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Cours8 - Channel coding and error correction in digital...

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Channel coding and error correction in digital transmission Prof. L. Vandendorpe UCL Communications and Remote Sensing Lab. 1
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Outline Basics of error correcting codes Block and convolutional codes ML hard and soft decoding; the Viterbi algorithm 2
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Basics of error correcting codes Error correcting codes improve system performance Emphasis not on code design but on receiver structure FEC: Forward Error Correction 3
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Basics of error correcting codes: assumption Classical codes: perform well when channel errors are indepen- dent Assumption: Interleaving makes the channel memoryless Burst of errors on the channel does not appear as a burst at the decoder input 4
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Basics of error correcting codes: system model 5
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Information required To compute the FEC peformance, one needs DMC (discrete memoryless channel) transition probabilities p [ y j | x j ,z j ] (1) y j decoder input at channel use j x j coder output or interleaver input at channel use j z j channel state information j 6
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Information required DMC inputs and outputs may belong to different alphabets Example: output may be ”0”, ”1”, or ”don’t know” (informa- tion abour reliability) Output may also be continuous When additional information wrt ”0” or ”1” is output: soft decision channel May provide Channel state information (CSI) (says if jammer active) Information on jammer on/off or even amplitude 7
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Elementary linear block code concepts Binary input and output assumed k input bits, n output bits; code rate k/n ; redundancy n k bits Hence 2 k possible codewords out off 2 n One-to-one mapping between input message ( k bits) and code- word Error correction capability: not all n -uples are possible Choice: a number of errors must occur to confuse a codeword with another 8
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Distance Codewords differing in d H positions are at Hamming distance d H For two such codewords d H errors may change x 1 into x 2 Minimum distance for all pairs of codewords: minimum distance of the code d min For a linear code: minimum weight of a non-zero codeword 9
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Properties of linear block codes Modulo-2 sum of 2 codewords is another codeword All zeros codeword is a codeword Take x a , x b at d H . Take any x c . Let x a = x a x c , x b = x b x c . Then d [ x a , x b ] = d H 10
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Example of weight distribution k = 4 , n = 7 , A 3 = 7 , A 4 = 7 , A 7 = 1 11
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Channel description Described by p [ y | x m , z ] where r = ( r 1 , · · · ,r n ) For a memoryless channel p [ y | x m , z ] = n j =1 p [ y j | x mj ,z j ] (2) If no jammer information available, p [ y | x m ] = n j =1 p [ y j | x mj ] (3) 12
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