Fourier Transforms

Fourier Transforms: Principles and Applications explains transform methods and their applications to electrical systems from circuits, antennas, and signal processors ably guiding readers from vector space concepts through the Discrete Fourier Transform (DFT), Fourier series, and Fourier transform to other related transform methods. Featuring chapter end summaries of key results, over two hundred examples and four hundred homework problems, and a Solutions Manual this book is perfect for graduate students in signal processing and communications as well as practicing engineers. Class-tested at Dartmouth Provides the same solid background as classic texts in the field, but with an emphasis on digital and other contemporary applications to signal and image processing Modular coverage of material allows for topics to be covered by preference MATLAB files and Solutions Manual available to instructors Over 300 figures, 200 worked examples, and 432 homework problems

Fourier Transforms: Principles and Applications explains transform methods and their applications to electrical systems from circuits, antennas, and signal processors--ably guiding readers from vector space concepts through the Discrete Fourier Transform (DFT), Fourier series, and Fourier transform to other related transform methods. Featuring chapter end summaries of key results, over two hundred examples and four hundred homework problems, and a Solutions Manual this book is perfect for graduate students in signal processing and communications as well as practicing engineers.* Class-tested at Dartmouth* Provides the same solid background as classic texts in the field, but with an emphasis on digital and other contemporary applications to signal and image processing* Modular coverage of material allows for topics to be covered by preference* MATLAB files and Solutions Manual available to instructors* Over 300 figures, 200 worked examples, and 432 homework problems

PREFACE xiCHAPTER 1 REVIEW OF PREREQUISITE MATHEMATICS 11.1 Common Notation 11.2 Vectors in Space 31.3 Complex Numbers 81.4 Matrix Algebra 111.5 Mappings and Functions 151.6 Sinusoidal Functions 201.7 Complex Exponentials 221.8 Geometric Series 241.9 Results from Calculus 251.10 Top 10 Ways to Avoid Errors in Calculations 33Problems 33CHAPTER 2 VECTOR SPACES 362.1 Signals and Vector Spaces 372.2 Finite-dimensional Vector Spaces 392.3 Infinite-dimensional Vector Spaces 642.4 Operators 862.5 Creating Orthonormal Bases-the Gram-Schmidt Process 942.6 Summary 99Problems 101CHAPTER 3 THE DISCRETE FOURIER TRANSFORM 1093.1 Sinusoidal Sequences 1093.2 The Discrete Fourier Transform 1143.3 Interpreting the DFT 1173.4 DFT Properties and Theorems 1263.5 Fast Fourier Transform 1523.6 Discrete Cosine Transform 1563.7 Summary 164Problems 165CHAPTER 4 THE FOURIER SERIES 1774.1 Sinusoids and Physical Systems 1784.2 Definitions and Interpretation 1784.3 Convergence of the Fourier Series 1874.4 Fourier Series Properties and Theorems 1994.5 The Heat Equation 2154.6 The Vibrating String 2234.7 Antenna Arrays 2274.8 Computing the Fourier Series 2334.9 Discrete Time Fourier Transform 2384.10 Summary 256Problems 259CHAPTER 5 THE FOURIER TRANSFORM 2735.1 From Fourier Series to Fourier Transform 2745.2 Basic Properties and Some Examples 2765.3 Fourier Transform Theorems 2815.4 Interpreting the Fourier Transform 2995.5 Convolution 3005.6 More about the Fourier Transform 3105.7 Time-bandwidth Relationships 3185.8 Computing the Fourier Transform 3225.9 Time-frequency Transforms 3365.10 Summary 349Problems 351CHAPTER 6 GENERALIZED FUNCTIONS 3676.1 Impulsive Signals and Spectra 3676.2 The Delta Function in a Nutshell 3716.3 Generalized Functions 3826.4 Generalized Fourier Transform 4046.5 Sampling Theory and Fourier Series 4146.6 Unifying the Fourier Family 4296.7 Summary 433Problems 436CHAPTER 7 COMPLEX FUNCTION THEORY 4547.1 Complex Functions and Their Visualization 4557.2 Differentiation 4607.3 Analytic Functions 4667.4 exp z and Functions Derived from It 4707.5 Log z and Functions Derived from It 4727.6 Summary 489Problems 490CHAPTER 8 COMPLEX INTEGRATION 4948.1 Line Integrals in the Plane 4948.2 The Basic Complex Integral: integral Gamma zndz 4978.3 Cauchy's Integral Theorem 5028.4 Cauchy's Integral Formula 5128.5 Laurent Series and Residues 5208.6 Using Contour Integration to Calculate Integrals of Real Functions 5318.7 Complex Integration and the Fourier Transform 5438.8 Summary 556Problems 557CHAPTER 9 LAPLACE, Z, AND HILBERT TRANSFORMS 5639.1 The Laplace Transform 5639.2 The Z Transform 6079.3 The Hilbert Transform 6299.4 Summary 652Problems 654CHAPTER 10 FOURIER TRANSFORMS IN TWO AND THREE DIMENSIONS 66910.1 Two-Dimensional Fourier Transform 66910.2 Fourier Transforms in Polar Coordinates 68410.3 Wave Propagation 69610.4 Image Formation and Processing 70910.5 Fourier Transform of a Lattice 72210.6 Discrete Multidimensional Fourier Transforms 73110.7 Summary 736Problems 737BIBLIOGRAPHY 743INDEX 747