lecture19

# lecture19 - EE455/591 MoreAdvance C d hanne C l oding More...

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EE455/591 EE455/591 More Advanced Channel Coding More Advanced Channel Coding Reading: Read Section 9.7 pages 623-625, 630-634 Reading: Read Section 9.7 pages 623-625, 630-634 Review: Hamming Codes Review: Hamming Codes Topics: Topics: 1. 1. Extended Hamming Codes Extended Hamming Codes 2. 2. Convolutional Codes Convolutional Codes 3. 3. Viterbi Decoding Viterbi Decoding

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Review: Generation of Hamming Code = = 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 u uG x Hamming (2 n -1, 2 n –n-1) code: x=uG=u[I A]; codes are in the vector space of G A has 2 n –n-1 rows of n bits The rows of A are the 2 n possible n-bit sequences, minus the n sequences with only 1 one, minus the all-0 sequence
Review: Decoding Hamming Code Decoding Hamming code with corrupted y=x+e : Syndrome s=yH points out where the single error in e is Parity matrix H=[A, I]’ so that GH=[I A][A I]’=A+A=0 xH=0; i.e. codes are in the null space of H Rows of H are all possible 2 n non-zero n-bit sequences Correcting single error (1 in i-th location of e): eH picks up i-th row of H Now the rows are unique Syndrome s=eH indicates error location OK to permute rows of H = = = × × 3 3 3 4 1 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 I A yH s

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1. Codebook for Extended Hamming Code Info Code x=uG Info Code x=uG Info Code x=uG Info Code x=uG 0000 0000000 0 (0) 0100 0100101 1 (4) 1000 1000011 1 (4) 1100 1100110 0 (4) 0001 0001111 0 (4) 0101 0101010 1 (4) 1001 1001100 1 (4) 1101 1101001
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lecture19 - EE455/591 MoreAdvance C d hanne C l oding More...

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