Module5part1

Module Overview

Module Title: Information Theory and Source Coding Part I

Instructor: Ms. Rashmita K. MohapatraPosition: Assistant Professor, E&TC at Thakur College of Engineering & Technology

Announcements

• SSIC Lecture Cancellation: Students in TE will have their SSIC lecture cancelled tomorrow by Amol Sir. Attendance will still be recorded for the lecture; however, students are encouraged to review the material independently to maintain continuity in their learning.

Uncertainty and Information

• The amount of information associated with a given event is inherently related to its probability of occurrence, which can be expressed as a numeric value (p_k).

• As the probability p_k decreases (meaning the event is less likely to occur), the information content increases. This misunderstanding can often lead to misconceptions about certainty and information.

• Information content is intricately linked to the prior knowledge possessed by the recipient about the event's occurrence, reflecting the uncertainty that the event carries for them.

• Example: Consider a scenario where the first statement is a known fact (i.e., it has a high probability of occurrence); thus, it conveys no new information. Conversely, subsequent sentences may present uncertain events, thereby increasing their information content significantly.

Introduction to Information Theory

• Communication systems exist to transfer information seamlessly from one location to another via various types of communication channels, be they physical or virtual.

• Information theory addresses fundamental communication limits that can significantly affect performance:

  • Data Compression Limits: What is the theoretical maximum compressibility for specific types of data without losing essential information?

  • Reliable Communication Over Noisy Channels: How many bits can be sent reliably per unit of time despite the unavoidable noise?

• This branch of probability theory employs mathematical modeling to understand and analyze how communication systems effectively work under different conditions and limitations.

Key Source Elements

• Binary Source (BS): Generates symbols '0' and '1' at a specified rate 'r' (measured in symbols/second), which is crucial for understanding the basic building blocks of information transmission.

• Binary Symmetric Channel (BSC): Models a key element of information theory, helping in the analysis of signal distortion and error rates in transmission.

Measures of Information

• Key Concepts:

  • Entropy: Represents a measure of uncertainty and provides insight into the average information contained within various sources. Understanding entropy is crucial for optimizing data encoding and transmission strategies.

  • Information Rate: Derived from entropy concepts; it quantifies the speed at which information is produced or consumed.

  • Other significant measures include differential entropy, joint entropy, and conditional entropy, each serving various purposes in data analysis and compression algorithms.

• Coding Methods: Familiarity with Huffman Source Coding and Shannon-Fano Source Coding methods is essential for encoding messages efficiently.

• Mutual Information and Channel Capacity: These concepts are fundamental principles in information theory, quantifying how much information is shared between two random variables and the maximum limit of information that can be transmitted over a channel.

Importance of Information Theory in Digital Communication

• Given the vast amounts of source data generated daily, there is a critical need for the development of efficient information description methods to:

  • Minimize Bandwidth Usage: Efficient transmission is vital for modern communication systems and mobile technologies.

  • Reduce Storage Memory: Compressed data takes up less space, leading to cost savings and improved performance in storage systems.

• Information theory also addresses challenges in maximizing data transmission over noisy channels by employing source coding techniques designed to eliminate redundancy and improve overall effectiveness.

Information Content of Symbols

• While information itself is dimensionless, it is often quantified in bits in the context of binary communication.

• The information content of a transmitted message can be represented mathematically using probabilities, as outlined in the formula:

• Definition of Information: I_k = log2(1/p_k)

• This highlights how specific probabilities correlate with the information conveyed by a given message.

Numerical Examples

1. Numerical 1: A message with a probability of occurrence denoted as p_k = 3/4 yields:

   • Information Content: I_k = log2(1/p_k) → Result: 0.415 bits

2. Numerical 2: When the probability is p_k = 1/4, the resulting information conveyed is:

   • Result: 2 bits

3. Numerical 3: In a Pulse Code Modulation (PCM) system with M = 2^N messages, each message corresponds to N bits of information optimizing signal representation.

4. Numerical 4: When a specific message, mk, is always transmitted with a probability of p_k = 1, the information conveyed is zero. This indicates the absence of uncertainty, reinforcing the concept that certainty equates to a lack of new information.

5. Numerical 5: For independent messages, the total information conveyed is the sum of the individual messages. Mathematically, this relationship can be represented through the multiplication of their respective probabilities.

Additional Concepts

• Properties of Information: The information content tends to decrease as the probability of an event's occurrence increases, indicating that certainty diminishes the information value. Furthermore, information is inherently a non-negative quantity, approaching zero when certainty is at its peak.

Introduction to Entropy

• Definition: Entropy measures the uncertainty associated with a random variable, specifically defined for discrete variables. It aids in understanding the average information across various source probabilities, illustrating how different events contribute to overall uncertainty.

Properties of Entropy

• The calculated entropy, H(X), is mathematically bounded between 0 and log2(m), where 'm' represents the number of symbols in the source.

• Maximum uncertainty occurs when all events are equally probable, allowing for optimal data compression techniques to be employed.

• H(X) reaches zero when only one symbol has a non-zero probability, indicating absolute certainty.

Variations on Entropy

• This section explores the relationship between variations in symbol probabilities and overall entropy, detailing how changes in probabilities impact the average information transmitted.

Resource References

• The module includes key textbooks on communication systems and encourages the utilization of online resources such as NPTEL for further exploration and mastery of the subject matter.

Conclusion

• This comprehensive study of Information Theory elucidates both its theoretical foundations and practical applications in communication technology, equipping students with the necessary insights to navigate the challenges of efficient data transmission in an increasingly digital world.