Fundamentals of Signal Processing in Generalized Metric Spaces

Algorithms and Applications

Andrey Popoff

• Publisher: CRC Press
• Year: 2022
• Print ISBN:
  9781032231259
• eBook ISBN:
  9781000572001
• Topic: Technology & Engineering
• Subtopic: Group Theory
• Index: Technology & Engineering

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.

1Introduction to Signal Processing on L-groups

DOI: 10.1201/​9781003275855-1

The chapter begins from considering the general questions concerning the application of L-group operations and the functions related with them and transforms to Signal Processing Theory. The goal of this chapter is acquainting the reader with those important parts of mathematical apparatus that have not yet found proper reflection in the university courses in mathematics intended for preparing specialists in signal processing for different branches of electronics. For these purposes the author uses the existing parallels between the signal spaces explored in the book and well-known linear spaces with the scalar (inner) product.

When studying the material of this chapter, it is important to understand the essence of main definitions, get accustomed to corresponding notations, learn performing operations with mathematical objects (signals and their samples) defined on L-groups. Reaching this goal is provided by examples exposed within every section whose consideration allows the reader to improve understanding of the stated material. In brief, this chapter contains information that is ...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Preface
• Introduction
• List of Abbreviations
• Notation System
• 1 Introduction to Signal Processing on L-groups
• 2 Estimation of Signal Parameters in Sample Spaces with L-group Properties
• 3 Signal Filtering Algorithms in Spaces with L–group Properties
• 4 Signal Detection, Classification, and Resolution in Spaces with L–group Properties
• 5 Spectral Estimation and Spectral Analysis on L-groups
• 6 Antenna Array Signal Processing Based on L–group Operations
• 7 Signal Processing L-group Algorithms for Communication Systems and Networks
• 8 Wavelets on L-groups
• Conclusion
• Bibliography
• Index