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Information Theory and Coding

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Information Theory and Coding Information Theory and Coding Subject Code : 10EC55 IA Marks : 25 No. of Lecture Hrs/Week : 04 Exam Hours : 03 Total no. of Lecture Hrs. : 52 Exam Marks : 100 PART - A Unit – 1: Information Theory: Introduction, Measure of information, Average information content of symbols in long independent sequences, Average information content of symbols in long dependent sequences. Mark-off statistical model for information source, Entropy and information rate of mark-off source. 6 Hours Unit – 2: Source Coding: Encoding of the source output, Shannon‟s encoding algorithm. Communication Channels, Discrete communication channels, Continuous channels. 6 Hours Unit – 3: Fundamental Limits on Performance: Source coding theorem, Huffman coding, Discrete memory less Channels, Mutual information, Channel Capacity. 6 Hours Unit – 4: Channel coding theorem, Differential entropy and mutual information for continuous ensembles, Channel capacity Theorem. 6 Hours PART - B Unit – 5: Introduction to Error Control Coding: Introduction, Types of errors, examples, Types of codes Linear Block Codes: Matrix description, Error detection and correction, Standard arrays and table look up for decoding. 7 Hours Unit – 6: Binary Cycle Codes, Algebraic structures of cyclic codes, Encoding using an (n-k) bit shift register, Syndrome calculation. BCH codes. 7 Hours Unit – 7: RS codes, Golay codes, Shortened cyclic codes, Burst error correcting codes. Burst and Random Error correcting codes. 7 Hours Unit – 8: Convolution Codes, Time domain approach. Transform domain approach. 7Hours Text Books: Digital and analog communication systems, K. Sam Shanmugam, John Wiley, 1996. Digital communication, Simon Haykin, John Wiley, 2003. Reference Books: ITC and Cryptography, Ranjan Bose, TMH, II edition, 2007 Digital Communications - Glover and Grant; Pearson Ed. 2nd Ed 2008 Page 1 INDEX SHEET Sl No. Unit & Topic of Discussion PAGE NO. PART - A 4 1 UNIT – 1: INFORMATION THEORY 2 Introduction 5 3 Measure of information 5 Average information content of symbols in long independent 8 4 Sequences Average information content of symbols in long dependent 9 5 Sequences 6 Mark-off statistical model for information source, 11 7 Entropy and information rate of mark-off source. 19 8 Review questions 27 UNIT – 2 29 9 SOURCE CODING 10 Encoding of the source output 30 11 Shannon‟s encoding algorithm 31 12 Communication Channels 44 13 Discrete communication channels 45 14 Review questions 73 UNIT – 3 74 15 FUNDAMENTAL LIMITS ON PERFORMANCE 16 Source coding theorem 75 17 Huffman coding 75 18 Discrete memory less Channels 81 19 Mutual information 88 20 Channel Capacity 90 21 Review questions 110 22 UNIT – 4 111 23 Continuous Channel 112 Differential entropy and mutual information for continuous 119 24 Ensembles 25 Channel capacity Theorem 121 26 Review questions 129 PART – B 27 UNIT – 5 130 INTRODUCTION TO ERROR CONTROL CODING 28 Introduction 131 29 Types of errors 133 30 Types of codes 133 31 Linear Block Codes: Matrix description. 136 32 Error detection and correction 146 33 Standard arrays and table look up for decoding 149 Page 2 34 Hamming codes 153 35 Review questions 155 36 UNIT – 6 156 37 Binary Cyclic Codes 157 38 Algebraic structures of cyclic codes 167 39 Encoding using an (n-k) bit shift register, 171 40 Syndrome calculation. 178 41 BCH codes 181 42 Review questions 182 43 UNIT – 7 183 44 Introduction 184 45 Golay codes and Shortened cyclic codes 188 46 R S codes 189 47 Burst error correcting codes 189 48 Burst and Random Error correcting codes 191 49 Review questions 196 50 UNIT – 8 197 51 Convolution Codes 198 52 Time domain approach 200 53 Transform domain approach. 206 54 Review questions 216 Page 3 www.allsyllabus.com Information Theory and Coding 10EC55 PART A Unit – 1: Information Theory Syllabus: Introduction, Measure of information, Average information content of symbols in long independent sequences, Average information content of symbols in long dependent sequences. Mark-off statistical model for information source, Entropy and information rate of mark-off source. 6 Hours Text Books: Digital and analog communication systems, K. Sam Shanmugam, John Wiley, 1996. Reference Books: Digital Communications - Glover and Grant; Pearson Ed. 2nd Ed 2008 Page 4 Unit – 1: Information Theory 1.1 Introduction: Communication Communication involves explicitly the transmission of information from one point to another, through a succession of processes. Basic elements to every communication system o Transmitter o Channel and o Receiver Communication System Source User of Transmitter CHANNEL Receiver of information information Message Transmitted Received Estimate of signal Signal signal message signal Information sources are classified as: INFORMATION SOURCE ANALOG DISCRETE Source definition Analog : Emit a continuous – amplitude, continuous – time electrical wave from. Discrete : Emit a sequence of letters of symbols. The output of a discrete information source is a string or sequence of symbols. 1.2 Measure the information: To measure the information content of a message quantitatively, we are required to arrive at an intuitive concept of the amount of information. Consider the following examples: A trip to Mercara (Coorg) in the winter time during evening hours, 1. It is a cold day 2. It is a cloudy day 3. Possible snow flurries Page 5 Amount of information received is obviously different for these messages. o Message (1) Contains very little information since the weather in coorg is „cold‟ for most part of the time during winter season. o The forecast of „cloudy day‟ contains more informat ion, since it is not an event that occurs often. o In contrast, the forecast of „snow flurries‟ convey s even more information, since the occurrence of snow in coorg is a rare event. On an intuitive basis, then with a knowledge of the occurrence of an event, what can be said about the amount of information conveyed? It is related to the probability of occurrence of the event. What do you conclude from the above example with regard to quantity of information? Message associated with an event „least likely to occur‟ contains most information. The information content of a message can be expressed quantitatively as follows: The above concepts can now be formed interns of probabilities as follows: Say that, an information source emits one of „q‟ po ssible messages m1, m2 …… m q with p1, p2 …… pq as their probs. of occurrence. th Based on the above intusion, the information content of the k message, can be written as 1 I (mk) p k Also to satisfy the intuitive concept, of information. I (mk) must zero as pk 1 Therefore, I (mk) > I (mj); if pk < pj I (mk) O (mj); if pk 1 -------------- I I (mk) ≥ O; when O < pk < 1 Another requirement is that when two independent messages are received, the total information content is – Sum of the information conveyed by each of the messages. Thus, we have I (mk & mq) I (mk & mq) = Imk + Imq ------ I We can define a measure of information as – 1 I (m ) = log ------ III k p k Page 6 10EC55 Unit of information measure Base of the logarithm will determine the unit assigned to the information content. Natural logarithm base : „nat‟ Base - 10 : Hartley / decit Base - 2 : bit Use of binary digit as the unit of information? Is based on the fact that if two possible binary digits occur with equal proby (p1 = p2 = ½) then the correct identification of the binary digit conveys an amount of information. I (m1) = I (m2) = – log 2 (½ ) = 1 bit One bit is the amount if information that we gain when one of two possible and equally likely events occurs. Illustrative Example A source puts out one of five possible messages during each message interval. The probs. of 1 1 1 1 1 these messages are p1 = ; p2 = ; p1 = : p1 = , p5 2 4 4 16 16 What is the information content of these messages? 1 I (m1) = - log2 = 1 bit 2 1 I (m2) = - log2 = 2 bits 4 1 I (m3) = - log = 3 bits 8 1 I (m4) = - log2 = 4 bits 16 1 I (m5) = - log2 = 4 bits 16 HW: Calculate I for the above messages in nats and Hartley Page 7 Digital Communication System: Estimate of the Source of Message signal Message signal User of information information Source Source Receiver Transmitter Encoder decoder Source Estimate of code word source codeword Channel Channel Encoder decoder Channel Estimate of code word channel codeword Modulator Demodulator Received Wavefo rm Channel signal Entropy and rate of Information of an Information Source / Model of a Mark off Source 1.3 Average Information Content of Symbols in Long Independence Sequences Suppose that a source is emitting one of M possible symbols s0, s1 ….. s M in a statically independent sequence Let p1, p2, …….. p M be the problems of occurrence of the M-symbols resply. suppose further that during a long period of transmission a sequence of N symbols have been generated. On an average – s 1 will occur NP1 times S will occur NP times 2 2 : : si will occur NPi times 1 The information content of the i th symbol is I (si) = log p bits i PiN occurrences of si contributes an information content of 1 i i i p i bitsPN.I(s)=PN.log Total information content of the message is = Sum of the contribution due to each of Page 8 M symbols of the source alphabet M 1 total NP 1 log bits i.e., I = ∑ p i 1 i I Averageinforamtion content total M 1 bits per H = ∑NP1 log ---- IV p per symbol in given by N i 1 i symbol This is equation used by Shannon Average information content per symbol is also called the source entropy. 1.4 The average information associated with an extremely unlikely message, with an extremely likely message and the dependence of H on the probabilities of messages consider the situation where you have just two messages of probs.
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