Understanding Digital Signal Processing
Third Edition
Richard G. Lyons
Upper Saddle River, NJ • Boston • Indianapolis • San Francisco
New York • Toronto • Montreal • London • Munich • Paris • Madrid
Capetown • Sydney • Tokyo • Singapore • Mexico City
Preface
This book is an expansion of previous editions of Understanding Digital Signal Processing. Like those earlier
editions, its goals are (1) to help beginning students understand the theory of digital signal processing (DSP)
and (2) to provide practical DSP information, not found in other books, to help working engineers/scientists
design and test their signal processing systems. Each chapter of this book contains new information beyond that
provided in earlier editions.
It’s traditional at this point in the preface of a DSP textbook for the author to tell readers why they should learn
DSP. I don’t need to tell you how important DSP is in our modern engineering world. You already know that.
I’ll just say that the future of electronics is DSP, and with this book you will not be left behind.
For Instructors
This third edition is appropriate as the text for a one- or two-semester undergraduate course in DSP. It follows
the DSP material I cover in my corporate training activities and a signal processing course I taught at the
University of California Santa Cruz Extension. To aid students in their efforts to learn DSP, this third edition
provides additional explanations and examples to increase its tutorial value. To test a student’s understanding
of the material, homework problems have been included at the end of each chapter. (For qualified instructors, a
Solutions Manual is available from Prentice Hall.)
For Practicing Engineers
To help working DSP engineers, the changes in this third edition include, but are not limited to, the following:
• Practical guidance in building discrete differentiators, integrators, and matched filters
• Descriptions of statistical measures of signals, variance reduction by way of averaging, and techniques for
computing real-world signal-to-noise ratios (SNRs)
• A significantly expanded chapter on sample rate conversion (multirate systems) and its associated filtering
• Implementing fast convolution (FIR filtering in the frequency domain)
• IIR filter scaling
• Enhanced material covering techniques for analyzing the behavior and performance of digital filters
• Expanded descriptions of industry-standard binary number formats used in modern processing systems
• Numerous additions to the popular “
Digital Signal Processing Tricks” chapter
For Students
Learning the fundamentals, and how to speak the language, of digital signal processing does not require
profound analytical skills or an extensive background in mathematics. All you need is a little experience with
elementary algebra, knowledge of what a sinewave is, this book, and enthusiasm. This may sound hard to
believe, particularly if you’ve just flipped through the pages of this book and seen figures and equations that
look rather complicated. The content here, you say, looks suspiciously like material in technical journals and
textbooks whose meaning has eluded you in the past. Well, this is not just another book on digital signal
processing.
In this book I provide a gentle, but thorough, explanation of the theory and practice of DSP. The text is not
written so that you may understand the material, but so that you must understand the material. I’ve attempted to
avoid the traditional instructor–student relationship and have tried to make reading this book seem like talking
to a friend while walking in the park. I’ve used just enough mathematics to help you develop a fundamental
understanding of DSP theory and have illustrated that theory with practical examples.
I have designed the homework problems to be more than mere exercises that assign values to variables for the
student to plug into some equation in order to compute a result. Instead, the homework problems are designed
to
be as educational as possible in the sense of expanding on and enabling further investigation of specific aspects
of DSP topics covered in the text. Stated differently, the homework problems are not designed to induce “death
by algebra,” but rather to improve your understanding of DSP. Solving the problems helps you become
proactive in your own DSP education instead of merely being an inactive recipient of DSP information.
The Journey
Learning digital signal processing is not something you accomplish; it’s a journey you take. When you gain an
understanding of one topic, questions arise that cause you to investigate some other facet of digital signal
processing.
†
Armed with more knowledge, you’re likely to begin exploring further aspects of digital signal processing
much like those shown in the diagram on page xviii. This book is your tour guide during the first steps of your
journey.
†
“You see I went on with this research just the way it led me. This is the only way I ever heard of research going. I asked a question,
devised some method of getting an answer, and got—a fresh question. Was this possible, or that possible? You cannot imagine what
this means to an investigator, what an intellectual passion grows upon him. You cannot imagine the strange colourless delight of these
intellectual desires” (Dr. Moreau—infamous physician and vivisectionist from H.G. Wells’ Island of Dr. Moreau, 1896).
You don’t need a computer to learn the material in this book, but it would certainly help. DSP simulation
software allows the beginner to verify signal processing theory through the time-tested trial and error process.
‡
In particular, software routines that plot signal data, perform the fast Fourier transforms, and analyze digital
filters would be very useful.
‡
“One must learn by doing the thing; for though you think you know it, you have no certainty until you try it” (Sophocles, 496–406
B.C.).
As you go through the material in this book, don’t be discouraged if your understanding comes slowly. As the
Greek mathematician Menaechmus curtly remarked to Alexander the Great, when asked for a quick
explanation of mathematics, “There is no royal road to mathematics.” Menaechmus was confident in telling
Alexander the only way to learn mathematics is through careful study. The same applies to digital signal
processing. Also, don’t worry if you need to read some of the material twice. While the concepts in this book
are not as complicated as quantum physics, as mysterious as the lyrics of the song “Louie Louie,” or as
puzzling as the assembly instructions of a metal shed, they can become a little involved. They deserve your
thoughtful attention. So, go slowly and read the material twice if necessary; you’ll be glad you did. If you show
persistence, to quote Susan B. Anthony, “Failure is impossible.”
Coming Attractions
Chapter 1 begins by establishing the notation used throughout the remainder of the book. In that chapter we
introduce the concept of discrete signal sequences, show how they relate to continuous signals, and illustrate
how those sequences can be depicted in both the time and frequency domains. In addition, Chapter 1 defines
the operational symbols we’ll use to build our signal processing system block diagrams. We conclude that
chapter with a brief introduction to the idea of linear systems and see why linearity enables us to use a number
of powerful mathematical tools in our analysis.
Chapter 2 introduces the most frequently misunderstood process in digital signal processing, periodic sampling.
Although the concept of sampling a continuous signal is not complicated, there are mathematical subtleties in
the process that require thoughtful attention. Beginning gradually with simple examples of lowpass sampling,
we then proceed to the interesting subject of bandpass sampling. Chapter 2 explains and quantifies the
frequency-domain ambiguity (aliasing) associated with these important topics.
Chapter 3 is devoted to one of the foremost topics in digital signal processing, the discrete Fourier transform
(DFT) used for spectrum analysis. Coverage begins with detailed examples illustrating the important properties
of the DFT and how to interpret DFT spectral results, progresses to the topic of windows used to reduce DFT
leakage, and discusses the processing gain afforded by the DFT. The chapter concludes with a detailed
discussion of the various forms of the transform of rectangular functions that the reader is likely to encounter in
the literature.
Chapter 4 covers the innovation that made the most profound impact on the field of digital signal processing,
the fast Fourier transform (FFT). There we show the relationship of the popular radix 2 FFT to the DFT,
quantify the powerful processing advantages gained by using the FFT, demonstrate why the FFT functions as it
does, and present various FFT implementation structures. Chapter 4 also includes a list of recommendations to
help the reader use the FFT in practice.
Chapter 5 ushers in the subject of digital filtering. Beginning with a simple lowpass finite impulse response
(FIR) filter example, we carefully progress through the analysis of that filter’s frequency-domain magnitude
and phase response. Next, we learn how window functions affect, and can be used to design, FIR filters. The
methods for converting lowpass FIR filter designs to bandpass and highpass digital filters are presented, and the
popular Parks-McClellan (Remez) Exchange FIR filter design technique is introduced and illustrated by
example. In that chapter we acquaint the reader with, and take the mystery out of, the process called
convolution. Proceeding through several simple convolution examples, we conclude Chapter 5 with a
discussion of the powerful convolution theorem and show why it’s so useful as a qualitative tool in
understanding digital signal processing.
Chapter 6 is devoted to a second class of digital filters, infinite impulse response (IIR) filters. In discussing
several methods for the design of IIR filters, the reader is introduced to the powerful digital signal processing
analysis tool called the z-transform. Because the z-transform is so closely related to the continuous Laplace
transform, Chapter 6 starts by gently guiding the reader from the origin, through the properties, and on to the
utility of the Laplace transform in preparation for learning the z-transform. We’ll see how IIR filters are
designed and implemented, and why their performance is so different from that of FIR filters. To indicate under
what conditions these filters should be used, the chapter concludes with a qualitative comparison of the key
properties of FIR and IIR filters.
Chapter 7 introduces specialized networks known as digital differentiators, integrators, and matched filters. In
addition, this chapter covers two specialized digital filter types that have not received their deserved exposure
in traditional DSP textbooks. Called interpolated FIR and frequency sampling filters, providing improved
lowpass filtering computational efficiency, they belong in our arsenal of filter design techniques. Although
these are FIR filters, their introduction is delayed to this chapter because familiarity with the z-transform (in
Chapter 6) makes the properties of these filters easier to understand.
Chapter 8 presents a detailed description of quadrature signals (also called complex signals). Because
quadrature signal theory has become so important in recent years, in both signal analysis and digital
communications implementations, it deserves its own chapter. Using three-dimensional illustrations, this
chapter gives solid physical meaning to the mathematical notation, processing advantages, and use of
quadrature signals. Special emphasis is given to quadrature sampling (also called complex down-conversion).
Chapter 9 provides a mathematically gentle, but technically thorough, description of the Hilbert transform—a
process used to generate a quadrature (complex) signal from a real signal. In this chapter we describe the
properties, behavior, and design of practical Hilbert transformers.
Chapter 10 presents an introduction to the fascinating and useful process of sample rate conversion (changing
the effective sample rate of discrete data sequences through decimation or interpolation). Sample rate
