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Characterization and Correction of Analog-To-Digital Converters

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Characterization and Correction of Analog-To-Digital Converters Characterization and Correction of Analog-to-Digital Converters HENRIK F. LUNDIN Doctoral Thesis Stockholm, Sweden 2005 TRITA–S3–SB-0575 ISSN 1103-8039 KTH School of Electrical Engineering ISRN KTH/SB/R--05/75--SE SE-100 44 Stockholm ISBN 91-7178-210-9 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i signalbehand- ling fredagen den 16 december 2005 klockan 13.00 i D3, Kungl Tekniska högskolan, Lindstedtsvägen 5, Stockholm. c Henrik Fahlberg Lundin, november 2005 Tryck: Universitetsservice US AB iii Abstract Analog-to-digital conversion and quantization constitute the topic of this thesis. Post- correction of analog-to-digital converters (ADCs) is considered in particular. ADCs usu- ally exhibit non-ideal behavior in practice. These non-idealities spawn distortions in the converters output. Whenever the errors are systematic, it is possible to mitigate them by mapping the output into a corrected value. The work herein is focused on problems associated with post-correction using look-up tables. All results presented are supported by experiments or simulations. The first problem considered is characterization of the ADC. This is in fact an esti- mation problem, where the transfer function of the converter should be determined. This thesis deals with estimation of quantization region midpoints, aided by a reference signal. A novel estimator based on order statistics is proposed, and is shown to have superior performance compared with the sample mean traditionally used. The second major area deals with predicting the performance of an ADC after post- correction. A converter with static differential nonlinearities and random input noise is considered. A post-correction is applied, but with limited (fixed-point) resolution in the corrected values. An expression for the signal-to-noise and distortion ratio after post- correction is provided. It is shown that the performance is dependent on the variance of the differential nonlinearity, the variance of the random noise, the resolution of the converter and the precision of the correction values. Finally, the problem of addressing, or indexing, the correction look-up table is dealt with. The indexing method determines both the memory requirements of the table and the ability to describe and correct dynamically dependent error effects. The work here is devoted to state-space–type indexing schemes, which determine the index from a num- ber of consecutive samples. There is a tradeoff between table size and dynamics: more samples used for indexing gives a higher dependence on dynamic, but also a larger ta- ble. An indexing scheme that uses only a subset of the bits in each sample is proposed. It is shown how the selection of bits can be optimized, and the exemplary results show that a substantial reduction in memory size is possible with only marginal reduction of performance. iv Sammanfattning Denna avhandling behandlar analog–digitalomvandling. I synnerhet behandlas post- korrektion av analog–digitalomvandlare (A/D-omvandlare). A/D-omvandlare är i prak- tiken behäftade med vissa fel som i sin tur ger upphov till distorsion i omvandlarens utsignal. Om felen har ett systematiskt samband med utsignalen kan de avhjälpas genom att korrigera utsignalen i efterhand. Detta verk behandlar den form av postkorrektion som implementeras med hjälp av en tabell ur vilken korrektionsvärden hämtas. Innan en A/D-omvandlare kan korrigeras måste felen i den mätas upp. Detta görs genom att estimera omvandlarens överföringsfunktion. I detta arbete behandlas speciellt problemet att skatta kvantiseringsintervallens mittpunkter. Det antas härvid att en refe- renssignal finns tillgänglig som grund för skattningen. En skattare som baseras på sorterade data visas vara bättre än den vanligtvis använda skattaren baserad på sampelmedelvärde. Nästa huvudbidrag visar hur resultatet efter korrigering av en A/D-omvandlare kan predikteras. Omvandlaren antas här ha en viss differentiell olinjäritet och insignalen antas påverkad av ett slumpmässigt brus. Ett postkorrektionssystem, implementerat med be- gränsad precision, korrigerar utsignalen från A/D-omvandlaren. Ett utryck härleds som beskriver signal–brusförhållandet efter postkorrektion. Förhållandet visar sig bero på den differentiella olinjäritetens varians, det slumpmässiga brusets varians, omvandlarens upp- lösning samt precisionen med vilken korrektionstermerna beskrivs. Till sist behandlas indexering av korrektionstabeller. Valet av metod för att indexera en korrektionstabell påverkar såväl tabellens storlek som förmågan att beskriva och korrigera dynamiska fel. I avhandlingen behandlas i synnerhet tillståndsmodellbaserade metoder, det vill säga metoder där tabellindex bildas som en funktion utav flera på varandra föl- jande sampel. Allmänt gäller att ju fler sampel som används för att bilda ett tabellindex, desto större blir tabellen, samtidigt som förmågan att beskriva dynamiska fel ökar. En in- dexeringsmetod som endast använder en delmängd av bitarna i varje sampel föreslås här. Vidare så påvisas hur valet av indexeringsbitar kan göras optimalt, och experimentella utvärderingar åskådliggör att tabellstorleken kan reduceras avsevärt utan att fördenskull minska prestanda mer än marginellt. De teorier och resultat som framförs här har utvärderats med experimentella A/D- omvandlardata eller genom datorsimuleringar. Acknowledgements As a PhD student, you sometimes get the response ‘That must be difficult!’ after having told what you do for a living. Well, I’ll tell you what the really difficult part of it all is: writing the acknowledgements. In this thesis that you now hold in you hands, I’ve spent some 190 pages elaborating at great length on a subject so narrow that almost nobody has ever heard of it. Meanwhile, I have less than two pages to express feelings such as joy and gratitude, and to declare the importance of friends and family and of great colleagues. And this is the one part that everyone reads—if you do not flip this pages to see what’s coming next, you will be in good company, but will perhaps miss out on some interesting results... First of all I’d like to express my deepest gratitude to my supervisors Peter Händel and Mikael Skoglund. From the first day of my Master’s thesis project, which is where my researching days started, I’ve always felt your support and your confidence in me. Without your skillful supervision and deep knowledge in the various aspects of signal processing, this thesis would not have been what it is today. I thank you for all this. The Signal Processing group, with its past and present members, has been a never-ending source of knowledge, inspiration and, most important, great fun. The number of lunches and coffee breaks that have extended way beyond reasonable time, simply because of interesting and amusing discussions, are uncountable. Al- though all of you contribute to the unique and warm atmosphere, I must extend my special thanks to a few of you: The first one out is Tomas Andersson, for putting up with me on trips around the globe, for joint research and also for being a great office neighbor. I know you are doing your best to “keep up that good Swedish tradition.” Although Niclas Björsell spends his days in Gävle, to me he is a significant part of the Signal Processing group. Thanks for good company on several conference trips, and for your valuable feedback on my work. I thank the computer support group, Nina Unkuri and Andreas Stenhall, for splendid computer support. Together with Tomas Andersson we have also shown that shouting is the best form of intra-office communication between neighboring rooms. Karin Demin deserves recognition for her impeccable administration of this re- search group – whatever you need to know, Karin finds it in her binders within v vi seconds. Reaching outside the department, I’d like to thank Professor Pasquale Daponte, Associate Professor Sergio Rapuano, Dr. Luca De Vito and the rest of the people at L.E.S.I.M., the Laboratory of Signal and Measurement Information Processing at the University of Sannio in Benevento, Italy. They make me feel very welcome when visiting Benevento, and the collaboration has lead to joint publications. I thank Professor Paolo Carbone, Associate Professor Ana Rusu, Dr. Jan-Erik Eklund and Professor Håkan Johansson for acting opponent and grading committee during my public defense of this thesis. I would like to thank my friends for keeping my mind off my work from time to time. Of course, I thank my parents for their support and for patiently waiting the twenty-two years it took me to finish school. Finally, I thank my wife Therese for her infinite support and love. I would not have made it without you. Henrik Fahlberg Lundin Stockholm, November 2005 Contents Contents vii 1 Introduction 1 1.1 Background................................ 2 1.2 TheAnalog-to-DigitalConverter . 3 1.3 ADCArchitectures............................ 9 1.4 DescribingA/DConverterPerformance . 12 1.5 ThePost-correctionProblem . 16 1.6 OutlineandContributions. 17 1.7 Notation.................................. 21 1.8 Abbreviations............................... 23 I ADCCorrectionMethods:AnOverview 25 2 Introduction to Part I 27 3 Look-Up Table Based Methods 29 3.1 ClassificationofLUTMethods . 29 3.2 IndexingSchemes............................. 30 3.3 CorrectionValues............................. 33 3.4 CalibrationofLUTs ........................... 36 4 Dithering Based Methods 39 4.1 Statistical Theory of Quantization
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