Exploring the interrelations between generalized metric spaces, lattice-ordered groups, and order statistics, the book contains a new algebraic approach to Signal Processing Theory. It describes mathematical concepts and results important in the development, analysis, and optimization of signal processing algorithms intended for various applications. The book offers a solution of large-scale Signal Processing Theory problems of increasing both signal processing efficiency under prior uncertainty conditions and signal processing rate that is provided by multiplication-free signal processing algorithms based on lattice-ordered group operations.
From simple basic relationships to computer simulation, the text covers a wide range of new mathematical techniques essential for understanding the proposed signal processing algorithms developed for solving the following problems: signal parameter and spectral estimation, signal filtering, detection, classification, and resolution; array signal processing; demultiplexing and demodulation in multi-channel communication systems and multi-station networks; wavelet analysis of 1D/ 2D signals. Along with discussing mathematical aspects, each chapter presents examples illustrating operation of signal processing algorithms developed for various applications.
The book helps readers understand relations between known classic and obtained results as well as recent research trends in Signal Processing Theory and its applications, providing all necessary mathematical background concerning lattice-ordered groups to prepare readers for independent work in the marked directions including more advanced research and development.
Author(s): Andrey Popoff
Publisher: CRC Press
Year: 2022
Language: English
Pages: 447
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Introduction
List of Abbreviations
Notation System
1. Introduction to Signal Processing on L-groups
1.1. Operations of L-group
1.2. Interrelation between Operations of L-group and Known Nonlinear Functions
1.2.1. Interrelation with Step Functions
1.2.2. Interrelation with Estimation and Influence Functions
1.2.3. Interrelation with Signal Limiters
1.2.4. Interrelation with Quantizers
1.2.5. Interrelation with Companders
1.3. Generalized Metric Spaces with L-group Properties
1.4. Measures of Statistical Interrelation between Signals in Space with L-group Properties
1.4.1. Normalized Measure of Statistical Interrelation between Stochastic Signals in Space with L-group Properties
1.4.2. Sample Normalized Measures of Statistical Interrelation between Signals in Space with L-group Properties
1.4.3. Sample Measures of Statistical Interrelation between Signals in Space with L-group Properties
1.4.4. Estimating Sample Measures of Statistical Interrelation between Signals in Space with L-group Properties
1.5. Metric Characteristics of Random Samples with Arbitrary Distributions
1.6. Hyperspectral Representation of Signals in Space with L-group Properties
1.6.1. Linear Space with Scalar Product
1.6.2. Hyperspectral Representation of Deterministic Signals Based on Generalized Metrics in L-group
1.6.3. Estimating Hyperspectral Density of Stochastic Signal on L-groups
1.7. Digital Filtering Based on L-group Operations
1.8. Statistical Demultiplexing Based on L-group Operations
2. Estimation of Signal Parameters in Sample Spaces with L-group Properties
2.1. Sample Ordering Algorithms Based on Lattice Operations
2.1.1. Method of Ordering by a Choice
2.1.2. Method of Ordering by Pairwise Successive Permutation
2.1.3. Method of Ordering by Serial-parallel 2m-union
2.1.4. Method of Ordering by Series-parallel Pairwise Union with Pairwise Permutation
2.1.5. Method of Ordering by Series-parallel Ternary Union with Ternary Permutation
2.2. Sample Median Calculation Algorithms Based on Lattice Operations
2.3. Algorithm of Forming M-estimator Based on L-group Operations
2.4. Algorithms of Forming L-estimators Based on L-group Operations
2.5. Algorithms of Forming R-estimators Based on L-group Operations
2.6. Comparative Efficiency of Some Estimators
2.7. Signal Parameter Estimation Algorithms in Spaces with L-group Properties
3. Signal Filtering Algorithms in Spaces with L–group Properties
3.1. Robust Signal Filtering Algorithms Based on L-group Operations
3.2. Adaptive Filtering Algorithms Based on L-group Operations
3.2.1. Signal Filtering Algorithms Based on Method of Mapping of Linear Space into Space with Lattice Properties
3.2.2. Adaptive Filtering Algorithms Based on MSI
3.3. Wiener Filter Based on L-group Operations
3.3.1. Wiener Filter Based on Method of Mapping of Linear Space into Space with Lattice Properties
3.3.2. Wiener Filter Based on MSI
3.3.2.1. Wiener Filter Based on MSI Concerned with Pseudometric
3.3.2.2. Wiener Filter Based on MSI Concerned with Semimetric
3.3.2.3. Wiener Filter Based on MSI Concerned with l1-metric
3.4. Least Mean Squares Filter Based on L-group Operations
3.4.1. Least Mean Squares Filter Based on the Method of Mapping of Linear Space into Space with Lattice Properties
3.4.2. Least Mean Squares Filter Based on MSI
3.4.2.1. Least Mean Squares Filter Based on MSI Concerned with Pseudometric
3.4.2.2. Least Mean Squares Filter Based on MSI Concerned with Semimetric
3.4.2.3. Least Mean Squares Filter Based on Msi Concerned With l1-metric
3.5. Recursive Least Squares Filter Based on L-group Operations
3.5.1. Recursive Least Squares Filter Based on the Method of Mapping of Linear Space into Space with Lattice Properties
3.5.2. Recursive Least Squares Filter Based on MSIs
3.6. Kalman Filter Based on Method of Mapping of Linear Space into Space with Lattice Properties
3.7. L-group Composite Filter
3.8. Robust Filtering Algorithms Based on MSIs
4. Signal Detection, Classification, and Resolution in Spaces with L–group Properties
4.1. Nonparametric Detection Algorithms in Signal Space with L–group Properties
4.1.1. Formulating Signal Detection Problem
4.1.2. Sign Detection Algorithm
4.1.3. Rank Detection Algorithm
4.1.4. Sign-rank Detection Algorithm
4.1.5. Short Summary Concerning Sign, Rank, and Sign-rank Detection Algorithms
4.2. Notion of Generalized Matched Filter
4.3. Detection Algorithm Based on MSI Concerned with Pseudometric
4.4. Detection Algorithm Based on MSI Concerned with l1-metric
4.5. Detection Algorithm Based on MSI Concerned with Semimetric
4.6. Classification Algorithms in Signal Spaces with L-group Properties
4.6.1. Classification Algorithm Based on MSI Concerned with Pseudometric
4.6.2. Classification Algorithm Based on MSI Concerned with l1-metric
4.6.3. Classification Algorithm Based on MSI Concerned with Semimetric
4.6.4. Asymptotic Relative Efficiency of L-group Classification Algorithms
4.7. Algorithms of Signal Resolution in Space with L-group Properties
4.7.1. Ambiguity Function and Mismatching Function of Signals
4.7.2. Signal Resolution. General Considerations
4.7.3. Signal Resolution in Linear Space
4.7.4. Signal Resolution Algorithm Based on MSI Concerned with Pseudometric
4.7.5. Signal Resolution Algorithm Based on MSI Concerned with l1-metric
4.7.6. Signal Resolution Algorithm Based on MSI Concerned with Semimetric
5. Spectral Estimation and Spectral Analysis on L-groups
5.1. Correlogram Method of Spectral Estimation Based on L–group Operations
5.2. Periodogram Method of Spectral Estimation Based on L–group Operations
5.2.1. Periodogram Method of Spectral Estimation Based on MSI
5.2.2. Periodogram Method of Spectral Estimation Based on MSI: Bartlett Approach
5.2.3. Periodogram Method of Spectral Estimation Based on MSI: Welch Approach
5.3. Spectral Estimation Methods Based on MSI Matrix Estimator
5.3.1. Correlogram Method of Spectral Estimation Based on MSI Matrix Estimator
5.3.2. Minimum Variance Spectral Estimation Method Based on MSI Matrix Estimator
5.4. Spectral Estimation Method Based on Eigenvectors of MSI Matrix Estimator
5.4.1. Spectral Estimation by Eigenvector Method Based on MSI Matrix Estimator
5.4.2. Spectral Estimation by MUSIC Method Based on MSI Matrix Estimator
5.4.3. Spectral Estimation by Minimum Norm Method Based on MSI Matrix Estimator
5.4.4. Spectral Estimation by ESPRIT Method Based on MSI Matrix Estimator
6. Antenna Array Signal Processing Based on L–group Operations
6.1. Antenna Systems with Logical Signal Processing Based on L–group Operations
6.1.1. Two-channel Antenna System with Logical Coherent Signal Processing Based on L–group Operations
6.1.2. Two-channel Antenna System with Logical Incoherent Signal Processing Based on L–group Operations
6.1.3. Logical Signal Processing in Antenna Arrays Based on L–group Operations
6.1.3.1. Logical Signal Processing in Linear Antenna Array Based on L–group Operations
6.1.3.2. Logical Signal Processing in Mills Cross Antenna Array Based on L–group Operations
6.2. Spatial Filtering: Signal Space Mapping Method
6.2.1. Spatial Filtering: Forming Vector of Weight Coefficients by Direct Inversion of Correlation Matrix
6.2.2. Spatial Filtering: Forming Vector of Weight Coefficients by Direct Inversion of Correlation Matrix with Decreasing Its Dimensionality
6.2.3. Spatial Filtering: Forming Vector of Weight Coefficients Based on Probability Distribution
6.2.4. Spatial Filtering Suboptimal L-group Algorithms: Forming Vector of Weight Coefficients by Direct Inversion of Correlation Matrix
6.2.5. Spatial Filtering Suboptimal L-group Algorithms: Forming Vector ofWeight Coefficients Based on Probability Distribution
6.2.6. Spatial Filtering Suboptimal L-group Algorithms: Nonlinearities of Amplitude Characteristics of Antenna Array Receiving Channels
6.3. Spatial Filtering: Method Based on MSI
6.3.1. Spatial Filtering in Linear Antenna Array: L-group Algorithm
6.3.2. Spatial Filtering in Circular Antenna Array: L-group Algorithm
6.4. Direction of Arrival Estimation Based on L-group Operations
6.4.1. Minimum Variance Direction of Arrival Estimation Based on MSI Matrix Estimator
6.4.2. Direction of Arrival Estimation Methods Based on Eigenvectors of MSI Matrix Estimator
6.4.2.1. Direction of Arrival Estimation by Eigenvector Method Based on MSI Matrix Estimator
6.4.2.2. Direction of Arrival Estimation by MUSIC Method
6.4.2.3. Direction of Arrival Estimation by Minimum Norm Method
6.4.2.4. Direction of Arrival Estimation by ESPRIT Method
6.5. Wideband Antenna Array Signal Processing Based on L-group Operations
6.5.1. Spatial Filtering in Wideband Linear Antenna Array: L-group Algorithm
6.5.2. Spatial Filtering in Wideband Circular Antenna Array: L-group Algorithm
6.5.3. Direction of Arrival Estimation in Wideband Linear Antenna Array: L-group Algorithm
6.5.4. Direction of Arrival Estimation in Wideband Circular Antenna Array: L-group Algorithm
6.6. Adaptive Algorithms of Spatial Filtering Based on L–group Operations
6.6.1. Least Mean Squares Algorithm of Spatial Filtering Based on L-group Operations
6.6.2. Recursive Least Squares Algorithm of Spatial Filtering Based on L-group Operations
6.6.3. Adaptive Spatial Filtering Algorithm Based on Method of Recursive Forming MSI Matrix Estimator
7. Signal Processing L-group Algorithms for Communication Systems and Networks
7.1. Multichannel Communication Systems and Multi-station Networks
7.2. Main Types of Orthogonal Signal Systems Used in Information Transmitting Systems
7.3. Demultiplexing/Demodulation L-group Algorithms for TDM(A) Systems
7.4. Demultiplexing/Demodulation L-group Algorithms for DS-CDM(A) Systems
7.5. Demultiplexing/Demodulation L-group Algorithms for MFSK-CDM(A) Systems
7.6. Demultiplexing/Demodulation L-group Algorithms for OFDM(A) Systems
7.7. Demultiplexing/Demodulation L-group Algorithms for OFDM-CDM(A) Systems
8. Wavelets on L-groups
8.1. Wavelet Signal Analysis on L-groups: Introduction
8.1.1. Continuous Wavelet Transform and Its Properties
8.1.2. Discrete Wavelet Transform
8.1.3. Discrete Wavelet Transform in Sample Spaces with Lattice Properties
8.1.4. Discrete Wavelet Transform in Sample Spaces with Pseudometric and Semimetric: Harmonic Signals
8.1.5. Discrete Wavelet Transform in Sample Space with l1-metric: Harmonic Signals. Improving a Resolution by a Combination with Pseudometric ans Semimetric Spaces
8.1.6. Discrete Wavelet Transform in Sample Space with l1-metric: BPSK, QPSK, V-LFM, and FSK Signals
8.1.7. Discrete Wavelet Transform in Sample Spaces with Pseudometric and Semimetric: Signals with Finite Durations
8.2. Multiscale Image Decomposition on L-groups
8.2.1. Multiscale Image Decompositions Based on Hadamard Matrix
8.2.1.1. Linear Multiscale Image Decomposition Based on Hadamard Matrix
8.2.1.2. Multiscale Image Decomposition Based on Hadamard Matrix and L-group Operations
8.2.2. Fast 2D Discrete Wavelet Transform Based on L-group Operations
8.2.2.1. Fast 2D Discrete Wavelet Transform Based on Linear Algorithms
8.2.2.2. Fast 2D Discrete Wavelet Transform Based on L-group Operations
Conclusion
Bibliography
Index
