1. Institute for Experimental Mathematics Ellernstrasse 29 45326 Essen - Germany Coding for a Terrible Channel A.J. Han Vinck July 3, 2005 COST 289

2. University Duisburg-Essendigital communications group 2 Content Motivation –Impulsive noise (broadband noise) –Broadcasters (narrowband noise) –Background noise –Frequency selective fading –High attenuation Permutation codes Block permutation code Convolutional permutation code Reed Solomon codes

3. University Duisburg-Essendigital communications group 3 Example: Power Line channel Maximum amplitude < 5 Volt ( CENELEC) –Use constant envelope modulation M-FSK ( or clipped OFDM) Impulsive noise (broad band) –Width < 100  Sec; interval average 0.1-1 sec. Permanent disturbances (narrow band) –Television sets, PC, broadcasters Back ground noise –Power Density function: Log 10 N(f) = K – 4*10 -5 f Frequency selective fading –Channel impedance mismatch High attenuation –< 100 dB/Km A.J. Han Vinck, “ Coded Modulation for PLC,” AEU, Vol. 2000, pp. 45-49

4. University Duisburg-Essendigital communications group 4 communication overview encoder K M-ary symbols N M-ary symbols M-FSK modulator non- coherent detection N x M envelopes decoder Message estimate of length K constant envelope Permutation codes Convolutional-permutation codes Reed Solomon Codes

5. University Duisburg-Essendigital communications group 5 Encoder: use permutation code Transmit messages as:  sequences (code words) of length M where all M symbols are different  minimum distance (# of differences) D p Example: M = 3 D p = 2 Code: 123 312 231 132 321 213 time f

6. University Duisburg-Essendigital communications group 6 Non-coherent detection with thresholds filter matched to f 1 Envelope detector y1y1 filter matched to f 2 Envelope detector y2y2 filter matched to f M Envelope detector yMyM  Quantize > Th = 1 < Th = 0 sample  1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 DetectPresence of code sequence X transmit ? ?

7. University Duisburg-Essendigital communications group 7 Block detection matrix structure 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 3 4 1 3 1 1 0 2 2 3 4 4 4 2 2 10 3 2 11 3 9 9 2 4 6 1 0 1 1 7 4 hard soft softer softest select above ranking calculate largest threshold T likelihoods 0 0 1 1 1 1 0 0 Needs full channel knowledge T=.6E 1/2 “like adding received energy” complexity

8. University Duisburg-Essendigital communications group 8 Effect of noise (simplified) Transmitted Background-insertion Background-deletion Impulsive-broadband narrowband-jammer frequency selective fading

9. University Duisburg-Essendigital communications group 9 Performance Permutation codewords: –M slots; M different symbols; minimum distance D p Error events agree with codewords in  1 position Hence:  D p events can create an additional codeword and a detection error may occur

10. University Duisburg-Essendigital communications group 10 Performance AWGN

11. University Duisburg-Essendigital communications group 11 Coding gain Coding gain (soft) for AWGN only: 0 3 6 dB P e 10 -1 10 -2 10 -3 uncoded coded M = 4, D p = 4 3 dB !

12. University Duisburg-Essendigital communications group 12 Simulation results: PLC 1 10 -2 10 -4 600m800m1000m Impulse + jammer + background Impulse + background Coded D = 3 Uncoded Pe distance

13. University Duisburg-Essendigital communications group 13 Code parameters Upperbound on cardinality of the code Q1: when do we achieve equality? Q2: if not, what is the upperbound References: - Ian Blake, Permutation codes for discrete channels (1975, IT) - P. Frankl and M. Deza, On the max. # of Permutations with given Max. Or Min. Distance (1977, Jrnl of Comb. Th.) - T. Klöve: |C| = 18 for M = 6, D = 5 instead of 30

14. University Duisburg-Essendigital communications group 14 From block to convolutional codes like coded modulation! Advantages: lower complexity decoding lower decoding error probability

15. University Duisburg-Essendigital communications group 15 Permutation convolutional codes mapping 0 1 213 D free ( permutation conv. code) = 5 + 3 = 8 IDEA: convert binary output to permutation codewords keep ( or increase) distanceif possible!

16. University Duisburg-Essendigital communications group 16 Example of the mapping (a) Original code (b) permutation trellis code.

17. University Duisburg-Essendigital communications group 17 Reason why conv. code outputpermutation code word 00 01 10 11231 213 132 123 00 0 1 1 2 231 0 2 2 3 01 1 0 2 1 213 2 0 3 2 10 1 2 0 1 132 2 3 0 2 11 2 1 1 0 123 3 2 2 0 Distance tables +1 per branch!

18. University Duisburg-Essendigital communications group 18 Mappings, M = 4 0000, 0001, 0010, 0011 1234, 1243, 1342, 1342 0100, 0101, 0110, 0111 1423, 1432, 2134, 2143 . 1000, 1001, 1010, 1011 3214, 3241, 2314, 2341 1100, 1101, 1110, 1111 3421, 3412, 3124, 3142 n = 4, distance conserving { 000,001,010,011}  { 4231,4213,4132,4123 } {100,101,110,111 }  { 1234,1243,1432,1423 } n = 3, distance increasing

19. University Duisburg-Essendigital communications group 19 Problems worked on Construct mappings: n bits to codewords from permutation code with at least the same distance or with distance increasing mappings Construct permutation convolutional codes increase thefree distance H.C. Ferreira and A.J. Han Vinck, Permutation Trellis Codes, to be published, IEEE Tr. on Comms.

20. University Duisburg-Essendigital communications group 20 Narrowband + background noise

21. University Duisburg-Essendigital communications group 21 Impulsive Free distance = 8

22. University Duisburg-Essendigital communications group 22 Encoder: use Reed Solomon code n 111... 1aa 2... k+1... = G c = x G 1a k a 2k... Property: minimum difference between codewords D = N – k maximum number of agreements = N-D = N – (N – k) = k THUS: THUS: since  ( 1, 1,...,1) is a codeword for all  other codewords do not have more than k symbols of type 

23. University Duisburg-Essendigital communications group 23 Channel disturbances (1) Envelope detector output HHHHHHHH HHHHHHHH ALL outputs HIGH for some time: these are ERASURES for the RS code HLLLHLLL non- coherent detection Random errors occur H  L and L  H O( p 2 ) Erasures H  L or L  H O( p ) HLLHHLLH LLLLLLLL LLHLLLHL non- coherent detection erasure error Impulsive noise background noise

24. University Duisburg-Essendigital communications group 24 Threshold detection: AWGN, RS(15,3) 10 -1 10 -2 10 -3 10 -4 SNR 25811 uncoded threshold Viterbi select best Prob. Symbol Error

25. University Duisburg-Essendigital communications group 25 Threshold detection: AWGN, RS(15,3) + impulsive noise ( av. 3 symbols) 10 -1 10 -2 10 -3 10 -4 SNR 25811 uncoded threshold Viterbi select best (no alternatives) errors Prob. Symbol Error AWGN- only D = 13 erasures

26. University Duisburg-Essendigital communications group 26 Channel disturbances (2) LLHLLLHL LLLLLLLL LHLLLHLL Output LOW for long time non- coherent detection Effect: causes erasures # erasures depends on # hits HLHLHLHL HLLLHLLL HHLLHHLL output HIGH for long time non- coherent detection Effect: causes many erasures narrowband noise freq. sel. fading

27. University Duisburg-Essendigital communications group 27 Reed Solomon: avoid constant symbol codeword 111... mother code 1aa 2... k+1... 1a k a 2k... subcode The sub-code does not contain the constant symbol codewords ( except for 0 ) Properties: Minimum distance N-k+1 maximum # of symbols of same type = k at least N / k different symbols in a codeword (  0)

28. University Duisburg-Essendigital communications group 28 IDEA Detection strategy N=7, k = 2, D = 6; narrowband noise/fading Detect: reset: H H H H H H H L L L L L L L L L L L L L LL L L L L L L L L L L H H L L L L L L L L L L L L L L H L L L L L L L Result: maximum of 4 erasures General: D = N - k + 1 > k * # narrowband disturbances 6 = 7 – 2 + 1 > 2 * 2 = 4

29. University Duisburg-Essendigital communications group 29 Threshold detection: modified RS(15,9) + narrow band noise 10 -1 10 -2 10 -3 10 -4 SNR 25811 uncoded AWGN only Prob. Symbol Error Modified RS No side info + side info + constant codewords

30. University Duisburg-Essendigital communications group 30 Threshold detection: modified RS(15,3) + narrow band noise 10 -1 10 -2 10 -3 10 -4 SNR 25811 uncoded + constant codewords + side info Prob. Symbol Error AWGN only modified RS no side info

31. University Duisburg-Essendigital communications group 31 Overall result Modified RS D > # narrowband noise errors (row insertion H) + # fading errors ( row deletion L ) + # impulsive noise errors (symbol insertion H) + # insertion/deletion errors (background noise) ERASURES !

32. University Duisburg-Essendigital communications group 32 Reed Solomon: avoid certain output symbols information Precoder avoid symbols from A M-ary RS- code avoid symbols from A |A| symbols from A forbidden Idea: Control the M-ary RS-output with r control bits r control sbls k info sblsn code sbls M- FSK G. Solomon, „A Note on Alphabet Codes and Fields of Computation,“ Inf. and Control, 1974, pp. 395-398

33. University Duisburg-Essendigital communications group 33 Reed Solomon: avoid certain output symbols M-ary RS Code in systematic form: ( info, control ) I k 0 P = C k r 0 I r n - forbidden set of output symbols A, cardinality |A| - information precoded ( in A c ) - control such that symbols in A do not occur in C - PERFORMANCE?

34. University Duisburg-Essendigital communications group 34 performance ( info, control ) I k 0 P = C k r 0 I r k+r n-(k+r) possible if # possible control vectors = (M-|A|) r > # forbidden vectors in checkpart = ( n – (k+r)) |A |(M-|A|) r-1 Or |A| < (M-|A|)/(n-(k+r)) For |A| = 1, r = 1,  M = n + 1 > n - k, which is true for all RS codes

35. University Duisburg-Essendigital communications group 35 CONCLUSIONS – Application of permutation codes for „terrible“ channels – Extend: block to convolutional – Constructions and simulations – Reed Solomon codes with restricted output

36. University Duisburg-Essendigital communications group 36 Envelope detector with threshold Example: transmit detect 1 error 0 f 3 0 0 00010001 01000100 non- coherent detection f 4 f 2 f 3 00100010 00110011 00000000 erasure output M = 4