Overview
- Provides an introduction to information theory, fundamental to the digitized world
- Accessible to mathematics and computer science undergraduates
- Includes numerous exercises, with solutions
Part of the book series: UNITEXT (UNITEXT, volume 144)
Part of the book sub series: La Matematica per il 3+2 (UNITEXTMAT)
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Table of contents (5 chapters)
Keywords
About this book
This book provides an introduction to information theory, focussing on Shannon’s three foundational theorems of 1948–1949. Shannon’s first two theorems, based on the notion of entropy in probability theory, specify the extent to which a message can be compressed for fast transmission and how to erase errors associated with poor transmission. The third theorem, using Fourier theory, ensures that a signal can be reconstructed from a sufficiently fine sampling of it. These three theorems constitute the roadmap of the book.
The first chapter studies the entropy of a discrete random variable and related notions. The second chapter, on compression and error correcting, introduces the concept of coding, proves the existence of optimal codes and good codes (Shannon's first theorem), and shows how information can be transmitted in the presence of noise (Shannon's second theorem). The third chapter proves the sampling theorem (Shannon's third theorem) and looks at its connections with other results, such as the Poisson summation formula. Finally, there is a discussion of the uncertainty principle in information theory.
Featuring a good supply of exercises (with solutions), and an introductory chapter covering the prerequisites, this text stems out lectures given to mathematics/computer science students at the beginning graduate level.
Authors and Affiliations
About the author
Antoine Chambert-Loir is a professor of mathematics at Université Paris Cité. His research addresses questions in algebraic geometry which are motivated by number theoretical problems. He is the author of two books published by Springer-Verlag: A Field Guide To Algebra, an introduction to Galois theory; and (Mostly) Commutative Algebra, an intermediate-level exposition of commutative algebra. With J. Nicaise and J. Sebag, he cowrote the research monograph Motivic Integration (published by Birkhäuser), which was awarded the 2017 Ferran Sunyer i Balaguer prize.
Bibliographic Information
Book Title: Information Theory
Book Subtitle: Three Theorems by Claude Shannon
Authors: Antoine Chambert-Loir
Series Title: UNITEXT
DOI: https://doi.org/10.1007/978-3-031-21561-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
Softcover ISBN: 978-3-031-21560-5Published: 16 March 2023
eBook ISBN: 978-3-031-21561-2Published: 15 March 2023
Series ISSN: 2038-5714
Series E-ISSN: 2532-3318
Edition Number: 1
Number of Pages: XII, 209
Number of Illustrations: 1 b/w illustrations
Topics: Mathematics of Computing, Coding and Information Theory