Topics in Signal Processing
Overview
Foundation
1. Set Theory
1.1. Sets
1.2. Relations
1.3. Functions
1.4. Cardinality
1.5. Sequences
1.6. General Cartesian Product
2. Elementary Real Analysis
2.1. Real Line
2.2. Topology of Real Line
2.3. Sequences and Series
2.4. The Extended Real Line
2.5. Real Valued Functions
2.6. Real Functions
2.7. Differentiable Functions
2.8. Some Important Inequalities
3. Metric Spaces
3.1. Introduction
3.2. Metric Topology
3.3. Boundedness
3.4. Sequences
3.5. Subspace Topology
3.6. Functions and Continuity
3.7. Completeness
3.8. Compactness
3.9. Real Valued Functions
3.10. Discrete Metric Space
3.11. Special Topics
4. Linear Algebra
4.1. Matrices I
4.2. Vector Spaces
4.3. Matrices II
4.4. Linear Transformations
4.5. Normed Linear Spaces
4.6. Inner Product Spaces
4.7. Dual Spaces
4.8. The Euclidean Space
4.9. Matrices III
4.10. Eigen Values
4.11. Singular Values
4.12. Important Vector Spaces
4.13. Matrix Norms
4.14. Sequence Spaces
4.15. Affine Sets and Transformations
5. Multivariate Calculus
5.1. Differentiation
5.2. Differentiation in Banach Spaces
6. Probability
6.1. Random Variables
6.2. Univariate Distributions
6.3. Basic Inequalities
6.4. Two Variables
6.5. Expectation
6.6. Random Vectors
6.7. Multivariate Gaussian Distribution
6.8. Subgaussian Distributions
7. Numerical Optimization
7.1. Mathematical Optimization
Convexity
8. Convex Sets and Functions
8.1. Real Vector Spaces
8.2. Convex Sets
8.3. Convex Subsets of
\(\RR^n\)
8.4. Cones
8.5. Cones II
8.6. Cones III
8.7. Generalized Inequalities
8.8. Convex Functions
8.9. Differentiability and Convex Functions
8.10. Function Operations
8.11. Topology of Convex Sets
8.12. Separation Theorems
8.13. Continuity
8.14. Recession Cones
8.15. Directional Derivatives
8.16. Subgradients
8.17. Conjugate Functions
8.18. Smoothness
8.19. Infimal Convolution
9. Convex Optimization
9.1. Convex Optimization
9.2. Projection on Convex Sets
9.3. Directions of Recession
9.4. Basic Duality
9.5. Constrained Optimization I
9.6. Linear Constraints
9.7. Constrained Optimization II
9.8. Lagrange Multipliers
9.9. Lagrangian Duality
9.10. Conjugate Duality
9.11. Linear Programming
9.12. Quadratic Programming
10. Subgradient Methods
10.1. Basic Subgradient Method
11. Proximal Algorithms
11.3. Proximal Mappings and Operators
Sparsity
12. Sparse Signal Models
12.3. Underdetermined Linear Systems
12.4. Sparsity in Orthonormal Bases
12.5. Sparse and Redundant Representations
12.6. Dictionaries
12.7. Compressive Sensing
12.8. Restricted Isometry Property
12.9. Dictionaries II
13. Compressive Sensing
13.1. Sensing Matrices
14. Sparse Approximation with Dictionaries
14.1. Stability of the Sparsest Solution
14.2. Basis Pursuit
14.3. Orthogonal Matching Pursuit
15. Sparse Recovery from Compressive Measurements
15.1. Stability of the Sparsest Solution
15.2. Basis Pursuit
15.3. Orthogonal Matching Pursuit
15.4. Compressive Sampling Matching Pursuit
16. Dictionary Learning
16.1. Introduction
Epilogue
Notation
Bibliographic Notes
Index
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