Introduction to Information Theory

Questions and Answers

What is the purpose of Shannon's entropy function?

To eliminate errors in data transmission.

To measure information in terms of bits. (correct)

To store information without compression.

To simplify source coding techniques.

How many bits does Shannon suggest are needed on average to store each outcome from tossing a dice ?

3 bits

8 bits

2.585 bits (correct)

1.5 bits

What is the size of the file when 1,000,000 dice outcomes are stored using ASCII representation?

2,585,000 bits

3,000,000 bits

1,000,000 bits

8,000,000 bits (correct)

What is the compression ratio achieved using Shannon's method when compressing a file originally 8,000,000 bits?

<p>32.31%</p> Signup and view all the answers

What do computer programs like Winzip and WinRAR achieve compared to Shannon's original compression method?

<p>Compression ratios lower than 32.31%.</p> Signup and view all the answers

Study Notes

Shannon Theory Overview

Established in 1948, foundational framework for Information Theory.
Consists of three main components: measurement of information, source coding theory, and channel coding theory.

Measurement of Information

Key inquiry: determining how to measure information in bits.
Events are inherently probabilistic, leading to the conclusion that the entropy function serves as the definitive measure of information.

Entropy Function Example

Case study: Tossing a die yields 6 outcomes (1, 2, 3, 4, 5, 6).
Each outcome has an equal probability (1/6).
The resulting information quantity (2.585 bits) is not an integer, highlighting the abstract nature of information measurement.

Source Coding Theorem

Shannon's assertion: to reliably store information from a random source X, an average of H(X) bits are needed for each outcome.
Example: After 1,000,000 dice tosses, storing outcomes with a minimum of 3 bits each totals 3,000,000 bits.
ASCII requires 8 bits (1 byte) for each outcome, leading to a total of 8,000,000 bits for data storage.

Compression Insights

Shannon's contribution suggests only 2.585 bits are necessary for data storage per outcome, allowing for significant compression to 2,585,000 bits for 1,000,000 outcomes.
Comparison of compression ratios reveals the efficiency of Shannon's optimal approach.

Compression Ratio Statistics

Original file size: 8,000,000 bits (100%).
Shannon's compressed file size: 2,585,000 bits (32.31% compression).
Winzip compression: 2,930,736 bits (36.63% compression).
WinRAR compression: 2,859,336 bits (35.74% compression).

Conclusion

Shannon's theoretical framework demonstrated substantial potential for data compression, validated mathematically over half a century ago.

Description

Explore the foundational concepts of Information Theory as introduced by Claude Shannon in 1948. This quiz covers measurement of information, source coding theory, and channel coding theory. Test your understanding of how information is quantified in bits and the probabilistic nature of events.