Fourier Transforms

Eric W. Hansen, PhD, received his MS and PhD in Electrical Engineering from Stanford University. He is a member of IEEE, OSA, and the ASEE. Dr Hansen has been on the Dartmouth faculty since 1979, and received the Excellence in Teaching Award from the Thayer School of Engineering.

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

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: ∫↺ Γ zndz 497

8.3 Cauchy’s Integral Theorem 502

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