Channel Coding I Exercises â WS 2012/2013 â - UniversitÃ¤t Bremen

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November 7, 2012 I Exercises 1 Introduction 1 Exercise 1.1: Design of a discrete channel . . . . . . . . . . . . . . . . . . . . . . . . 1 Exercise 1.2: Statistics of the discrete channel . . . . . . . . . . . . . . . . . . . . . . 1 Exercise 1.3: Binary symmetric channel (BSC) . . . . . . . . . . . . . . . . . . . . . 1 Exercise 1.4: Serial concatenation of two BSC . . . . . . . . . . . . . . . . . . . . . . 1 Exercise 1.5: Transmission of a coded data over BSCs . . . . . . . . . . . . . . . . . . 2 2 Information theory 3 Exercise 2.1: Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Exercise 2.2: Channel capacity of a discrete memory-free channel . . . . . . . . . . . 3 Exercise 2.3: Channel capacity of a BSC . . . . . . . . . . . . . . . . . . . . . . . . . 3 Exercise 2.4: Channel capacity of the AWGN . . . . . . . . . . . . . . . . . . . . . . 4 3 Linear block codes 5 3.2 Finite field algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exercise 3.1: 2-out-of-5-code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exercise 3.2: Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Distance properties of block codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exercise 3.3: Error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exercise 3.4: Sphere-packing bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.5 Matrix description of block codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exercise 3.5: Generator and parity check matrices . . . . . . . . . . . . . . . . . . . . 6 Exercise 3.6: Expansion, shortening, and puncturing of codes . . . . . . . . . . . . . . 6 Exercise 3.7: Coset decomposition and syndrome decoding . . . . . . . . . . . . . . . 7 Exercise 3.8: Coding program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.6 Cyclic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exercise 3.9: Polynomial multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exercise 3.10: Polynomial division . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercise 3.11: Generator polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercise 3.12: Syndrome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercise 3.13: Primitive polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercise 3.14: CRC codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.7 Reed-Solomon- and BCH-codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercise 3.15: Reed-Solomon-codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
November 7, 2012 II Exercise 3.16: BCH-codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Exercise 3.17: Bit- and word error curves of BCH-codes . . . . . . . . . . . . . . . . 10 4 Convolution codes 11 4.1 Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Exercise 4.1: Convolution code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Characterization of convolution codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Exercise 4.2: Catastrophic codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Distance properties of convolution codes . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Exercise 4.3: Distance properties of convolution codes . . . . . . . . . . . . . . . . . 11 Exercise 4.4: Comparison of NSC- and RSC-codes . . . . . . . . . . . . . . . . . . . 11 4.4 Puncturing of convolution codes of the rate 1/n . . . . . . . . . . . . . . . . . . . . . . 12 Exercise 4.5: Puncturing of convolution codes* . . . . . . . . . . . . . . . . . . . . . 12 4.5 Optimal decoding with Viterbi-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 13 Exercise 4.6: Viterbi-decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Exercise 4.7: Viterbi-decoding with puncturing . . . . . . . . . . . . . . . . . . . . . 13 Exercise 4.8: Coding and decoding of a RSC code . . . . . . . . . . . . . . . . . . . . 13 Exercise 4.9: Investigation to Viterbi-decoding . . . . . . . . . . . . . . . . . . . . . . 14 Exercise 4.10: Simulation of a convolution-coder and -decoder . . . . . . . . . . . . . 14
Page 1: Channel Coding I Exercises - WS 201
Page 5 and 6: 1 INTRODUCTION November 7, 2012 1 1
Page 7 and 8: 2 INFORMATION THEORY November 7, 20
Page 9 and 10: 3 LINEAR BLOCK CODES November 7, 20
Page 15 and 16: 4 CONVOLUTION CODES November 7, 201
Page 17 and 18: 4 CONVOLUTION CODES November 7, 201

Channel Coding I Exercises â WS 2012/2013 â - UniversitÃ¤t Bremen

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Channel Coding I Exercises â WS 2012/2013 â - UniversitÃ¤t Bremen