1. Classical Coding for Forward Error Correction Prof JA Ritcey Univ of Washington

2. Parity Check Codes Coding and Modulation are separate Additional bits are added to data stream provides checks for Error Detection and Error Correction Goal is to increase distance yet maintain reasonable decoding complexity

3. System Block Diagram

4. Block Codes Can be binary or non-binary symbols Field for arithmetic is required GF(2) usual binary field where we can add, subtract multiply and divide and have 0 and 1. Addition is XOR, Multiplication is AND Bitwise operations Can be extended to other Finite Fields

5. Example of GF(2)

6. Hamming Distance Given 2 vectors x & y, HD(x,y) = # of places in which they disagree HD(x,y) = Wgt(x+y), recall x+y = x-y in GF(2) Wgt = # of non-zero elements For the BSC(p), HD plays the role of Euclidean distance |x-y|

7. Linear Encoding Info bits u 1xk row k-tuple Channel bits x 1xn row n-tuple R_c = k/n<1 Code Rate Linear x = uG, Generator matrix G Forms a code word x. A code is collection of all codewords |C| = 2^k points our of 2^n n-tuples

8. Systematic Encoding Codeword explicitly lists k info bits G = [ I | P ] is a partitioned matrix kxn = kxk kx(n-k), Here x is “cross” (row x col) (6,3) example

9. Parity Check matrix H = [ P’| I’]’ using transpose Output y = x+e This is the bitwise BSC(p)

11. Maximum Likelihood Decoding Channel Dependent – BSC(p), p<1/2 Intuition – Decoder looks for a nearby codeword to y = x+e = codeword + error sequence Bits are flipped rarely, since p<1/2

13. Hamming Codes k = 2^m –m+1 n = 2^m -1, n-k =m Dmin = 3 (also min weightcodeword) Can correct any single bit error As m increases, code rate k/n improves Perfect code –all n-tuples in a decoding sphere

14. Richard Hamming Bell Labs

15. Hamming con’t

16. Single Error Correcting

17. Syndrome decoding with Std Array

20. Standard Array Decoding

21. Error Correcting Capabilty

22. Bounded Distance Decoder Codeword Error Prob

25. Asymptotic coding gain Competition between Code Rate R = k/n and Error Correcting Capability Here t = Floor (d_min -1)/2 Can correct any t-bit error pattern

26. Code Hamming Distance

27. BER Curves BER curves

28. Trellis Codes Convolutional Codes Trellis Coded Modulation TCM Bit Interleaved Coded modulation BICM Turbo Codes – concatenated trellis codes Commonly employed in wireless standards May use an outer block code (often Reed Solomon) for Burst error correction

29. Trellis Codes First application of the Viterbi Algorithm Dynamic programming applied to decoding Converts decoding to a shortest path problem Can easily incorporate soft decision (multi-bit quantization) information. Complexity rises with constraint length

30. Convolutional Code

31. Trellis

32. State Transition Diagram