Complex and Hypercomplex Analytic Signals

Name: Complex and Hypercomplex Analytic Signals | Theory and Applications
Brand: Artech House
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Theory and Applications

Stefan L. Hahn(Author)

Artech House (Publisher)

1st Edition

Published on 1. January 2016

240 pages

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978-1-63081-438-0 (ISBN)

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Intro
Complex and Hypercomplex Analytic Signals: Theory and Applications
Preface
Contents
1 Introduction and Historical Background
1.1 Introduction
1.1.1 The Signal Domain Method
1.1.2 The Frequency Domain Method
1.2 A Historical Survey
References
2 Survey of Chosen Hypercomplex Algebras
2.1 Cayley-Dickson Algebras
2.1.1 The Cayley-Dickson Construction
2.1.2 The Cayley-Dickson algebra of quaternions
2.1.3 The Cayley-Dickson Algebra of Octonions
2.2 Selected Clifford Algebras
2.2.1 The Clifford Algebra of Biquaternions
2.2.2 The Clifford Algebra of Bioctonions
2.3 Comparison of Algebras
2.4 Applications of Hypercomplex Algebras in Signal Processing
2.5 Summary
References
3 Orthants of the n-Dimensional Cartesian Space and Single-Orthant Operators
3.1 The Notion of an Orthant
3.2 Single-Orthant Operators
3.3 Decomposition of Real Functions into Even and Odd Terms
References
4 Fourier Transformation in Analysis of n-Dimensional Signals
4.1 Complex n-D Fourier Transformation
4.1.1 Spectrum of a 1-D Real Signal in Terms of its Even and Odd Components
4.1.2 Spectrum of a 2-D Real Signal in Terms of its Even and Odd Components
4.1.3 Spectrum of a 3-D Real Signal in Terms of its Even and Odd Components
4.2 Cayley-Dickson Fourier Transformation
4.2.1 General Formulas
4.2.2 Quaternion Fourier Spectrum in Terms of its Even and Odd Components
4.2.3 Octonion Fourier Spectrum in Terms of its Even and Odd Components
4.3 Relations Between Complex and Hypercomplex Fourier Transforms
4.3.1 Relation Between QFT and 2D FT
4.3.2 Relation Between OFT and 3D FT
4.4 Survey of Applications of Complex and Hypercomplex Fourier Transformations
4.4.1 Applications in the Domain of Analytic Signals
4.5 Summary
References
5 Complex and Hypercomplex Analytic Signals
5.1 1-D Analytic Signals as Boundary Distributions of 1-D Analytic Functions
5.2 The nD Analytic Signal
5.2.1 The 2-D Complex Analytic Signals
5.2.2 3-D Complex Analytic Signals
5.3 Hypercomplex n-D Analytic Signals
5.3.1 2-D Quaternion Signals
5.3.2 3-D Hypercomplex Analytic Signals
5.4 Monogenic 2-D Signals
5.5 A Short Survey of the Notions of Analytic Signals with Single Orthant Spectra
5.6 Survey of Application of nD Analytic Signals
5.6.1 Applications Presented in Other Chapters of this Book
5.6.2 Applications Described in Hahn's Book on Hilbert Transforms [7]
5.6.3 Selected Applications
References
6 Ranking of Analytic Signals
6.1 Definition of a Suborthant
6.1.1 Subquadrants in 2-D
6.1.2 Suboctants in 3-D
6.2 Ranking of Complex Analytic Signals
6.2.1 Ranking of 2-D Complex Analytic Signals
6.2.2 Ranking of 3-D Complex Analytic Signals
6.3 Sanking of Hypercomplex Analytic Signals
6.3.1 Ranking of 2-D Cayley-Dickson Analytic Signals
6.3.2 Ranking of 3-D Cayley-Dickson Analytic Signals
6.4 Summary
References
7 Polar Representation of Analytic Signals
7.1 Introduction
7.2 Polar Representation of Complex Numbers
7.3 Polar Representation of 1-D Analytic Signals
7.3.1 Representation of the Instantaneous Complex Frequency using the Wigner Distribution
7.4 Polar Representation of 2-D Analytic Signals
7.4.1 2-D Complex Analytic Signals with Single-Quadrant Spectra
7.4.2 2-D Hypercomplex Quaternion Analytic Signals with Single-Quadrant Spectra
7.4.3 Relations between the Analytic and Quaternion 2-D Phase Functions
7.4.4 Polar Representation of the Monogenic 2-D Signal
7.4.5 Common Examples for 2-D Polar Representations of Analytic, Quaternion, and Monogenic Signals
7.5 Polar Representation of 3-D Analytic Signals
7.5.1 3-D Complex Signals with Single Octant Spectra
7.5.2 3-D Octonion Signals with Single Octant Spectra
References
8 Quasi-Analytic Signals
8.1 Definition of a Quasi-Analytic Signal
8.2 1-D Quasi-Analytic Signals
8.3 Phase Signals
8.4 n-D Quasi-Analytic Signals
References
9 Space-Frequency Representations of n-D Complex and Hypercomplex Analytic Signals
9.1 Wigner Distributions and Woodward Ambiguity Functions of Complex Analytic Signals
9.1.1 WDs and AFs of 1-D Signals
9.1.2 WDs and AFs of 2-D Complex Signals
9.1.3 WDs and AFs of 2-D Complex Analytic Signals
9.2 Wigner Distributions and Woodward Ambiguity Functions of Quaternion and Monogenic Signals
9.2.1 The WDs of Quaternion Signals
9.2.2 AFs of Quaternion Signals
9.2.3 WDs of Monogenic Signals
9.2.4 AFs of Monogenic Signals
9.3 Double-Dimensional Wigner Distributions
9.4 Applications of Space-Frequency Distributions in Signal Processing
9.4.1 Wigner Distribution in Noise Analysis
9.4.2 Wigner Distribution in Image Processing
References
10 Causality of Signals
10.1 Kramers-Kronig Relations
10.2 Extension of the Notion of Causality to Higher Dimensions
10.2.1 Derivation of the Dispersion Relations
10.3 Summary
References
11 Summary
References
Appendix A Table of Properties of 1-D Fourier Transformation
Selected Bibliography
Appendix B Table of Chosen 1-D Fourier Pairs
Selected Bibliography
Appendix C Table of Properties of 2-D Fourier Transformations
Selected Bibliography
Appendix D Chosen 2-D Fourier Pairs
Selected Bibliography
Appendix E Table of Properties of Quaternion Fourier Transformation of Real Signals
Selected Bibliography
Appendix F Properties of 1-D Hilbert Transformations
Appendix G 1-D Hilbert Pairs
Appendix H 2-D Hilbert Quadruples
List of Symbols
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Complex and Hypercomplex Analytic Signals

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