Module5part2

Chapter 5: Information Theory and Source Coding Part II

Author: Mrs. Rashmita Kumari Mohapatra

Asst. Prof. – E&TC Thakur College of Engineering & Technology

Overview of Content

In this chapter, we delve into critical topics in information theory that are fundamental for understanding modern communication systems. Key topics covered include:

• Measure of Information: Discussing key concepts such as entropy and its properties which encapsulate the amount of uncertainty involved in random variables.

• Information Rate: Analyzing joint and conditional entropy to measure information shared between different variables and the uncertainty remaining after one variable is known.

• Source Coding Techniques: Examining Huffman coding and Shannon-Fano coding, two widely used practical techniques for efficient data compression.

• Mutual Information and Channel Capacity: Exploring the concepts of mutual information and channel capacity that define the limits of effective data transmission across a communication channel.

• The Channel Coding Theorem: Introducing the channel coding theorem, which expresses the maximum reliable data rate achievable over a noisy channel.

Information Source

An information source is characterized as an entity that produces events randomly selected according to defined probability distributions. Key elements include:

• Source Alphabet: The complete set of symbols that can be generated by the source is referred to as its source alphabet.

• Memory Types:

  • Memory: Here, the current symbol generated depends on previous symbols, indicating a stateful process.

  • Memoryless Source: In this scenario, the symbol produced is entirely independent of any previously produced symbols, allowing for simplicity in analysis.

  • Discrete Memoryless Source (DMS): This is defined by a set list of symbols along with their probability distributions, allowing for the analysis of probabilities in symbol generation without memory dependencies.

Information Content of a DMS

The quantity of information associated with a particular event is inherently linked to its uncertainty. To properly measure information, it must adhere to the following axioms:

• Measurement of Information Scales with Uncertainty: The more unpredictable an event is, the more information it contains when observed.

• Additivity of Information: The total information from independent outcomes must be additive, i.e., the information of two events combined should equal the sum of the information contained in each individual event.

Discrete Memoryless Channels (DMCs)

A DMC is statistically defined as:

• A communication channel characterized by discrete input (X) and output (Y) with memoryless properties, where the output depends solely on the present input, independent of past inputs.

Channel Matrix (Transition Matrix)

This matrix organizes transition probabilities for a DMC, illustrating the relationships between different inputs and their corresponding outputs. Each entry describes the likelihood of state transitions under specific conditions.

Joint and Conditional Entropy

• Joint Entropy: This denotes the measure of uncertainty associated with two random variables being analyzed together, reflecting their combined uncertainty.

• Conditional Entropy (H(x|y)): It quantifies the remaining uncertainty regarding the input (X) after the output (Y) is observed. The interplay between H(x|y) and H(y|x) provides insights into how information is exchanged in communication processes.

Shannon’s Theorems on Information Theory

The chapter discusses several pivotal theorems:

• First Theorem: Encompasses the source encoding theorem, explaining optimal encoding strategies for information sources.

• Second Theorem: Describes channel encoding theorem as discussed in Chapter 4, which is critical in understanding how to encode information for transmission over channels effectively.

• Third Theorem: The Shannon-Hartley theorem describes the capacity of an Additive White Gaussian Noise (AWGN) channel, outlining the relationship between bandwidth, signal power, and noise.

Shannon’s Limit and Performance

Different designs can yield varied performance outcomes in real-world scenarios. Shannon's theory is critical to optimizing communication channel performance, detailing the inherent trade-off between information capacity, noise, and processing capabilities of the channel.

Channel Capacity and Communication Systems

Channel capacity, closely related to bandwidth and noise power, determines the maximum data transmission rate over a channel. The efficiency and effectiveness of design choices are directly influenced by:

• Signal-to-Noise Ratio (SNR): Increasing SNR results in enhanced capacity.

• Bandwidth Adjustments: Altering the bandwidth can also increase channel capacity, making it a critical factor in communication system design decisions.

Shannon-Hartley Theorem

This theorem articulates the implications of channel capacity in Gaussian communication channels, emphasizing its dependence on the relationship between overall bandwidth and SNR, providing essential insights for effective communication system design.

Source Coding Techniques

Source coding involves transforming data from a DMS into binary sequences to facilitate efficient communication. Important concepts include:

• Codeword Length: Refers to the length of the binary representation of a symbol.

• Average Codeword Length: The mean length of the codewords used to represent symbols.

• Code Rate: It is calculated as the ratio of entropy to average code length, directly influencing the efficiency of the coding process.

• Kraft Inequality: This principle is essential to ensure the existence of an instantaneous or prefix-free binary code, crucial for lossless data encoding.

Huffman vs. Shannon-Fano Coding

• Huffman Coding: Recognized as an optimal variable-length encoding scheme, Huffman coding achieves greater efficiency than Shannon-Fano by minimizing the code lengths assigned to more frequently used symbols.

• Shannon-Fano Coding: Although foundational and straightforward to implement, it does not always provide optimal encoding results compared to Huffman coding.

Examples and Numerical Exercises

This section presents several numerical exercises to illustrate practical applications, including calculations for channel capacity, code efficiency, and average codeword length. The exercises emphasize the effectiveness of Huffman encoding, particularly in scenarios involving symbol probability distributions and average transmission rates, illustrating its integral role in lossless data compression.

Conclusion

In conclusion, information theory plays a pivotal role in improving communications systems. Understanding the principles of data transfer and effective coding techniques is fundamental for navigating the limits imposed by noise and signal conditions, ensuring optimal data transmission in various applications.

References

• Taub, H. & Schilling, D. (2012). Principles of Communication Systems. Tata McGraw Hill.

• Lathi, B. P. & Ding, Z. (2009). Modern Digital and Analog Communication Systems. Oxford university Press.

• Haykin, S. (2014). Digital Communication Systems. John Wiley and Sons.

• Sklar, B. & Ray, P. K. (2009). Digital Communication: Fundamentals and Applications. Pearson.

Acknowledgements

The content presented in this chapter is derived from lectures and various source materials as cited above, aiming to encapsulate the essential principles of Information Theory applicable to modern communication systems.