal, since it is known that |s| = cos(Δψ /2), cf. Figure 5.12. The absolute phase change in each branch is given by arg(yk,P ) = arg(sk ) + hk (Δψ ) + f k (Δψ (s1,P , s2,P )) (7.5) for k = 1, 2. We now have a model structure describing how the phases of each outphasing signal, and thus the amplitude and phase of the output, depend on the characteristics g1 , g2 , f 1 and f 2 of the pa and the predistorter functions h1 and h2 . 7.2 The Ideal DPD As mentioned above, the pa output signal y(t) is a distorted version of the input signal. An ideal dpd should compensate for this distortion and result in a normalized output signal yP (t) = y1,P (t) + y2,P (t) that is equal to the input signal s(t) = 0.5s1 (t) + 0.5s2 (t). In the ideal case when g1 = g2 = g0 = 0.5, this is obtained when y1 (t) = 0.5s1 (t) and y2 (t) = 0.5s2 (t). However, this is not possible to achieve when gk 0.5, k = 1, 2. In this case, the ideal values for y1,P (t) and y2,P (t) are instead s˜1 (t) and s˜2 (t), as described in (6.3). These signals deﬁne an alternative decomposition of s(t) such that the gain mismatch is accounted for. Assume now that an ideal dpd (7.1) is used together with the pa (6.9). In this case, the equalities (7.6) y1,P (t) = s˜1 (t) and y2,P (t) = s˜2 (t) (7.7) hold, which results in yP (t) = y1,P (t) + y2,P (t) = s˜1 (t) + s˜2 (t) = s(t). That is, when the ideal dpd is applied to the pa, the original input will be retrieved. This assumes that the model perfectly describes the pa. Some more conclusions can be drawn about the ideal dpd by looking at the amplitudes and the phases of the input and the output. In order not to distort the amplitude at 88 7 Predistortion the output, the phase diﬀerence between y1,P (t) and y2,P (t) must be equal to the one between s˜1 (t) and s˜2 (t), i.e., Δψ (y1,P , y2,P ) = Δψ (s1,P , s2,P ) = arg(s˜1 ) − arg(s˜2 ) = = arg(s1 ) + ξ1 (Δψ ) − arg(s2 ) + ξ2 (Δψ ) Δ = Δψ + ξ1 (Δψ ) − ξ2 (Δψ ) = ξ̃(Δψ ). (7.8) Hence, inserting (7.8) into (7.4) gives f˜(h̃(Δψ )) = ξ̃(Δψ ) ⇔ h̃(Δψ ) = f˜−1 (ξ̃(Δψ )), (7.9) assuming that f˜ is invertible. Furthermore, for (7.6) and (7.7) to hold, that is, y1,P = s˜1 and y2,P = s˜2 , we require that the phases of the two signals are equal, arg(yk,P ) = arg(s˜k ), k = 1, 2. (7.10) Now, we have a description of how the predistorter will aﬀect the output as well as of how the gain factors g1 and g2 changes the desired outphasing output signals. The phase condition (7.10) combined with (7.3), (7.5) as well as (6.6) or (6.7), respectively (for each branch), gives arg(sk ) + hk (Δψ ) + f k (h̃(Δψ )) = arg(sk ) + ξk (Δψ ), k = 1, 2. That is, the predistorter functions hk is the only unknown in each branch and can be solved for. This results in hk (Δψ ) = −f k (h̃(Δψ )) + ξk (Δψ ) = −f k (f˜−1 (ξ̃(Δψ ))) + ξk (Δψ ) (7.11) for k = 1, 2. Here, (7.9) has been used in the last equality. Hence, using the predistorters (7.11) in (7.1), the output y(t) will be an ampliﬁed replica of the input signal s(t), despite the gain mismatch and nonlinear behavior of the ampliﬁers. 7.3 Nonconvex DPD Estimator A ﬁrst approach to identify the predistorter is to notice that the goal is to minimize the diﬀerence between the normalized input and the normalized predistorted output. This can be written down in a straightforward way as solving the minimization criterion θ̂DPD = argmin N 2 s(t) − ŷ (t, θ P DPD ) , (7.12) θDPD ŷP (t, θDPD ) = ĝ1 e t=1 j p(η̂1 ,Δψ (s1,P ,s2,P )) s1,P (t) + ĝ2 e j p(η̂2 ,Δψ (s1,P ,s2,P )) s2,P (t), (7.13) 7.4 89 Analytical DPD Estimator where sk,P (t) = e j p(ηk,DPD ,Δψ (s1 ,s2 )) sk (t), k = 1, 2, (7.14) T T η2,DPD ]T ∈ R2n+2 . The signal ŷP (t) is the output from a and θDPD = [η1,DPD pa model, using a predistorted input, as in Figure 7.1, where the ampliﬁers are replaced by the obtained models thereof. The dpd is thus identiﬁed based on a model of the forward system, according to Method B in Section 4.1. The forward model was approximated by polynomials, fˆk (Δψ ) = p(ηk , Δψ ), according to (6.11), and this is used in (7.12)-(7.13) to explicitly point out the dependence on the model parameters. When identifying the dpd model, the model structure was assumed to be the same as for the pa model, see (6.11), motivated by the StoneWeierstrass theorem (Theorem 7.26 in Rudin [1976]), so that ĥk (Δψ ) = p(ηk,DPD , Δψ ) = nh ηk,i,DPD Δiψ , k = 1, 2, (7.15) i=0 where ηk,DPD = ηk,0,DPD ηk,1,DPD . . . ηk,n,DPD T . The resulting estimated parameter vector θ̂DPD contains the dpd model parameters. This formulation leads to a nonconvex optimization problem and is thus at a risk of obtaining a suboptimal solution if the optimization algorithm ﬁnds a local minimum. To restart the algorithm at diﬀerent initial points is a possible way to reduce the risk of getting stuck in a local minimum instead of the global minimum, but this solution would not be useful in an online implementation, see also Section 6.5 for a discussion on convex and nonconvex optimization. 7.4 Analytical DPD Estimator The ideal dpd outlined in Section 7.2 requires knowledge of the pa model, and once the pa characteristics g1 , g2 , f 1 and f 2 are known (or estimated), the predistorter functions can be determined. The ﬁrst step to construct a dpd is thus to obtain a model of the pa, as described in Chapter 6. This method follows Method A in Section 4.1, where a model of the system itself is used to analytically produce an inverse. The parameter estimates η̂k deﬁne function estimates (6.20) fˆk (z) = p(η̂k , z), from which an estimate k = 1, 2, f˜ˆ(z) = z + fˆ1 (z) − fˆ2 (z) (7.16) of the function f˜ from (7.4) can be computed. Provided that this function can be inverted numerically, estimates ĥk of the ideal phase correction functions can be computed as in (7.11), i.e., ĥk (Δψ ) = −fˆk (f˜ˆ−1 (ξ̃(Δψ ))) + ξk (Δψ ) (7.17) 90 7 Predistortion for k = 1, 2, where Δψ is given by (6.2) and ξ̃, ξ1 and ξ2 by (7.8), (6.6) and (6.7), respectively. Hence, the complete dpd estimator consists of the selection of gain factors g1 and g2 , see Sections 6.1 and 6.3. Also, the two least-squares estimators given by (6.18), a numerical function inversion in order to obtain f˜ˆ−1 and the expressions for the phase correction functions in (7.17) make part of the complete dpd estimator. The dpd estimation can either be done at each point in time, or (as has been done here) by evaluating the function for the range of possible Δψ and saving this nonparametric, piecewise constant function. The dpd estimator will result in two functions ĥ1 and ĥ2 which take Δψ as argument, and by using these as in (7.1), the predistorted input signals s1,P (t) and s2,P (t) can be calculated for arbitrary data. Measurement results for a validation data set, not used during the modeling, will be presented in Chapter 8. The algorithm thus consists of two main parts, A – Estimation of pa model and B – Calculation of dpd functions. Part A consists of three subparts where the ﬁrst, A.I, produces candidates for the gain factors g1 and g2 by either using the dr by gridding possible values. A.II produces ls estimates of the nonlinear functions fˆ1 and fˆ2 for each pair of g1 and g2 and in A.III, the best performing model is chosen among all the candidates. In Part B, the dpd functions ĥ1 and ĥ2 are calculated. The diﬀerent steps are described in more detail in Algorithm 1. 7.5 Inverse Least-Squares DPD Estimator In the deduction of the predistorter described above, the ideal dpd was deduced using analytical relationships between the input and the desired output, following the basic Method A described in Section 4.1, page 34. By instead choosing Method C, we want to estimate the inverse directly. This means that the system input s(t) (or rather s1 (t) and s2 (t)) will be considered as output to the identiﬁcation, and y(t) (or y1 (t) and y2 (t)) as input. Since g1 and g2 can be found rather easily (through the dynamic range or gridding), these can still be assumed to be known, so the decomposition of y(t) into y1 (t) and y2 (t) can be performed using (6.3). In each branch k = 1, 2 we thus have arg(sk ) = arg(yk ) − hk Δψ (y1 , y2 ) . (7.18) The left hand side is the input, which is known. The ﬁrst term on the right hand side represents what we have measured, using the decomposition (6.3). The second term represents how the outphasing outputs should be modiﬁed to match the input, a postdistorter. The only unknown is thus the predistorter functions h1 and h2 in the two branches. By approximating these as polynomials, ĥk ≈ p(ζk , Δψ (y1 , y2 )) = nh ζk,i Δiψ (y1 , y2 ), i=0 where ζk = ζk,0 ζk,1 . . . ζk,n T . k = 1, 2, (7.19) 7.5 91 Inverse Least-Squares DPD Estimator Algorithm 1 ls modeling and analytical dpd method Require: model order n, method for choice of g1 and g2 , precision of pa model (p M ) and inverse (p I ), estimation data. {A – Estimation of pa model} y(t) 1: Normalize the output y(t) = max(|y(t)|) 2: Calculate Δ ψ ∀t according to (6.2). {A.I – Estimation of gain factor candidates g1 and g2 } if Use Dynamic Range to determine g1 and g2 then Calculate cDR using (5.13), and Δg using (6.17). Calculate possible choices of g1 , g2 according to (6.15). 6: else {g1 and g2 over a range of values} 7: Grid g1 ∈ [gmin , gmax ] with precision p M and let g2 = 1 − g1 . 8: end if 3: 4: 5: {A.II – Estimation of nonlinearity function candidates fˆ1 and fˆ2 } 9: for all pairs of g1 , g2 do 10: Create s˜k = gk e j arg(s˜k ) and yk = gk e j arg(yk ) , k = 1, 2 using (6.4) to (6.7). 11: Find ηk using (6.18) and calculate fˆk , k = 1, 2 using (6.20). ˆ ˆ Simulate the output ŷg1 ,g2 (t) = g1 e j f 1 (Δψ ) s1 (t) + g2 e j f 2 (Δψ ) s2 (t). 13: Calculate error Vg (g1 , g2 ) = t |y(t) − ŷg1 ,g2 (t)|2 . 14: end for 12: {A.III – Choose best forward model, ĝ1 , ĝ2 , fˆ1 and fˆ2 } 15: Select ĝ1 = arg ming Vg (g1 , 1 − g1 ), ĝ2 = 1 − ĝ1 and the corresponding fˆ1 and 1 fˆ2 . 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: {B – Calculation of dpd functions ĥ1 and ĥ2 } {Create a Look Up Table (lut) for diﬀerent values of Δψ by creating an intermediate signal s} Grid Δψ ∈ [0, π] with precision p I . for each value of Δψ do Create s = cos(Δψ /2) according to (5.10) assuming α = 0 and rmax = 1 (ϕ = Δψ /2). Create s1 and s2 according to (5.11) and s˜1 and s˜2 using (6.3) to (6.5). Find ξ̃ using (7.8), (6.6) and (6.7). Calculate f˜ˆ(ξ̃) using (7.16). end for Invert f˜ˆ(ξ̃) numerically to get f˜ˆ−1 . This can e.g. be done by calculating f˜(ξ̃) for a number of values of ξ̃ ∈ [0, π], grid f˜ˆ(ξ̃) and match with the ξ̃ that gives the closest value. for each value of Δψ in line 16 do Find estimate ĥk (Δψ ) according to (7.17). end for 92 7 Predistortion as was done for the pa model, the parameters corresponding to the hk -functions can be found. The estimates ζ̂k,i of the model parameters have been computed by minimizing a quadratic cost function, i.e., ζ̂k = arg min Vkh (ζk ), k = 1, 2, (7.20) ζk where Vkh (ζk ) = N 2 arg (yk (t)) − arg (sk (t)) − p ζk , Δψ (y1 (t), y2 (t)) . (7.21) t=1 The parameter estimates ζ̂k deﬁne inverse function estimates ĥk (z) = p(ζ̂k , z), k = 1, 2, (7.22) that can be used as a dpd. As discussed in Chapter 3, this method assumes commutativity of the two systems (system and inverse), so that the inverse which was estimated at the output of the power ampliﬁer, a postdistorter, can also be used at the input as a predistorter. The method is summarized in Algorithm 2. 7.5 Inverse Least-Squares DPD Estimator 93 Algorithm 2 Inverse ls dpd method Require: model order nh , method for choice of g1 and g2 , precision of gain factors (p M ), estimation data. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: Normalize the output y(t) = y(t) max(|y(t)|) {I – Estimation of gain factor candidates g1 and g2 } if Use Dynamic Range to determine g1 and g2 then Calculate cDR using (5.13), and Δg using (6.17). Calculate possible choices of g1 , g2 according to (6.15). else {g1 and g2 over a range of values} Grid g1 ∈ [gmin , gmax ] with precision p M and let g2 = 1 − g1 . end if {II – Estimation of nonlinearity function candidates ĥ1 and ĥ2 } for all pairs of g1 , g2 do Create yk = gk e j arg(yk ) using (6.4) to (6.7) and sk using (5.11). Calculate Δψ (y1 , y2 ) ∀t according to (6.2). Find ζ̂k using (7.20) and calculate ĥk , k = 1, 2 using (7.22). Simulate the input sˆg1 ,g2 (t) = e j ĥ1 (Δψ (y1 ,y2 )) y1 (t) + e j ĥ2 (Δψ (y1 ,y2 )) y2 (t) Calculate error Vg (g1 , g2 ) = t |s(t) − sˆg1 ,g2 (t)|2 end for {III – Choose best inverse model, ĥ1 and ĥ2 } Select ĝ1 = arg ming1 Vg (g1 , 1 − g1 ), ĝ2 = 1 − ĝ1 and the corresponding ĥ1 and ĥ2 . 94 7 Predistortion Table 7.1: dpd Model Validation Method Analytical ls Gain factors only 7.6 |s − ŷp |22 0.0532 1.008 65.5 Simulated Evaluation of Analytical and LS Predistorter The goal here is to evaluate the performance of the predistorter methods in simulations, and determine how well the diﬀerent methods achieve an inversion. One way is to look at the am-am modulation to assess how much the amplitude of the predistorted output is distorted. For an outphasing pa, this is connected to the phase diﬀerence Δψ (y1,P , y2,P ) of the outphasing outputs y1,P and y2,P . The predistorter methods in Sections 7.4 and 7.5 are evaluated using a model of the ampliﬁer as “the truth”. The model is presented in Chapter 6, where the gain factors were estimated using the dr and the nonlinearities using the ls approach, see Section 6.3 and the model validation in Section 6.4 and Figure 6.5. The same validation data have been used in order to evaluate the diﬀerent predistorter methods. Evaluation on a real pa will be presented in Chapter 8. Test 1 – Inversion Evaluation We will start by looking at the am-am modulation to determine how much the amplitude of the predistorted output is changed. The deviation from the ideal phase diﬀerence at the output (i.e., the output amplitude) with and without predistortion is presented in Figure 7.2. Both the analytical method and the ls method clearly reduce the phase shift introduced by the pa. Figure 7.3 shows the estimated deviation from the ideal phase for each signal branch with and without predistortion, with rather similar performance for the two dpd methods. The values of the cost function (7.12) are presented in Table 7.1 for the two methods. The result using only the estimation of the gain factors and the alternative decomposition (using the knowledge of the nonequal gain factors) is also presented. It is clear that incorporating the nonlinearities improves the performance. For cases when the gain factors diﬀer more from the ideal g1 = g2 = 0.5 than in this case (g1 = 0.4986 and g2 = 0.5014), the alternative decomposition (6.3) will have a larger improvement on the modeling than in this case, when the diﬀerence is small. For the ls method, the ﬁt is almost perfect in the middle range, which is to be expected since a polynomial is used (see discussion in Section 6.4, page 79). Also the number of measurements is unevenly spread out over Δψ with most data in the middle, only 0.9% of the estimation data have Δψ < 0.8 or Δψ > 3. For the 95 Simulated Evaluation of Analytical and LS Predistorter Deviation from ideal phase difference [rad] 7.6 · 10−2 2 0 −2 −4 −6 −8 0 0.5 1 1.5 ∆ψ (s1 , s2 ) 2 2.5 3 Figure 7.2: Simulated predistorter evaluation for a model with polynomial degree n = 5 using the wcdma input signal (see Chapter 8). The signals are generated using the dpd functions and the pa model. For an ideal pa, there is no amplitude distortion, that is, the phase difference of the outphasing signals is the same at the output and the input. The deviation from this ideal phase difference for the (modeled, not predistorted) output signal ŷ is shown in dotted green and the predistorted output signals ŷP in pink and blue. The pink line shows the result using the analytical inversion as described in Sections 7.2 and 7.4 and the dashed blue line shows the result of the ls approach in Section 7.5, with predistorter degree nh = 5 in (7.19). The two methods both perform very well in a large interval. analytical solution, one can see an inversion error close to ∆ψ = π. This is a consequence of the nonequal gain factors, ∆ψ = π should represent a complete opposition of the two outphasing signals such that the output amplitude is zero. If g1 , g2 however, this is not possible and no phase combination of the two outphasing signals will lead to a zero-amplitude output. A power amplifier with a large dynamic range (dr, difference between the gain factors g1 and g2 ) will have a very small distortion close to ∆ψ = π, whereas a pa with a small dr will show this distortion in a larger region. The errors of the two methods when compared to validation data are shown in Figure 7.4. Also in this plot, it can be seen that both methods reduce the power amplifier distortion, and that the analytical inversion performs slightly better. 96 7 Predistortion Deviation from ideal phase [rad] 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 0 0.5 1 1.5 ∆ψ (s1 , s2 ) 2 2.5 3 Figure 7.3: Simulated predistorter evaluation for a model with polynomial degree n = 5 using the wcdma input signal. The signals are generated using the dpd functions and the pa model. The deviation from the ideal phase for the (modeled, not predistorted) output outphasing signals ŷ1 and ŷ2 are shown in green and the predistorted output signals ŷ1,P and ŷ2,P in pink and blue. The pink lines show the results using the analytical inversion as described in Sections 7.2 and 7.4 and the blue lines show the result of the ls approach in Section 7.5. Branch one is plotted in solid lines and branch two in dashed lines. 7.6 97 Simulated Evaluation of Analytical and LS Predistorter Amplitude 8 · 10−2 6 4 2 Phase [rad] 0 0 −1 −2 282 284 286 288 Time [µs] 290 292 Figure 7.4: The upper plot shows the amplitude error, |s − ŷp |, and the lower plot shows the phase error, arg(s) − arg(ŷp ), for the two dpd methods. The analytical method is in pink and the ls in blue. As a comparison, the errors for the original, unpredistorted signal y(t) are also plotted in green. 98 7 Predistortion Figure 7.5: Simulated aclr at 5 MHz and 10 MHz oﬀset with dpd (solid line) and without (dashed line) for the wcdma signal. Test 2 – Impact of ACLR on Predistorter Performance As previously explained, the result of a limited dynamic range is that all amplitude and phase errors occurring outside the dr cannot be corrected. The signal clipping in an outphasing pa occurs at small amplitudes, while in a conventional linear pa, the peak amplitudes are clipped. Thus, the dr in an outphasing pa limits the spectral performance when amplifying modulated signals. To investigate the performance limits of the predistorter, simulations have been done using two ampliﬁers with a given dr (no phase distortion), with and without dpd. In Figure 7.5, the aclr over dr at 5 MHz and 10 MHz for the wcdma signal are plotted with and without dpd. Here, the phase error between the outphasing signals is assumed to be zero. For a pa with a dr of 25 dB the diﬀerences in aclr between the nonpredistorted and predistorted outputs are 8-13 dB. When the dr is 25 dB the optimal theoretical aclr is achieved after dpd. For a pa with 45 dB of dr, the diﬀerence between when a dpd is used or not is negligible. Summary In this simulated evaluation, both dpd methods achieve an improvement, compared to the original power ampliﬁer output. The analytical inversion leads to slightly better results at the cost of a higher computational complexity. The lookup table for the analytical dpd has 2 ∗ 3142 elements (with precision p I = 0.001 in Algorithm 1), and the polynomials contain 2 ∗ 6 coeﬃcients (nh = 5 in (7.19)). Using a higher polynomial degree could lead to improved results for the ls method, and a smaller lut might lead to a small degradation of the analytical method. As implementation issues are out of scope for this thesis, the methods are not optimized for implementation and therefore these considerations have not been further pursued. 7.7 7.7 Recursive Least-Squares and Least Mean Squares 99 Recursive Least-Squares and Least Mean Squares Here, a few aspects of a possible future implementation of the dpd methods are presented. In addition to the guaranteed convergence, least-squares formulations also have the advantage that there are many eﬃcient numerical methods for solving this type of problems. They can be solved recursively by, for example, the recursive least-squares (rls) method [Björck, 1996] making them suitable for an online implementation. An even less complex parameter estimation algorithm is the least mean square (lms) method, which can make use of the linear regression structure of the optimization problem, developed here in (6.11) and (6.19). lms has been used for rf pa linearization in Montoro et al. [2007] and implemented in ﬁeld programmable gate array (fpga) technology, as shown in Gilabert et al. [2009]. With a recursive implementation of the algorithm, it is even more important that the algorithm can be proved to converge to good values, as no monitoring of the performance should be necessary in order for the method to be useful in practice. This also means that a nonconvex solution as in (6.12)-(6.13) is not suitable for online implementation since it cannot guarantee convergence to good enough minima. In an oﬄine application, the possibility to restart the optimization could be added but, together with the lack of a bound on the number of iterations, this does not seem like a good solution for an online version. Using well explored methods like rls or lms would result in a low-complexity implementation, and though it is hard to judge the exact complexity of the iterative implementation that would be needed for the online version of nonconvex solution, it is clear that it would be very hard to ﬁnd a simpler one than for the low-complexity lms version of the convex method. Since circuitry will behave diﬀerently depending on the settings under which it operates, it is important to be robust to such conditions. This is covered in the concept of process, voltage and temperature variations (pvt variations). One way to handle the pvt variations and changes in the setting, such as aging, would be to use a method with a forgetting factor, reducing the inﬂuence of older measurements [Ljung, 1999]. The rls and lms solutions assume the changes in the operating conditions to be slow. 8 Predistortion Measurement Results The models presented in Chapter 6 and the predistorters in Chapter 7 are based on measured data from a power ampliﬁer. In Chapter 7, the methods’ ability to invert the nonlinearities was investigated, using a forward model as a “true” system. In this chapter, the methods will be evaluated on real measurements. The predistorters are applied to a new data set, validation data, that is not the same as the signal used for estimation. To start oﬀ, a short introduction to the signal types used and the measurement setup will be presented. 8.1 Signals Used for Evaluation The predistortion methods have been evaluated for the diﬀerent signal types edge, wcdma and lte. Mobile communication technologies are often divided into generations, and the new devices of today are the fourth generation, 4G. The ﬁrst generation, 1G, was the ﬁrst analog mobile radio systems of the 1980s. 2G was the ﬁrst digital mobile systems and 3G the ﬁrst mobile systems handling broadband data. Enhanced data rates for gsm evolution (edge) is a mobile phone technology with higher bit rates than general packet radio service (gprs) [Ahlin et al., 2006], and has been called 2.75G since it did not quite reach the 3G standards. The carrier frequency used is 2 GHz, and the bandwidth is 200 kHz. Wideband code division multiple access (wcdma) is a third generation (3G) mobile phone technology, and is one of the 3G mobile communications standards [Frenzel, 2003]. The carrier frequency used is 2 GHz, and the bandwidth is 5 MHz. The bandwidth of the long term evolution (lte) signal is variable, and can be adjusted between 1 and 20 MHz. It is sometimes called 4G or 3.9G since it does not completely satisfy the 4G requirements Dahlman et al. [2011]. The wcdma and lte have large peak-to-minimum power ratio, i.e., the pa 101 102 8 Predistortion Measurement Results 1 0 -1 -1 0 edge 1 -1 0 wcdma 1 -1 0 lte 1 Figure 8.1: iq plots (imaginary part, q, vs real part, i) of signal realizations of the edge, wcdma and lte standards in the complex plane. The sampling frequency in the modeling data sets is four times higher than that shown here. 0.3 0.2 0.1 0 0 0.5 edge 1 0 0.5 wcdma 1 0 0.5 lte 1 Figure 8.2: Histograms of the distribution of the input amplitude of signal realizations of the edge, wcdma and lte standards. This diﬀerence in input distribution eﬀects the peak-to-average power ratio, and it also implicitly determines the weighting of the ﬁt of the polynomials, see also the discussion on polynomial ﬁtting on page 79. output signals include the minimum and maximum amplitudes (the full dynamic range). For these signals, the dr of the pa will eﬀect the output signal, by clipping the smallest amplitudes. For edge, the signal amplitude is never close enough to zero to be eﬀected by the pa dr, and no clipping will occur. Realizations of each signal type (edge, wcdma and lte) are shown in Figure 8.1 as iq-plots. Histograms of the distribution of the input amplitude are shown in Figure 8.2. The distribution also implicitly determines the weighting of the ﬁt of the polynomials, see also the discussion on polynomial ﬁtting on page 79. One characteristic of a signal is the peak-to-average power ratio (papr). A signal with a high papr sets high standards on the linearity of the pa, since a large range of input signal amplitudes has to be ampliﬁed. The signals used are created as random signals with predeﬁned characteristics. 8.2 Measurement Setup 103 Figure 8.3: Measurement setup for iq-data with two Master-Slaveconﬁgured SMBV signal generators [Rohde & Schwarz]. 8.2 Measurement Setup The measurements that will be discussed in Section 8.3 have been performed using an SMU200A signal generator with two phase-coherent rf outputs and an arbitrary waveform generator where the input signals (s1 (t) and s2 (t)) and the predistorted input signals (s˜1 (t) and s˜2 (t)) were stored. For the measurements that will be discussed in Section 8.4, two R&S SMBV100A signal generators with phase-coherent rf outputs and arbitrary waveform generators with maximum iq sample rate of 150 MHz have been used. Figure 8.3 shows the measurement setup. The outphasing power ampliﬁers used in the measurements have been developed by Jonas Fritzin et al. and are brieﬂy described in Appendix A and in more detail in Fritzin [2011]. Sampling The sampling rate in Section 8.4 was 92.16 MHz in the measurements, six times the original sampling frequency of the signal. The impact of baseband ﬁltering and limited bandwidth is investigated in Gerhard and Knöchel [2005a,b], where it was concluded that to obtain an optimal signal/distortion ratio over the entire bandwidth, a compromise between the sampling frequency and the ﬁlter characteristics has to be made. Here, we have evaluated the required bandwidth/sampling rate based on measurements with two signal generators and one combiner, no pa was used. Increasing the sampling frequency from the original 15.36 MHz to 30.72 MHz and 61.44 MHz, the aclr is improved, see Table 8.1. Thus, for the speciﬁc tests performed here, the aclr at 5 and 10 MHz can be improved by 6-9 dB and 4-8 dB, respectively, when increasing the sampling rate up to four times the original sampling rate of 15.36 MHz. Further increasing the sampling frequency, up to 92.16 MHz, shows no signiﬁcant change. 104 8 Predistortion Measurement Results Table 8.1: Measured Spectral Performance at 1.95 GHz for wcdma and lte Uplink Signals for Diﬀerent Sampling Frequencies. wcdma lte 8.3 Measured Parameter aclr @ 5 MHz [dBc] aclr @ 10 MHz [dBc] aclr @ 5 MHz [dBc] 15.36 MHz -44 -48 -34 30.72 MHz -50 -52 -43 61.44 MHz -52 -56 -46 Evaluation of Nonconvex Method In this section, the nonconvex approach presented in Sections 6.2 and 7.3 has been evaluated. The pa model has been obtained by minimizing the nonconvex cost function in (6.13) and the corresponding dpd by minimizing (7.12). The method involves solving two nonconvex optimization problems, and corresponds to Method B in Section 4.1. This method has been evaluated on the pa described in Appendix A.1 and Fritzin [2011]. The predistortion methods were evaluated on a physical chip. The measurement setup was optimized and the branch ampliﬁers were tuned to achieve the best performance possible. The phase oﬀset between s1 (t) and s2 (t) in the baseband was adjusted to minimize phase mismatch (ideally 180 ◦ between the two rf inputs for nonmodulated s1 (t) and −s2 (t) in Figure A.2, i.e. maximum output power for a continuous signal). Since this is not a reasonable assumption in a real-life application, an additional phase error of 3 ◦ was added in one of the branches. Measurements of input s(t) and output y(t) of length Nid were collected K times, and an average was taken to avoid the inﬂuence of measurement noise. This data was used to model the power ampliﬁer. Based on this pa model, a predistorter model was produced. Polynomials with order n have been used as parameterized versions of the pa nonlinearities and of order nh for the predistorter functions. The predistorted input signals, s1,P and s2,P , were then computed (in Matlab) for a validation input signal of length Nval . The predistorted outphasing input signals were sent to the pa, resulting in a predistorted output. The additional phase error was still applied during the predistorter validation. For the computation of the model parameters, a large number of algorithms are available for solving a nonlinear optimization problem. Here, the Matlab routine fminsearch, based on the Nelder-Mead simplex method, was used. The estimation and validation data sets contain Nid and Nval samples, respectively. The input and output sampling frequencies are denoted f s and f s,out , respectively. To minimize the inﬂuence of measurement noise, the signals were measured K times, and a mean was calculated. The data collection parameters are shown in Table 8.2. Since the wcdma is a more wide-band signal than the edge signal, the number of samples Nid and Nval were chosen larger. 8.3 105 Evaluation of Nonconvex Method Table 8.2: Data Collection, Nonconvex Method edge wcdma Nid 40 001 153 600 Nval 80 001 153 600 fs 8.67 MHz 61.44 MHz f s,out 34.68 MHz 61.44 MHz K 150 200 Table 8.3: Measured Spectral Performance of the edge Signal (a) With no phase error and no dpd. (b) For a 3 ◦ phase error and no dpd. (c) When dpd is applied to (b). Freq. 2 GHz 8.3.1 Freq. oﬀset 400 kHz 600 kHz Spec. -54 dB -60 dB Meas. (a) -54.4 dB -60.3 dB Meas. (b) -53.5 dB -59.9 dB Meas. (c) -65.9 dB -68.2 dB Measured Performance of EDGE Signal edge is a rather narrow-band signal with a peak-to-average power ratio (papr) of 3.0 dB. The spectrum of the estimation input data set is shown in Figure 8.4(d). The output of a perfectly matched pa in Figure 8.4(a) fulﬁlls the requirements, but without any margins to the spectral mask. The spectral mask is a nonlinearity measurement that describes the amount of power that is allowed to be spread to the neighboring channels. The requirements for an edge signal are summarized in Table 5.1 and illustrated in Figure 8.4. As the phase error cannot be assumed to be 0 ◦ in a transceiver, a phase error of 3 ◦ was added and led to a violated spectral mask as in Figure 8.4(b). When predistortion was applied to a validation data set, not used for estimation, the linearity improves, as seen in Figure 8.4(c). The pa model was of order n = 5 and the predistorter of order nh = 5. The measured power at 400 and 600 kHz oﬀsets were -65.9 and -68.2 dB, with margins of 11.9 and 8.2 dB, respectively. The average power at 2 GHz was +7 dBm with 22 % pae and root mean square (rms) evm of 2 %. The measured performance of the ampliﬁer for an edge signal is summarized in Table 8.3. 8.3.2 Measured Performance of WCDMA Signal The papr of the wcdma signal was 3.2 dB and the spectrum of the estimation data set is shown in Figure 8.5(d). Figure 8.5(a) shows the measured wcdma spectrum at 2 GHz, with minimized phase mismatch and no predistortion. When the same phase error of 3 ◦ as for the edge signal was added to simulate reasonable phase settings, a distorted spectrum as in Figure 8.5(b) was measured. The aclr is an integrated measure that describes the power spread to adjacent channels. 106 8 Predistortion Measurement Results EDGE Relative spectral density for RBW = 30 kHz [dB] 0 −10 ← spectral mask −20 −30 −40 −50 −60 −70 ←a ←c ←d b ↓ −80 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Offset from carrier frequency [MHz] at 2 GHz Figure 8.4: Measured edge spectrum at 2 GHz. (a) Output spectrum without phase error between s1 (t) and s2 (t). (b) Output spectrum with 3 ◦ phase error between s1 (t) and s2 (t). (c) Output spectrum when dpd is applied to (b). (d) Output spectrum of the estimation signal. The spectrum of the validation signal was similar. 8.3 107 Evaluation of Nonconvex Method Table 8.4: Measured Spectral Performance of the wcdma Signal (a) With no phase error and no dpd. (b) For a 3 ◦ phase error and no dpd. (c) When dpd is applied to (b). Freq. 1 GHz 2 GHz aclr 5 MHz 10 MHz 5 MHz 10 MHz Spec. -33 dBc -43 dBc -33 dBc -43 dBc Meas. (a) -40.6 dBc -59.8 dBc -43.4 dBc -53.9 dBc Meas. (b) -39.4 dBc -56.2 dBc -38.0 dBc -50.9 dBc Meas. (c) -53.6 dBc -60.3 dBc -50.2 dBc -52.2 dBc At 1 GHz and 2 GHz, the power ampliﬁer fulﬁlls the requirements, also with the additional phase error, as seen in Table 8.4. The phase predistortion method, with n = 5 and nh = 4, for a validation signal, improves the measured aclr. A spectrum is shown in Figure 8.5(c). The channel power at 2 GHz was +6.3 dBm with pae of 22 % and rms composite evm of 1.4 % (0.6 % after dpd). The measured performance of the ampliﬁer for a wcdma signal is summarized in Table 8.4. 8.3.3 Summary The nonconvex predistortion method clearly improves the pa performance for both edge and wcdma signals, even when an extra phase error is added. The measured spectral performance at 400 kHz oﬀset and the aclr at 5 MHz is comparable to state-of-the-art edge [Mehta et al., 2010] and wcdma [Huang et al., 2010] transmitters. 108 8 Predistortion Measurement Results WCDMA Relative spectral density for RBW = 30 kHz [dB] 0 −10 −20 −30 −40 −50 −60 −70 ←a ←b ←c ←d −80 −10.0 −5.0 0.0 5.0 10.0 Offset from carrier frequency [MHz] at 2 GHz Figure 8.5: Measured wcdma spectrum at 2 GHz. (a) Output spectrum without phase error between s1 (t) and s2 (t). (b) Output spectrum with 3 ◦ phase error between s1 (t) and s2 (t). (c) Output spectrum when dpd is applied to (b). (d) Output spectrum of the estimation signal. The spectrum of the validation signal was similar. 8.4 109 Evaluation of Least Squares PA and Analytical Inversion Method Table 8.5: Data Collection, Least-Squares and Analytical Method wcdma lte 8.4 Nid 100 000 100 000 Nval 100 000 100 000 fs 92.1 MHz 92.1 MHz f s,out 92.1 MHz 92.1 MHz K 10 10 Evaluation of Least Squares PA and Analytical Inversion Method In this section, the least-squares modeling of the pa, using the dr to estimate g1 and g2 , has been applied. An analytical inversion has been used to construct the predistorter functions, as in Method A in Section 4.1. The pa modeling is described in Section 6.3, the dpd in Section 7.4 and the method is summarized in Algorithm 1, page 91. This method has been evaluated on the pa described in Appendix A.2 and Fritzin et al. [2011c]. The measurement setup was optimized and the branch ampliﬁers were tuned to achieve the best performance possible. For the measurements without predistortion, the phase oﬀset between s1 (t) and s2 (t) in the baseband was adjusted to minimize phase mismatch (ideally 0 ◦ between nonmodulated s1 (t) and s2 (t), that is, maximum output power for a continuous signal). Moreover, the iq-delay between the signal generators was adjusted for optimal performance [Rohde & Schwarz]. Measurements of input s(t) and output y(t) were collected K times, and an average was taken to avoid the inﬂuence of measurement noise. This averaged data set was used to model the pa, and based on the pa model, a predistorter model was produced. Polynomials with order n have been used as parameterized versions of the pa nonlinearities and based on this model, an approximation of the ideal predistorter has been constructed. The predistorted input signals, s1,P and s2,P , were then computed (in Matlab) for a validation input signal. The predistorted outphasing input signals were sent to the pa, resulting in a predistorted output. The estimation and validation data sets contain Nid and Nval samples, respectively. The input and output sampling frequencies are denoted f s and f s,out , respectively. The data collection parameters are shown in Table 8.5. In all following experiments, the dpd estimates ĥk , k = 1, 2, have been calculated for 3142 uniformly distributed points (p I = 0.001 in Algorithm 1). This lut has been used in the construction of the predistorted outphasing input signals. For each input phase diﬀerence Δψ , the outphasing input signals s1 (t) and s2 (t) were adjusted according to the nearest neighbor principle. 110 8 Predistortion Measurement Results Figure 8.6: Measured wcdma spectrum at 1.95 GHz. (a) Measured wcdma spectrum without dpd. The measured aclr is printed in gray. (b) When dpd is applied to (a). The measured aclr is printed in black. (c) Spectrum of estimation signal. Spectrum of validation signal was similar. 8.4.1 Measured Performance of WCDMA Signal The papr of the wcdma uplink signal was 3.5 dB. The spectrum of the estimation data is shown in Figure 8.6(c). For the wcdma signal at 1.95 GHz without predistortion, the measured aclr at 5 MHz and 10 MHz oﬀsets were -35.5 dBc and -48.1 dBc, respectively. The spectrum is shown in Figure 8.6(a). The estimation output data y(t) were used in the predistortion method to extract the model parameters, with n = 5. The aclr is a measure describing the amount of leakage into adjacent channels that can be tolerated, and the standards for wcdma are -33 dBc and -43 dBc at 5 MHz and 10 MHz oﬀsets, respectively. The predistorted input signals, s1,P (t) and s2,P (t), were computed for the validation input signal, resulting in an output spectrum as shown in Figure 8.6(b). The power spectral densities of the predistorted input is similar to that of the nonpredistorted input signal, and therefore not included (similarly for the lte signal). With predistortion, the measured aclr at 5 MHz and 10 MHz oﬀsets were -46.3 dBc and -55.6 dBc, respectively. Thus, the measured aclr at 5 MHz and at 10 MHz oﬀsets were improved by 10.8 dB and 7.5 dB, respectively. The average power at 1.95 GHz was +26.0 dBm with 16.5 % pae. It is clear that the predistortion reduces the spectral leakage. Figure 8.7 shows the measured am-am (output amplitude vs. input amplitude) and am-pm (phase change vs. input amplitude) characteristics with and without dpd for the wcdma signal. The upper ﬁgure shows the amplitude mod- 8.4 Evaluation of Least Squares PA and Analytical Inversion Method 111 ulation, and should ideally be a straight line from lower left corner (0,0) to the upper right (1,1), such that the output amplitude equals the input amplitude for the whole range of the signal. If this is not the case, there will be amplitude distortions. Here, the improvement can be seen in normalized amplitudes smaller than 0.4. The lower plot shows the phase distortion, and the ideal is zero. It can be seen that the dpd reduces the phase distortion for normalized amplitudes in the range 0.05 |s| 0.95. For amplitudes close to one, the distortion is slightly worse with a predistorter than without. This is due to the polynomial ﬁt of the pa model, which has a best ﬁt in the middle region where the density of data points is largest. 8.4.2 Measured Performance of LTE Signal The papr of the lte uplink signal was 6.2 dB and the spectrum of the estimation data sets is shown in Figure 8.8(c). For the lte signal at 1.95 GHz without predistortion, the measured aclr at 5 MHz oﬀset was -34.1 dBc. The spectrum is shown in Figure 8.8(a). The estimation output data y(t) were used in the predistortion method to extract the model parameters with n = 5. The predistorted input signals, s1,P (t) and s2,P (t), were computed for the validation input signal, resulting in an output spectrum as shown in Figure 8.8(b). With the predistorted spectrum in Figure 8.8(b), a small asymmetry can be observed, which was expected due to the asymmetrical frequency spectrum of the reference signal. With predistortion, the measured aclr at 5 MHz oﬀset was -43.5 dBc. Thus, the measured aclr at 5 MHz oﬀset was improved by 9.4 dB. The average power at 1.95 GHz was +23.3 dBm with 8.0 % pae. Figure 8.9 shows the measured am-am and am-pm characteristics with and without dpd for the lte signal. The amplitude mapping in the upper ﬁgure should ideally be a straight line from the lower left corner to the upper right one, and the bottom ﬁgure should be zero always. The ﬁgure shows that the amplitude and phase errors are signiﬁcantly reduced for small amplitudes, with a normalized amplitude |s| 0.4. 112 8 Predistortion Measurement Results (a) (b) Figure 8.7: (a) Measured am-am characteristics (output amplitude vs. input amplitude) with dpd (black) and without dpd (gray) for wcdma signal. (b) Measured am-pm characteristics (phase change vs. input amplitude) with dpd (black) and without dpd (gray) for wcdma signal. 8.4 Evaluation of Least Squares PA and Analytical Inversion Method 113 Figure 8.8: Measured lte spectrum at 1.95 GHz. (a) Measured lte spectrum without dpd. The measured aclr is printed in gray. (b) When dpd is applied to (a). The measured aclr is printed in black. (c) Spectrum of estimation signal. Spectrum of validation signal was similar. 114 8 Predistortion Measurement Results (a) (b) Figure 8.9: (a) Measured am-am characteristics (output amplitude vs. input amplitude) with dpd (black) and without dpd (gray) for lte signal. (b) Measured am-pm characteristics (phase change vs. input amplitude) with dpd (black) and without dpd (gray) for lte signal. 8.4 Evaluation of Least Squares PA and Analytical Inversion Method 115 Figure 8.10: Measured aclr depending on the polynomial degree n of the pa model. Degree n = 0 represents the performance without predistortion. The nonlinear modeling and distortion clearly improves the performance by reducing the aclr. 8.4.3 Evaluation of Polynomial Degree A small evaluation of the impact of polynomial degree in the pa model has been performed, and the result is presented in Figure 8.10. It is clear that the added nonlinear terms improves the aclr and reduces the spectral leakage. Polynomials with orders above n = 5 did not further improve the results signiﬁcantly. A discussion on the impact of the choice of data points used in the ls problem can be found in Section 6.4 on page 79. 8.4.4 Summary The measured performance of the pa for modulated signals is summarized in Table 8.6. The table shows measured aclr with dpd, without dpd, and the required (Req) aclr for the wcdma [3GP] and the lte [3GPP] standards. In measurements at 1.95 GHz, the dpd proved to be successful and improved the wcdma aclr at 5 MHz and 10 MHz oﬀsets by 10.8 dB and 7.5 dB, respectively. The lte aclr at 5 MHz oﬀset was improved by 9.4 dB. Thus, the predistortion method improves the measured aclr to have at least 12.6 dB of margin to the requirements [3GP, 3GPP]. The measured aclr at 5 MHz is comparable to stateof-the-art wcdma transceivers [Huang et al., 2010]. To compare the dpd performance to the achievable aclr, a small simulation study has been performed. Assuming a pa with 35 dB of dynamic range (neglecting phase distortions), i.e. assuming g1 = 0.509 and g2 = 0.491, and a polynomial degree of n = 5, the computed achievable aclr at 5 MHz and 10 MHz is ∼3 dB better compared to the measurements with the wcdma signal. Similarly, the computed achievable aclr at 5 MHz is ∼2 dB better compared to the measurements with the lte signal. 116 8 Predistortion Measurement Results Table 8.6: Measured Spectral Performance at 1.95 GHz for wcdma and lte Uplink Signals with Predistortion (using n = 5) and without. wcdma lte Measured Parameter aclr @ 5 MHz [dBc] aclr @ 10 MHz [dBc] aclr @ 5 MHz [dBc] Req -33 -43 -30 Without dpd -35.5 -48.1 -34.1 With dpd -46.3 -55.6 -43.5 As discussed in Section 6.4 on page 79, the polynomial ﬁt is best in the middle, and in intervals where there is most data points. For the signals in this thesis, that is in the center of the interval, see Figure 8.2 for the distribution of the different signal types used. As seen in Figures 8.7 and 8.9, this is where the predistorter improves the performance. The predistorter is based on inversion of the pa models estimated using least squares. Since the inversion is almost perfect, see Figure 7.2 for the analytical inversion, the misﬁt at the smallest and largest input amplitudes can be assumed to be correlated with the polynomial ﬁt of the pa model. The nonlinearity functions can be compared for diﬀerent signal types, and though the overall appearance is very similar, a small shift can be seen, such that the ﬁt has been adapted to the signal type. That is, for an lte signal, the functions fˆk diﬀer a bit from the ones estimated for a wcdma signal. This can be seen for lower amplitudes in particular, where the lte signal has a higher signal density than the wcdma. 9 Concluding Remarks In this chapter, conclusions and some discussions on possible research ideas for the future are provided. 9.1 Conclusions In this thesis, some diﬀerent aspects concerning the estimation of inverse models have been discussed. In system identiﬁcation, the model should typically be estimated in the setting in which it will be used. This idea has been further investigated for the inverse model estimation, where diﬀerent approaches are used in applications. Here, an inverse model has been estimated with the purpose of using it in cascade with the system itself, as an inverter. A good inverse model in this setting would be one that, when used in series with the original system, reconstructs the original input. This problem has been treated in applications, such as power ampliﬁer predistortion, but theoretical insights as to how it should be done have been lacking. Diﬀerent methods can lead to good results, but it can be shown that the characteristics captured by the methods diﬀer. It is important to know how the choice of method eﬀects the inverse model. For a noise-free linear timeinvariant system, it is shown here that the weighting of the identiﬁcation will be adjusted to better reﬂect this intended use when the inverse is estimated directly instead of based on a forward model. This has also been illustrated by a small example. Inverse systems are used in many applications, and here, outphasing power ampliﬁer predistortion has been investigated. The goal is to obtain a predistorter that counteracts the nonlinearities introduced by the ampliﬁer. In outphasing pas, the signal is decomposed into two branches, so that highly eﬃcient, nonlinear ampliﬁers can be used. This structure makes it hard to use conventional 117 118 9 Concluding Remarks predistortion methods, but enables a theoretical description of the outphasing pa and the matching ideal predistorter. Here, a ﬁrst method based on two nonconvex optimization problems has been further developed using the structure of the outphasing ampliﬁer. The improved method basically consists of two leastsquares problems and an analytic inversion, and can be adapted to online implementation. It has been shown that the methods reduce the nonlinearities and the leakage into adjacent channels. 9.2 Open Questions For ampliﬁer predistortion, an interesting extension to the methods presented here is to include dynamics. In this thesis, two diﬀerent approaches have been mentioned that did not improve the modeling results. Since nonlinear systems is “everything that is not linear”, there are many ways to include nonlinear dynamics, and one of them not working (improving the modeling) does not mean that some other way will not. Even though the measurements did not indicate a large dynamic inﬂuence, extending the method to include possible dynamics would extend the ﬁeld of application. Since the measurements were performed in a rather ideal setup and then averaged over multiple realizations, the noise inﬂuence was minor, but the inﬂuence under less ideal conditions could be evaluated. Now, the noise has a large impact in the normalization and the estimation of gain factors in the two branches, since these depend on only one and two measurements, respectively. This could be made more robust by looking at multiple measurements. The construction of a least-squares method is one step towards a possible online implementation, but further adaption could also be done. This includes the choice of whether the dpd should be implemented as a polynomial or in a lut solution. Furthermore, if the method can be allowed to use some calibration time to adapt the parameters to the device at hand, this calls for another solution than if the method has to ﬁnd the parameters during operation. However, this would more concern hardware implementation questions. Concerning the research on estimation of inverse systems, only a small ﬁrst investigation has been performed, and many open questions remain. These include a more thorough analysis of the properties of the estimators, as well as the noise inﬂuence in the diﬀerent approaches. An extension to the nonlinear case would also be an interesting, but challenging topic. A Power Ampliﬁer Implementation The outphasing power ampliﬁers used for the measurements presented in Chapter 8 and the power ampliﬁer modeling in Chapter 6 have been constructed by Jonas Fritzin, Christer Svensson and Atila Alvandpour at the Division of Electronic Devices, Linköping University, Linköping, Sweden. The results and pictures in this chapter are all measured and reproduced with the authors’ permission and are published here for sake of completeness. As described in Section 5.2, a power ampliﬁer can be characterized by diﬀerent measures, such as the eﬃciency and the gain. For the pa beginner, a quick review of these concepts and the others in Section 5.2 could be useful. See also the Glossary in the preamble (page xvi). The power ampliﬁers are of outphasing-type. The ampliﬁer in each branch is a Class D ampliﬁer, based on inverters, that switches between VDD and GND. A.1 +10.3 dBm Class-D Outphasing RF Ampliﬁer in 90 nm CMOS The chip used for validation of the nonconvex method in Section 8.3 can be seen in the chip photo in Figure A.1 and the sketch in Figure A.2. The pa is a Class D outphasing ampliﬁer with an inverter-based output stage and an on-chip transformer as power combiner. More speciﬁcs can be found in Fritzin [2011] and Fritzin et al. [2011a]. Figure A.3a shows the measured maximum output power (Pout ), the drain eﬃciency (de) and the power-added eﬃciency (pae) over frequency for the power ampliﬁer. VDD and Vbias were 1.3 V and 0.65 V, respectively. The 3 dB bandwidth was 2 GHz (1-3 GHz). The output power at 2 GHz was +10.3 dBm with de and pae of 39 % and 33 %, respectively, with a gain of 23 dB from the buﬀers to the 119 120 A Power Ampliﬁer Implementation Figure A.1: Photo of the chip with size 1x1mm2 . Figure A.2: Implemented outphasing ampliﬁer with inverters in the output stage. 121 +30 dBm Class-D Outphasing RF Ampliﬁer in 65 nm CMOS 12.5 10 Pout [dBm] 50 ← Pout 40 DE → 7.5 30 PAE → 5 20 2.5 10 0 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 Carrier frequency [GHz] 3 PAE, DE [%] A.2 0 (a) 50 10 45 ←P out,max 0 40 −10 35 −20 30 −30 −40 −50 0.75 DR → ← Pout,min 1 25 20 1.25 1.5 1.75 2 2.25 2.5 2.75 Carrier frequency [GHz] 3 Dynamic range [dB] Output power [dBm] 20 15 (b) Figure A.3: (a) Measured output power (Pout ), de and pae over frequency. (b) Measured maximum output power, Pout,max , minimum output power, Pout,min , and dynamic range, dr, over frequency. output. The minimum and maximum output power and dr of the pa are plotted in Figure A.3b, where Pout,max = Pout in Figure A.3a. A.2 +30 dBm Class-D Outphasing RF Ampliﬁer in 65 nm CMOS The pa used for validation in Section 8.4 is described in more detail in Fritzin et al. [2011c] , but some basic characteristics can be found here. The chip photo can be seen in Figure A.4. Figure A.5 shows the outphasing pa, based on a Class D ampliﬁer stage utilizing a cascode conﬁguration illustrated in Figure A.6a. This conﬁguration improves the life-time of the transistors by achieving a low on-resistance in the on-state and distributing the voltage stress in the oﬀ state which assures that the root mean square (rms) electric ﬁelds across the gate oxide is kept low. The output stage is driven by an ac-coupled low-voltage driver operating at 1.3 V, VDD1 , to allow a 5.5 V, VDD2 , supply without excessive device voltage stress as discussed in Fritzin et al. [2011b] and Fritzin et al. [2011c]. The chip was attached 122 A Power Ampliﬁer Implementation Figure A.4: Photo of the chip with size 2.5x1.0mm2 . The photo has the same orientation as the simpliﬁed PA schematic in Figure A.5. Figure A.5: The implemented Class-D outphasing RF PA using two transformers to combine the outputs of four ampliﬁer stages. to an FR4 PCB and connected with bond-wires. The measured output power, drain eﬃciency and power-added eﬃciency over frequency and outphasing angle, ϕ in (5.11) (where ϕ = 2Δψ ), for VDD1 = 1.3 V and VDD2 = 5.5 V is shown in Figures A.7. The output power at 1.95 GHz was +29.7 dBm with a pae of 26.6 % (including all drivers). The pa had a peak to minimum power ratio of ∼35 dB and the gain was 26 dB from the drivers to the output. The dc power consumption of the smallest drivers was considered as input power. A.2 +30 dBm Class-D Outphasing RF Ampliﬁer in 65 nm CMOS (a) 123 (b) Figure A.6: (a) The Class-D stage used in the outphasing PA Fritzin et al. [2011c]. C1 -C4 are MIM capacitors. 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Thesis No. 1599, 2013.

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