Information Theory webpage offers an in-depth presentation of information theory and coding, providing a thorough understanding of key principles and their real-world implementations. Delve into the intricacies of data transmission and compression, the cornerstone of modern communication systems, and explore how information theory shapes our interconnected world.

• Coding Problem, Entropy and Mutual Information
  • Coding Problem in Claude Shannon Information-Theory: In Claude Shannon’s groundbreaking research on information theory, the coding problem is a pivotal challenge for the efficient transfer of information. Our lecture delves deep into this issue, examining its key elements: source coding, channel coding, and their synergy in joint source-channel coding.
  • Shannon’s Entropy and Conditional Entropy: This lecture offers a clear presentation of the principles quantifying uncertainty in data. This lecture is organized into four enlightening sections, providing an in-depth analysis of Shannon’s entropy and conditional entropy, and illuminating their theoretical foundations as well as their practical applications in diverse fields.
  • Mutual Information and Kullback-Leibler Divergence: This session provides a deep dive into the critical concept of mutual information, a key element in information theory. It quantifies the amount of information shared between random variables, playing a vital role in disciplines such as statistics, machine learning, and cryptography. This lecture is thoughtfully divided into three detailed sections, each thoroughly exploring different aspects of mutual information.
• Source Coding
  • Source Coding Theorem: We delve into the essential concept of the source coding theorem in information theory. Grasping this theorem is crucial for understanding the fundamental limits and capabilities of data compression. We start with an exploration of source coding capacity and its characterization, progressing to Shannon’s source coding theorem. The lecture then examines the principles of data compression and typical sequences, culminating in a detailed proof of the source coding theorem.
  • Error-Free Source Coding: We explore the essential principles and methods for encoding information efficiently and without loss. We begin by examining variable-length source coding and coding rates, laying the foundation for error-free encoding techniques. We clarify the role of uniquely decipherable codes, including codebooks and codewords, emphasizing the need for clear decoding in information transmission. We then investigate the use of tree-based codes and explore Kraft’s inequality, a key metric for code efficiency. The lecture culminates with a thorough analysis of the error-free source coding theorem, discussing optimization issues, bounds on average codeword length, and the role of Kraft’s inequality in devising optimal error-free encoding strategies.
  • Optimal Source Codes: This lecture, « Optimal Source Codes, » examines two fundamental approaches: Huffman’s optimal coding algorithm and the Shannon-Fano-Elias coding method. These strategies are essential for crafting efficient, lossless data compression codes. We will dissect the underlying principles of these methods, providing insights into the development of highly effective encoding systems for diverse data types.
  • Parsing-Translation and Tunstall’s Codes: This session explores advanced methods for attaining low coding rates through optimal parsing—grouping source symbols into variable-length blocks—and encoding these blocks using straightforward, non-optimized codes. Introduced by Tunstall in 1967, Tunstall’s code is a prime example of a wider category of codes that employ parsing followed by translation. We will conduct an in-depth examination of parsing-translation codes and Tunstall’s approach, gaining an understanding of the efficient encoding strategies and their real-world applications.
  • Universal Source Coding: We explore data compression and encoding, examining the concept of universal source coding and its impact. This approach aims to create compression techniques that can effectively condense any data type, regardless of prior knowledge of its statistical characteristics.
  • Lempel-Ziv Source Code: We examine the Lempel-Ziv algorithm, a key technique for error-free encoding of any source sequence, achieving universal source coding. We’ll break down how the algorithm works, explore its variations, assess its efficiency using automata theory, and demonstrate why it’s considered optimal compared to other encoding methods.
• Channel Coding
  • Channel Information Capacity: This lecture explores the core ideas of how much information communication channels can transmit dependably. Starting with the basics of channel properties, we build the essential knowledge needed to grasp information capacity. We then define and discuss the theory behind information capacity, and bring these ideas to life with practical examples, demonstrating capacity in actual communication situations.
  • Channel Coding Theorem: In the « Channel Coding Theorem » lecture, we investigate key information theory concepts, emphasizing the link between achievable transmission rates and a channel’s coding capacity. We start with the theorem’s basic principles, examining how to maximize transmission rates despite noisy channel limitations. Next, we lay the foundation for the theorem’s proof, introducing typical sequences for two random variables and crucial mutual information inequalities.
• Exercises with Solutions on Information Theory: This section offers a carefully selected set of problems aimed at strengthening your grasp of information theory. For each exercise, you’ll find an in-depth solution that walks you through the process, ensuring you not only understand the theory but can also apply it practically.