FOURIER TRANSFORMS

FOURIER TRANSFORMSPrinciples and Applications

ERIC W. HANSENThayer School of Engineering, Dartmouth College

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To my family

CONTENTS

PREFACE xi

CHAPTER 1 REVIEW OF PREREQUISITE MATHEMATICS 1

1.1 Common Notation 1

1.2 Vectors in Space 3

1.3 Complex Numbers 8

1.4 Matrix Algebra 11

1.5 Mappings and Functions 15

1.6 Sinusoidal Functions 20

1.7 Complex Exponentials 22

1.8 Geometric Series 24

1.9 Results from Calculus 25

1.10 Top 10 Ways to Avoid Errors in Calculations 33

Problems 33

CHAPTER 2 VECTOR SPACES 36

2.1 Signals and Vector Spaces 37

2.2 Finite-dimensional Vector Spaces 39

2.3 Infinite-dimensional Vector Spaces 64

2.4 ⋆ Operators 86

2.5 ⋆ Creating Orthonormal Bases–the Gram–Schmidt Process 94

2.6 Summary 99

Problems 101

CHAPTER 3 THE DISCRETE FOURIER TRANSFORM 109

3.1 Sinusoidal Sequences 109

3.2 The Discrete Fourier Transform 114

3.3 Interpreting the DFT 117

3.4 DFT Properties and Theorems 126

3.5 Fast Fourier Transform 152

3.6 ⋆ Discrete Cosine Transform 156

3.7 Summary 164

Problems 165

CHAPTER 4 THE FOURIER SERIES 177

4.1 Sinusoids and Physical Systems 178

4.2 Definitions and Interpretation 178

4.3 Convergence of the Fourier Series 187

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viii CONTENTS

4.4 Fourier Series Properties and Theorems 199

4.5 The Heat Equation 215

4.6 The Vibrating String 223

4.7 Antenna Arrays 227

4.8 Computing the Fourier Series 233

4.9 Discrete Time Fourier Transform 238

4.10 Summary 256

Problems 259

CHAPTER 5 THE FOURIER TRANSFORM 273

5.1 From Fourier Series to Fourier Transform 274

5.2 Basic Properties and Some Examples 276

5.3 Fourier Transform Theorems 281

5.4 Interpreting the Fourier Transform 299

5.5 Convolution 300

5.6 More about the Fourier Transform 310

5.7 Time–bandwidth Relationships 318

5.8 Computing the Fourier Transform 322

5.9 ⋆ Time–frequency Transforms 336

5.10 Summary 349

Problems 351

CHAPTER 6 GENERALIZED FUNCTIONS 367

6.1 Impulsive Signals and Spectra 367

6.2 The Delta Function in a Nutshell 371

6.3 Generalized Functions 382

6.4 Generalized Fourier Transform 404

6.5 Sampling Theory and Fourier Series 414

6.6 Unifying the Fourier Family 429

6.7 Summary 433

Problems 436

CHAPTER 7 COMPLEX FUNCTION THEORY 454

7.1 Complex Functions and Their Visualization 455

7.2 Differentiation 460

7.3 Analytic Functions 466

7.4 exp z and Functions Derived from It 470

7.5 Log z and Functions Derived from It 472

7.6 Summary 489

Problems 490

CHAPTER 8 COMPLEX INTEGRATION 494

8.1 Line Integrals in the Plane 494

8.2 The Basic Complex Integral: ∫↺Γzndz 497

8.3 Cauchy’s Integral Theorem 502

CONTENTS ix

8.4 Cauchy’s Integral Formula 512

8.5 Laurent Series and Residues 520

8.6 Using Contour Integration to Calculate Integrals of Real Functions 531

8.7 Complex Integration and the Fourier Transform 543

8.8 Summary 556

Problems 557

CHAPTER 9 LAPLACE, Z, AND HILBERT TRANSFORMS 563

9.1 The Laplace Transform 563

9.2 The Z Transform 607

9.3 The Hilbert Transform 629

9.4 Summary 652

Problems 654

CHAPTER 10 FOURIER TRANSFORMS IN TWO AND THREE DIMENSIONS 669

10.1 Two-Dimensional Fourier Transform 669

10.2 Fourier Transforms in Polar Coordinates 684

10.3 Wave Propagation 696

10.4 Image Formation and Processing 709

10.5 Fourier Transform of a Lattice 722

10.6 Discrete Multidimensional Fourier Transforms 731

10.7 Summary 736

Problems 737

BIBLIOGRAPHY 743

INDEX 747

PREFACE

The mathematical techniques known as “transform methods” have long been a basictool in several branches of engineering and science, and no wonder. Fourier’s simpleidea, radical in its time, that a function can be expressed as a sum of sine waves,is ubiquitous. It underlies fields as diverse as communications, signal and imageprocessing, control theory, electromagnetics, and acoustics. Electrical engineers typ-ically encounter the rudiments of Fourier transforms in undergraduate systems andcircuits courses, for modeling the spectral content of signals and designing frequencyselective circuits (filters). The Laplace transform, a close cousin of the Fourier trans-form, enables the efficient analytical solution of ordinary differential equations andleads to the popular “S plane” and “root locus” methods for analyzing linear systemsand designing feedback controllers. Discrete-time versions of the Fourier and Laplacetransforms model spectra and frequency responses for digital signal processing andcommunications. Physics and engineering students meet the Fourier series whenlearning about harmonic motion or solving partial differential equations, for exam-ple, for waves and diffusion. The Fourier transform also models wave propagationfrom acoustics to radio frequencies to optics to X-ray diffraction. The widespreaddissemination of the fast Fourier transform algorithm following its publication in1965 added a computational dimension to all of these applications, from everydayconsumer electronics to sophisticated medical imaging devices.

My purpose in writing this textbook is to pull these threads together and presenta unified development of Fourier and related transforms for seniors and graduatestudents in engineering and physics—one that will deepen their grasp of how andwhy the methods work, enable greater understanding of the application areas, andperhaps motivate further pursuit of the mathematics in its own right. Drafts of the bookhave been used by myself and others for a 10-week course in Fourier transforms andcomplex variable theory at the Thayer School of Engineering, Dartmouth College.The prerequisites are an introductory course in lumped parameter systems (includingthe Laplace transform) or differential equations. Our course is itself prerequisiteto courses in signal and image processing and a more advanced course in appliedanalysis taken by engineering and physics graduate students.

Philosophy and Distinctives

The book is more mathematically detailed and general in scope than a sophomoreor junior level signals and systems text, more focused than a survey of mathematicalmethods, and less rigorous than would be appropriate for students of advancedmathematics. In brief, here is the approach I have taken.

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xii PREFACE

1. The four types of Fourier transform on discrete and continuous domains—discrete Fourier transform (DFT), Fourier series, discrete-time Fourier trans-form (DTFT), and Fourier transform—are developed as orthogonal expansionswithin a vector space framework. They are introduced sequentially, startingwith the DFT and working up to the continuous-time Fourier transform. Thesame important properties and theorems are revisited for each transform inturn, reinforcing the basic ideas as each new transform is introduced. This is incontrast with an approach that either begins with the continuous-time transformand works down to the others as special cases, or develops all four in parallel.

2. The early presentation of the DFT makes it immediately available as a tool forcomputing numerical approximations to the Fourier series and Fourier trans-form. Several homework problems give the student practice using numericaltools. Matlab® is used throughout the book to demonstrate numerical meth-ods and to visualize important ideas, but whether to use Matlab or some othercomputational tool in the course is up to the instructor.

3. The fundamentals of complex analysis and integration are included as a bridgeto a more thorough understanding of the Laplace and Z transforms, and as anadditional way to calculate Fourier transforms.

4. Physical interpretations and applications are emphasized in the examples andhomework problems. My hope is that the student will cultivate intuition forhow the mathematics work as well as gain proficiency with calculation andapplication.

5. Starred sections, which may be skipped on a first reading, give brief introduc-tions to more advanced topics and references for further reading.

6. Each chapter has a table of key results, which should be particularly helpful forreference after the course is completed.

Any author of an applied mathematics book must decide the extent to which thedevelopment of the material will be supported by proofs. The level of rigor requiredby a mathematician generally exceeds what is needed to justify the trustworthinessof a result to an engineer. Moreover, to prove all the key theorems of Fourier analysisrequires a facility with real analysis and even functional analysis that exceeds theusual mathematical background of an undergraduate engineer. The approach takenhere, for the most part, is to include proofs when they build intuition about how themathematics work or contribute to the student’s ability to make calculations and applythe transforms. Otherwise, I will usually substitute informal plausibility arguments,derivations of weaker results either in the text or in the end-of-chapter problems, orcomputational illustrations of the principles involved. Footnotes refer the interestedreader to detailed treatments in more advanced texts.

Flow of the Book

The book has 10 chapters, which are described briefly here to show how the book’smain ideas are developed. Chapter 1 is a review of the topics from geometry,

PREFACE xiii

trigonometry, matrix algebra, and calculus that are needed for this course. Chapter 2then develops some fundamentals of vector spaces, particularly the generalizationsof the geometric ideas of norm, inner product, orthogonality and orthogonal expan-sion from vectors to functions. This provides the unifying framework for the Fourierfamily and acquaints the student with concepts of broad importance in engineeringmathematics.

Chapters 3–5 introduce, in sequence, the DFT, the Fourier series, the DTFT,and the (continuous-time) Fourier transform. The DFT has the easiest vector spaceinterpretation of the four transforms, since it expands finite-dimensional vectors interms of orthogonal finite-dimensional vectors. Some basic Fourier theorems (lin-earity, shift, energy conservation, convolution) are first presented here, then reappearlater for the other transforms. Chapter 3 includes a derivation of the fast Fouriertransform (FFT) algorithm and the discrete cosine transform (DCT), a close relativeof the DFT that is the mathematical foundation of JPEG image compression.

The Fourier series, Chapter 4, is a representation of a periodic function as aninfinite series of orthogonal sines and cosines. The appearance of the infinite seriesraises the question of convergence and leads to the important connections amongconvergence of the series, asymptotic behavior of the spectrum, and smoothness ofthe original function. The chapter includes applications to the diffusion and waveequations and to antenna arrays, and shows how to use the DFT to compute Fouriercoefficients and partial sums of Fourier series. Swapping the time and frequencydomains, the Fourier series becomes the DTFT, the basic tool for discrete-timesystem analysis and signal processing.

The Fourier transform, Chapter 5, expands an aperiodic function as a contin-uum of orthogonal sines and cosines rather than a set of discrete oscillatory modes.Despite the additional mathematical complication, it has many of the same propertiesas the DFT and the Fourier series and intuition developed earlier for these transformswill carry over to the Fourier transform. The chapter emphasizes using Fourier theo-rems for modeling systems (impulse response and transfer function) and performingcalculations. It also shows how to use the DFT to compute transforms and convo-lutions. A brief introduction to time-frequency transforms and wavelet transformsconcludes the chapter.

Chapter 6 begins by placing the impulse (delta) function on a more securefooting than the informal notion of “infinite height, zero width, unit area” that studentssometimes bring with them from earlier classes. This is followed by development of acommon, generalized framework for understanding ordinary functions together withimpulses and other singular functions. Sampling theory, introduced informally inChapter 3, is studied here in depth. It is also used to unify the four Fourier transforms,via the observation that sampling in the time domain produces periodicity in thefrequency domain, and vice versa.

Chapters 7 and 8 are devoted to the theory of complex functions and methods ofcomplex integration, with a focus on ultimately applying the theory to understandingand calculating transforms. Numerical calculations of the basic complex integral

∫↺ zn dz on different closed contours are used to help students visualize why the