Mathematics for Circuits and Filters

Table of contents

1. Cover
2. Half Title
3. Title
4. Copyright
5. Preface
6. Contributors
7. Contents
8. 1 Linear Operators and Matrices
   1. 1.1 Introduction
   2. 1.2 Vector Spaces Over Fields
   3. 1.3 Linear Operators and Matrix Representations
   4. 1.4 Matrix Operations
   5. 1.5 Determinant, Inverse, and Rank
   6. 1.6 Basis Transformations
   7. 1.7 Characteristics: Eigenvalues, Eigenvectors, and Singular Values
   8. 1.8 On Linear Systems
9. 2 Bilinear Operators and Matrices
   1. 2.1 Introduction
   2. 2.2 Algebras
   3. 2.3 Bilinear Operators
   4. 2.4 Tensor Product
   5. 2.5 Basis Tensors
   6. 2.6 Multiple Products
   7. 2.7 Determinants
   8. 2.8 Skew Symmetric Products
   9. 2.9 Solving Linear Equations
   10. 2.10 Symmetric Products
   11. 2.11 Summary
10. 3 The Laplace Transform
    1. 3.1 Introduction
    2. 3.2 Motivational Example
    3. 3.3 Formal Developments
    4. 3.4 Laplace Transform Analysis of Linear Systems
    5. 3.5 Conclusions and Further Reading
    6. 3.6 Appendix A: The Dirac Delta (Impulse) Function
    7. 3.7 Appendix B: Relationships among the Laplace, Fourier, and z-Transforms
11. 4 Fourier Series, Fourier Transforms and the DFT
    1. 4.1 Introduction
    2. 4.2 Fourier Series Representation of Continuous Time Periodic Signals
    3. 4.3 The Classical Fourier Transform for Continuous Time Signals
    4. 4.4 The Discrete Time Fourier Transform
    5. 4.5 The Discrete Fourier Transform
    6. 4.6 Family Tree of Fourier Transforms
    7. 4.7 Selected Applications of Fourier Methods
    8. 4.8 Summary
12. 5 z-Transform
    1. 5.1 Introduction
    2. 5.2 Definition of the z-Transform
    3. 5.3 Inverse z-Transform
    4. 5.4 Properties of the z-Transform
    5. 5.5 Role of the z-Transform in Linear Time-Invariant Systems
    6. 5.6 Variations on the z-Transform
    7. 5.7 Concluding Remarks
13. 6 Wavelet Transforms
    1. 6.1 Introduction
    2. 6.2 Signal Representation Using Basis Functions
    3. 6.3 The Short-Time Fourier Transform
    4. 6.4 Digital Filter Banks and Subband Coders
    5. 6.5 Deeper Study of Wavelets, Filter Banks, and Short-Time Fourier Transforms
    6. 6.6 The Space of L1 and L2 Signals
    7. 6.7 Riesz Basis, Biorthogonality, and Other Fine Points
    8. 6.8 Frames in Hilbert Spaces
    9. 6.9 Short-Time Fourier Transform: Invertibility, Orthonormality, and Localization
    10. 6.10 Wavelets and Multiresolution
    11. 6.11 Orthonormal Wavelet Basis from Para unitary Filter Banks
    12. 6.12 Compactly Supported Orthonormal Wavelets
    13. 6.13 Wavelet Regularity
    14. 6.14 Concluding Remarks
14. 7 Graph Theory
    1. 7.1 Introduction
    2. 7.2 Basic Concepts
    3. 7.3 Cuts, Circuits, and Orthogonality
    4. 7.4 Incidence, Circuit, and Cut Matrices of a Graph
    5. 7.5 Orthogonality Relation and Ranks of Circuit and Cut Matrices
    6. 7.6 Spanning Tree Enumeration
    7. 7.7 Graphs and Electrical Networks
    8. 7.8 Tellegen’s Theorem and Network Sensitivity Computation
    9. 7.9 Arc Coloring Theorem and the No-Gain Property
15. 8 Signal Flow Graphs
    1. 8.1 Introduction
    2. 8.2 Adjacency Matrix of a Directed Graph
    3. 8.3 Coates’ Gain Formula
    4. 8.4 Mason’s Gain Formula
16. Index