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Coding for a Terrible Channel

Coding for a Terrible Channel. A.J. Han Vinck July 3, 2005 COST 289. Content. Motivation Impulsive noise (broadband noise) Broadcasters (narrowband noise) Background noise Frequency selective fading High attenuation Permutation codes Block permutation code

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Coding for a Terrible Channel

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  1. Coding for a Terrible Channel A.J. Han Vinck July 3, 2005 COST 289

  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. 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: Log10N(f) = K – 4*10-5f • 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. communication overview • constant envelope M-FSK modulator K M-ary symbols encoder N M-ary symbols • Permutation codes • Convolutional-permutation codes • Reed Solomon Codes Message estimate of length K non-coherent detection decoder N x M envelopes

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

  6. Non-coherent detectionwith thresholds Envelope detector y1 filter matched to f1 1 Quantize > Th = 1 < Th = 0 Envelope detector y2 X filter matched to f2 0   Envelope detector yM 0 filter matched to fM sample transmit 1 0 0 0 ? ? ? ? Detect Presence of code sequence 0 0 1 0 ? ? ? ? 0 1 0 0 ? ? ? ? 0 0 0 1 ? ? ? ?

  7. Blockdetection matrix structure hard soft softer softest 1 0 0 1 1 0 0 1 1 3 4 1 10 3 2 11 0 0 1 0 1 1 1 0 3 1 1 0 3 9 9 2 0 1 0 0 0 1 0 0 2 2 3 4 4 6 1 0 0 0 0 0 0 0 1 1 4 4 2 2 1 1 7 4 select above ranking calculate largest threshold Tlikelihoods “like adding received energy” 0 0 0 0 T=.6E1/2 Needs full channel knowledge 0 0 0 0 1 1 1 1 0 0 0 0 complexity

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

  9. Performance • Permutation codewords: • M slots; M different symbols; minimum distance Dp • Error events agree with codewords in  1 position Hence:  Dp events can create an additional codeword and a detection error may occur

  10. Performance AWGN

  11. Coding gain • Coding gain (soft) for AWGN only: Pe 10-1 10-2 10-3 M = 4, Dp = 4 3 dB ! uncoded coded 0 3 6 dB

  12. Simulation results: PLC 1 Impulse + jammer + background Pe 10-2 Impulse + background Coded D = 3 Uncoded 10-4 600m 800m 1000m distance

  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. From block to convolutional codeslike coded modulation! Advantages: lower complexity decoding lower decoding error probability

  15. Permutation convolutional codes 0 213 mapping 1 Dfree ( permutation conv. code) = 5 + 3 = 8 IDEA: convert binary output to permutation codewords keep ( or increase) distance if possible!

  16. Example of the mapping (a) Original code (b) permutation trellis code.

  17. Reason why Distance tables conv. code output permutation code word 00 01 10 11 231 213 132 123 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 +1 per branch!

  18. Mappings, M = 4 0000, 0001, 0010, 00111234, 1243, 1342, 1342 0100, 0101, 0110, 01111423, 1432, 2134, 2143 . 1000, 1001, 1010, 10113214, 3241, 2314, 2341 1100, 1101, 1110, 11113421, 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. 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. Narrowband + background noise

  21. Impulsive Free distance = 8

  22. Encoder: use Reed Solomon code n 1 1 1 ... 1 a a2 ... k+1 ... = G c = x G 1 ak a2k ... Property: minimum difference between codewords D = N – k maximum number of agreements = N-D = N – (N – k) = k THUS:since ( 1, 1, ...,1) is a codeword for all  other codewords do not have more than k symbols of type 

  23. Channel disturbances (1) Impulsive noise H H H H H H H H H L L L non-coherent detection Envelope detector output ALL outputsHIGH for some time: these are ERASURESfor the RS code background noise H L L H L L L L L L H L non-coherent detection erasure error Random errors occur H  L and L  H O( p2 ) ErasuresH  L or L  H O( p)

  24. Threshold detection:AWGN, RS(15,3) 10-1 Prob. Symbol Error uncoded 10-2 threshold Viterbi 10-3 select best 10-4 2 5 8 11 SNR

  25. Threshold detection:AWGN, RS(15,3)+impulsive noise ( av. 3 symbols) 10-1 uncoded Prob. Symbol Error 10-2 select best (no alternatives) errors Viterbi threshold 10-3 D = 13 AWGN-only erasures 10-4 2 5 8 11 SNR

  26. Channel disturbances (2) freq. sel. fading L L H L L L L L L H L L non-coherent detection Output LOW for long time Effect: causes erasures # erasures depends on # hits narrowband noise H L H L H L L L H H L L non-coherent detection output HIGH for long time Effect: causesmanyerasures

  27. Reed Solomon: avoid constant symbol codeword 1 1 1 ... mother code 1 a a2 ... k+1 ... 1 ak a2k ... subcode The sub-codedoes not containthe 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. 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 H H H H H H H L L L L L L L L L L L L L L L L L L L L L L L L L HH L L L L L HH L L L L L L L L L L L L L L L L L L L L L H L L L L L L H L L L L L L L L L L L L L L L L L L L L L 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. Threshold detection:modifiedRS(15,9)+narrow band noise 10-1 uncoded Prob. Symbol Error 10-2 + side info + constant codewords AWGN only 10-3 Modified RS No side info 10-4 2 5 8 11 SNR

  30. Threshold detection:modifiedRS(15,3)+narrow band noise 10-1 uncoded Prob. Symbol Error 10-2 + constant codewords + side info modified RS no side info 10-3 AWGN only 10-4 2 5 8 11 SNR

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

  33. Reed Solomon: avoid certain output symbols M-ary RS Code in systematic form: ( info, control ) Ik 0 P = C k r 0 Ir 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. performance ( info, control ) Ik 0 P = C k r 0 Ir 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. CONCLUSIONS • Application of permutation codes for „terrible“ channels • Extend: block to convolutional • Constructions and simulations • Reed Solomon codes with restricted output

  36. Envelope detector with threshold output M = 4 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 non-coherent detection f4 f2 f3 erasure Example: transmit detect 1 error 0 f3 0 0

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