Channel Coding â Exercise I

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Exercise 4 Modification of linear block codes A (5, 4) linear block code is constructed by adapting a (7, 4) block code. The (7, 4) code can be described by the following generator matrix ⎛ ⎞ 0 1 0 1 1 0 0 G = ⎜ 1 0 1 0 1 0 0 ⎟ ⎝ 0 1 1 0 0 1 0 ⎠ . 1 1 0 0 0 0 1 a) Construct the codewords of the (5, 4) code and list them! b) What is the minimum distance of the (5, 4) code! Exercise 5 Syndrome decoding a) State the number of syndromes of the (7, 4, 3)-Hamming code and compare it with the number of correctable error patterns. b) The word d = (1 1 0 1 0 0 1) is found at the receiver. Which information word i was sent with the greatest probability? Exercise 6 Coding program Write a matlab-programm which codes and again decodes a certain number of input data bits. Besides it shall be possible to insert errors before the decoding. The (5, 2, 3)- Hamming code shall be used. Hint: The (5, 2, 3)-Hamming code maps m = 2 information symbols onto n = 5 code symbols. Within the matlab-program, m information symbols are randomly chosen using randint and encoded by the generator matrix. A randomly determined error vector is added and the syndrome is calculated. Please notice: all calculations have to be executed within GF (2).
Channel Coding – Exercise II Hochschule Wismar, University of Technology, Business and Design, Prof. Dr.-Ing. habil. A. Ahrens Baltic Summer School 2008, Tartu, Estonia, 08.08.–23.08.2008 Exercise 7 Convolution code Given is a convolution code with the code rate R = 1/3, memory (K − 1) = 2 and the generator polynomials g 1 (x) = 1 + x + x 2 and g 2 (x) = 1 + x 2 and g 3 (x) = 1 + x + x 2 . a) Determine the output sequence for the input sequence i = (0 1 1 0 1 0). b) Sketch the corresponding Trellis chart in the case of the under a) given input sequence. c) Sketch the state diagram of the encoder. d) Determine the corresponding free distance h free . Exercise 8 Catastrophic codes Given is the convolution code with the generator polynomial g(x) = (1 + x 2 , 1 + x). Show that this is a catastrophic code and explain the consequences. Exercise 9 Puncturing of convolution codes of the rate 1/n Given is the non-recursive convolution code with the generators g 1 (x) = 1 + x + x 3 and g 2 (x) = 1 + x + x 2 + x 3 . The non-recursive convolution code shall now be punctured to the code rate R = 2/3. Give several puncturing patterns and find out, if the puncturing results in a catastrophic code.
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Channel Coding â Exercise I