Sampling Theory and Analog-to-Digital Conversion (Abridged)

204 Pages·2016·8.31 MB·English

by  Patrick Jungwirth

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By
Patrick Jungwirth, PhD

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Sampling Theory, and AnaloCg-otnov-eDritgeirtAalD C 0x41
fs
Analog-to-Digital Conversion (Abridged)

The abridged edition of Sampling Theory and Analog-to-Digital Conversion
is a shortened version of the original book.

! Disclaimer !

! Caution: legal disclaimer, for educational use only, no formal reviews, no formal testing, no
government certifications, no industry certifications, etc., use at your own risk! Property damage,
environmental damage, injury, serious injury or worse may result! Don't receive a Darwin Award! !

! Not for any illegal/immoral use(s)

! Sampling Theory and Analog-to-Digital Conversion book does contain errors. Use at your own risk!

! Engineering humor. Microwave
Popcorn
! No user serviceable parts inside. Anyone?

! Only virtual groundhogs were used in the textbook!

! Check with a patent attorney, about potential patent rights.

! Trademark(s) are the property of their respective owners.

! You (your organization) agree to hold the author and all associates harmless.

! Webpage and email address subject to change without notice.

! Data and audio files are not included is this edition.

Sampling Theory, and
Email the Author
Analog-to-Digital Conversion
[email protected]
(Abridged) Edition 0

You may email the author at [email protected]. Why begin with edition 0?
Name a computer language, still in common use today, that starts with array index = 1!

All submissions become the property of the author.
ii
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By
Patrick Jungwirth, PhD

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Sampling Theory and Analog-to-Digital Conversion (Abridged)

Copyright
© 2016

by Patrick Jungwirth, PhD

ALL RIGHTS RESERVED

iii
In Memory of
Coach Douglas Rudolf
July 24, 1936 - January 27, 2012

Thank you for your
43 years of coaching
the Maroon Machine at Mt. Whitney

They that hope in the LORD will renew their strength, they will soar with eagles’ wings;
They will run and not grow weary, walk and not grow faint.
Isaiah 40:30-31

iv
Tiomnaithe do

Banphrionsa uasal Sorcha
Ethan mac uasal

v
Special Thanks

to

Professor Emeritus Alexander Poularikas
University of Alabama in Huntsville

Professor Kaveh Heidary
Alabama A&M University

vi
Sampling Theory, and AnaloCg-otnov-eDritgeirtAalD C 0x41
fs
Analog-to-Digital Conversion (Abridged)

The abridged edition of Sampling Theory and Analog-to-Digital Conversion
is a shortened version of the original book.

Preface

Why another book on sampling theory and analog-to-digital conversion? This book takes a linear
system theory approach to analog-to-digital conversion. From linear systems theory, we introduce
sampling theory and use the tools from linear system theory to prove Shannon's sampling theorem.
Shannon's sampling theorem shows how an analog signal can be converted to a discrete time signal and
how the original can be exactly recovered from the discrete time signal. Digital is an approximation to
discrete time so exact reconstruction is not possible; however, practically speaking the reconstructed
signal is identical to the original analog signal. Digital is more than just 1's and 0's. Digital technology is
universal. Once data is in digital form, it can be converted from one digital format to another without any
additional loss of information (excluding lossy data compression).
We use linear system theory and the sampling theorem to derive the model for an ideal analog to
digital converter. We derive performance metrics from the ideal model. The performance metrics are put
to good use to illustrate how to test and evaluate an analog-to-digital converter. Applications motivate the
reader to apply concepts learned. More complex applications lead the reader to an introduction to
software defined radios. Appendices provide summaries of the sampling theorem, communications
engineering units, transform tables, and ADC model.
Chapter 1 begins by introducing the digital world. Money is used to introduce analog, discrete,
and digital. Chapter 2 starts off with a simple description of linear systems. Chapter 2 takes the reader
from algebra, calculus, differential equations, Fourier transforms, and back to algebra. We want the
reader to have a basic understanding of signal processing. Linear system theory provides the tools to
prove the sampling theorem in Chapter 3. A graphical proof and analytical proof of the sampling theorem
are presented in Chapter 4. We show why wagon wheels turn backwards in Western movies.
Chapter 5 covers the binary math, we need to work with analog-to-digital converters. Code
examples are provided for the "software interface" for an analog-to-digital converter. A useful part of
Chapter 5 is how to solve the problem of a signed 10 bit analog-to-digital converter connected to a 16 bit
microprocessor. (not included in abridged edition).
The ideal analog-to-digital converter model is developed in Chapter 6. Chapter 7 introduces
some common analog-to-digital converters: flash, pipeline, successive approximation register, and delta
sigma (ΔΣ). Performance metrics and testing of analog-to-digital converters are found in Chapter 8.
Chapter 9 presents sampling and analog-to-digital conversion applications. Chapter 10 covers a brief
introduction to analog-to-digital converter datasheets focused on software defined radio applications. (not
included in abridged edition).
Chapter 11 presents an introduction to radio receiver block diagrams and finishes with a short
introduction to software defined radios. Chapter 11 completes the journey from linear systems, to
sampling theory, to analog-to-digital converters, and then the most useful part, applications. We hope this
book serves as a good stepping stone to more complex applications. As computer power continues to
increase and costs continue to drop, new applications will be found for the future. Be part of developing
the future. If you have any suggestions for improvements, or find errors please email the book author (see
book preface).

Patrick Jungwirth
March 2016
vii
Sampling Theory, and AnaloCg-otnov-eDritgeirtAalD C 0x41
fs
Analog-to-Digital Conversion (Abridged)

The abridged edition of Sampling Theory and Analog-to-Digital Conversion
is a shortened version of the original book.

Table of Contents

1.0 Sampling Theory and Analog-to-Digital Conversion Introduction........................................2
1.1 Early Digital Communication Systems .............................................................................................. 3
1.2 What are the Advantages and Costs of Digital Technology? .............................................................. 4
1.3 Pulse Code Modulation ..................................................................................................................... 7
1.4 Shannon's Sampling Theorem ........................................................................................................... 8
1.5 Chapter Summary .............................................................................................................................. 9
1.6 Book Organization ............................................................................................................................ 9
1.7 Chapter 1 Questions (not included in abridged edition) ................................................................... 11
1.8 Chapter 1 Suggested Reading.......................................................................................................... 11
1.9 Chapter 1 References ...................................................................................................................... 11

2.0 Linear Systems Theory .......................................................................................................13
2.1 Linear Systems Theory Introduction ................................................................................................ 13
2.2 Definition of a Linear System ......................................................................................................... 16
2.3 Some Examples of Linear Systems ................................................................................................. 18
2.3.1 Discrete Time Filter Example ...................................................................................................................... 20
2.3.2 Linear System Example .............................................................................................................................. 21
2.4 Introduction to Calculus ................................................................................................................... 22
2.5 Introduction to Differential Equations .............................................................................................. 28
2.6 Frequency Domain Transforms ........................................................................................................ 30
2.7 Introduction to Sampling Theory .................................................................................................... 35
2.8 Chapter 2 Questions (not included in abridged edition) ................................................................... 37
2.9 Chapter 2 Suggested Reading.......................................................................................................... 37
2.10 Chapter 2 References .................................................................................................................... 37

3.0 Linear Systems Tools and Sampling Theory Introduction ..................................................39
3.1 Sampling Function .......................................................................................................................... 41
3.2 Impulse Response of a Linear System ............................................................................................. 44
3.3 Convolution Integral ....................................................................................................................... 45
3.4 Power Spectral Density ................................................................................................................... 46
3.5 Fourier Transform........................................................................................................................... 49
3.6. Fourier transform identities. ........................................................................................................... 51
3.7 Linear Systems: Block Diagram Algebra (not included in abridged edition) ................................... 52
3.8 Frequency Domain Filtering (not included in abridged edition) ....................................................... 52
3.9 Spatial Domain Filtering -- Image Processing (not included in abridged edition) ............................. 52
viii
3.10 Summary -- Tools needed for Chapter 4. ........................................................................................ 52
3.11 Chapter 3 Questions (not included in abridged edition) .................................................................. 53
3.12 Chapter 3 Suggested Reading ........................................................................................................ 53
3.13 Chapter 3 References ..................................................................................................................... 53

4.0 Shannon's Sampling Theorem............................................................................................55
4.1 Sampling Theory Introduction ........................................................................................................ 55
4.2 Sampling Theory Introduction ........................................................................................................ 58
4.3 Discontinuous and Continuous Functions ........................................................................................................ 58
4.4 Time and Bandlimited Functions .................................................................................................................... 61
4.5 Sampling Theory and Analog-to-Digital Conversion ...................................................................................... 62
4.6 Shannon's Sampling Theorem ......................................................................................................... 65
4.6.1 Graphical Proof of Shannon's Sampling Theorem ........................................................................................ 66
4.6.2 Frequency Aliasing ..................................................................................................................................... 67
4.6.3 Frequency Aliasing and Wagon Wheels ...................................................................................................... 68
4.6.4 Shannon's Sampling Theorem ..................................................................................................................... 69
4.6.5 Proof of Shannon's Sampling Theorem (not included in abridged edition) .................................................... 70
4.6.6 Ideal Low Pass Filter is Noncausal .............................................................................................................. 70
4.6.7 Bandlimited and Almost Bandlimited .......................................................................................................... 71
4.7 Analog-to-Digital Converter Model ................................................................................................ 71
4.8 Two Dimensional Sampling (Image Processing) (not included in abridged edition) ......................... 72
4.9 Summary ........................................................................................................................................ 72
4.10 Questions.(not included in abridged edition) ................................................................................. 73
4.11 Suggested Reading........................................................................................................................ 73
4.12 References: ................................................................................................................................... 73

5.0 Binary Arithmetic (not included in abridged edition)

6.0 Quantization and Coding ...................................................................................................76
6.1 Analog-to-Digital Converter Model ................................................................................................ 78
6.2 Quantization ......................................................................................................................... 79
6.2.1 Round Down Quantization .......................................................................................................................... 82
6.2.2 Half Step Offset Quantization ..................................................................................................................... 82
6.3 Quantization Error Model ................................................................................................................ 87
6.4 Signal-to-Quantization Noise Ratio ................................................................................................. 90
6.5 Oversampling (not included in abridged edition) .............................................................................. 91
6.6 Coding............................................................................................................................................. 92
6.7 Analog-to-Digital Converters ........................................................................................................... 94
6.8 Nonlinear Quantization (not included in abridged edition) ............................................................... 96
6.9 Summary ........................................................................................................................................ 96
6.10 Chapter 6 Questions (not included in abridged edition) ................................................................. 97
6.11 Chapter 6 Suggested Reading ........................................................................................................ 97
6.12 Chapter 6 References ..................................................................................................................... 97

ix
7.0 Analog-to-Digital Converters .............................................................................................99
7.1 Flash Analog-to-Digital Converter ................................................................................................ 102
7.2 Pipeline Analog-to-Digital Converters .......................................................................................... 108
7.3 Successive Approximation Register ADC ..................................................................................... 108
7.4 Delta Sigma (∆Σ) ADC ................................................................................................................. 110
7.4.1 Review of Oversampling ........................................................................................................................... 110
7.4.2 ΔΣ Analog-to-Digital Converter Linear Model (not included in abridged edition) ...................................... 115
7.5 ADC Summary .............................................................................................................................. 115
7.6 Chapter 7 Questions (not included in abridged edition) ................................................................. 115
7.7 Suggested Reading ....................................................................................................................... 115
7.8 References: ................................................................................................................................... 116

8.0 Analog-to-Digital Conversion Performance and Testing .................................................. 118
8.1 Quantization Error Review (not included in abridged edition) ....................................................... 119
8.2 Static Analog-to-Digital Converter Metrics ................................................................................... 120
8.2.1 DC Transfer Function ............................................................................................................................... 121
8.2.2 ADC Gain Errors ...................................................................................................................................... 122
8.2.3 ADC Offset Errors .................................................................................................................................... 122
8.2.4 ADC Transfer Curve Nonlinearities ........................................................................................... 123
8.2.5 Monotonic Transfer Curve ........................................................................................................................ 123
8.2.6 Differential Nonlinearity ........................................................................................................................... 124
8.2.7 Differential Nonlinearity Example ............................................................................................................. 132
8.2.8 Sine Wave Differential Nonlinearity Test .................................................................................................. 133
8.3 AC Performance Characteristics ................................................................................................... 138
8.3.1 Why do we need a Windowing Function? (not included in abridged edition) ............................................. 138
8.3.2 AC Performance Metrics: SNR, SINAD, and SFDR ................................................................................. 139
8.3.3 Effective Number of Bits (ENOB) ............................................................................................................. 141
8.3.4 Intermodulation Distortion (not included in abridged edition) .................................................................... 141
8.4 Chapter 8 Questions (not included in abridged edition) ................................................................. 142
8.5 Chapter 8 Suggested Reading........................................................................................................ 142
8.6 References .................................................................................................................................... 142

9.0 Sampling and ADC Applications. ...................................................................................... 144
9.1 Optical Chopper Quantization ....................................................................................................... 144
9.1.1 Wind Speed Sensor ................................................................................................................................... 145
9.1.2 Crank Shaft Angle Sensor (not included in abridged edition) ..................................................................... 147
9.2 Sampling Application -- Phase Recovery (not included in abridged edition) .................................. 147
9.3 Heart Monitor -- 60 Hz Notch Filter (not included in abridged edition) ......................................... 147
9.4 Microwave Popcorn System .......................................................................................................... 148
9.5 Microwave Popcorn System Version 2 (not included in abridged edition) ..................................... 163
9.6 Simple FSK Modem ...................................................................................................................... 164
9.7 Automatic Frequency Control (Drift in Morse Code) (not included in abridged edition) ................ 166
9.8 3 Axis Accelerometer Example (not included in abridged edition)................................................. 166
9.9 Chapter 9 Questions (not included in abridged edition) ................................................................. 166
9.10 Chapter 9 References .................................................................................................................. 167

10.0 (not included in abridged edition)
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