This textbook offers a fresh approach to digital signal processing (DSP), combining heuristic reasoning and physical appreciation with sound mathematical methods. It uses metaphors, analogies and creative explanations and exercises. There are 500 figures, over 170 fully worked examples, hundreds of end-of-chapter problems, and over 150 drill exercises with student solutions.
This textbook offers a fresh approach to digital signal processing (DSP) that combines heuristic reasoning and physical appreciation with sound mathematical methods to illuminate DSP concepts and practices. It uses metaphors, analogies and creative explanations, along with examples and exercises to provide deep and intuitive insights into DSP concepts. Practical DSP requires hybrid systems including both discrete- and continuous-time components. This book follows a holistic approach and presents discrete-time processing as a seamless continuation of continuous-time signals and systems, beginning with a review of continuous-time signals and systems, frequency response, and filtering. The synergistic combination of continuous-time and discrete-time perspectives leads to a deeper appreciation and understanding of DSP concepts and practices. * For upper-level undergraduates * Illustrates concepts with 500 high-quality figures, more than 170 fully worked examples, and hundreds of end-of-chapter problems, more than 150 drill exercises, including complete and detailed solutions * Seamlessly integrates MATLAB throughout the text to enhance learning

Since its emergence as a field of importance in the 1970s, digital signal processing (DSP) has grown in exponential lockstep with advances in digital hardware. Today's digital age requires that under-graduate students master material that was, until recently, taught primarily at the graduate level. Many DSP textbooks remain rooted in this graduate-level foundation and cover an exhaustive (and exhausting!) number of topics. This book provides an alternative. Rather than cover the broadest range of topics possible, we instead emphasize a narrower set of core digital signal processing con-cepts. Rather than rely solely on mathematics, derivations, and proofs, we instead balance necessary mathematics with a physical appreciation of subjects through heuristic reasoning, careful examples, metaphors, analogies, and creative explanations. Throughout, our underlying goal is to make digital signal processing as accessible as possible and to foster an intuitive understanding of the material.
Practical DSP requires hybrid systems that include both discrete-time and continuous-time com-ponents. Thus, it is somewhat curious that most DSP textbooks focus almost exclusively on discrete-time signals and systems. This book takes a more holistic approach and begins with a review of continuous-time signals and systems, frequency response, and filtering. This material, while likely familiar to most readers, sets the stage for sampling and reconstruction, digital filtering, and other aspects of complete digital signal processing systems. The synergistic combination of continuous-time and discrete-time perspectives leads to a deeper and more complete understanding of digital signal processing than is possible with a purely discrete-time viewpoint. A strong foundation of continuous-time concepts naturally leads to a stronger understanding of discrete-time concepts.
Notable Features
Some notable features of this book include the following:
1. This text is written for an upper-level undergraduate audience, and topic treatment is appro-priately geared to the junior and senior levels. This allows a sufficiently detailed mathematical. treatment to obtain a solid foundation and competence in DSP without losing sight of the basics.
2. An underlying philosophy of this textbook is to provide a simple and intuitive understanding of essential DSP concepts without sacrificing mathematical rigor. Much attention has been paid to provide clear, friendly, and enjoyable writing. A physical appreciation of the topics is attained through a balance of intuitive explanations and necessary mathematics. Concepts are illustrated using nearly 500 high-quality figures and more than 170 fully worked examples. Further reinforcement is provided through more than 150drill exercises, complete detailed solutions of which are provided as an appendix to the book. Hundreds of end-of-chapter problems provide students with additional opportunities to learn and practice.
3. Unlike most DSP textbooks, this book maintains strong tiesto continuous-time signals and systems concepts, which helps readers to better understand complete DSP systems. Further, by leveraging off a solid background of continuous-time concepts, discrete-time concepts are more easily and completely understood. Since the continuous-time background material is included, readers have immediate access to as much or as little background material as neces-sary, all in a notation ally consistent format.
4. MATLAB is effectively utilized throughout the text to enhance learning. This MATLAB ma-terial is tightly and seamlessly integrated into the text so as to seem a natural part of the material and problem solutions rather than an added afterthought. Unlike many DSP texts, this book does not have specific "MATLAB Examples" or "MATLAB Problems" any more than it has "Calculator Examples" or "Calculator Problems." Modern DSP has evolved to the point that sophisticated computer packages (such as MATLAB) should be used every bit as naturally as calculus and calculators, and it is this philosophy that guides the manner with which MATLAB is incorporated into the book.
Many DSP books relyon canned MATLAB functions to solve various digital signal processing problems. While this produces results quickly and with little effort, students often miss how problem solutions are coded or how theory is translated into practice. This book specifically avoids high-level canned functions and develops code from a more basic level; this approach reinforces connections to the underlying theory and develops sound skills in the practice of DSP. Every piece of MATLAB code precisely conforms with book concepts, equations, and notations.
Book Organization and Use
Roughly speaking, this book is organized into five parts.
1. Review of continuous-time signals and systems (Ch. 1) and continuous-time (analog) filtering (Ch. 2)
2. Sampling and reconstruction (Ch. 3)
3. Introduction to discrete-time signals and systems (Ch. 4) and the time-domain analysis of discrete-time systems (Ch. 5)
4. Frequency-domain analysis of discrete-time systems using the discrete-time Fourier transform (Ch. 6) and the z-transform (Ch. 7)
5. Discrete-time (digital) filtering (Ch. 8) and the discrete-Fourier transform (Ch. 9)
The first quarter of this book (Chs. 1 and 2, about 150 pages) focuses on continuous-time concepts, and this material can be scanned or skipped by readers who possess a solid background in these areas. The last three quarters of the book (Chs. 3 through 9, about 450 pages) cover traditional discrete-time concepts that form the backbone of digital signal processing. The majority of the book can be covered over a semester in atypical 3-or 4-credit-hour undergraduate-level course, which corresponds to around 45 to 60 lecture-hours of contact.
As with most textbooks, this book can be adapted to accommodate a range of courses and student backgrounds. Students with solid backgrounds in continuous-time signals and systems can scan or perhaps altogether skip the first two chapters. Students with knowledge in the time-domain analysis of discrete-time signals and systems can scan or skip Chs. 4 and 5. Courses that do not wish to emphasize filtering operations can eliminate coverage of Chs. 2 and 8. Many other options exist as well. For example, students enter the 3-credit Applied Digital Signal Processing and Filtering course at North Dakota State University having completed a 4-credit Signals and Systems course that covers both continuous-time and discrete-time concepts, including Laplace and z-transforms but not including discrete-time Fourier analysis. Given this student background, the NDSU DSP course covers Chs. 2, 3, 6, 8, and 9, which leaves enough extra time to introduce (and use) digital signal processing hardware from Texas Instruments; Chs. 1, 4, 5, and 7 are recommended for reading, but not required.
Acknowledgments
We would like to offer our sincere gratitude to the many people who have generously given their time and talents to the creation, improvement, and refinement of this book. Books, particularly sizable ones such as this, involve a seemingly infinite number of details, and it takes the combined efforts of a good number of good people to successfully focus these details into a quality result. During the six years spent preparing this book, we have been fortunate to receive valuable feedback and recommendations from numerous reviewers, colleagues, and students. We are grateful for the reviews provided by Profs. Zekeriya Aliyazicioglu of California State Polytechnic University-Pomona, Mehmet Celenk of Ohio University, Liang Dong of Western Michigan University, Jake Gunther of Utah State University, Joseph P. Hoff beck of the University of Portland, Jianhua Liu of Embry-Riddle Aeronautical University, Peter Mathys of the University of Colorado, Phillip A. Mlsna of Northern Arizona University, S. Hossein Mousavinezhad of Idaho State University, Kalyan Mondal of Fairleigh Dickinson University, Anant Sahai of UC Berkeley, Jose Sanchez of Bradley University, and Xiao mu Song of Widener University. We also offer our heartfelt thanks for the thoughtful comments and suggestions provided by the many anonymous reviewers, who outnumbered the other reviewers more than two-to-one. We wish that we could offer a more direct form of recognition to these reviewers. Some of the most thoughtful and useful comments came from students taking the Applied Digital Signal Processing and Filtering course at North Dakota State University. Two students in particular-Kyle Kraning and Michael Boyko-went above the call of duty, providing more than one hundred corrections and comments. For their creative contributions of cartoon ideas, we also give thanks to NDSU students Stephanie Rosen (Chs. 1, 4, and 5) and Tanner Voss (Ch. 2). Book writing is a time-consuming activity, and one that inevitably causes hardship to those who are close to an author. Thus, we offer our final thanks to our families for their sacrifice, support, and love.
B. P. Lathi
Roger A. Green

Preface xi

1 Review of Continuous-Time Signals and Systems 1

1.1 Signals and Signal Categorizations 2

      1.1.1 Continuous-Time and Discrete-Time Signals 3

      1.1.2 Analog and Digital Signals 3

1.2 Operations on the Independent CT Variable 4

      1.2.1 CT Time Shifting 4

      1.2.2 CT Time Scaling 5

      1.2.3 CT Time Reversal 5

      1.2.4 Combined CT Time Shifting and Scaling 6

1.3 CT Signal Models 7

      1.3.1 CT Unit Step Function u(t) 7

      1.3.2 CT Unit Gate Function II(t) 8

      1.3.3 CT Unit Triangle Function A(t) 8

      1.3.4 CT Unit Impulse Function o(t) 9

      1.3.5 CT Exponential Function est 12

      1.3.6 CT Interpolation Functions inc(t) 13

1.4 CT Signal Classifications 15

      1.4.1 Causal, Noncausal, and Anti-Causal CT Signals 15

      1.4.2 Real and Imaginary CT Signals 16

      1.4.3 Even and Odd CT Signals 18

      1.4.4 Periodic and Aperiodic CT Signals 21

      1.4.5 CT Energy and Power Signals 21

      1.4.6 Deterministic and Probabilistic Signals 25

1.5 CT Systems and Properties 25

      1.5.1 Linearity 26

      1.5.2 Time Invariance 27

      1.5.3 The Zero-State Response of an LTIC System 28

      1.5.4 Causality 29

      1.5.5 Stability 29

1.6 Foundations of Frequency-Domain Analysis 30

      1.6.1 LTIC System Response to an Everlasting Exponential est 30

1.7 The Fourier Series 33

      1.7.1 Exponential Form of the Fourier Series 34

      1.7.2 Trigonometric and Compact Trigonometric Forms 37

      1.7.3 Convergence of a Series 41

1.8 The Fourier Transform 45

1.9 Fourier Transform Properties 50

      1.9.1 Duality Property 51

      1.9.2 Linearity Property 52

      1.9.3 Complex-Conjugation Property 52

      1.9.4 Scaling Property 53

      1.9.5 Time-Shifting Property 54

      1.9.6 Time-Differentiation and Time-Integration Properties 59

      1.9.7 Time-Domain Convolution Property 59

      1.9.8 Correlation and the Correlation Property 61

      1.9.9 Extending Fourier Transform Properties to the Fourier Series 66

1.10 The Laplace Transform 68

      1.10.1 Connection between the Fourier and Laplace Transforms 70

      1.10.2 Laplace Transform Properties 72

1.11 Summary 73

2 Continuous-Time Analog Filters 85

2.1 Frequency Response of an LTIC System 85

      2.1.1 Pole-Zero Plots 89

2.2 Signal Transmission through LTIC Systems 92

      2.2.1 Distortion less Transmission 94

      2.2.2 Real Bandpass Systems and Group Delay 97

2.3 Ideal and Realizable Filters 100

2.4 Data Truncation by Windows 104

      2.4.1 Impairments Caused by Windowing 104

      2.4.2 Lowpass Filter Design Using Windows 106

      2.4.3 Remedies for Truncation Impairments 109

      2.4.4 Common Window Functions 109

2.5 Specification of Practical Filters 112

2.6 Analog Filter Transformations 113

      2.6.1 Lowpass-to-Lowpass Transformation 115

      2.6.2 Lowpass-to-Highpass Transformation 116

      2.6.3 Lowpass-to-Bandpass Transformation 117

      2.6.4 Lowpass-to-Bandstop Transformation 118

2.7 Practical Filter Families 120

      2.7.1 Butterworth Filters 120

      2.7.2 Chebyshev Filters 129

      2.7.3 Inverse Chebyshev Filters 139

      2.7.4 Elliptic Filters 144

      2.7.5 Bessel-Thomson Filters 147

2.8 Summary 149

3 Sampling: The Bridge from Continuous to Discrete 155

3.1 Sampling and the Sampling Theorem 155

      3.1.1 Practical Sampling 161

3.2 Signal Reconstruction 164

3.3 Practical Dificulties in Sampling and Reconstruction 168

      3.3.1 Aliasing in Sinusoids 173

3.4 Sampling of Bandpass Signals 176

3.5 Time-Sampling Dual: The Spectral Sampling Theorem 181

3.6 Analog-to-Digital Conversion 185

      3.6.1 Analog-to-Digital Converter Transfer Characteristics 189

      3.6.2 Analog-to-Digital Converter Errors 194

      3.6.3 Analog-to-Digital Converter Implementations 196

3.7 Digital-to-Analog Conversion 199

      3.7.1 Sources of Distortion in Signal Reconstruction 200

3.8 Summary 202

4 Discrete-Time Signals and Systems 212

4.1 Operations on the Independent DT Variable 214

      4.1.1 DT Time Shifting 214

      4.1.2 DT Time Reversal 215

      4.1.3 DT Time Scaling: Sampling Rate Conversion 216

4.2 DT Signal Models 219

      4.2.1 DT Unit Step Function u[n] 219

      4.2.2 DT Unit Impulse Function o[n] 220

      4.2.3 DT Exponential Function zn 222

4.3 DT Signal Classifications 231

      4.3.1 Causal, Noncausal, and Anti-Causal DT Signals 231

      4.3.2 Real and Imaginary DT Signals 232

      4.3.3 Even and Odd DT Signals 232

      4.3.4 Periodic and Aperiodic DT Signals 233

      4.3.5 DT Energy and Power Signals 236

4.4 DT Systems and Examples 238

      4.4.1 The Order and General Form of Difference Equations 245

      4.4.2 Kinship of Difference Equations to Differential Equations 246

      4.4.3 Advantages of Digital Signal Processing 248

4.5 DT System Properties 248

      4.5.1 Time Invariance 248

      4.5.2 Linearity 250

      4.5.3 The Zero-State Response of an LTID System 252

      4.5.4 Causality 254

      4.5.5 Stability 256

      4.5.6 Memory 256

      4.5.7 Invertibility 257

4.6 Digital Resampling 257

4.7 Summary 261

5 Time-Domain Analysis of Discrete-Time Systems 270

5.1 Iterative Solutions to Difference Equations 270

5.2 Operator Notation 275

5.3 The Zero-Input Response 277

      5.3.1 Insights into the Zero-Input Behavior of a System 282

5.4 The Unit Impulse Response 284

      5.4.1 Closed-Form Solution of the Impulse Response 285

5.5 The Zero-State Response 288

      5.5.1 Convolution Sum Properties 291

      5.5.2 Graphical Procedure for the Convolution Sum 294

      5.5.3 Interconnected Systems 300

      5.5.4 LT ID System Response to an Everlasting Exponential zn 303

5.6 Total Response 304

5.7 System Stability 305

      5.7.1 External (BIBO) Stability 306

      5.7.2 Internal (Asymptotic) Stability 306

5.8 Intuitive Insights into System Behavior 311

      5.8.1 Dependence of System Behavior on Characteristic Modes 311

      5.8.2 Response Time of a System: The System Time Constant 312

      5.8.3 Time Constant and Rise Time of a System 314

      5.8.4 Time Constant and Filtering 314

      5.8.5 Time Constant and Pulse Dispersion 315

      5.8.6 The Resonance Phenomenon 315

5.9 Classical Solution of Linear Difference Equations 317

5.10 Summary 322

6 Discrete-Time Fourier Analysis 331

6.1 The Discrete-Time Fourier Transform 331

      6.1.1 The Nature of Fourier Spectra 337

      6.1.2 Obtaining the DTFT from the CTFT 338

      6.1.3 DTFT Tables and the Nuisance of Periodicity 340

6.2 Properties of the DTFT 343

6.2.1 Duality 343

6.2.2 Linearity Property 343

6.2.3 Complex-Conjugation Property 343

6.2.4 Time Scaling and the Time-Reversal Property 344

6.2.5 Time-Shifting Property 345

6.2.6 Frequency-Differentiation Property 350

6.2.7 Time-Domain and Frequency-Domain Convolution Properties 351

6.2.8 Correlation and the Correlation Property 354

6.3 LTID System Analysis by the DTFT 355

      6.3.1 Distortion less Transmission 359

      6.3.2 Ideal and Realizable Filters 362

6.4 Connection between the DTFT and the CTFT 364

6.5 Digital Processing of Analog Signals 370

      6.5.1 A Mathematical Representation 371

      6.5.2 Time-Domain Criterion: The Impulse Invariance Method 373

6.6 Digital Resampling: A Frequency-Domain Perspective 379

      6.6.1 Using Bandlimited Interpolation to Understand Resampling 380

      6.6.2 Downsampling and Decimation 383

      6.6.3 Interpolation and Upsampling 387

      6.6.4 Time-Domain Characterizations 391

      6.6.5 Fractional Sampling Rate Conversion 394

6.7 Generalization of the D TFT to the z-Transform 395

6.8 Summary 397

7 Discrete-Time System Analysis Using the z-Transform 410

7.1 The z-Transform 410

      7.1.1 The Bilateral z-Transform 410

      7.1.2 The Unilateral z-Transform 416

7.2 The Inverse z-Transform 419

      7.2.1 Inverse z-Transform by Power Series Expansion 425

7.3 Properties of the z-Transform 427

      7.3.1 Linearity Property 427

      7.3.2 Complex-Conjugation Property 427

      7.3.3 Time Scaling and the Time-Reversal Property 428

      7.3.4 Time-Shifting Property 428

      7.3.5 z-Domain Scaling Property 432

      7.3.6 z-Domain Differentiation Property 433

      7.3.7 Time-Domain Convolution Property 433

      7.3.8 Initial and Final Value Theorems 435

7.4 z-Transform Solution of Linear Difference Equations 436

      7.4.1 Zero-State Response of LTID Systems: The Transfer Function 439

7.5 Block Diagrams and System Realization 445

      7.5.1 Direct Form Realizations 447

      7.5.2 Transposed Realizations 451

      7.5.3 Cascade and Parallel Realizations 453

7.6 Frequency Response of Discrete-Time Systems 457

      7.6.1 Frequency Response from Pole-Zero Locations 462

7.7 Finite Word-Length Effects 469

      7.7.1 Finite Word-Length Effects on Poles and Zeros 469

      7.7.2 Finite Word-Length Effects on Frequency Response 472

7.8 Connection between the Laplace and z-Transforms 474

7.9 Summary 476

8 Digital Filters 485

8.1 Infinite Impulse Response Filters 485

      8.1.1 The Impulse Invariance Method Revisited 486

      8.1.2 The Bilinear Transform 491

      8.1.3 The Bilinear Transform with Prewarping 497

      8.1.4 Highpass, Bandpass, and Bandstop Filters 501

      8.1.5 Realization of IIR Filters 508

8.2 Finite Impulse Response Filters 511

      8.2.1 Linear Phase FIR Filters 511

      8.2.2 Realization of FIR Filters 515

      8.2.3 Windowing in FIR Filters 517

      8.2.4 Time-Domain Methods of FIR Filter Design 521

      8.2.5 Window Method FIR Filter Design forgiven Specifications 529

      8.2.6 Frequency-Domain Methods of FIR Filter Design 537

      8.2.7 Frequency-Weighted Least-Squares FIR Filter Design 544

8.3 Summary 552

9 Discrete Fourier Transform 559

9.1 The Discrete Fourier Transform 560

      9.1.1 The Picket Fence Effect and Zero Padding 563

      9.1.2 Matrix Representation of the DFT and Its Inverse 565

      9.1.3 DFT Interpolation to Obtain the DTFT 567

9.2 Uniqueness: Why Confinea x[n]to 0≤n≤N-1? 569

      9.2.1 Modulo-N Operation 572

      9.2.2 Circular Representation of an N-Length Sequence 573

9.3 Properties of the DFT 579

      9.3.1 Duality Property 579

      9.3.2 Linearity Property 579

      9.3.3 Complex-Conjugation Property 580

      9.3.4 Time-Reversal Property 580

      9.3.5 Circular Shifting Properties 580

      9.3.6 Circular Convolution Properties 581

      9.3.7 Circular Correlation Property 582

9.4 Graphical Interpretation of Circular Convolution 583

      9.4.1 Circular and Linear Convolution 585

      9.4.2 Aliasing in Circular Convolution 588

9.5 Discrete-Time Filtering Using the DFT 590

      9.5.1 Block Convolution 593

9.6 Goertzel's Algorithm 600

9.7 The Fast Fourier Transform 603

      9.7.1 Decimation-in-Time Algorithm 604

      9.7.2 Decimation-in-Frequency Algorithm 609

9.8 The Discrete-Time Fourier Series 612

9.9 Summary 617

A MATLAB 625

B Useful Tables 640

C Drill Solutions 646

Index 731

Roger A. Green is an Associate Professor of Electrical and Computer Engineering at North Dakota State University. He holds a PhD from the University of Wyoming. He is co-author, with B. P. Lathi, of Signal Processing and Linear Systems, 2nd edition.

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