The Discrete Charm of the Machine: Why the World Became Digital
()
About this ebook
The genesis of the digital idea and why it transformed civilization
A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation.
The spark of individual genius shines through this story of innovation: the stored program of Jacquard’s loom; Charles Babbage’s logical branching; Alan Turing’s brilliant abstraction of the discrete machine; Harry Nyquist’s foundation for digital signal processing; Claude Shannon’s breakthrough insights into the meaning of information and bandwidth; and Richard Feynman’s prescient proposals for nanotechnology and quantum computing. Ken Steiglitz follows the progression of these ideas in the building of our digital world, from the internet and artificial intelligence to the edge of the unknown. Are questions like the famous traveling salesman problem truly beyond the reach of ordinary digital computers? Can quantum computers transcend these barriers? Does a mysterious magical power reside in the analog mechanisms of the brain? Steiglitz concludes by confronting the moral and aesthetic questions raised by the development of artificial intelligence and autonomous robots.
The Discrete Charm of the Machine examines why our information technology, the lifeblood of our civilization, became digital, and challenges us to think about where its future trajectory may lead.
Related to The Discrete Charm of the Machine
Related ebooks
How Transistor Area Shrank by 1 Million Fold Rating: 0 out of 5 stars0 ratingsOptical Transistor: Computing at the Speed of Light Rating: 5 out of 5 stars5/5Turing's Man: Western Culture in the Computer Age Rating: 4 out of 5 stars4/5Pro TypeScript: Application-Scale JavaScript Development Rating: 4 out of 5 stars4/5MATLAB Optimization Techniques Rating: 0 out of 5 stars0 ratingsBeginning Backbone.js Rating: 3 out of 5 stars3/5Visual Studio Condensed: For Visual Studio 2013 Express, Professional, Premium and Ultimate Editions Rating: 0 out of 5 stars0 ratingsProgramming and Customizing the Multicore Propeller Microcontroller: The Official Guide Rating: 4 out of 5 stars4/5Statistical and Thermal Physics: With Computer Applications, Second Edition Rating: 0 out of 5 stars0 ratingsAutomated Theorem Proving in Software Engineering Rating: 0 out of 5 stars0 ratingsPython for Probability, Statistics, and Machine Learning Rating: 0 out of 5 stars0 ratingsObjective-C for Absolute Beginners: iPhone, iPad and Mac Programming Made Easy Rating: 0 out of 5 stars0 ratingsEmbedded Software Design and Programming of Multiprocessor System-on-Chip: Simulink and System C Case Studies Rating: 0 out of 5 stars0 ratingsPointers in C Programming: A Modern Approach to Memory Management, Recursive Data Structures, Strings, and Arrays Rating: 0 out of 5 stars0 ratingsSingular Spectrum Analysis for Time Series Rating: 0 out of 5 stars0 ratingsCreative Education and Dynamic Media Rating: 0 out of 5 stars0 ratingsThe Coming Robot Revolution: Expectations and Fears About Emerging Intelligent, Humanlike Machines Rating: 0 out of 5 stars0 ratingsAssessing and Improving Prediction and Classification: Theory and Algorithms in C++ Rating: 0 out of 5 stars0 ratings.NET IL Assembler Rating: 0 out of 5 stars0 ratingsA++ and the Lambda Calculus: Principles of Functional Programming Rating: 3 out of 5 stars3/5IoT Development for ESP32 and ESP8266 with JavaScript: A Practical Guide to XS and the Moddable SDK Rating: 0 out of 5 stars0 ratingsBeginning PyQt: A Hands-on Approach to GUI Programming Rating: 0 out of 5 stars0 ratingsMastering the Raspberry Pi Rating: 3 out of 5 stars3/5Python for Marketing Research and Analytics Rating: 0 out of 5 stars0 ratingsModern C for Absolute Beginners: A Friendly Introduction to the C Programming Language Rating: 0 out of 5 stars0 ratings.NET DevOps for Azure: A Developer's Guide to DevOps Architecture the Right Way Rating: 0 out of 5 stars0 ratingsZappa Gear: The Unique Guitars, Amplifiers, Effects Units, Keyboards and Studio Equipment Rating: 0 out of 5 stars0 ratingsFast Circuit Boards: Energy Management Rating: 0 out of 5 stars0 ratingsPro ASP.NET 4.5 in C# Rating: 0 out of 5 stars0 ratings
Computers For You
The Invisible Rainbow: A History of Electricity and Life Rating: 5 out of 5 stars5/5Procreate for Beginners: Introduction to Procreate for Drawing and Illustrating on the iPad Rating: 0 out of 5 stars0 ratingsSlenderman: Online Obsession, Mental Illness, and the Violent Crime of Two Midwestern Girls Rating: 4 out of 5 stars4/5The Innovators: How a Group of Hackers, Geniuses, and Geeks Created the Digital Revolution Rating: 4 out of 5 stars4/5Elon Musk Rating: 4 out of 5 stars4/5Standard Deviations: Flawed Assumptions, Tortured Data, and Other Ways to Lie with Statistics Rating: 4 out of 5 stars4/5CompTIA IT Fundamentals (ITF+) Study Guide: Exam FC0-U61 Rating: 0 out of 5 stars0 ratingsMastering ChatGPT: 21 Prompts Templates for Effortless Writing Rating: 5 out of 5 stars5/5Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game - Updated Edition Rating: 4 out of 5 stars4/5The Professional Voiceover Handbook: Voiceover training, #1 Rating: 5 out of 5 stars5/5The Complete Powershell Training for Beginners Rating: 0 out of 5 stars0 ratings101 Awesome Builds: Minecraft® Secrets from the World's Greatest Crafters Rating: 4 out of 5 stars4/5The ChatGPT Millionaire Handbook: Make Money Online With the Power of AI Technology Rating: 4 out of 5 stars4/5The Hacker Crackdown: Law and Disorder on the Electronic Frontier Rating: 4 out of 5 stars4/5Uncanny Valley: A Memoir Rating: 4 out of 5 stars4/5CompTIA Security+ Get Certified Get Ahead: SY0-701 Study Guide Rating: 5 out of 5 stars5/5SQL QuickStart Guide: The Simplified Beginner's Guide to Managing, Analyzing, and Manipulating Data With SQL Rating: 4 out of 5 stars4/5Everybody Lies: Big Data, New Data, and What the Internet Can Tell Us About Who We Really Are Rating: 4 out of 5 stars4/5How to Create Cpn Numbers the Right way: A Step by Step Guide to Creating cpn Numbers Legally Rating: 4 out of 5 stars4/5The Huffington Post Complete Guide to Blogging Rating: 3 out of 5 stars3/5Going Text: Mastering the Command Line Rating: 4 out of 5 stars4/5Deep Search: How to Explore the Internet More Effectively Rating: 5 out of 5 stars5/5Grokking Algorithms: An illustrated guide for programmers and other curious people Rating: 4 out of 5 stars4/5
Reviews for The Discrete Charm of the Machine
0 ratings0 reviews
Book preview
The Discrete Charm of the Machine - Ken Steiglitz
The Discrete Charm of the Machine
The
Discrete
Charm
of the
Machine
Why the World
Became Digital
Ken
Steiglitz
Princeton University Press / Princeton and Oxford
Copyright © 2019 by Princeton University Press
Published by Princeton University Press
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press
6 Oxford Street, Woodstock, Oxfordshire OX20 1TR
press.princeton.edu
All Rights Reserved
Library of Congress Control Number: 2018936381
ISBN 978-0-691-17943-8
eISBN 978-0-691-18417-3 (ebook)
Version 1.0
British Library Cataloging-in-Publication Data is available
Editorial: Vickie Kearn
Production Editorial: Leslie Grundfest
Text Design: Chris Ferrante
Production: Jacquie Poirier
Publicity: Sara Henning-Stout
Copyeditor: Jennifer McClain
To my daughter Bonnie
Contents
To the Reader xiii
Part I / A Century of Valves 1
1 The Discrete Revolution 3
1.1 My Golden Age of Garbage 3
1.2 Nostalgia and the Aesthetics of Technology 4
1.3 Some Terminology 6
2 What’s Wrong with Analog? 10
2.1 Signals and Noise 10
2.2 Reproduction and Storage 11
2.3 The Origins of Noise 11
2.4 Thermal Noise in Electronics 12
2.5 Other Noise in Electronics 13
2.6 Digital Immunity 17
2.7 Analog Rot 19
2.8 Caveats 20
3 Signal Standardization 22
3.1 A Reminiscence 22
3.2 Ones and Zeros 22
3.3 Directivity of Control 24
3.4 Gates 24
3.5 The Electron 25
3.6 Edison’s Lightbulb Problems 26
3.7 De Forest’s Audion 27
3.8 The Vacuum Tube as Valve 30
3.9 The Rest of Logic 33
3.10 Clocks and Doorbells 34
3.11 Memory 35
3.12 Other Ways to Build Valves 36
4 Consequential Physics 43
4.1 When Physics Became Discrete 43
4.2 The Absolute Size of Things 46
4.3 The Heisenberg Uncertainty Principle 47
4.4 Explaining Wave-Particle Duality 49
4.5 The Pauli Exclusion Principle 50
4.6 Atomic Physics 53
4.7 Semiconductors 54
4.8 The P-N Junction 56
4.9 The Transistor 58
4.10 Quantum Tunneling 59
4.11 Speed 60
5 Your Computer Is a Photograph 62
5.1 Room at the Bottom 62
5.2 The Computer as Microphotograph 64
5.3 Heisenberg in the Chip Foundry 66
5.4 Moore’s Law and the Time of Silicon:ca. 1960–? 68
5.5 The Exponential Wall 74
Part II / Sound and Pictures 77
6 Music from Bits 79
6.1 The Monster in 1957 79
6.2 A Chance Encounter with a D-to-A Converter 81
6.3 Sampling and Monsieur Fourier 82
6.4 Nyquist’s Sampling Principle 83
6.5 Another Win for Digital 86
6.6 Another Isomorphism 88
7 Communication in a Noisy World 90
7.1 Claude Shannon’s 1948 Paper 90
7.2 Measuring Information 91
7.3 Entropy 94
7.4 Noisy Channels 96
7.5 Coding 97
7.6 The Noisy Coding Theorem 101
7.7 Another Win for Digital 102
Part III / Computation 105
8 Analog Computers 107
8.1 From the Ancient Greeks 107
8.2 More Ingenious Devices 111
8.3 Deeper Questions 117
8.4 Computing with Soap Films 119
8.5 Local and Global 121
8.6 Differential Equations 123
8.7 Integration 125
8.8 Lord Kelvin’s Research Program 126
8.9 The Electronic Analog Computer 129
9 Turing’s Machine 132
9.1 The Ingredients of a Turing Machine 132
9.2 The All-Analog Machine 133
9.3 The Partly Digital Computer 134
9.4 A Reminiscence: The Stored-Program Loom in New Jersey 135
9.5 Monsieur Jacquard’s Loom 136
9.6 Charles Babbage 138
9.7 Babbage’s Analytical Engine 141
9.8 Augusta Ada Byron, Countess of Lovelace 143
9.9 Turing’s Abstraction 144
10 Intrinsic Difficulty 148
10.1 Being Robust 148
10.2 The Polynomial/Exponential Dichotomy 149
10.3 Turing Equivalence 151
10.4 Two Important Problems 153
10.5 Problems with Easily Checked Certificates (NP) 154
10.6 Reducing One Problem to Another 156
10.7 Yes/No Problems 157
10.8 Cook’s Theorem: 3-SAT Is NP-Complete 158
10.9 Thousands More NP-Complete Problems 160
11 Searching for Magic 164
11.1 Analog Attacks on NP-Complete Problems 164
11.2 The Missing Law 169
11.3 The Church-Turing Thesis 170
11.4 The Extended Church-Turing Thesis 171
11.5 Locality: From Einstein to Bell 172
11.6 Behind the Quantum Curtain 176
11.7 Quantum Hacking 178
11.8 The Power of Quantum Computers 179
11.9 Life Itself 180
11.10 The Uncertain Limits of Computation 182
Part IV / Today and Tomorrow 183
12 The Internet, Then the Robots 185
12.1 Ideas 185
12.2 The Internet: Packets, Not Circuits 187
12.3 The Internet: Photons, Not Electrons 189
12.4 Enter Artificial Intelligence 193
12.5 Deep Learning 194
12.6 Obstacles 196
12.7 Enter Robots 202
12.8 The Problem of Consciousness 204
12.9 The Question of Values 206
Epilogue 209
Notes 211
Bibliography 221
Index 229
To the Reader
What this book is about
The machines we call computers have reshaped our lives, and may in the end transform humanity itself. The revolution is based on just one idea: build devices that store and manipulate information in the form of discrete bits. My aim in this book is to explain why this seemingly simple idea is so powerful.
It happens without trying that in pinpointing the virtues of the discrete, digital form, questions arise about the limits of the spectacular progress in technology we’ve seen in the past half century. Computers are cramming more and more components into smaller and smaller spaces, operating faster and faster. Can this go on forever? Computer programs are getting more and more clever. Are there problems that will always be beyond the reach of computers? Will computers become more clever than we? Will they replace us?
At the end of the book, we return to the opening theme and pose a further fundamental question: Will digital computers always be superior to analog computers, which use information in continuous, nondiscrete form, or is there some magic
that remains hidden in the analog world, beyond the reach of the digital computer? The human brain uses both digital and analog forms of information—is Nature keeping some secrets to herself about the ultimate nature of computation?
Who is the intended reader?
Briefly: My ideal reader is interested in science generally, perhaps computers in particular, but is not technically trained. And she just might be curious about why computers are digital. This book is not by any means an introduction to computer science, nor is it about how to program or use computers. There are no equations and no computer code. The reader will not escape, however, without some knowledge of how today’s computers are built at the most basic, microscopic level, and an appreciation of why they got that way.
A quick tour
There are a number of reasons why computers are digital. Some are physical in nature, and these naturally tend to be more concrete and intuitively clear. For example, the inevitable presence of noise, everywhere in nature, tends to obscure information. Similarly, electrical current consists of the flow of discrete particles called electrons. This means that electrical signals are, at the microscopic level, necessarily granular. We begin, in part I, by discussing these physical obstacles to reliable computation and how they are circumvented by storing and using information in digital form.
We next show how the familiar notion of a valve can provide a building block for all computation. The transistor is a valve in silicon, and the explosive development of solid-state technology reflected in Moore’s law has given us the integrated circuit chip that today holds more than a billion transistors. We shall see that the limits of this progress will ultimately be determined by quantum mechanics and, in particular, by Heisenberg’s uncertainty principle.
Part II is devoted to two fundamental ideas that emerge from the study of communication rather than physics. Their development resulted in digital signal processing, high-speed networking, and the internet. The resulting ability to share sound and images almost instantly across the globe has changed our lives profoundly in just one generation.
The first idea, Fourier analysis, tells us that we can treat any signal as being composed of a collection of different frequencies. This insight leads to Nyquist’s principle, which determines just how fast we need to sample audio and video signals to preserve all their information, and is behind the concept of bandwidth, now a commonly recognized—and critical—resource in our modern world.
The second idea is the use of coding to protect information in a noisy environment. The empirical practice of using redundancy for safely transmitting and storing signals inspired an elegant and influential theory of information, which sprang fully fledged from the brain of Claude Shannon. The crown jewel of the theory is his remarkable (and surprising) noisy coding theorem, which reveals the full depth and significance of the concept of bandwidth.
In part III we move on to yet more sophisticated and challenging territory, ending up, in fact, at the limits of current scientific knowledge. Returning to analog machines for computation, we develop the notion of a problem that is intrinsically difficult. At this point we get a taste of modern complexity theory, the concept of an NP-complete problem, and the most important open problem in computer science.
Finally, we ask if there might be ways to escape the limits of the computers we use today. This leads naturally to the Church-Turing thesis, which asserts that the hypothetical machine invented by Alan Turing essentially captures the concept of computation; and the extended Church-Turing thesis, which takes this one step further, proposing that the Turing machine is the embodiment of all practical computation (including analog). We will see that neither thesis is purely mathematical in nature, and neither can ever be proved. From here it is a short step to questions about the ultimate power of computers that take advantage of quantum mechanics.
In the concluding chapter, which constitutes part IV, we review the six main ideas that, in barely a half century, transformed our information technology from analog to digital and led to today’s packet-switched and optically delivered internet. We arrive at the edge of the unknown: Are NP-complete problems intrinsically difficult? (Probably yes.) Do Turing machines capture the notion of all practical computation? (Probably yes, with a quantum-mechanical upgrade.) Can machines be conscious, and can they suffer? (Quite up in the air.) Whatever the answers to these questions, and regardless of whether their brains will be able to tap unknown analog or quantum power, the current accelerating development of discrete machines is attending the birth of autonomous robots. Ready or not, the robots are coming! How will we face our responsibility to our heirs and successors? Will our human cultural values survive?
A personal note
I grew up on masterpieces of popular science like Gamow (1947), Courant and Robbins (1996), and, later, Russell (2009) and Feynman (2006). These books share one essential feature: they simplify and at times may cut corners, but they never, ever lie. As Ralph Leighton says to the hypothetical student in his preface to Feynman (2006), There is nothing in this book that has to be ‘unlearned.’
With this book I have tried, with my limited resources and all the humility I can muster, to follow these heroes.
Finally, I must confess to a nostalgic attachment to the analog/digital theme. I was born at just about the same time that the first functioning digital computers were being built, but I grew up listening to remarkably practical analog radio. My first paycheck was for summer work writing assembler code for a vacuum-tube digital computer, but I used analog computers in some of my undergraduate courses. My dissertation was on the correspondence between analog and digital signal processing. Throughout this book-in-sonata-form, we develop the analog/digital theme from a variety of points of view; I invite you to chapter 1, the exposition section.
Acknowledgments
There is always the fear that acknowledging the aid or inspiration of colleagues and friends will implicate them in my mischief. Nevertheless, I thank the following for help of all sorts; they are appreciated and blameless: Andrew Appel, Sanjeev Arora, David August, György Buzsáki, Bernard Chazelle, David Dobkin, Mike Fredman, Jack Gelfand, Mike Honig, Andrea LaPaugh, Kai Li, Richard Lipton, Christos Papadimitriou, Mona Singh, Olga Troyanskaya, Kevin Wayne, Andy Yao.
For encouragement, expert guidance, and good cheer, I am also indebted to the staff at Princeton University Press, including acquisitions editor Vickie Kearn, production editor Leslie Grundfest, and editorial assistant Lauren Bucca, as well as to copyeditor Jennifer McClain.
Part I
A Century of Valves
1 The Discrete Revolution
1.1 My Golden Age of Garbage
What is usually called the computer revolution
is really about much more—it’s about a radical conversion of our view of the world from continuous to discrete. As for your author, my entrance into this world couldn’t have been timed better to observe the apparently sudden transformation. I arrived in 1939, a few months before Hitler invaded Poland. At that time the stage had been set, rather subtly and gradually, for the development of things digital, and the pressure of the ensuing war years propelled us all, not so subtly and not so gradually, into what we now know as the Digital Age. This book is about the most basic ideas and principles behind the change. Why did the world change in such a fundamental way from analog to digital, and where might we humans—a species itself built along both analog and digital lines—be headed?
I apologize for the rather dark beginning, but it’s a fact that the dirty fingers of war have never failed to leave their prints on the annals of what we term progress.
The dawn of the computer age is closely linked to decryption efforts in World War II, as well as to the development of the atomic bomb.
On August 6, 1945, I was only dimly aware of the fact that I was in New Jersey and not Japan, where bombardier Thomas Ferebee was watching Hiroshima’s Aioi Bridge in the crosshairs of his Norden bombsight. The bombsight, which subsequently released the first uranium-fission atomic bomb and began the end of World War II, was an analog computer. It solved the equations of motion that determined the path of the bomb, using things like cams and gears, a gyroscope, and a telescope, all mechanical devices. But it was a computer nevertheless, although applying the term to a mess of moving steel parts might surprise some people today. Well into the 1950s there were two kinds of computers: analog and digital. In fact, analog computers of the electronic sort were the only way to solve certain kinds of complicated problems, and were, in a handful of situations, very useful. Electronic analog computers were programmed by plugging wires into a patch panel, which was like a telephone switchboard (you may have seen one in an old movie), and by the time any interesting problem was running, the patch panel was a rat’s nest.
But before the mid-twentieth century everything was analog; digital just hadn’t been invented.¹ The most important piece of information technology I knew as a child was the radio, very analog at the time, and it was my remarkable piece of good fortune when the postwar engines of production turned to consumer goods, and consumers bought new, streamlined, plastic radios. Garbage night meant that the monstrous mahogany console radios of the 1930s could often be found curbside—with booming bass, hardly any treble because of the limitations of AM broadcasting, and all manner of interesting electronic parts inside.² That was how I learned to love the glow of vacuum tubes and the aroma of hot rosin-core solder congealing around the twisted leads of condensers (as capacitors were called), resistors, coils, and other more exotic components. Sometimes it was an autopsy that I performed on these found radios, but often it was a vivi-section, since many of them worked, or could be made to work, excellently. Some of these lucky finds even had shortwave bands, and garbage night turned out to be my gateway to the world at large.
It was all analog. When television came, that, too, was all analog. So were telephones. There just wasn’t anything else.
1.2 Nostalgia and the Aesthetics of Technology
Video and audio signals fly in and out of our brains all day long, and devices that process those signals—radio, television, recorded film and music players, telephones—were all digitized in the latter half of the twentieth century; that is, within my lifetime. One consequence is that the devices we use every day for what is now called digital signal processing have more or less converged to the same, rather dull-looking machine—essentially a small chip behind a screen, in a plastic case, occasionally with a couple of wires hanging out. In contrast, in the good old days radios were radios, television sets were television sets, cameras cameras, telephones telephones. You could tell what a device did by looking at it. And sometimes you would need an elephant to make it portable: the Stromberg Carlson console radio I lugged home with the help of my friends was crafted with a sturdy wooden cabinet, housing a loudspeaker with a huge electromagnet, a large lit dial, and hefty knobs that gave the operator the feeling of controlling an important piece of equipment—to a child, and perhaps to a grown-up as well, a spaceship.
My favorite effect was the magic eye tuning indicator, usually a 6E5 vacuum tube that had a fluorescent screen at its end, visible in a circular hole on the front panel of the radio. It glowed green with a dark crescent that contracted in proportion to the signal strength. Carefully tuning a station to reduce the crescent to a narrow slit was a joyful experience, especially in a dark room where the eerie glow did seem magical for sure. Punching in the frequency (or URL) of a radio station just does not provide the same tactile and visual pleasure. If your childhood came after such electronic apparatus, you don’t know what I’m talking about; such is the nature of nostalgia. No doubt the iPhone will stimulate similar feelings fifty years from now, when signals may very well go directly to our brains without the need for any beautiful little intermediary machines.
Of course there is a lively market for retro style and retro devices; certain cults have grown around the disappearance of, for example, shellac, vinyl, and analog tape recordings, or film cameras and the once pervasive technology of chemical-based photography. It’s common to hear that vacuum-tube amplifiers have a warmer
sound, although it’s not certain how much of the warmth is due to distortion from the inherent nonlinearity of the vacuum-tube analog technology, or the psychological glow from the hot tubes themselves.
Sometimes the nostalgic longing approaches the mystical. Water Lily Acoustics produces superb recordings of Indian classical music, and they go through great pains to keep the sound recording free of the digital taint until the very last step in the process. For example, the booklet for a compact disc recording of Ustad Imrat Khan offers the following assurance:³
This is a pure analog recording done exclusively with custom-built vacuum-tube electronics. The microphone set-up was the classic Blumlein arrangement. No noise reduction, equalization, compression, or limiting of any sort was used in the making of this recording.
The booklet goes on to describe the microphones (which use tubes), recorder (Ampex MR70, half-inch, two-track, 15-inch-per-second tape, using vacuum tubes called nuvistors), and so on.
Spiritual values aside, a good analog sound recording, or, for that matter, a good analog photograph taken with film and printed well, can be, technically, a lot better than a bad digital recording or a bad digital photograph. We have much more to say about the ultimate and practical limitations of analog and digital technology as we go along.
1.3 Some Terminology
So far, we’ve been using the terms digital and analog rather loosely. Before going further, we need to clarify this terminology. For our purposes, digital means that a signal of interest is being represented by a sequence or array of numbers; analog means that a signal is represented by the value of some continuously variable quantity. This variable can be the voltage or current in an electrical circuit, say, or the brightness of a scene at some point, or temperature, pressure, velocity, and so on, as long as its value is continuously variable. All the possible values of a digital signal can be counted, and there is a definite gap between them; those of an analog variable cannot be counted, and there is no definite gap between them. Generally, we use discrete (actually discrete-valued
) to mean digital and continuous (actually continuous-valued
) to mean analog, although this overlooks some distinctions that are not important at this point.
When you buy a wristwatch or a clock, for example, you have a choice between an analog display
and a digital display.
This is exactly the sense in which we use the terms—but take note of the fact that we refer to the display and not the internal mechanism of the timekeeper. A clock with an analog display has hands that can move continuously, whereas a digital display shows numbers that change discontinuously, which is another way to say suddenly. The hands of a clock actually represent time by the rotational position of gears. These days, the usual clock with an analog display has an internal timekeeping mechanism that is digital (except for old-fashioned windup clocks). But at one point there were the opposite kinds of clocks, with analog mechanisms and digital displays—usually using gears and cams to flip displays with numbers printed on them.
On the morning of Pi Day
(March 14) of 2015, there was a moment a bit after 9:26 and 53 seconds when the time could be written 3.14159265358979...; that is, π. The moment was fleeting to say the least; it was infinitesimally brief. And it will never occur again. Ever. If you were watching the hands of a clock with an analog display, you might have tried to take a photo at the exact moment of π, but the photo would have taken some finite time, and you would have necessarily blurred the second hand. That is an inevitable consequence of measuring an analog quantity of any kind.
Very commonly, audio and video signals are represented by voltages, either in a computer, smartphone, copper cable, or some kind of electrical circuit like those in an amplifier. This is the usual way that such signals are recorded by microphones and video cameras, and the resulting signals are transmitted and reproduced using voltages in electrical circuits. A microphone converts a sound pressure wave in the air to a time-varying voltage. A video camera converts a light image into an array of time-varying voltages. These audio and video signals usually start their lives out as analog signals and are converted to digital form after their initial capture, assuming that they are going to be processed in some way in digital form.
The device that converts an analog signal to digital form is called, naturally, an analog-to-digital converter (A-to-D converter), and the opposite operation is performed by a digital-to-analog converter (D-to-A converter). Thus, for example, the light-sensitive screen in a digital camera is really an A-to-D converter, whereas your computer monitor is really a D-to-A converter.
I’ll try to be clear about what I mean when we use the terms digital, analog, discrete, and continuous, but I should mention some possible sources of confusion. First, it often happens that it is time itself that is thought of as discrete or continuous, rather than the values of a signal. When there is any possible confusion, I will state explicitly that time is being considered. Second, there is the awkward fact that standard mathematical terminology uses the term continuous in a slightly different way. Mathematically speaking, a curve is continuous
if it does not jump suddenly from one value to another but rather changes smoothly.
The reader who has studied calculus will be aware of this alternate interpretation, but will not be confused by it.
Finally, the term discrete is used by physicists in another sense. A most important example of this usage comes up when we ask the question, What is light?
The question has puzzled scientists for