数字信号处理：英文版

　　本书将理论与实践完美地结合起来，通过具有启发性的解释和精心挑选的例子，采用读者可以理解的数学表示方法，对数字信号处理 （DSP） 技术进行了解释，帮助读者从整体上掌握DSP，并逐步掌握较高层次的DSP概念和应用。本书第1版就很畅销，可读性强，受到专业人员的广泛喜爱。第2版以此为基础，除包括周期采样、离散傅里叶变换 （DFT） 、快速傅里叶变换 （FFT） 、数字滤波器、离散希尔伯特变换、采样速率转换、量化、信号取均值等内容外，还扩展了无限脉冲响应滤波器 （IIR） 、脉冲频率采样和内插式有限脉冲响应滤波器 （FIR） 等内容，增添了多种新的速率系统，包括多相滤波器和串级积分器-梳状有限脉冲响应滤波器 （FIR） 。本书是工程师以及专业技术人员掌握和应用DSP技术的最佳参考书。本书的主要内容·包含实际的、直观的、丰富的DSP例子。·强调实际的、日常的DSP应用和解决方案。·提供了全新的正交信号处理内容，包括易于理解的三维空间图。·包括即使是经验丰富的专业人士也可能忽略的技术方法。·涵盖了频率采样、内插式FIR、CIC等重要滤波器。·提供了流行的数字信号处理技巧。Richard G. Lyons 毕业于美国加州阿克隆大学， 获得BSEE学位. 他是美国加州Mount View市Besser公司的咨询系统工程师和讲师， 为许多著名公司举办过DSP方面的技术培训， 这些公司包括：摩托罗拉. 洛克希德·马丁. 霍尼韦尔. 德州仪器. 诺思洛普·格鲁曼. 朗迅. 诺基亚. 高通. 国家半导体. 通用动力. 西门子 （现为英飞灵） 等. 他做过国家安全局和TRW公司的众多DSP系统的首席硬件工程师， 负责系统设计. 开发. 测试和安装. Lyons曾在加利福尼亚大学圣克鲁兹分校任教， 并发表过很多有关DSP的文章. 作为IEEE Signal Processing杂志的副主编， 他创建和编辑了"DSP Tips & Tricks"专栏.This book is an expansion of the original Understanding Digital Signal Process-ing textbook published in 1997 and， like the first edition， its goal is to help be-ginners understand this relatively new technology of digital signal processing （DSP）. Additions to this second edition include：· Expansion and clarification of selected spectrum analysis and digital ill-tering topics covered in the first edition making that material more valu-able to the DSP beginner.· Expanded coverage of quadrature （complex I/Q） signals. In many cases we used three-dimension time and frequency plots to enhance the de-scription of， and give physical meaning to， these two-dimensional sig-nals.· With the new emphasis on quadrature signals， material was added de-scribing the Hilbert transform and how it‘s used in practice to generate quadrature signals.· Discussions of Frequency Sampling， Interpolated FIR， and CIC filters;giving these important filters greater exposure than they‘ve received in past DSP textbooks.· A significant expansion of the popular "Digital Signal Processing Tricks" chapter.· Revision of the terminology making it more consistent with the modern day language of DSP. It‘s traditional at this point in the preface of a DSP textbook for the au-thor 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. LEARNING DIGITAL SIGNAL PROCESSING 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 al-gebra， 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 the material in technical journals and textbooks that， in the past， have successfully resisted your attempts to understand. Well， this is not just another book on digital signal processing. This book‘s goal is to gently provide explanation followed by illustration， not so that you may understand the material， but that you must understand the material？ Remember the first time you saw two people playingchess？ The game probably appeared to be mysterious and confusing. As younow know， no individual chess move is complicated. Given a little patience， the various chess moves are easy to learn. The game‘s complexity comes from deciding what combinations of moves to make and when to make them. So it is with understanding digital signal processing. First we learn the fundamental rules and processes， and then practice using them in combination. If learning digital signal processing is so easy， then why does the subject have the reputation of being hard to understand？ The answer lies partially in how the material is typically presented in the literature. It‘s difficult to convey technical information， with its mathematical subtleties， in written form. It‘s one thing to write equations， but it‘s another matter altogether to explain what those equations really mean from a practical standpoint， and that‘s the goal of this book. Too often， written explanation of digital signal processing theory appears in one of two forms： either mathematical miracles occur and the reader is simply given a short and sweet equation without further explanation， or the reader is engulfed in a flood of complex variable equations and phrases such as "it is obvious that，" and "with judicious application of the homogeneity property." In their defense， authors usually do provide the needed information， but too often the reader must figuratively grab a pick and shovel， put on a miner‘s helmet， and try to dig the information out of a mountain of mathematical expressions. （This book presents the results of several fruitful mining expeditions.） How many times have you followed the derivation of an equation， after which the author states they‘re going to illustrate that equation with an examplewhich turns out to be just another equation？ Although mathematics is necessary to describe digital signal processing， I‘ve tried to avoid overwhelming the reader with math because a recipe for technical writing that‘s too rich in equations is hard for the beginner to digest. The intent of this book is expressed by a popular quote from E.B. White in the introduction of his Elements of Style （Macmillan Publishing， New York， 1959）： "Will （Strunk） felt that the reader was in serious trouble most of the time， a man floundering in a swamp， and that it was the duty of anyone attempting to write English to drain the swamp quickly and get his man up on dry ground， or at least throw him a rope." I‘ve attempted to avoid the traditional instructor-student relationship， but rather to make reading this book like talking to a friend while walking in the park. I‘ve used just enough mathematics to develop a fundamental understanding of the theory， and then illustrate that theory with practical examples. THE JOURNEY Learning digital signal processing is not something you accomplish; it‘s a journey you take. When you gain an understanding of some 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 following diagram. This book is your tour guide during the first steps of your journey. You don‘t need a computer to learn the material in this book， but it would sure 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. 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 have to read some of the material twice. While the concepts in thisbook are not as complicated as quantum physics， as mysterious as the lyricsof the song Louie Louie， or as puzzling as the assembly instructions of a metal shed， they do get a little involved. They deserve your attention and thought. So go slow and read the material twice if you have to; you‘ll be glad you did. If you show persistence， to quote a phrase from 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 it‘s straightforward to grasp the concept of sampling a continuous signal， there are mathematical subtleties in the process that require thoughtful attention. Beginning gradually with simple examples of low-pass sampling， and progressing to the interesting subject of bandpass sampling， Chapter 2 explains and quantifies the frequencydomain 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 beginner is likely to encounter in the literature. That last topic is included there to clarify and illustrate the DFT of both real and complex sinusoids. 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 low-pass finite impulse response （FIR） filter example， we carefully progress through the analysis of that filter‘s frequency domain magnitudeand phase response. Next we learn how window functions affect， and can be used to design， FIR filters. The methods for converting low-pass FIR filter designs to bandpass and high pass digital filters are presented， and the popular Remez Exchange （Parks McClellan） FIR filter design technique is intro duced 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 FIR filters. To indicate under what conditions these filters should be used， that chapter concludes with a qualitative comparison of the key properties of FIR and IIR filters. Chapter 7 introduces two specialized digital filter types that have not re- ceived their deserved exposure in traditional DSP textbooks. Called frequency sampling and interpolated FIR filters， and providing enhanced 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 communication 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 a brief introduction to the fascinating， and very use- ful， process of sample rate conversion （changing the effective sample rate of discrete data sequences through decimation or interpolation）. Sample rate conversion：so useful in improving the performance and reducing the computational complexity of many signal processing operations：is essentially an exercise in lowpass filter design. As such， polyphase and cascaded integratorcomb filters are also described in this chapter. Chapter 11 covers the important topic of signal averaging. There we learn how averaging increases the accuracy of signal measurement schemes by reducing measurement background noise. This accuracy enhancement is called processing gain， and that chapter shows how to predict the processing gain associated with averaging signals in both the time and frequency domains. In addition， the key differences between coherent and incoherent averaging techniques are explained and demonstrated with examples. To complete that chapter the popular scheme known as exponential averaging is covered in some detail. Chapter 12 presents an introduction to the various binary number formats the reader is likely to encounter in modern digital signal processing. We establish the precision and dynamic range afforded by these formats along with the inherent pitfalls associated with their use. Our exploration of the critical subject of binary data word width （in bits） naturally leads us to a discussion of the numerical resolution limitations of analog to digital （A/D） converters and how to determine the optimum A/D converter word size for a given application. The problems of data value overflow roundoff errors are covered along with a statistical introduction to the two most popular remedies for overflow， truncation， and rounding. We end that chapter by covering the interesting subject of floating point binary formats that allow us to overcome most of the limitations induced by fixed point binary formats， particularly in reducing the iii effects of data overflow. Chapter 13 provides a collection of tricks of the trade used to make digital signal processing algorithms more efficient. Those techniques are compiled into a chapter at the end of the book for two reasons. First， it seems wise to keep our collection of tricks in one chapter so that we‘ll know where to find them in the future. Second， many of these schemes require an understanding of the material from the previous chapters， so the last chapter is an appropriate place to keep our arsenal of clever tricks. Exploring these techniques in detail verifies and reiterates many of the important ideas covered in previous chapters. The appendices include a number of topics to help the beginner under- stand the nature and mathematics of digital signal processing. A comprehensive description of the arithmetic of complex numbers is covered in Appendix A， while Appendix B derives the often used， but seldom explained， closed form of a geometric series. The subtle aspects and two forms of time reversal in discrete systems （of which zero-phase digital filtering is an application） are explained in Appendix C. The statistical concepts of mean， variance， and standard deviation are introduced and illustrated in Appendix D， while Appendix E provides a discussion of the origin and utility of the logarithmic decibel scale used to improve the magnitude resolution of spectral representations. Appendix F， in a slightly different vein， provides a glossary of the ter-minology used in the field of digital filters. Preface iii 1 DISCRETE SEQUENCES AND SYSTEMS 1.1Discrete Sequences and Their Notation 2 1.2Signal Amplitude， Magnitude， Power 8 1.3Signal Processing Operational Symbols 9 1.4Introduction to Discrete Linear Time-Invariant Systems12 1.5Discrete Linear Systems 12 1.6Time-Invariant Systems 17 1.7The Commutative Property of Linear Time-Invariant Systems 18 1.8Analyzing Linear Time-Invariant Systems 19 2 PERIODIC SAMPLING 21 2.1Aliasing： Signal Ambiquity in the Frequency Domain 21 2.2Sampling Low-Pass Signals 26 2.3Sampling Bandpass Signals 30 2.4Spectral Inversion in Bandpass Sampling 39 3 THE DISCRETE FOURIER TRANSFORM45 3.1Understanding the DFT Equation 46 3.2DFT Symmetry 58 3.3DFT Linearity 60 3.4DFT Magnitudes 61 3.5DFT Frequency Axis 62 3.6DFT ShiftingTheorem 63 3.7Inverse DFT 65 3.8DFT Leakage 66 3.9Windows 74 3.10DFT Scalloping Loss 82 3.11DFT Resolution， Zero Padding， and Frequency-Domain Sampling 83 3.12DFT Processing Gain 88 3.13The DFT of Rectangular Functions 91 3.14The DFT Frequency Response to a Complex Input 112 3.15The DFT Frequency Response to a Real Cosine Input 116 3.16The DFT Single-Bin Frequency Response to a Real Cosine Input 117 3.17Interpreting the DFT 120 4 THE FAST FOURIER TRANSFORM125 4.1Relationship of the FFT to the DFT 126 4.2Hints on Using FFTs in Practice 127 4.3FFT Software Programs 131 4.4Derivation of the Radix-2 FFT Algorithm 132 4.5FFT Input/Output Data Index Bit Reversal 139 4.6Radix-2 FFT Butterfly Structures 141 5 FINITE IMPULSE RESPONSE FILTERS 151 5.1An Introduction to Finite Impulse Response FIR Filters 152 5.2Convolution in FIR Filters 157 5.3Low-Pass FIR Filter Design 167 5.4Bandpass FIR Filter Design 183 5.5Highpass FIR Filter Design 184 5.6Remez Exchange FIR Filter Design Method 186 5.7Half-Band FIR Filters 188 5.8Phase Response of FIR Filters 190 5.9A Generic Description of Discrete Convolution 195 6 INFINITE IMPULSE RESPONSE FILTERS 211 6.1An Introduction to Infinite Impulse Response Filters 212 6.2The Laplace Transform 215 6.3The z-Transform 228 6.4Impulse Invariance IIR Filter Design Method 243 6.5Bilinear Transform IIR Filter Design Method 259 6.6Optimized IIR Filter Design Method 270 6.7Pitfalls in Building IIR Digital Filters 272 6.8Improving IIR Filters with Cascaded Structures 274 6.9A Brief Comparison of IIR and FIR Filters 279 SPECIALIZED LOWPASS FIR FILTERS283 7.1Frequency Sampling Filters： The Lost Art 284 7.2Interpolated Lowpass FIR Filters 319 8 QUADRATURE SIGNALS335 8.1Why Care About Quadrature Signals 336 8.2The Notation of Complex Numbers 336 8.3Representing Real Signals Using Complex Phasors 342 8.4A Few Thoughts on Negative Frequency 346 8.5Quadrature Signals in the Frequency Domain 347 8.6Bandpass Quadrature Signals in the Frequency Domain 350 8.7Complex Down-Conversion 352 8.8A Complex Down-Conversion Example 354 8.9An Alternate Down-Conversion Method 358 9 THE DISCRETE HILBERTTRANSFORM361 9.1Hilbert Transform Definition 362 9.2Why Care About the Hilbert Transform？ 364 9.3Impulse Response of a Hilbert Transformer 369 9.4Designing a Discrete Hilbert Transformer 371 9.5Time-Domain Analytic Signal Generation 377 9.6Comparing Analytical Signal Generation Methods 379 10 SAMPLE RATE CONVERSION381 10.1Decimation 382 10.2Interpolation 387 10.3Combining Decimation and Interpolation 389 10.4Polyphase Filters 391 10.5Cascaded Integrator-Comb Filters 397 11 SIGNAL AVERAGING411 11.1Coherent Averaging 412 11.2Incoherent Averaging 419 11.3Averaging Multiple Fast Fourier Transforms 422 11.4Filtering Aspects of Time-Domain Averaging 430 11.5Exponential Averaging 432 12 DIGITAL DATA FORMATS AND THEIR EFFECTS 439 12.1Fixed-Point Binary Formats 439 12.2Binary Number Precision and Dynamic Range 445 12.3Effects of Finite Fixed-Point Binary Word Length 446 12.4Floating-Point Binary Formats 462 12.5Block Floating-Point Binary Format 468 13 DIGITAL SIGNAL PROCESSING TRICKS 471 13.1Frequency Translation without Multiplication 471 13.2High-Speed Vector-Magnitude Approximation 479 13.3Frequency-Domain Windowing 484 13.4Fast Multiplication of Complex Numbers 487 13.5Efficiently Performing the FFT of Real Sequences 488 13.6Computing the Inverse FFT Using the Forward FFT 500 13.7Simplified FIR Filter Structure 503 13.8Reducing A/D Converter Quantization Noise 503 13.9A/D Converter Testing Techniques 510 13.10Fast FIR Filtering Using the FFT 515 13.11Generating Normally Distributed Random Data 516 13.12Zero-Phase Filtering 518 13.13Sharpened FIR Filters 519 13.14Interpolating a Bandpass Signal 521 13.15Spectral Peak Location Algorithm 523 13.16Computing FFT Twiddle Factors 525 13.17Single Tone Detection 528 13.18The Sliding DFT 532 13.19The Zoom FFT 541 13.20A Practical Spectrum Analyzer 544 13.21An Efficient Arctangent Approximation 547 13.22Frequency Demodulation Algorithms 549 13.23DC Removal 552 13.24Improving Traditional CIC Filters 556 13.25Smoothing Impulsive Noise 561 13.26Efficient Polynomial Evaluation 563 13.27Designing Very High-Order FIR Filters 564 13.28Time-Domain Interpolation Using the FFT 568 13.29Frequency Translation Using Decimation 571 13.30Automatic Gain Control （AGC） 571 13.31Approximate Envelope Detection 574 13.32A Quadrature Oscillator 576 13.33Dual-Mode Averaging 578 APPENDIX A. THE ARITHMETIC OF COMPLEX NUMBERS 585 A.1Graphical Representation of Real and Complex Numbers 585 A.2Arithmetic Representation of Complex Numbers 586 A.3Arithmetic Operations of Complex Numbers 588 A.4Some Practical Implications of Using Complex Numbers 593 APPENDIX B. CLOSED FORM OF A GEOMETRIC SERIES595 APPENDIX C. TIME REVERSAL AND THE DFT599 APPENDIX D. MEAN， VARIANCE， AND STANDARD DEVIATION603 D.1Statistical Measures 603 D.2Standard Deviation， or RMS， of a Continuous Sinewave 606 D.3The Mean and Variance of Random Functions 607 D.4The Normal Probability Density Function 610 APPENDIX E. DECIBELS （DB AND DBM） 613 E.1Using Logarithms to Determine Relative Signal Power 613 E.2Some Useful Decibel Numbers 617 E.3Absolute Power Using Decibels 619 APPENDIX F. DIGITAL FILTER TERMINOLOGY621 APPENDIX G. FREQUENCY SAMPLING FILTER DERIVATIONS633 G.1Frequency Response of a Comb Filter 633 G.2Single Complex FSF Frequency Response 634 G.3Multisection Complex FSF Phase 635 G.4Multisection Complex FSF Frequency Response 636 G.5Real FSF Transfer Function 638 G.6Type-IV FSF Frequency Response 640 APPENDIX H. FREQUENCY SAMPLING FILTER DESIGN TABLES643 INDEX 657 ABOUT THE AUTHOR667

　　Richard G. Lyons，毕业于美国加州阿克隆大学，获得BSEE学位。他是美国加州Mount View市Besser公司的咨询系统工程师和讲师，为许多著名公司举办过DSP方面的技术培训，这些公司包括：摩托罗拉、洛克希德·马丁、霍尼韦尔、德州仪器、诺思洛普·格鲁曼、朗迅、诺基亚、高通、国家半导体、通用动力、西门子 （现为英飞灵） 等。他做过国家安全局和TRW公司的众多DSP系统的首席硬件工程师，负责系统设计、开发、测试和安装。Lyons曾在加利福尼亚大学圣克鲁兹分校任教，并发表过很多有关DSP的文章。作为IEEE Signal Processing杂志的副主编，他创建和编辑了“DSP Tips & Tricks”专栏。

Preface iii
 1 DISCRETE SEQUENCES AND SYSTEMS
 1.1  Discrete Sequences and Their Notation 2
 1.2  Signal Amplitude, Magnitude, Power 8
 1.3  Signal Processing Operational Symbols 9
 1.4  Introduction to Discrete Linear Time-Invariant Systems  12
 1.5  Discrete Linear Systems 12
 1.6  Time-Invariant Systems 17
 1.7  The Commutative Property of Linear Time-Invariant Systems 18
 1.8  Analyzing Linear Time-Invariant Systems 19
 2 PERIODIC SAMPLING   21
 2.1  Aliasing: Signal Ambiquity in the Frequency Domain 21
 2.2  Sampling Low-Pass Signals 26
 2.3  Sampling Bandpass Signals 30
 2.4  Spectral Inversion in Bandpass Sampling 39
 3 THE DISCRETE FOURIER TRANSFORM  45
 3.1  Understanding the DFT Equation 46
 3.2  DFT Symmetry 58
 3.3  DFT Linearity 60
 3.4  DFT Magnitudes 61
 3.5  DFT Frequency Axis 62
 3.6  DFT ShiftingTheorem 63
 3.7  Inverse DFT 65
 3.8  DFT Leakage 66
 3.9  Windows 74
 3.10  DFT Scalloping Loss 82
 3.11  DFT Resolution, Zero Padding, and Frequency-Domain Sampling 83
 3.12  DFT Processing Gain 88
 3.13  The DFT of Rectangular Functions 91
 3.14  The DFT Frequency Response to a Complex Input 112
 3.15  The DFT Frequency Response to a Real Cosine Input 116
 3.16  The DFT Single-Bin Frequency Response to a Real Cosine Input 117
 3.17  Interpreting the DFT 120
 4 THE FAST FOURIER TRANSFORM                125
 4.1  Relationship of the FFT to the DFT 126
 4.2  Hints on Using FFTs in Practice 127
 4.3  FFT Software Programs 131
 4.4  Derivation of the Radix-2 FFT Algorithm 132
 4.5  FFT Input/Output Data Index Bit Reversal 139
 4.6  Radix-2 FFT Butterfly Structures 141
 5 FINITE IMPULSE RESPONSE FILTERS               151
 5.1  An Introduction to Finite Impulse Response FIR Filters 152
 5.2  Convolution in FIR Filters 157
 5.3  Low-Pass FIR Filter Design 167
 5.4  Bandpass FIR Filter Design 183
 5.5  Highpass FIR Filter Design 184
 5.6  Remez Exchange FIR Filter Design Method 186
 5.7  Half-Band FIR Filters 188
 5.8  Phase Response of FIR Filters 190
 5.9  A Generic Description of Discrete Convolution 195
 6 INFINITE IMPULSE RESPONSE FILTERS             211
 6.1  An Introduction to Infinite Impulse Response Filters 212
 6.2  The Laplace Transform 215
 6.3  The z-Transform 228
 6.4  Impulse Invariance IIR Filter Design Method 243
 6.5  Bilinear Transform IIR Filter Design Method 259
 6.6  Optimized IIR Filter Design Method 270
 6.7  Pitfalls in Building IIR Digital Filters 272
 6.8  Improving IIR Filters with Cascaded Structures 274
 6.9  A Brief Comparison of IIR and FIR Filters 279
 SPECIALIZED LOWPASS FIR FILTERS              283
 7.1  Frequency Sampling Filters: The Lost Art 284
 7.2  Interpolated Lowpass FIR Filters 319
 8 QUADRATURE SIGNALS                    335
 8.1  Why Care About Quadrature Signals 336
 8.2  The Notation of Complex Numbers 336
 8.3  Representing Real Signals Using Complex Phasors 342
 8.4  A Few Thoughts on Negative Frequency 346
 8.5  Quadrature Signals in the Frequency Domain 347
 8.6  Bandpass Quadrature Signals in the Frequency Domain 350
 8.7  Complex Down-Conversion 352
 8.8  A Complex Down-Conversion Example 354
 8.9  An Alternate Down-Conversion Method 358
 9 THE DISCRETE HILBERTTRANSFORM              361
 9.1  Hilbert Transform Definition 362
 9.2  Why Care About the Hilbert Transform? 364
 9.3  Impulse Response of a Hilbert Transformer 369
 9.4  Designing a Discrete Hilbert Transformer 371
 9.5  Time-Domain Analytic Signal Generation 377
 9.6  Comparing Analytical Signal Generation Methods 379
 10 SAMPLE RATE CONVERSION                  381
 10.1  Decimation 382
 10.2  Interpolation 387
 10.3  Combining Decimation and Interpolation 389
 10.4  Polyphase Filters 391
 10.5  Cascaded Integrator-Comb Filters 397
 11 SIGNAL AVERAGING                      411
 11.1  Coherent Averaging 412
 11.2  Incoherent Averaging 419
 11.3  Averaging Multiple Fast Fourier Transforms 422
 11.4  Filtering Aspects of Time-Domain Averaging 430
 11.5  Exponential Averaging 432
 12 DIGITAL DATA FORMATS AND THEIR EFFECTS         439
 12.1  Fixed-Point Binary Formats 439
 12.2  Binary Number Precision and Dynamic Range 445
 12.3  Effects of Finite Fixed-Point Binary Word Length 446
 12.4  Floating-Point Binary Formats 462
 12.5  Block Floating-Point Binary Format 468
 13 DIGITAL SIGNAL PROCESSING TRICKS             471
 13.1  Frequency Translation without Multiplication 471
 13.2  High-Speed Vector-Magnitude Approximation 479
 13.3  Frequency-Domain Windowing 484
 13.4  Fast Multiplication of Complex Numbers 487
 13.5  Efficiently Performing the FFT of Real Sequences 488
 13.6  Computing the Inverse FFT Using the Forward FFT 500
 13.7  Simplified FIR Filter Structure 503
 13.8  Reducing A/D Converter Quantization Noise 503
 13.9  A/D Converter Testing Techniques 510
 13.10  Fast FIR Filtering Using the FFT 515
 13.11  Generating Normally Distributed Random Data 516
 13.12  Zero-Phase Filtering 518
 13.13  Sharpened FIR Filters 519
 13.14  Interpolating a Bandpass Signal 521
 13.15  Spectral Peak Location Algorithm 523
 13.16  Computing FFT Twiddle Factors 525
 13.17  Single Tone Detection 528
 13.18  The Sliding DFT 532
 13.19  The Zoom FFT 541
 13.20  A Practical Spectrum Analyzer 544
 13.21  An Efficient Arctangent Approximation 547
 13.22  Frequency Demodulation Algorithms 549
 13.23  DC Removal 552
 13.24  Improving Traditional CIC Filters 556
 13.25  Smoothing Impulsive Noise 561
 13.26  Efficient Polynomial Evaluation 563
 13.27  Designing Very High-Order FIR Filters 564
 13.28  Time-Domain Interpolation Using the FFT 568
 13.29  Frequency Translation Using Decimation 571
 13.30  Automatic Gain Control (AGC) 571
 13.31  Approximate Envelope Detection 574
 13.32  A Quadrature Oscillator 576
 13.33  Dual-Mode Averaging 578
 APPENDIX A. THE ARITHMETIC OF COMPLEX NUMBERS         585
 A.1  Graphical Representation of Real and Complex Numbers 585
 A.2  Arithmetic Representation of Complex Numbers 586
 A.3  Arithmetic Operations of Complex Numbers 588
 A.4  Some Practical Implications of Using Complex Numbers 593
 APPENDIX B. CLOSED FORM OF A GEOMETRIC SERIES          595
 APPENDIX C. TIME REVERSAL AND THE DFT  599
 APPENDIX D. MEAN, VARIANCE, AND STANDARD DEVIATION  603
 D.1  Statistical Measures 603
 D.2  Standard Deviation, or RMS, of a Continuous Sinewave 606
 D.3  The Mean and Variance of Random Functions 607
 D.4  The Normal Probability Density Function 610
 APPENDIX E. DECIBELS (DB AND DBM) 613
 E.1  Using Logarithms to Determine Relative Signal Power 613
 E.2  Some Useful Decibel Numbers 617
 E.3  Absolute Power Using Decibels 619
 APPENDIX F. DIGITAL FILTER TERMINOLOGY  621
 APPENDIX G. FREQUENCY SAMPLING FILTER DERIVATIONS        633
 G.1  Frequency Response of a Comb Filter 633
 G.2  Single Complex FSF Frequency Response 634
 G.3  Multisection Complex FSF Phase 635
 G.4  Multisection Complex FSF Frequency Response 636
 G.5  Real FSF Transfer Function 638
 G.6  Type-IV FSF Frequency Response 640
 APPENDIX H. FREQUENCY SAMPLING FILTER DESIGN TABLES      643
 INDEX   657
 ABOUT THE AUTHOR  667