Fourier, Laplace, and the Tangled Love Affair with Transforms

Description

Unlock the intricate language of signals and systems with this in-depth exploration of Fourier and Laplace transforms. Designed for advanced undergraduates, graduate students, and professionals in engineering, physics, and applied mathematics, this book unravels the foundations of signal processing with a rigorous yet engaging approach. Beginning with the fundamentals and building to advanced topics, each chapter guides you through the Fourier series, Fourier, and Laplace transform and into the realms of discrete Fourier and Z transforms, multi-dimensional analysis, and applications of the Fourier Transform in solving PDE, ODE, and Integral equations. The text brings mathematical theory to life through real-world applications in signal synthesis, systems engineering, and differential equations, making complex topics accessible and inspiring. With its unique blend of historical insights, practical applications, and intuitive explanations, this book offers a comprehensive yet approachable journey into the world of transforms. Whether you're a student building your foundation or a professional seeking to deepen your expertise, this book invites you to discover the elegance and utility of transforms in a way that bridges theory with the demands of modern engineering and science. Preface Acknowledgments Contents 1 Introduction 1.1 Historical Development of the Subject 1.2 The Sinusoids 1.3 Orthogonal Functions 1.3.1 Set of Orthogonal Functions 1.4 Odd and Even Mathematical Functions 1.5 Series Expansion of a Function 1.6 Complex Variables 1.7 Differentiation and Integration of Complex Numbers 1.8 Useful Mathematical Formulae 1.8.1 Trigonometrical Relations 1.8.2 Hyperbolic Relations 1.8.3 Taylor Series Expansion 1.8.4 Standard Derivatives 1.8.5 Standard Integrals 1.9 Closure Reference 2 Signals and Systems 2.1 Signals 2.2 Basic Signal Examples 2.2.1 Sinusoids 2.2.2 Exponential and Geometric Signals 2.2.3 Box and rect signals 2.2.4 Kronecker Delta Function 2.2.5 Heaviside Step Function 2.2.6 Signum Function 2.2.7 sinc Function 2.2.8 (x) Function 2.2.9 Gaussian Function 2.2.10 Bessel Function 2.2.11 Random Signals 2.3 Systems 2.3.1 Linearity 2.3.2 Time-Invariant Systems 2.3.3 Causality 2.3.4 Stability 2.3.5 Memory 2.4 Noise 2.5 Closure Reference 3 The Fourier Series 3.1 Observation from Nature 3.2 Fourier Series Expansion of Odd and Even Functions 3.3 Complex Number Representation of Fourier Series 3.4 The Double Fourier Series 3.5 Fourier Series Expansion of Some Common Signals 3.6 Closure References 4 The Fourier Analysis 4.1 Fourier Series to Fourier Analysis 4.2 Existence of Fourier Integral 4.3 Properties of Fourier Transform 4.3.1 Physical Realization of Fourier Transform 4.3.2 Transform Pair and Duality in Fourier Transform 4.3.3 Linearity in Fourier Transform 4.3.4 Constant Multiplication with Fourier Transform 4.3.5 Symmetric Properties of Fourier Pair 4.3.6 Time Delay or Space Shift in Fourier Transform 4.3.7 Frequency or Wavenumber Shift in Fourier Transform 4.3.8 Stretching and Shrinking Phenomenon in Fourier Transform 4.3.9 Differentiation of Fourier Transform Pair 4.4 Closure References 5 Fourier Transform of Generalized Functions 5.1 The Universality of the Fourier Transform 5.1.1 Rapidly Decreasing Function 5.1.2 Dirac's Delta Function 5.1.3 Distribution Induced by Functions 5.1.4 Derivative of Distributions 5.1.5 Few More Properties of Distributions 5.1.6 Dirac's Comb or Sha Function 5.2 Fourier Transform of Some Standard Functions 5.3 Closure References 6 The Laplace Transform 6.1 History of Laplace Transform 6.2 Definition of the Laplace Transform 6.3 Existence and Uniqueness of Laplace Transforms 6.4 Important Properties of Laplace Transform 6.4.1 Linearity of the Laplace Transform 6.4.2 Heaviside's First Shifting Theorem 6.4.3 Time Shifting (t-Shifting): Second Shifting Theorem 6.4.4 Scaling Property of the Laplace Transform 6.4.5 Laplace Transform of Impulse Response 6.5 The Inverse Laplace Transform 6.5.1 Partial Fraction Decomposition Method 6.5.2 Contour Integration of the Laplace Inversion Integral 6.5.3 Heaviside's Expansion Theorem for Inverse Laplace Transform 6.6 Derivatives and Integration of Laplace Transforms 6.6.1 Laplace Transform of Derivatives of a Function 6.6.2 Laplace Transform of the Integral of a Function 6.6.3 Derivative of the Laplace Transform of a Function 6.6.4 Integral of the Laplace Transform of a Function 6.7 Initial and Final Value Theorems of the Laplace Transform 6.7.1 Initial Value Theorem (IVT) 6.7.2 Final Value Theorem (FVT) 6.8 Laplace Transform for System Modeling 6.9 The Bilateral Laplace Transform 6.10 Laplace Transform of Some Standard Functions 6.11 Closure Reference 7 Discrete Fourier Transform 7.1 Introduction 7.1.1 Signal Processing 7.1.2 Communications 7.1.3 Medical Imaging 7.1.4 Scientific Research 7.1.5 Economic and Financial Analysis 7.2 Discrete Representation of Data 7.3 From Continuous to Discrete Fourier Transform 7.3.1 Linearity of DFT 7.3.2 Circular Shift of a Sequence 7.3.3 Circular Shift of a Spectrum 7.3.4 Circular Time Reversal 7.3.5 Symmetry Relations of DFT 7.3.6 Difference of Sequence and Its DFT 7.3.7 Simultaneous Calculation of Real Transforms 7.3.8 Upsampling of a Sequence 7.4 Fast Fourier Transform 7.4.1 Cooley-Tukey Algorithm 7.4.2 Mixed Radix Fast Transforms 7.5 Spectrum Analysis 7.6 Sampling and Interpolation 7.7 Windowing 7.7.1 Rectangular Window 7.7.2 Hann (or Hanning) Window 7.7.3 Hamming Window 7.7.4 Blackman Window 7.7.5 Gaussian Window 7.8 Noise Characterization 7.8.1 White Noise 7.8.2 Pink Noise 7.8.3 Brown Noise (Red Noise) 7.8.4 Blue Noise (Azure Noise) 7.8.5 Violet Noise (Purple Noise) 7.8.6 Gray Noise 7.8.7 Black Noise 7.9 Quantization 7.10 Practicality in DFT Analysis 7.11 Closure References 8 Convolution, Cross-correlation, and Stochastic Analysis 8.1 Introduction 8.2 Convolution 8.3 Discrete Convolution 8.4 Application of Convolution 8.4.1 Convolution to Calculate Inverse Laplace Transform 8.4.2 Convolution for Solution of Volterra Integral 8.4.3 Convolution for Signal Filtering 8.4.4 Convolution for Time Series Analysis 8.4.5 Image Processing 8.5 Circular Convolution 8.6 Introduction to Probability Theory 8.6.1 History of Probability Theory 8.6.2 Grammar of Probability Theory 8.6.2.1 Uniform Distribution 8.6.2.2 Normal (Gaussian) Distribution 8.6.2.3 Exponential Distribution 8.6.2.4 Gamma Distribution 8.6.2.5 Beta Distribution 8.6.2.6 Chi-Square Distribution 8.6.2.7 Log-Normal Distribution 8.6.2.8 Weibull Distribution 8.6.3 Joint Probability Distributions 8.6.4 Correlations and Random Signals 8.6.5 Ensembles and Expected Values of a Random Signal 8.6.6 Systems and Random Signal 8.7 Convolution and Probability Distribution 8.7.1 Central Limit Theorem 8.8 Cross-correlation 8.8.1 Mathematical Properties of Cross-correlation 8.9 Closure References 9 Linear Systems 9.1 Description of a System 9.2 Linear System 9.3 Translation or Shifting 9.4 Cascading Linear Systems 9.5 The Impulse Response 9.6 Linear Time-Invariant (LTI) Systems 9.7 Eigenfunctions 9.8 Translating in Time and Plugging into L 9.9 The Fourier Transform and LTI Systems 9.10 Causality 9.11 System Stability 9.12 Matched Filters 9.13 Closure References 10 Z Transform 10.1 History of Z Transform 10.2 Definition of the Z Transform 10.3 Basic Operational Properties of Z Transforms 10.3.1 Linearity of Z Transforms 10.3.2 Translation 10.3.3 Multiplication 10.3.4 Division 10.3.5 Convolution 10.3.6 Parseval's Formula 10.3.7 Initial Value Theorem 10.3.8 Final Value Theorem 10.3.9 The Z Transform of Partial Derivatives 10.4 The Inverse Z Transform 10.5 Applications of Z Transforms 10.5.1 Solution of Difference Equation 10.5.2 Transfer Function 10.5.3 Characterization of a System by Its Poles and Zeros 10.5.4 Frequency Response and the Locations of the Poles and Zeros 10.5.5 System Stability 10.5.6 Summation of Infinite Series 10.6 Closure References 11 Higher-Dimensional Fourier Analysis 11.1 Introduction 11.2 Properties of Multidimensional Fourier Transform 11.3 Multidimensional Fourier Transform of Separable Functions 11.4 Radial Functions in 2D and Their Fourier Transforms 11.5 Higher-Dimensional Impulse Response 11.6 Higher-Dimensional Sampling 11.7 Closure Reference 12 Fourier Analysis for the Solution of Differential and Integral Equations 12.1 Introduction 12.2 Fourier Transforms for the Solution of Ordinary Differential Equations 12.3 Fourier Transforms for the Solution of Integral Equations 12.4 Fourier Transforms for the Solution of Partial Differential Equations 12.4.1 The Principle of Superposition of Solutions 12.4.1.1 Case 1: Distinct Real Roots (> 0) 12.4.1.2 Case 2: Repeated Real Roots (= 0) 12.4.1.3 Case 3: Complex Conjugate Roots (< 0) 12.4.2 The Concept of Variable Separation Method in PDE Solutions 12.4.3 One-Dimensional Heat Equation 12.4.4 Two-Dimensional Heat Equation 12.4.5 Solutions for the 2D Laplace Equations with Boundary Conditions 12.4.6 The One-Dimensional Wave Equation 12.4.7 Classification of Partial Differential Equations 12.5 Closure References Index