Analog-to-Digital Conversion

Marcel J.M. Pelgrom

Analog-to-DigitalConversion

Marcel J.M. PelgromNXP SemiconductorsHTC-32Eindhoven 5656AEThe Netherlandsmarcel.pelgrom@nxp.com

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Preface

A book is like a window that allows you to look into the world. The window isshaped by the author and that makes that every window presents a unique view ofthe world. This is certainly true for this book. It is shaped by the topics and theprojects throughout my career. Even more so, this book reflects my own style ofworking and thinking.

That starts already in Chap. 2. When I joined Philips Research in 1979, many ofmy colleagues used little paper notebooks to keep track of the most used equationsand other practical things. This notebook was the beginning for Chap. 2: a collectionof topics that form the basis for much of the other chapters. Chapter 2 is not intendedto explain these topics, but to refresh your knowledge and help you when you needsome basics to solve more complex issues.

In the chapters discussing the fundamental processes of conversion, you will rec-ognize my preoccupation with mathematics. I really enjoy finding an equation thatproperly describes the underlying mechanism. Nevertheless mathematics is not agoal on its own: the equations help to understand the way the variables are connectedto the result. Real insight comes from understanding the physics and electronics. Inthe chapters on circuit design I have tried to reduce the circuit diagrams to the sim-plest form, but not simpler. . . I do have private opinions on what works and whatshould not be applied. Most poor solutions have simply been left out, sometimesyou might read a warning in the text on a certain aspect of an interesting circuit.

Another of my favorites is the search for accuracy. In Chap. 11 you will find a de-tailed description, but also in the earlier chapters, there is a lot of material referringto accuracy.

Circuit design and analog-to-digital circuit design is about bridging the gap be-tween technology and systems. Both aspects have been treated less than they de-serve. Still I hope it will be sufficient to create an interest to probe further.

This book is based on my lectures for graduate students who are novice in analog-to-digital design. In the classes my aim is to bring the students to a level where theycan read and interpret the literature (such as IEEE Journal of Solid-State Circuits)and judge the reported results on their merits. Still that leaves a knowledge gap withthe designer of analog-to-digital converters. For those designers this book may serveas a reference of principles and background.

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Inevitably this book has some strong points but also weak points. There are stillso many wonderful ideas, that are not addressed here but certainly would deservesome space, but simply did not fit in this volume. Still I hope this book will letyou experience the same thrill that all analog-to-digital designers feel, when theytalk about their passion. Because that is the goal of this book: to encourage you toproceed on the route towards even better analog-to-digital converters.

Acknowledgments

Archimedes said: “Give me one fixed point and I will move the Earth”. Home hasalways served for me as the fixed point from which I could move forward in mywork. I owe my wife Elisabeth a debt of gratitude for creating a wonderful home.She herself was once part of this semiconductor world and understands its crazyhabits. Yet, the encouragement and support she gave me is invaluable.

This book reflects parts of my 30 years of work in the Philips Natuurkundig Lab-oratorium and its successor. If there is anything I would call “luck” in my life, itwas the opportunity to work in this place. The creativity, energy, opportunities, andpeople in this laboratory are unique. It is not trivial to create such research freedomin a financially driven industry. My seven years as a mixed-signal department headhave taught me that. Therefore I am truly grateful to those who served in the man-agement of Philips Research and backed me when working on things outside theproject scope or looking in unusual directions. Just naming here: Theo van Kessel,Kees Wouda, Gerard Beenker, Hans Rijns and Leo Warmerdam.

A laboratory is just as good as the people that work in it. In my career I met a lotof extraordinary people. They formed and shaped my way of thinking and analyz-ing problems. They challenged my ideas, took the time to listen to my reasoning andpointed me in promising directions. I am grateful for being able to use the insightsand results of the Mixed-signal circuits and systems group. Without the useful dis-cussions and critical comments of the members of this group this book would notexist. However, there are many more colleagues that have contributed in some form.Without the illusion of being complete, I want to express my gratitude for a pleasantcollaboration with: Carel Dijkmans, Rudy van der Plassche, Eduard Stikvoort, Robvan der Grift, Arthur van Roermund, Erik van der Zwan, Peter Nuijten, Ed van Tu-ijl, Maarten Vertregt, Pieter Vorenkamp, Johan Verdaasdonk, Anton Welbers, AadDuimaijer, Jeannet van Rens, Klaas Bult, Govert Geelen, Stephane Barbu, LaurentGiry, Robert Meyer, Othmar Pfarkircher, Ray Speer, John Jennings, Joost Briaire,Raf Roovers, Lucien Breems, Robert van Veldhoven, Kathleen Philips, Bram Nauta,Hendrik van der Ploeg, Kostas Doris, Erwin Janssen, Robert Rutten, Violeta Pe-trescu, Harry Veendrick, Hans Tuinhout, Jan van der Linde, Peter van Leeuwen andmany others.

This book is based on the lectures in the Philips Center for Technical Training,at universities and in the MEAD/EPFL courses. I want to thank prof. Bram Nautaand prof. Kofi Makinwa for giving me the opportunity to teach at the universitiesof Twente and Delft, prof. Bruce Wooley and prof. Boris Murmann of Stanford

Preface vii

University for their collaboration, prof. Gabor Temes and dr. Vlado Valence forinviting me to lecture in the MEAD and EPFL courses.

A special word of thanks goes to all the students for their questions, remarks andstimulating discussions.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 About this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Components and Definitions . . . . . . . . . . . . . . . . . . . . . . . 52.1 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . 92.1.2 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 152.1.5 The z-transform . . . . . . . . . . . . . . . . . . . . . . . 192.1.6 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.2 Voltage and Temperature Coefficient . . . . . . . . . . . . 292.2.3 Measuring Resistance . . . . . . . . . . . . . . . . . . . . 292.2.4 Electromigration . . . . . . . . . . . . . . . . . . . . . . . 302.2.5 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.2 Energy in a Coil . . . . . . . . . . . . . . . . . . . . . . . 382.3.3 Straight Wire Inductance . . . . . . . . . . . . . . . . . . 382.3.4 Skin Effect and Eddy Current . . . . . . . . . . . . . . . . 402.3.5 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.6 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3.7 Energy in Capacitors . . . . . . . . . . . . . . . . . . . . . 432.3.8 Partial Charging . . . . . . . . . . . . . . . . . . . . . . . 442.3.9 Digital Power Consumption . . . . . . . . . . . . . . . . . 452.3.10 Coaxial Cable . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.1 Semiconductor Resistivity . . . . . . . . . . . . . . . . . . 482.4.2 Voltage and Temperature Coefficient . . . . . . . . . . . . 49

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2.4.3 Matching of Resistors . . . . . . . . . . . . . . . . . . . . 502.4.4 MOS Capacitance . . . . . . . . . . . . . . . . . . . . . . 512.4.5 Capacitance Between Layers . . . . . . . . . . . . . . . . 542.4.6 Voltage and Temperature Coefficient . . . . . . . . . . . . 562.4.7 Matching of Capacitors . . . . . . . . . . . . . . . . . . . 562.4.8 The pn-junction . . . . . . . . . . . . . . . . . . . . . . . 562.4.9 The Bipolar Transistor . . . . . . . . . . . . . . . . . . . . 60

2.5 The MOS Transistor . . . . . . . . . . . . . . . . . . . . . . . . . 622.5.1 Weak Inversion . . . . . . . . . . . . . . . . . . . . . . . 662.5.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5.3 Drain Voltage Influence . . . . . . . . . . . . . . . . . . . 692.5.4 Large Signal and Small Signal . . . . . . . . . . . . . . . 702.5.5 High-frequency Behavior . . . . . . . . . . . . . . . . . . 712.5.6 Gate Leakage . . . . . . . . . . . . . . . . . . . . . . . . 732.5.7 Temperature Coefficient . . . . . . . . . . . . . . . . . . . 732.5.8 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5.9 Latch-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.5.10 Enhancement and Depletion . . . . . . . . . . . . . . . . . 772.5.11 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.6 Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.6.1 Kirchhoff’s Laws . . . . . . . . . . . . . . . . . . . . . . 792.6.2 Two-port Networks . . . . . . . . . . . . . . . . . . . . . 802.6.3 Energy and Power . . . . . . . . . . . . . . . . . . . . . . 812.6.4 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.6.5 Opamps and OTAs . . . . . . . . . . . . . . . . . . . . . . 862.6.6 Differential Design . . . . . . . . . . . . . . . . . . . . . 882.6.7 Switched-capacitor Circuits . . . . . . . . . . . . . . . . . 902.6.8 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.7 Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 982.7.1 Classification of Amplifiers . . . . . . . . . . . . . . . . . 982.7.2 One-transistor Amplifier . . . . . . . . . . . . . . . . . . . 1002.7.3 The Inverter . . . . . . . . . . . . . . . . . . . . . . . . . 1022.7.4 Source Follower . . . . . . . . . . . . . . . . . . . . . . . 1032.7.5 The Differential Pair . . . . . . . . . . . . . . . . . . . . . 1042.7.6 Degeneration . . . . . . . . . . . . . . . . . . . . . . . . . 1072.7.7 Current Mirror . . . . . . . . . . . . . . . . . . . . . . . . 1072.7.8 Darlington Pair . . . . . . . . . . . . . . . . . . . . . . . . 1092.7.9 Cascode and Regulated Cascode . . . . . . . . . . . . . . 1102.7.10 Single-stage Amplifier . . . . . . . . . . . . . . . . . . . . 1132.7.11 Miller Amplifier . . . . . . . . . . . . . . . . . . . . . . . 1142.7.12 Choosing the W/L Ratios in a Miller Opamp . . . . . . . . 1182.7.13 Dominant-pole Amplifier . . . . . . . . . . . . . . . . . . 1192.7.14 Feedback in Electronic Circuits . . . . . . . . . . . . . . . 1202.7.15 Bias Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 1222.7.16 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.1 Sampling in Time and Frequency . . . . . . . . . . . . . . . . . . 133

3.1.1 Folding Back of Spectra . . . . . . . . . . . . . . . . . . . 1373.1.2 Sampling and Modulation . . . . . . . . . . . . . . . . . . 1403.1.3 Sampling of Noise . . . . . . . . . . . . . . . . . . . . . . 1413.1.4 Jitter of the Sampling Pulse . . . . . . . . . . . . . . . . . 143

3.2 Time-discrete Filtering . . . . . . . . . . . . . . . . . . . . . . . . 1463.2.1 FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.2.2 Half-band Filters . . . . . . . . . . . . . . . . . . . . . . . 1513.2.3 Down Sample Filter . . . . . . . . . . . . . . . . . . . . . 1523.2.4 IIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4 Sample and Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.1 Track-and-Hold and Sample-and-Hold Circuits . . . . . . . . . . . 1554.2 Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.3 Capacitor and Switch Implementations . . . . . . . . . . . . . . . 160

4.3.1 Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.3.2 Switch Topologies . . . . . . . . . . . . . . . . . . . . . . 1614.3.3 Bottom Plate Sampling . . . . . . . . . . . . . . . . . . . 1644.3.4 The CMOS Bootstrap Technique . . . . . . . . . . . . . . 1654.3.5 Buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.4 Track-and-Hold Circuit Topologies . . . . . . . . . . . . . . . . . 1684.4.1 Basic Configurations . . . . . . . . . . . . . . . . . . . . . 1684.4.2 Amplifying Track-and-Hold Circuit . . . . . . . . . . . . . 1714.4.3 Correlated Double Sampling . . . . . . . . . . . . . . . . 1714.4.4 A Bipolar Example . . . . . . . . . . . . . . . . . . . . . 1724.4.5 Distortion and Noise . . . . . . . . . . . . . . . . . . . . . 173

5 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.1.1 Integral Linearity . . . . . . . . . . . . . . . . . . . . . . 1775.1.2 Differential Linearity . . . . . . . . . . . . . . . . . . . . 178

5.2 The Quantization Error . . . . . . . . . . . . . . . . . . . . . . . 1805.2.1 One-bit Quantization . . . . . . . . . . . . . . . . . . . . 1805.2.2 2–6 bit Quantization . . . . . . . . . . . . . . . . . . . . . 1815.2.3 7-bit and Higher Quantization . . . . . . . . . . . . . . . . 183

5.3 Signal-to-Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.3.1 Related Definitions . . . . . . . . . . . . . . . . . . . . . 1875.3.2 Non-uniform Quantization . . . . . . . . . . . . . . . . . . 1875.3.3 Dither . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.3.4 DNL and SNR . . . . . . . . . . . . . . . . . . . . . . . . 189

6 Reference Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.1 General Requirements . . . . . . . . . . . . . . . . . . . . . . . . 1916.2 Bandgap Reference Circuits . . . . . . . . . . . . . . . . . . . . . 192

6.2.1 Bipolar Bandgap Circuit . . . . . . . . . . . . . . . . . . . 196

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6.2.2 CMOS Bandgap Circuit . . . . . . . . . . . . . . . . . . . 1976.2.3 Low-voltage Bandgap Circuits . . . . . . . . . . . . . . . 200

6.3 Alternative References . . . . . . . . . . . . . . . . . . . . . . . . 201

7 Digital-to-Analog Conversion . . . . . . . . . . . . . . . . . . . . . . 2037.1 Unary and Binary Representation . . . . . . . . . . . . . . . . . . 203

7.1.1 Digital Representation . . . . . . . . . . . . . . . . . . . . 2057.1.2 Physical Domain . . . . . . . . . . . . . . . . . . . . . . . 207

7.2 Digital-to-Analog Conversion Schemes . . . . . . . . . . . . . . . 2097.2.1 DA Conversion in the Voltage Domain . . . . . . . . . . . 2097.2.2 R-2R Ladders . . . . . . . . . . . . . . . . . . . . . . . . 2137.2.3 Digital-to-Analog Conversion in the Current Domain . . . 2147.2.4 Semi-digital Filter/Converters . . . . . . . . . . . . . . . . 2207.2.5 DA Conversion in the Charge Domain . . . . . . . . . . . 2217.2.6 DA Conversion in the Time Domain . . . . . . . . . . . . 2247.2.7 Class-D Amplifiers . . . . . . . . . . . . . . . . . . . . . 227

7.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.3.1 Limits to Accuracy . . . . . . . . . . . . . . . . . . . . . 227

7.4 Methods to Improve Accuracy . . . . . . . . . . . . . . . . . . . . 2317.4.1 Current Calibration . . . . . . . . . . . . . . . . . . . . . 2337.4.2 Dynamic Element Matching . . . . . . . . . . . . . . . . . 2347.4.3 Data-weighted Averaging . . . . . . . . . . . . . . . . . . 235

7.5 Digital-to-Analog Conversion: Implementation Examples . . . . . 2397.5.1 Resistor Ladder Digital-to-Analog Converter . . . . . . . . 2397.5.2 Current Domain Digital-to-Analog Conversion . . . . . . . 2427.5.3 A Comparison . . . . . . . . . . . . . . . . . . . . . . . . 2437.5.4 An Algorithmic Charge-based Digital-to-Analog Converter 244

8 Analog-to-Digital Conversion . . . . . . . . . . . . . . . . . . . . . . 2498.1 The Comparator . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

8.1.1 The Dynamics of Transistor Comparator . . . . . . . . . . 2538.1.2 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . 2548.1.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 2568.1.4 Metastability and Bit-Error Rate . . . . . . . . . . . . . . 2588.1.5 Kick-back . . . . . . . . . . . . . . . . . . . . . . . . . . 2598.1.6 Comparator Schematics . . . . . . . . . . . . . . . . . . . 2608.1.7 Auto-zero Comparators . . . . . . . . . . . . . . . . . . . 262

8.2 Full-flash Converters . . . . . . . . . . . . . . . . . . . . . . . . . 2648.2.1 Ladder Implementation . . . . . . . . . . . . . . . . . . . 2678.2.2 Comparator Yield . . . . . . . . . . . . . . . . . . . . . . 2678.2.3 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . 2728.2.4 Averaging and Interpolation . . . . . . . . . . . . . . . . . 2748.2.5 Technology Scaling for Full-flash Converters . . . . . . . . 2778.2.6 Folding Converter . . . . . . . . . . . . . . . . . . . . . . 277

8.3 Sub-ranging Methods . . . . . . . . . . . . . . . . . . . . . . . . 280

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8.4 Pipeline Converters . . . . . . . . . . . . . . . . . . . . . . . . . 2848.4.1 Error Sources in Pipeline Converters . . . . . . . . . . . . 286

8.4.2 Digital Calibration . . . . . . . . . . . . . . . . . . . . . . 2888.5 1.5 Bit Pipeline Analog-to-Digital Converter . . . . . . . . . . . . 289

8.5.1 Design of a Stage . . . . . . . . . . . . . . . . . . . . . . 2918.5.2 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . 2938.5.3 Pipeline Variants . . . . . . . . . . . . . . . . . . . . . . . 294

8.6 Successive Approximation Converters . . . . . . . . . . . . . . . 2968.6.1 Charge-redistribution Conversion . . . . . . . . . . . . . . 2988.6.2 Algorithmic Converters . . . . . . . . . . . . . . . . . . . 300

8.7 Linear Approximation Converters . . . . . . . . . . . . . . . . . . 3048.8 Time-interleaving Time-discrete Circuits . . . . . . . . . . . . . . 3058.9 An Implementation Example . . . . . . . . . . . . . . . . . . . . 308

8.9.1 An Auto-zero Comparator . . . . . . . . . . . . . . . . . . 3098.9.2 Full-flash Analog-to-Digital Converter . . . . . . . . . . . 3108.9.3 Successive-approximation Analog-to-Digital Converter . . 3118.9.4 Multi-step Analog-to-Digital Converter . . . . . . . . . . . 3128.9.5 A Comparison . . . . . . . . . . . . . . . . . . . . . . . . 313

8.10 Other Conversion Ideas . . . . . . . . . . . . . . . . . . . . . . . 3148.10.1 Level-crossing Analog-to-Digital Conversion . . . . . . . . 3148.10.2 Asynchronous Conversion . . . . . . . . . . . . . . . . . . 3158.10.3 Time-related Conversion . . . . . . . . . . . . . . . . . . . 3168.10.4 The Vernier/Nonius Principle . . . . . . . . . . . . . . . . 3188.10.5 The Floating-point Converter . . . . . . . . . . . . . . . . 318

9 Sigma-delta Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 3219.1 Oversampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.2 Noise Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3259.3 Sigma-delta Modulation . . . . . . . . . . . . . . . . . . . . . . . 329

9.3.1 Sigma-delta Digital-to-Analog Conversion . . . . . . . . . 3349.4 Time-discrete Sigma-delta Modulation . . . . . . . . . . . . . . . 334

9.4.1 A First Order Modulator . . . . . . . . . . . . . . . . . . . 3349.4.2 A Second Order Modulator . . . . . . . . . . . . . . . . . 3379.4.3 Cascaded Sigma-delta Modulator . . . . . . . . . . . . . . 339

9.5 Time-continuous Sigma-delta Modulation . . . . . . . . . . . . . 3419.5.1 A First-order Modulator . . . . . . . . . . . . . . . . . . . 3419.5.2 Higher Order Sigma-delta Converters . . . . . . . . . . . . 3459.5.3 Time-discrete and Time-continuous Sigma Delta

Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 3489.6 Multi-bit Sigma-delta Conversion . . . . . . . . . . . . . . . . . . 3509.7 Various Forms of Sigma-delta Modulation . . . . . . . . . . . . . 353

9.7.1 Complex Sigma-delta Modulation . . . . . . . . . . . . . . 3539.7.2 Asynchronous Sigma-delta Modulation . . . . . . . . . . . 3539.7.3 Input Feed-forward Modulator . . . . . . . . . . . . . . . 3549.7.4 Band-pass Sigma-delta Converter . . . . . . . . . . . . . . 355

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9.7.5 Sigma Delta Loop with Noise-shaping . . . . . . . . . . . 3569.7.6 Incremental Sigma-delta Converter . . . . . . . . . . . . . 356

10 Characterization and Specification . . . . . . . . . . . . . . . . . . . 35910.1 The Test Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . 35910.2 Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . 363

10.2.1 INL and DNL . . . . . . . . . . . . . . . . . . . . . . . . 36310.2.2 Harmonic Behavior . . . . . . . . . . . . . . . . . . . . . 365

10.3 Self Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

11 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36911.1 Technology Roadmap . . . . . . . . . . . . . . . . . . . . . . . . 369

11.1.1 Power Supply and Signal Swing . . . . . . . . . . . . . . . 37011.1.2 Feature Size . . . . . . . . . . . . . . . . . . . . . . . . . 37111.1.3 Process Options . . . . . . . . . . . . . . . . . . . . . . . 372

11.2 Variability: an Overview . . . . . . . . . . . . . . . . . . . . . . . 37311.3 Deterministic Offsets . . . . . . . . . . . . . . . . . . . . . . . . 375

11.3.1 Offset Caused by Electrical Differences . . . . . . . . . . . 37611.3.2 Offset Caused by Lithography . . . . . . . . . . . . . . . . 37711.3.3 Proximity Effects . . . . . . . . . . . . . . . . . . . . . . 37811.3.4 Temperature Gradients . . . . . . . . . . . . . . . . . . . . 38011.3.5 Offset Caused by Stress . . . . . . . . . . . . . . . . . . . 38111.3.6 Offset Mitigation . . . . . . . . . . . . . . . . . . . . . . . 385

11.4 Random Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 38611.4.1 Random Fluctuations in Devices . . . . . . . . . . . . . . 38611.4.2 MOS Threshold Mismatch . . . . . . . . . . . . . . . . . 38911.4.3 Current Mismatch in Strong and Weak Inversion . . . . . . 39211.4.4 Mismatch for Various Processes . . . . . . . . . . . . . . . 39411.4.5 Application to Other Components . . . . . . . . . . . . . . 39611.4.6 Modeling Remarks . . . . . . . . . . . . . . . . . . . . . . 397

11.5 Consequences for Design . . . . . . . . . . . . . . . . . . . . . . 39811.5.1 Analog design . . . . . . . . . . . . . . . . . . . . . . . . 39811.5.2 Digital Design . . . . . . . . . . . . . . . . . . . . . . . . 39911.5.3 Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40011.5.4 Limits of Power and Accuracy . . . . . . . . . . . . . . . 401

11.6 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40311.7 Substrate Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

12 System Aspects of Conversion . . . . . . . . . . . . . . . . . . . . . . 41312.1 System Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

12.1.1 Specification of Functionality . . . . . . . . . . . . . . . . 41612.1.2 Signal Processing Strategy . . . . . . . . . . . . . . . . . 41812.1.3 Input Circuits . . . . . . . . . . . . . . . . . . . . . . . . 42012.1.4 Conversion of Modulated Signals . . . . . . . . . . . . . . 422

12.2 Comparing Converters . . . . . . . . . . . . . . . . . . . . . . . . 42312.3 Limits of Conversion . . . . . . . . . . . . . . . . . . . . . . . . 427

Contents xv

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4331 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4332 Components and Definitions . . . . . . . . . . . . . . . . . 4333 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4364 Sample-and-Hold . . . . . . . . . . . . . . . . . . . . . . . 4365 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 4376 Reference Circuits . . . . . . . . . . . . . . . . . . . . . . . 4377 Digital-to-Analog Conversion . . . . . . . . . . . . . . . . . 4388 Analog-to-Digital Conversion . . . . . . . . . . . . . . . . . 4409 Sigma-delta Conversion . . . . . . . . . . . . . . . . . . . . 443

10 Characterization and Specification . . . . . . . . . . . . . . 44411 Physical Restrictions . . . . . . . . . . . . . . . . . . . . . 44412 System Aspects . . . . . . . . . . . . . . . . . . . . . . . . 446

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

List of Symbols

A General variableA Area [cm2]C Capacitance [F]Cox Oxide capacitance [F/cm2]Dn Diffusion coefficient of electrons [cm2/s]dox Oxide thickness [cm = 108 Å]E Electric field [V/cm]EFn Fermi energy level of electrons [eV]EFp Fermi energy level of holes [eV]EG Band gap energy [1.205 eV]Ei Energy level of an intrinsic semiconductor [eV]f Frequency [Hz]fc Clock frequency [Hz]fi Frequency of input signal [Hz]fs Sample rate [Hz]H Transfer function [1]I Large signal or DC current [A]i Small signal current [A]J Current density [A/cm2]Jn Electron current density [A/cm2]Jp Hole current density [A/cm2]K Substrate voltage influence on the threshold voltage [

√V]

k Boltzmann’s constant [1.38 × 10−23 J/K]L Length of transistor gate [cm]Lw Inductance of a wire [H]M Multiplex factor [1]N Resolution [1]Na Substrate doping concentration [cm−3]n Volume density of electrons [cm−3]ni Intrinsic charge volume density [1.4 × 1010 cm−3 (300 K)]

p Volume density of holes [cm−3]

xvii

xviii List of Symbols

pp0 Volume density of holes in a p-substrate in equilibrium [cm−3]R Resistance [�]Q Charge [C]q Electron charge [1.6 × 10−19 C]T Time period [s]T Absolute temperature [K or (T − 273.15) °Celsius]T0 Reference temperature [K]Ts Sample period [s]t Time as a running variable [s]V Bias or DC potential [V]v Small signal voltage [V]VDD Positive power supplyVDS Drain potential with respect to the source potential [V]VFB Flat-band voltage [V]VG Gate potential with respect to ground [V]VGS Gate potential with respect to the source potential [V]VT MOS transistor threshold voltage [V]W Width of transistor channel [cm]X,Y General input and output variable [1]x Dimension perpendicular to the wafer surface [cm]y Dimension parallel to the transistor current [cm]Z Complex impedance [�]β Current gain factor of MOS transistor: Wβ�/L [A/V2]β� Current gain factor of a square MOS transistor [A/V2]ε Permittivity in vacuum [8.854 × 10−14 F/cm]εoxε Permittivity in silicon dioxide [3.45 × 10−13 F/cm]εsε Permittivity in silicon [10.5 × 10−13 F/cm]φF Potential difference between intrinsic and hole Fermi level [V]μ0 Magnetic permeability in vacuum [4π × 10−7 H/m or N/A−2]μn,μp Mobility of electrons and holes [cm2/Vs]π Angular constant [3.14159]ψ Electrostatic potential [V]ψB Electrostatic potential at which strong inversion starts [V]ψs Electrostatic potential at the interface [V]σP Standard deviation of P

σn Electron capture cross-section [5 × 10−15 cm2]τ Time-constant [s]ω = 2πf Angular or radian frequency [rad/s]

Reference Tables and Figures

Table 1.1 Key functions in analog-to-digital conversion 3Table 2.1 Multiplier abbreviations 6Table 2.2 Elementary algebraic functions 6Table 2.5 Goniometrical relations 7Table 2.6 Standard manipulations for derivatives and integrals

of functions 8Table 2.7 Taylor series expansions 8Table 2.8 Fourier series expansions 12Figure 2.4 Distortion relations: HD2, HD3 and IM3 16Table 2.9 Laplace transforms 17Table 2.10 Probability of Gauss distribution 22Figure 2.13 Color coding for discrete resistors 26Table 2.11 Resistivity of (semi-)conductors 27Table 2.14 Electrostatic properties of semiconductors 49Figure 2.24 Resistivity of doped silicon 49Table 2.15 Resistors in 0.18 µm to 90 nm CMOS 50Table 2.16 Diffusion capacitances in a 0.25 µm process and 65 nm CMOS 54Table 2.17 Gate capacitance from 0.8 µm to 65 nm CMOS 54Table 2.18 Passive capacitances for 0.18 µm to 90 nm CMOS 55Table 2.19 Data for vertical pnp transistors 62Figure 2.37 Current factor for various MOS processes 66Table 2.20 Transistor parameters 0.8 µm to 65 nm CMOS 66Figure 2.41 The voltage gain of a transistor versus gate length and process 71Table 2.24 Low-pass filter functions 96Figure 2.63 Amplifiers classes: A, B, AB, C, D, E 99Figure 3.4 Suppression of alias filters 139Figure 3.7 kT /C noise 141Figure 3.11 The signal-to-noise ration as a function of jitter and bandwidth 145Figure 5.4 Definition of Integral non-linearity 178Figure 5.5 Definition of Differential non-linearity 179Table 5.1 Thermal noise and quantization error properties 184

xix

xx Reference Tables and Figures

Equation 5.13 Ratio between signal and quantization error 185Table 7.1 Digital representation 206Figure 7.17 SFDR versus bandwidth for current-steering DACs 219Figure 8.22 Yield on monotonicity versus the standard deviation

of the comparator random offset 269Table 8.1 Mismatch in full-flash conversion 271Figure 9.9 Signal-to-noise gain in noise-shapers and sigma-delta 328Table 10.1 Characterization parameters of analog-to-digital conversion 360Table 11.1 Excerpt from the ITRS 2005 370Table 11.3 Classification of variance 374Table 11.6 Guide lines for low-offset layout 385Figure 11.20 Mismatch factor AVT versus oxide thickness/process

generation 394Table 11.8 Matching coefficients of various devices 396Table 11.10 Package names 405Table 12.1 Analog-to-digital system requirements 416Table 12.4 Power efficiency of ISSCC published converters 425Figure 12.12 Figure of merit of analog-to-digital converters 426Figure 12.15 Limits to analog-to-digital conversion 428

Chapter 1Introduction

Analog-to-digital conversion is everywhere around us. In all forms of electronicequipment the analog-to-digital converter links our physical world to digital com-puting machines. This development has enabled all the marvelous functionality thathas been introduced over the last thirty years, from mobile phone to internet, frommedical imaging machines to hand-held television.

Pure analog electronics circuits can do a lot of signal processing in a cheap andwell-established way. Many signal processing functions are so simple that analogprocessing serves the needs (audio amplification, filtering, radio). In more complexsituations, analog processing however lacks required functionality. There digital sig-nal processing offers crucial extensions of this functionality. The most importantadvantages of digital processing over analog processing are a perfect storage ofdigitized signals, unlimited signal-to-noise ratio, the option to carry out complexcalculations, and the possibility to adapt the algorithm of the calculation to chang-ing circumstances. If an application wants to use these advantages, analog signalshave to be converted with high quality into a digital format in an early stage of theprocessing chain. And at the end of the digital processing the conversion has to becarried out in the reverse direction. The digital-to-analog translates the outcome ofthe signal processing into signals that can be rendered as a picture or sound. Thismakes analog-to-digital conversion a crucial element in the chain between our worldof physical quantities and the rapidly increasing power of digital signal process-ing. Figure 1.1 shows the analog-to-digital converter (abbreviated A/D-converter orADC) as the crucial element in a system with combined analog and digital function-ality.

The analog-to-digital converters and digital-to-analog converters discussed inthis book convert high resolution and high speed signals to and from the digitaldomain. The basics of the conversion process is shown in Fig. 1.2. In the analogdomain a ratio exists between the actual signal and a reference quantity. This ratiois reflected in the digital domain, where the digital code is a fraction of the availableword width. The analog-to-digital converter tries to find an optimum match betweenthese ratios at any moment in time. However, an essential rounding error must beaccepted.

M.J.M. Pelgrom, Analog-to-Digital Conversion,DOI 10.1007/978-90-481-8888-8_1, © Springer Science+Business Media B.V. 2010

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2 1 Introduction

Fig. 1.1 The analog-to-digital and digital-to-analog converters are the ears and eyes of a digitalsystem

Fig. 1.2 In analog-to-digitalconversion a connection ismade between the analogworld of physical quantitiesand the digital world ofnumbers and bits

Fig. 1.3 Functions of the analog-to-digital converter: sampling, quantizing and linking to a refer-ence

Signals in the digital domain differ from analog signals, which exist in the phys-ical world, because digital signal are sampled and quantized, Fig. 1.3. Sampled sig-nals only have meaning at their sample moments as given by the sample frequency.Moreover digital signals are arithmetic quantities, which are only meaningful inthe physical world while there is somewhere an assignment that relates the digitalnumber range to a physical reference value. These three main functions characterizethe analog-to-digital converter, see Table 1.1. These functions will be visible in eachstage of the discussion of analog-to-digital conversion and are reflected in the set-upof this book.

1.1 About this Book 3

Table 1.1 Key functions inanalog-to-digital conversion Analog-to-digital Digital-to-analog

Time discretization Holding the signal

Amplitude discretization Amplitude restoration

Reference to Reference from

A conversion unit A conversion unit

1.1 About this Book

An analog-to-digital converter and a digital-to-analog converter are electronics cir-cuits that are designed and fabricated in silicon IC technology. The main focus inthis book is on CMOS realizations. Chapter 2 summarizes the main physics andmathematics for understanding the operation of analog-to-digital converters. Thischapter is meant to refresh existing knowledge.

In Chaps. 3 to 6 the three basic functions for conversion are analyzed. Chapter 3describes the sampling process and give guidelines for the different choices aroundsampling in analog-to-digital conversion design. The design challenges around thesampling process are discussed in the design of sample-and-hold circuits in Chap. 4.Both sampling and quantization operation are non-linear, resulting in undesired be-havior. The combination of sampling and quantization results in a fundamental er-ror: the quantization error. The attainable performance of every analog-to-digitalconversion is fundamentally limited by this error as is described in Chap. 5. Chap-ter 6 deals with the generation and handling of reference voltages.

The main task of a designer is to construct circuits. Chapter 7 describes the basicsof digital-to-analog converter circuit design and some implementations of digital-to-analog converters. The design of analog-to-digital converters is detailed in Chap. 8.Oversampling and sigma delta conversion are a special class of conversion tech-niques and are discussed in Chap. 9.

Next to theory and circuit design, the proper operation of a converter relies onsome additional aspects. The measurement methods for analog-to-digital convertersand specification points are the subject of Chap. 10. Chapter 11 deals with some ofthe boundary conditions in conversion due to technological and physical limitations.Finally Chap. 12 deals with system aspects of the application of analog-to-digitalconversion like sample frequency choices and the various forms of input handling.This section also introduces a Figure-of-Merit for conversion and compares the var-ious implementation forms. In this way an optimal converter for a given systemsituation can be chosen.

Several books have been published in the field of analog-to-digital conversion.One of the first books was published by Seitzer [1] in 1983 describing the basicprinciples. Van der Plassche [2] in 1994 and 2003 and Razavi [3] in 1994 discussextensively bipolar and CMOS realizations. Jespers [4] and Maloberti [5] addressthe theme on a graduate level. These text books review the essential aspects andfocus on general principles, circuit realizations, and their merits.

Chapter 2Components and Definitions

An electronic engineer combines in his/her work elements of mathematics, physics,network theory and other scientific disciplines. The creative combination of the ele-ments allows the engineer to bridge fundamental theoretical insights with practicalrealizations. Often these theoretical disciplines are phrased in mathematical descrip-tions. Therefore it is relevant to start this book with a summary of these disciplines.

2.1 Mathematical Tools

Events and processes in semiconductor devices span a large range of numbers. Ab-breviations for these numbers are shown in Table 2.1. The words “billion” and “tril-lion” must be avoided as they refer to different quantities in Europe and the USA.1

Mathematical expressions are built from functions of variables: y = f (x). Ta-ble 2.2 lists some elementary mathematical functions and equations. Extensive listsof mathematical functions for engineers are found in [6–8]. When calculations intwo or three dimensions are too complicated, it can help to map the problem on acircle or a sphere. Especially time-repetitive signals and electromagnetic field cal-culations use cyclic and spherical functions to simplify the analysis. Table 2.3 listssome mathematical properties of spheres and circles. Also the use of complex nota-tion can help to visualize rotation, see Table 2.4.

Many events in nature have a cyclic and repetitive character. Sinusoidal waveforms describe these events and Table 2.5 gives a number of regular goniometricalexpressions.

Derivatives of functions represent the way a function changes from one set of val-ues of its variables to a next set. In electronics the derivative helps to understand thebehavior of complex functions that are used in a small range of the main variables.

1This confusion is related to the use of the “long scale” numbering system in continental Europeand the “short scale” numbering system in the USA and UK.

M.J.M. Pelgrom, Analog-to-Digital Conversion,DOI 10.1007/978-90-481-8888-8_2, © Springer Science+Business Media B.V. 2010

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6 2 Components and Definitions

Table 2.1 Multiplierabbreviations Name Abbreviation Multiplier

Googol 10100

Exa E 1018

Peta P 1015

Tera T 1012

Giga G 109

Mega M 106

kilo k 103

hecto h 102

deca da 10

unity 1

deci d 10−1

centi c 10−2

milli m 10−3

micro μ 10−6

nano n 10−9

pico p 10−12

femto f 10−15

atto a 10−18

Table 2.2 Elementary algebraic functions [6–8]

n! = 1 × 2 × 3 × · · · × n 0! = 1, 1! = 1(n

m

)= n!

m!(n − m)!(

n

n

)=(

n

0

)= 1

(a + b)n = an +(

n

1

)an−1b +

(n

2

)an−2b2

+ · · · +(

n

n − 1

)abn−1 + bn

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

an − bn = (a − b)(an−1 + an−2b

+ · · · + abn−2 + bn−1)

a2 − b2 = (a − b)(a + b)

ax2 + bx + c = 0, x1,2 = −b ± √b2 − 4ac

2a

n∑i=0

i = n(n + 1)

2

n∑i=0

ri = 1 − rn+1

1 − r

∞∑i=0

ri = 1

1 − r, |r| < 1

ab = eb lna a0 = e0 = 1, e = 2.71828

ln(b) = ln(a) ×a log(b) ln(10) = 2.303, 10log(10n) = n

10log(2) = 0.301, 10log(3) = 0.477

2.1 Mathematical Tools 7

Table 2.3 Circle and spherefunctions [6–8] Perimeter of a circle 2πr

Area of a circle πr2

Surface of a sphere 4πr2

Volume of a sphere4

3πr3

Table 2.4 Complex notation [6–8]

j2 = −1 |a + jb|2 = a2 + b2

z = a + jb conjugate(z): z∗ = a − jb Re(z) = Re(z∗) = a, Im(z) = −Im(z∗) = b

e±jα = cos(α) ± j sin(α) sin(α) = ejα − e−jα

2j, cos(α) = ejα + e−jα

2

Table 2.5 Goniometrical relations used in this book. The argument of goniometric formulas isexpressed in radians (0 . . .2π ) not in degrees [6–8]

sin(−α) = − sin(α) cos(−α) = cos(α)

sin(α) = cos(π

2− α) sin

(π

4

)= cos

(π

4

)= 1√

2

tan(α) = sin(α)

cos(α)= a arctan(a) = α

sin2(α) + cos2(α) = 1

sin(2α) = 2 sin(α) cos(α) cos(2α) = cos2(α) − sin2(α) = 2 cos2(α) − 1

sin(3α) = −4 sin3(α) + 3 sin(α) cos(3α) = 4 cos3(α) − 3 cos(α)

sin(α + β) = sin(α) cos(β) + cos(α) sin(β) cos(α + β) = cos(α) cos(β) − sin(α) sin(β)

sin(α − β) = sin(α) cos(β) − cos(α) sin(β) cos(α − β) = cos(α) cos(β) + sin(α) sin(β)

2 sin(α) sin(β) = − cos(α + β) + cos(α − β) 2 cos(α) cos(β) = cos(α + β) + cos(α − β)

2 sin(α) cos(β) = sin(α + β) + sin(α − β) 2 cos(α) sin(β) = sin(α + β) − sin(α − β)

sinh(α) = eα − e−α

2cosh(α) = eα + e−α

2

tanh(α) = sinh(α)

cosh(α)= eα − e−α

eα + e−α, tanh(α) = −j tan(jα)

Integration of functions is the main mathematical method to form summationsover time, area, etc. Elementary manipulation of derivatives and integrals of func-tions are given in Table 2.6.

In many applications the function f (x) is known, however the behavior of this(perhaps) complicated function is required over only a small fraction of the entirerange of x. If the function’s behavior close to point x = a is needed, the derivativeof the function gives the direction in which the function will change. By adding

8 2 Components and Definitions

Table 2.6 Standard manipulations for derivatives and integrals of functions [6–8]

du(v(t))

dt= du(v)

dv

dv(t)

dt

du(at)

dt= a

du(t)

dt

d

dt(u(t)v(t)) = u(t)

dv(t)

dt+ v(t)

du(t)

dt

d

dtun(t) = un−1(t)

u(t)

dt

d

dt

(u(t)

v(t)

)=

v(t)du(t)

dt− u(t)

dv(t)

dt

v2(t)

d

dtln(at) = a

t

d

dtsin(at) = a cos(at)

d

dtcos(at) = −a sin(at)

∫xn dx = xn+1

n + 1n �= 1

∫1

xdx = ln(x)

∫1

a2 + x2dx = 1

aarctan(x/a)

∫ ∞

x=0

1

a2 + x2dx = π

2a

∫f (x)δ(x − x0) dx = f (x0)

∫ x=L

x=0sin(2πnx/L) sin(2πmx/L)dx = 0,

if m �= n

∫ x=L

x=0sin2(2πnx/L)dx = 0.5 n,m : integer

Table 2.7 Taylor series expansions [6–8]

(1 + a)n ≈ 1 + na + n(n − 1)

2! a2 + · · · , |a| � 1√

1 + a ≈ 1 + 0.5a, |a| � 1

e1+a ≈ 1 + a + +a2

2! + a3

3! , |a| � 1 ln(1 + a) ≈ a − a2

2+ a3

3, |a| � 1

sin(α) ≈ α − α3

3! + α5

5! , α in radians, |α| � 1 cos(α) ≈ 1 − α2

2! + α4

4! , |α| � 1

higher order derivatives a series expansion is formed, that is useful to represent acomplicated function. A function as defined by its Taylor series, looks like:

f (x) = f (a) + (x − a)

1!df (x)

dx

∣∣∣∣x=a

+ (x − a)2

2!d2f (x)

dx2

∣∣∣∣x=a

+ (x − a)3

3!d3f (x)

dx3

∣∣∣∣x=a

+ · · · . (2.1)

Table 2.7 gives some Taylor series expansions for regular functions.The Taylor series expands a function at one moment in time. For static signals this

representation is the basis for non-linear analysis. The Volterra series is a methodfor including time-dependent effects.

2.1 Mathematical Tools 9

f (x, t) =∞∑i=1

1

i!∫ ∞

−∞· · ·∫ ∞

−∞gi(τ1, . . . , τi)x(t − τ1) × · · · × x(t − τi) dτ1 · · ·dτi

= g0 + 1

1!∫ ∞

−∞g1(τ1)x(t − τ1) dτ1

+ 1

2!∫ ∞

−∞

∫ ∞

−∞g1(τ1)g2(τ1, τ2)x(t − τ1)x(t − τ2) dτ1 dτ2

+ 1

3!∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞g1(τ1)g2(τ1, τ2)g3(τ1, τ2, τ3)x(t − τ1)x(t − τ2)

× x(t − τ3) dτ1 dτ2 τ3 . . .

The first term in the Volterra series is the convolution function as in (2.144).Laplace and Fourier analogies of the Volterra series are useful to evaluate time-dependent distortion. Various techniques exist to estimate the coefficients gi .

2.1.1 The Fourier Transform

The Fourier transform is used to analyze the behavior of time repetitive signals h(t).The Fourier transform and its inverse transform are defined as:

H(ω) =∫ ∞

t=−∞h(t)e−jωt dt =

∫ ∞

t=−∞h(t)e−j2πf t dt

h(t) = 1

2π

∫ ∞

ω=−∞H(ω)ejωt dω =

∫ ∞

f =−∞H(f )ej2πf t df (2.2)

In case a sinusoidal current is applied, the steady-state voltage and current relationsfor coils, capacitors and resistors are described in the Fourier domain, resulting in:

v(ω) = jωLi(ω) = j2πf Li(ω)

v(ω) = Ri(ω) = i(ω)/g

v(ω) = i(ω)

jωC= i(ω)

j2πf C

where the complex notation “j = √−1” is used to indicate that a 90° phase shiftexists between the current and the terminal voltage.

Some special relations exist between the time-domain and the Fourier domain.A physical signal is a real signal and its imaginary part2 equals 0. The Fourier trans-form converts the real time-continuous function A(t) in a complex function A(ω)

2In modern communication theory, single side-band modulation coins its two signal componentswith a 90° phase relation (I/Q) as the “real part” and an “imaginary part”. This is a form of notationon a higher level than straight-forward Fourier analysis of a physical signal.

10 2 Components and Definitions

Fig. 2.1 The transform from a box function in the time domain leads to a sin(x)/x shape in thefrequency domain. Three different rise and fall times show the transition from a block to a triangleshaped pulse

with ω = 2πf , describing the signal in the frequency domain. After transformationa real signal results in a Hermitian function with the following properties:

A(ω) = A∗(−ω) or equivalently

Re(A(ω)) = Re(A(−ω)) and

Im(A(ω)) = −Im(A(−ω))

where “Re” and “Im” define the real and imaginary parts of a function.Parseval’s energy conservation theorem states that if two time functions x(t) and

y(t) exist in the frequency domain as X(ω) and Y(ω) then:∫ ∞

t=−∞x(t)y∗(t) dt = 1

2π

∫ ∞

ω=−∞X(ω)Y ∗(ω)dω (2.3)

The substitution of x(t) = y(t) results in the “Energy theorem”:∫ ∞

t=−∞|x(t)|2 dt = 1

2π

∫ ∞

ω=−∞|X(ω)|2 dω =

∫ ∞

f =−∞|X(f )|2 df (2.4)

The energy of a signal over infinite time equals the energy over infinite frequencyrange.

The Fourier transform links the time domain to the frequency domain, Fig. 2.1.A box function in the time domain leads to a sin(x)/x function in the frequencydomain. If the slope of the time signal is made less steep, this its Fourier transformmoves to a (sin(x)/x)2 function in case of a triangle shape. As the Fourier transformis symmetrical to and from each domain, a box function in the frequency domainwill result also in a sin(x)/x function in the time domain. This is e.g. visible in thedefinition of time-discrete filter coefficients, Sect. 3.2.1.

2.1.2 Fourier Analysis

The sine wave describes in time the position of an object following a repetitivecircular motion. The mathematical technique to split up any form of repetitive signal

2.1 Mathematical Tools 11

in sine waves is the Fourier series expansion. The expansion for a function f (x) thatis repetitive over a distance L is:

f (x) = 1

2a0 +

∞∑n=1

an cos(2πnx/L) +∞∑

n=1

bn sin(2πnx/L) (2.5)

with

a0 = 2

L

∫ L/2

x=−L/2f (x)dx

an = 2

L

∫ L/2

x=−L/2f (x) cos(2πnx/L)dx

bn = 2

L

∫ L/2

x=−L/2f (x) sin(2πnx/L)dx

As an example consider a square wave in time f (t), defined as:

−T/2 < t < 0 f (t) = −1

0 < t < T/2 f (t) = 1

which repeats over a period T . Evaluation shows that: a0 = 0, and an = 0 for all n.Only the bn coefficients are unequal to zero:

bn = 2

T

[∫ 0

t=−T/2(−1) sin(2πnt/T )dt +

∫ T/2

t=0(+1) sin(2πnt/T )dt

]

bn = −1

nπ

[(−1)(cos(0) − cos(nπ)) + (+1)(cos(nπ) − cos(0))

]

= 2(1 − cos(nπ))

nπ

The resulting terms bn equal zero for even n and bn = 4/nπ for odd n. Con-sequently the Fourier expansion for a square wave is a sum of sine waves withfrequencies that are odd multiples of the fundamental n = 1 frequency, Fig. 2.2:

f (t) =∞∑

n=1,3,5,...

4

nπsin(2πnt/T )

The result of a Fourier expansion may seemingly differ if another starting pointon the curve is chosen. In this case the same square wave f (t) could also be definedsymmetrically around t = 0:

−T/2 < t < −T/4 f (t) = −1−T/4 < t < T/4 f (t) = 1T/4 < t < T/2 f (t) = −1

resulting in a0 = 0, and bn = 0 for all n. Only the odd an coefficients are nowunequal to zero:

f (t) =∞∑

n=1,3,5,...

4 sin(nπ/2)

nπcos(2πnt/T )

12 2 Components and Definitions

Fig. 2.2 A square wavesignal is decomposed in sinewaves with frequencies thatare odd integers of thefundamental frequency. Thefirst five components areadded into an approximationof a square-wave

Table 2.8 Fourier series expansions of signals repeating at a period T [6–8]

Square wave (transition at t = 0)

f (t) ={

−1 −T/2 < t < 0

+1 0 < t < T/2f (t) =

∞∑n=1,3,5,...

4

nπsin(2πnt/T )

Square wave symmetrical around t = 0

f (t) =

⎧⎪⎨⎪⎩

−1 −T/2 < t < −T/4

+1 −T/4 < t < T/4

−1 T/4 < t < T/2

f (t) =∞∑

n=1,3,5,...

4 sin(nπ/2)

nπcos(2πnt/T )

Square wave with Tc high period

f (t) =

⎧⎪⎨⎪⎩

−1 −T/2 < t < −Tc/2

+1 −Tc/2 < t < Tc/2

−1 Tc/2 < t < T/2

f (t) =∞∑

n=1

4(−1)n+1 sin(πnTc/T )

nπcos(2πnt/T )

Triangle

f (t) =⎧⎨⎩

− t

T−T/2 < t < 0

+ t

T0 < t < T/2

f (t) = 1

2−

∞∑n=1,3,5,...

4

n2π2cos(2πnt/T )

Saw tooth

f (t) = t

T0 < t < T f (t) = 1

2−

∞∑n=1

1

nπsin(2πnt/T )

The even coefficients are zero. Just as for the square wave the resulting Fourierseries experiences a 90° phase shift with respect to the first analysis.

Some more Fourier expansions are given in Table 2.8 [8, p. 405].Fourier series expansions are used for determining distortion components. In this

example the square wave is composed of a fundamental sine wave and odd harmon-ics. In order to find the power ratio between the fundamental and the harmonics, thepower content of both must be established.

2.1 Mathematical Tools 13

P = 1

T

∫ T/2

t=−T/2(f (t))2 dt

= 1

T

∫ T/2

t=−T/2

∞∑n=1,3,5,...

(4

nπsin(2πnt/T )

)2

dt

= 1

T

∫ T/2

t=−T/2

∞∑n=1,3,5,...

(4

nπ

)2

sin2(2πnt/T )dt =∞∑

n=1,3,5,...

8

n2π2

The last formula conversion uses the fact that an integral over a period T of anyproduct sin(2πnt/T ) sin(2πmt/T ) with n �= m is equal to zero and only the DCterms of the squared sinusoidal terms result in non-zero contributions. On the otherhand the power of the square wave can be easily calculated: the amplitude is either−1 or +1, which both result in a power of 1. Consequently:

∞∑n=1,3,5,...

8

n2π2= 8

π2+

∞∑n=3,5,...

8

n2π2= 0.81 + 0.19 = 1 (2.6)

from which the ratio between the first harmonic and the sum of the remaining com-ponents in a square wave can be derived:

1010log

(1

π2

8 − 1

)= 6.31 dB

2.1.3 Distortion

In signal processing ratios between various quantities (signals, noise, distortion) aremostly specified as power ratios, e.g. the total harmonic distortion (THD):

THD = Pdistortion

Pfundamental

In audio engineering the THD is expressed in %. As these ratios can amount manyorders of magnitude, a logarithmic notation often replaces the exponential notation3:

THD = 10 10log

(Pdistortion

Pfundamental

)

The unit of ratio is called “decibel” or “dB”, indicating the “deci” or one-tenthfraction of the unit “Bell”.4 In many cases a relation to the signal in the voltage orcurrent domain is required:

THD = 1010log

(V 2

distortion/R

V 2fundamental/R

)= 2010log

(Vdistortion

Vfundamental

)(2.7)

343.8 dB is a short hand for 4.167 × 10−5 power ratio. Use the exponential notation in complexcalculations.4The signal-to-noise ratio is defined in the opposite way: signal power divided by noise power.Therefore the minus sign that normally precedes the THD number is sometimes omitted.

14 2 Components and Definitions

The popular 2010log(voltage ratio) is in fact a derived power ratio. In case of doubtalways use power ratios.

Example Assume a transfer function of the form:

y = x + ax2

where a � 1 is the generating term for second order distortion. This type of distor-tion is called “soft distortion” in contrast to “hard distortion” where a discontinuousjump in the signal or transfer is present. With an input signal x(t) = V sin(ωt) andusing some goniometric equivalences from Table 2.5, the distortion is calculated5:

y(t) = 1

2aV 2 + V sin(ωt) − aV 2

2cos(2ωt)

Consequently the second order distortion component relative to the first order com-ponent is:

HD2 = aV 2/2

V= aV

2

The second order component goes up quadratically if the signal amplitude riseslinearly. Similar for a third order distortion:

y = x + bx3

y = V sin(ωt) + 3bV 3

4sin(ωt) − bV 3

4sin(3ωt)

The third order distortion relative to the first order component is:

HD3 = bV 2

4 + 3bV 2

The third order component goes up with a third power when the input amplitudeincreases linearly. In the RF field this observation has led to a somewhat deviat-ing formulation of the third order distortion: the third-order intercept point IP3,Fig. 2.3. In this point the extrapolated first order amplitude equals the extrapo-lated third order distortion amplitude. Using the previous analysis: VIP3 = 2/

√b.

The third-order intercept point can be related to the input axis (IIP3) or the out-put level (OIP3). Values of IIP3 exceeding 1 V are normally considered rathergood.

In the previous analysis the stimulus consisted out of a single tone at a fundamen-tal frequency. Its distortion products are situated at multiples of that frequency. Insystems with filters, the transfer function for the distortion products can be differentfrom the processing of the fundamental frequency. In that case an intermodulationmethod is used to determine the third order distortion. The input is chosen as the

5Using goniometric equivalences is an engineering short cut. The Fourier series expansion givesthe same results.