Basic Transforms for Electrical Engineering

Foreword
Contents
List of Symbols and Abbreviations
Part I Background
	1 Complex Numbers
		1.1 Representation of Complex Numbers
		1.2 Euler's Identity
			1.2.1 Complex Exponential
			1.2.2 Conjugate of a Complex Number
		1.3 Mathematical Operations
			1.3.1 Identity
			1.3.2 Addition and Subtraction
			1.3.3 Multiplication and Division
			1.3.4 Rotating a Number in Complex Plane
		1.4 Roots of a Complex Number
		1.5 Applications of Complex Numbers
			1.5.1 Complex Numbers Versus Trigonometry
			1.5.2 Integration
			1.5.3 Phasors
			1.5.4 3-Phase Electric Circuits
			1.5.5 Negative Frequency
			1.5.6 Complex Numbers in Mathematics Software
			1.5.7 Roots of a Polynomial
		Further Reading
		Problems
	2 Functions of a Complex Variable
		2.1 Limit of a Complex Function
		2.2 Derivative of Complex Functions and Analyticity
		2.3 Cauchy–Riemann Conditions
		2.4 Rules of Differentiation
		2.5 Harmonic Functions
		2.6 Applications of Complex Functions and Analyticity
			2.6.1 Elementary Functions
				Polynomial and Rational Functions
				Exponential Function of a Complex Variable
				Logarithm of a Complex Number
				Trigonometric Functions of a Complex Variable
				Hyperbolic Functions of a Complex Variable
				A Worked-Out Example: The Inverse Cosine Function
			2.6.2 Conformal Mapping
			2.6.3 Fractals
		Further Reading
		Problems
	3 Complex Integration
		3.1 Integrating Complex Functions of a Real Variable
		3.2 Contours
		3.3 Integrating Functions of a Complex Variable
		3.4 Numerical Computation of the Complex Integral
		3.5 Properties of the Complex Integral
		3.6 The Cauchy–Goursat Theorem
			3.6.1 Integrating Differentiable Functions
			3.6.2 The Principle of Contour Deformation
			3.6.3 Cauchy's Integral for Multiply Connected Domains
		3.7 Cauchy's Integral Formula
		3.8 Higher-Order Derivatives of Analytic Functions
		3.9 Complex Sequences and Series
		3.10 Power Series Expansions of Functions
			3.10.1 Taylor and Maclaurin Series
			3.10.2 Differentiation and Integration of Power Series
		3.11 Laurent Series
		3.12 Residues
			3.12.1 Residue Theorem
			3.12.2 Residue at Infinity
			3.12.3 Finding Residues
		3.13 Residue Integration of Real Integrals
		3.14 Fourier Integrals
		Further Reading
		Problems
Part II Transforms
	4 The Laplace Transform
		4.1 Motivation to Use Laplace Transform
		4.2 Definition of the Laplace Transform
		4.3 Properties of the Laplace Transform
			4.3.1 Linearity
			4.3.2 Real Differentiation
			4.3.3 Real Integration
			4.3.4 Differentiation by s
			4.3.5 Real Translation
			4.3.6 Complex Translation
			4.3.7 Periodic Functions
			4.3.8 Laplace Transform of Convolution
			4.3.9 Initial Value Theorem
			4.3.10 Final Value Theorem
		4.4 The Inverse Laplace Transform
			4.4.1 Real Poles
			4.4.2 Complex Poles
			4.4.3 Multiple Poles
		4.5 More on Poles and Zeros
			4.5.1 Factoring Polynomials
			4.5.2 Poles and Time Response
			4.5.3 An Alternative Way to Solve Differential Equations
		4.6 Inverse Laplace Transform by Contour Integration
		4.7 Applications of Laplace Transform
			4.7.1 Electrical Systems
			4.7.2 Inverse LTI Systems
			4.7.3 Evaluation of Definite Integrals
		Problems
	5 The Fourier Series
		5.1 Vectors and Signals
		5.2 The Fourier Series
		5.3 Calculating Fourier Series Coefficients
		5.4 Properties of the Fourier Series
			5.4.1 Linearity
			5.4.2 Symmetry Properties
		Even Symmetry
		Odd Symmetry
		Half-Period Symmetry
			5.4.3 Shifting in Time
			5.4.4 Time Reversal
			5.4.5 Differentiation
			5.4.6 Integration
		5.5 Parseval's Relation
		5.6 Convergence of Fourier Series
		5.7 Gibbs Phenomenon
		5.8 Discrete-Time Fourier Series
			5.8.1 Periodic Convolution
			5.8.2 Parseval's Relation for Discrete-Time Signals
		5.9 Applications of Fourier Series
		Problems
	6 The Fourier Transform
		6.1 Introduction
		6.2 Definition of the Fourier Transform
		6.3 Fourier Transform Versus Fourier Series
		6.4 Convergence of the Fourier Transform
		Dirichlet Conditions
		6.5 Properties of the Fourier Transform
			6.5.1 Symmetry Issues
			6.5.2 Linearity
			6.5.3 Time Scaling
			6.5.4 Time Reversal
			6.5.5 Time Shift
			6.5.6 Frequency Shift (Amplitude Modulation)
			6.5.7 Differentiation with Respect to Time
			6.5.8 Integration with Respect to Time
			6.5.9 Duality
			6.5.10 Convolution
			6.5.11 Multiplication in Time Domain
			6.5.12 Parseval's Relation
			6.5.13 Two-way Transform: Fourier Integral Theorem
			6.5.14 Fourier Transform of a Periodic Time Function
		6.6 Sampling
			6.6.1 Impulse-Sampling and Aliasing
			6.6.2 Natural Sampling: The Zero-Order Hold
			6.6.3 Undersampling
		6.7 Fourier Transform Versus Laplace Transform
		6.8 Discrete-Time Signals
		6.9 Fourier Transform of Discrete Signals
			6.9.1 The Discrete Fourier Transform
		6.10 Two-Dimensional Fourier Transform
		6.11 Applications
			6.11.1 Signal Processing
				Spectrogram
				Cepstrum Analysis
				Correlation and Energy Spectrum
				Filtering
			6.11.2 Circuit Applications
			6.11.3 Communication
				Propagation
				Time-Division Multiplexing (TDM)
				Frequency-Division Multiplexing (FDM)
				Amplitude Modulation and Demodulation
				FM Slope Detectors
			6.11.4 Instrumentation
		Further Reading
		Problems
	7 Short-Time-Fourier Transform
		7.1 Short-Time Fourier Transform
			7.1.1 Frequency Resolution
			7.1.2 Inverse Short-Time Fourier Transform
			7.1.3 Discrete-Time STFT
		Windowing
		7.2 Gabor Transform
		7.3 STFT in LabVIEW
		Problems
	8 Fast Fourier Transform
		8.1 Radix-2 FFT Algorithms
			8.1.1 Decimation in Time
			8.1.2 Decimation in Frequency
		8.2 Computer Implementation
			8.2.1 LabVIEW Implementation
			8.2.2 Implementing FFT in C
		Further Reading
		Assignments
	9 z-Transform
		9.1 Definition of the z-Transform
		9.2 Region of Convergence for the z-Transform
		9.3 z-Transform Properties
			9.3.1 Linearity
			9.3.2 Time Shifting
			9.3.3 Multiplication by an Exponential Sequence
			9.3.4 Multiplication by n
			9.3.5 Division by n
			9.3.6 Conjugate of a Complex Sequence
			9.3.7 Convolution of Sequences
			9.3.8 Time Reversal
			9.3.9 Initial Value Theorem
		9.4 The Inverse z-Transform
			9.4.1 Inversion by Partial Fraction Expansion
				Complex Roots
				Multiple Roots
			9.4.2 Inverse z-Transform Using Contour Integration
		9.5 Complex Convolution Theorem
		9.6 Parseval Theorem
		9.7 One-Sided z-Transform
		9.8 Difference Equations
		9.9 Conversions Between Laplace Transform and z–Transform
		9.10 Fourier Transform of Discrete-Time Signals
		9.11 Applications of the z-Transform
			9.11.1 Digital Oscillator
				A 3-Tap FIR Filter
				Derivative in Discrete-Time
				Fibonacci Sequence in Closed Form
		9.12 Discrete Signal Processing vs Digital Technologies: A Historical Retrospect
		Further Reading
		Problems
	10 Discrete Cosine Transform
		10.1 From DFT to DCT
			10.1.1 One-Dimensional Signal
			10.1.2 Two-Dimensional Signal
		10.2 DCT Implementation
		10.3 DCT Applications
		Further Reading
		Problems
References
Index