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Entropy and Information Theory Aida Austin 4/24/2009.

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Presentation on theme: "Entropy and Information Theory Aida Austin 4/24/2009."— Presentation transcript:

1 Entropy and Information Theory Aida Austin 4/24/2009

2 Overview  What is information theory?  Random variables and entropy  Entropy in information theory  Applications  Compression  Data Transmission

3 Information Theory  Developed in 1948 by Claude E. Shannon at Bell Laboratories  Introduced in “A Mathematical Theory of Communication''  Goal: Efficient transmission of information over a noisy network  Defines fundamental limits on compression needed for reliable data communication Claude E. Shannon

4 Random Variables  A random variable is a function.  Assigns numerical values to all possible outcomes (events)  Example: A fair coin is tossed. Let X represent a random variable. Possible outcomes:

5 Entropy in Information Theory  Entropy is the measure of the average information content missing from a set of data when the value of the random variable is not known.  Helps determine the average number of bits needed for storage or communication of a signal.  As the number of possible outcomes for a random variable increases, entropy increases.  As entropy increases, information decreases  Example: MP3 sampled at 128 kbps has higher entropy rate than 320 kbps MP3

6 Applications  Data Compression  MP3 (lossy)  JPEG (lossy)  ZIP (lossless)  Cryptography  Encryption  Decryption  Signal Transmission Across a Network   Text Message  Cell phone

7 Data Compression  “Shrinks” the size of a signal/file/etc. to reduce cost of storage and transmission  Smaller data size reduces the possible outcomes of the associated random variables, thus decreasing the entropy of the data.  Entropy - minimum number of bits needed to encode with a lossless compression.  Lossless (no data lost) if the rate of compression = entropy rate

8 Signal/Data Transmission  Channel coding reduces bit error and bit loss due to noise in a network.  As entropy increases, the likelihood of valuable information transmitted decreases.  Example: Consider a signal composed of random variables. We may know the probability of certain values being transmitted, but we do not know the exact values will be received unless the transmission rate = entropy rate.

9 Questions?

10 Resources ormation.html

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