Information Entropy in Information Theory

Information entropy, a key concept in information theory, measures the average amount of information in a message. Source entropy and binary source entropy are explained with examples, along with maximum source entropy for both binary and non-binary sources. Learn how to calculate entropy for different sources and grasp the concept of uncertainty in information transmission.

• Information Theory
• Entropy
• Source Entropy
• Binary Source
• Uncertainty

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1. Al-Mustaqbal University College Department of Computer Engineering Techniques Information Theory and coding Fourth stage By: MSC. Ridhab Sami

2. Lecture 5 Average information (entropy)

3. Average information (entropy): In information theory, entropy is the average amount of information contained in each message received. 1. Source Entropy: If the source produces not equal probability messages then ?(?? ),?= 1, 2, 3, n are different. Then the statistical average of ?(?? ) over i will give the average amount of uncertainty associated with source X. This average is called source entropy and denoted by ?(?), given by: ? ) ? ? = ? ?? ? ?? ? ?? = ?????(?? ?=1 ? ?? ? ?????? ? ? = ? ?? ????? ?? ?=1

4. Example: Find the entropy of the source producing the following messages: ??1 = 0.25, ??2 = 0.1, ??3 = 0.15, ?????4 = 0.5 Solution: 4 ? ? = ? ?? ???2? ?? ?=1 H ? = 0.25???20.25 + 0.1???20.1 + 0.15???20.15 + 0.5???20.5 0.25??0.25 + 0.1??0.1 + 0.15??0.15 + 0.5??0.5 ??2 = ?(?) = 1.742 ?? ? ??????

5. 2. Binary Source entropy: In information theory, the binary entropy function, denoted or H(X) or Hb(X), is defined as the entropy of a Bernoulli process with probability p of one of two values. Mathematically, the Bernoulli trial is modeled as a random variable X that can take on only two values: 0 and 1: ?(0) + ?(1) = 1 ? We have: ? ? = ? ?? ????? ?? ?=1 2 ??? = ? ?? ????? ?? ?=1 Then ??? = ? 0 ???2? 0 + ? 1 ???2? 1 ?? ? ??????

6. Example: Example: Find the entropy for binary source if P(0)=0.2. Solution: Solution: P(1) = 1-P(0) = 1-0.2 =0.8 Then 2 ??? = ? ?? ????? ?? ?=1 ??? = 0.2???20.2 + 0.8???20.8 0.2??0.2 + 0.8??0.8 ??2 = = 0.7 ?? ? ??????

7. 3. Maximum Source Entropy: For binary source, if?(0) = ?(1) = 0.5, , then the entropy is: ??? = 0.5???20.5 + 0.5???20.5 1 2 = ???2 = ???22 = 1 ???

8. 3. Maximum Source Entropy: For any non-binary source, if all messages are equiprobable ?(??) = 1/? Then 1 ????? 1 ? so that: ) ? ? = ?(????= ? 1 ? = ???? ) ?(????= ????? ????/?????? Which is the maximum value of source entropy. Also, ?(?) = 0 if one of the message has the probability of a certain event or p(x) =1.

9. Example: A source emits 8 characters with equal probability, Find the max entropy ?(?)???. Solution: n=8 ) ?(????= ???2? = ???28 = 3 ?? ? ??????

10. 4. Source Entropy Rate: It is the average rate of amount of information produced per second. ?(x) = ?(x) ???? ?? ????????? ? ? ??????? = bits/sec =bps The unit of H(X) is bits/symbol and the rate of producing the symbols is symbol/sec, so that the unit of R(X) is bits/sec. ) ?(? ? ? ? = Where ? ? = ??? ?? ?=1 ? is the average time duration of symbols, ?? is the time duration of the symbol ??.

11. Example : A source produces dots . And dashes - with P(dot)=0.65. If the time duration of dot is 200ms and that for a dash is 800ms. Find the average source entropy rate. Solution: ) ?(? ? ?(??? ) = 1 ?(???) = 1 0.65 = 0.35 ? ? = ? ? ? = ? ?? ????? ?? ?=1 ? ? = 0.65???20.65 + 0.35???20.35 =0.934 bits/symbol ? ? = ??? ?? ?=1 200 1000 =0.2 sec ? = 0.2 0.65 + 0.8 0.35 = 0.41 ??c ) ?(? ? 0.934 0.41= 2.278 ??? ? ? = =

12. Example :A discrete source emits one of five symbols once every millisecond. The symbol probabilities are 1/2, 1/4, 1/8, 1/16 and 1/16 respectively. Calculate the information rate. Solution: 5 ? ? = ? ?? ????? ?? ?=1 1 2log2 1 2+1 1 4+1 1 8+ 1 1 1 1 = 4log2 8???2 16???2 16+ 16???2 16 1 2log22 +1 4log24 +1 1 1 = 8???28 + 16???216 + 16???216 = 0.5 + 0.5 + 0.375 + 0.25 + 0.25 = 1.875 ?? ? ?????? ) ?(? ? 1.875 10 3= 1.875 ???? ? ? = =