Information Theory

Three Theorems by Claude Shannon
1. Auflage 2022
ISBN: 978-3-031-21560-5
Verlag: Springer International Publishing

Buch, Englisch, Band 144, 209 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 388 g

Reihe: UNITEXT

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 entropyof 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.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Chambert-Loir, Antoine

Fachgebiete

• Mathematik | Informatik EDV | Informatik Informatik Mathematik für Informatiker
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Informationstheorie, Kodierungstheorie

Inhaltsverzeichnis

Elements of Theory of Probability.- Entropy and Mutual Information.- Coding.- Sampling.- Solutions to Exercises.- Bibliography.- Notation.- Index.

Über Autor(innen)

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.