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Source Coding

Source Coding. Source Coding. The purpose of source coding is to reduce the number of bits required to convey the information provided by the information source. For the file of the source data this means compression For the transmission, this is a bit rate reduction. Classification.

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Source Coding

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  1. Source Coding Dr.E.Regentova

  2. Source Coding The purpose of source coding is to reduce the number of bits required to convey the information provided by the information source. • For the file of the source data this means compression • For the transmission, this is a bit rate reduction Dr.E.Regentova

  3. Classification • There are two broad categories of data compression • Lossless compression, such as zip, gzip, gif • Lossy compression, such as mpeg, jpeg .. Dr.E.Regentova

  4. Lossless Compression • An exact copy of the original data is obtained after decompression • Structured data can be compressed to 40-60% of their original size Dr.E.Regentova

  5. Lossy Compression • Original information content is lost • Any data can be compressed significantly (e.g., 100 times) Dr.E.Regentova

  6. What is information The color of the hat is red The color of the table is red The color of the lamp is red The book, hat, and a lamp are red Dr.E.Regentova

  7. INFORMATION THEORY The area of information theory explores the information content of data We can predict the size of the content by modeling data Within a given model we can obtain lower bounds on the number of bits required to represent data Dr.E.Regentova

  8. Modeling-1 Consider the following message of x1, x2,….x12 9,11,11,11,14,13,15,17,16,17,20,21 Using binary encoding scheme, we can store each of these numbers using 5 bits per symbol Dr.E.Regentova

  9. Modeling Information-2 Original message 9,11,11,11,14,13,15,17,16,17,20,21 We can store instead the following 0,2,2,2,5,4,6,8,7,8,11,12 Each of these values represents the number n+9, thus, we can use 4 bits per number . Dr.E.Regentova

  10. Modeling Information-3 Original: 9,11,11,11,14,13,15,17,16,17,20,21 Or, we can store instead the following 9,2,0,0,3,-1,2,2,-1,2,2,-1,1,3,1 The first number is stored as is. To encode xi, we use xi –x i-1. There are only 5 distinct values which can be encoded with 3 bits each Dr.E.Regentova

  11. Probabilistic Modeling-1 An ensemble ‘X’ is a random variable x with a set of possible outcomes Ax ={a1,a2,…. aI}, having probabilities {p1,p2,…pI}, with P(x=aI)=Pi, Pi>=0 and ΣxєAx P(x)=1. A is called Alphabet Dr.E.Regentova

  12. Probabilistic Modeling-2 Source outputs alphabet {1,2,3,4}. Each sample is modeled as a random variable with the probabilities P(1)= 8/15 P(2) = 4/15 P(3)= 2/15 P(4) = 1/15 Dr.E.Regentova

  13. Probabilistic Modeling-3 If no compression takes place, then the simple code with the average 2 bit/symbol is as follows • 1  00 • 2  01 •  10 •  11 Dr.E.Regentova

  14. Problem: Suppose the channel capacity is 1.75 b/sec Dr.E.Regentova

  15. Variable Length Code-1 Intuition: Less bits for higher probability symbols P(1)= 8/15 P(2) = 4/15 P(3)= 2/15 P(4) = 1/15 • 1  0 • 2  10 •  110 •  111 Dr.E.Regentova

  16. Variable Length Code-2 The expected number of bits per code sample (average length l ) is 1*p(1) + 2*p(2) +3*p(3)+3*p(3)= 5/3 That is 1.67. Thus, the channel will transmit the code perfectly. Dr.E.Regentova

  17. Measure of Information Shannon defined a quantity: self-information associated with a symbol in a message. The self-information of a symbol a is given by the formula Dr.E.Regentova

  18. Intuition The higher the probability that a symbol occurs the lower the information content. At the extreme if P(a)=1, then there is nothing learned when receiving an a, since that is the only possibility. Dr.E.Regentova

  19. Entropy Shannon also defined a quantity: ENTROPY associated with a message, representing the average self information per symbol in the message expressed in radix b Dr.E.Regentova

  20. Information Entropy is a measure of information (b/symbol) that gives a low bound for coding. Dr.E.Regentova

  21. Entropy Coding-1 The noiseless source coding theorem (also called Shannon's first theorem) states that an instantaneous code can be found that encodes a source of entropy H(x) with an average number of bits per symbol B such that B=H(x) Assumption: all symbols are independent and identically distributed (iid) Dr.E.Regentova

  22. Entropy Coding-3 P(1)= 8/15 P(2) = 4/15 P(3)= 2/15 P(4) = 1/15 H(x) = -8/15 *log(8/15)-4/15* log(4/15) – 2/15*log(2/15) –1/15 *log(1/15) = 1.5b/symbol Dr.E.Regentova

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