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On Optical Orthogonal Codes

On Optical Orthogonal Codes. or Cyclically Permutable Error-Correcting Codes (Gilbert). A.J. Han Vinck. content. 1. Optical Orthogonal codes properties 2. OOC transmission codes 3. Super OOCs 4. Alternatives. signature. Other users. noise. OPTICAL matched filter TRANSMITTER/RECEIVER.

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On Optical Orthogonal Codes

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  1. On Optical Orthogonal Codes or Cyclically Permutable Error-Correcting Codes (Gilbert) A.J. Han Vinck

  2. content • 1. Optical Orthogonal codes • properties • 2. OOC transmission codes • 3. Super OOCs • 4. Alternatives A.J. Han Vinck

  3. signature Other users noise OPTICAL matched filter TRANSMITTER/RECEIVER A.J. Han Vinck

  4. why Collect all the ones in the signature: 0 0 0 1 0 1 1 delay 0 0 0 0 1 0 1 1 delay 2 0 0 0 1 0 1 1 delay 3 weight w A.J. Han Vinck

  5. We want: • weight w large high peak • side peaks  1 • for other signatures cross correlation  1 A.J. Han Vinck

  6. Several possibilities A or 0 B or shifted C or another A.J. Han Vinck

  7. note For situation A: or 0 A sequence might look like: x 0 x x 0 0    For situation C: or another A sequence might look like: x y y x y    A.J. Han Vinck

  8. Optical Orthogonal Codes: definition • Property: x, y  {0, 1} AUTO CORRELATION CROSS CORRELATION x x y y cross shifted x x A.J. Han Vinck

  9. Important properties (for code construction) 1) All intervals between two ones must be different 1000001  = 1, 6 1000010  = 2, 5 1000100  = 3, 4 C(7,2,1) 2) Cyclic shifts give cross correlation  1 they are not in the OOC A.J. Han Vinck

  10. autocorrelation w = 3 0 0 0 1 0 1 1 signature x 0 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 3 1 1 1 side peak > 1 impossible correlation  2 A.J. Han Vinck

  11. Cross correlation 0 0 0 1 0 1 1signature x * * * 1 * * * signature y * * * 1 * * * * * * 1 * * ? Suppose that ? = 1 then cross correlation with x = 2 y contains same interval as x  impossible A.J. Han Vinck

  12. conclusion Signature in sync: peak of size w All other situations contributions  1 What about code parameters? A.J. Han Vinck

  13. Code size for code words of length n • # different intervals < n • must be different otherwise correlation  2 • For weight w vector: w(w-1) intervals • 1 1 0 1 0 0 0 1 1 0 1 0 00 • |C(n,w,1)|  (n-1)/w(w-1) ( = 6/6 = 1) 1, 2, 3, 4, 5, 6 A.J. Han Vinck

  14. Example C(7,2,1) 1000001  = 1, 6 1000010  = 2, 5 1000100  = 3, 4 A.J. Han Vinck

  15. Construction (n,w,1)-OOC IDEA: starting word 110100000 w=3, length n0 =9 1 2 Blow up intervals 1 1 0 1 0 0 0 0 0 0 *** 4 5 Parameter 1 0 0 0 1 0 0 0 0 1 0 *** m = 3 7 8 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 *** Proof OOC property: all intervals are different  correlation =1 A.J. Han Vinck

  16. Problem in construction • find good starting word • Find small value for blow up parameter A.J. Han Vinck

  17. result 1. Code construction: |C(n,w,1)| > 2n/(w-1)w3 2. Using difference sets as starting word: Code construction |C(n,w,1)| > 2n/(w-1)w2 problem: existance of difference sets Reference: IEICE, January 2002 Upperbound: |C(n,w,1)|  (n-1)/w(w-1) A.J. Han Vinck

  18. Difference set A difference set is : an ( n = w(w-1) + 1, w, 1 ) – OOC with a single code vector X0 Example: n = 7; w = 3 1 1 0 1 0 0 0 A.J. Han Vinck

  19. references Mathematical design solutions: ·projective geometry ( Chung, Salehi, Wei, Kumar) ·balanced incomplete block designs (R.N.M. Wilson) ·difference sets ( Jungnickel) Japanese reference: Tomoaki Ohtsuki ( Univ. of Tokyo) A.J. Han Vinck

  20. application All optical transmitter/ receiver is fast Use signature of OOC to transmit information A.J. Han Vinck

  21. Transmission of 1 bit/user User 1: 1000001 or 0000000 User 2: 1000010 or 0000000 (OOO) User 3: 1000100 or 0000000 2 users can lead to wrong decision at sample moment +: simple transmitter -: not balanced A.J. Han Vinck

  22. Model for UWB ( EWO) 1 or 0 * +3 = or -3 A.J. Han Vinck

  23. Transmitter / receiver(ref: Tomoaki Ohtsuki) data Data selector encoder laser Tunable optical delay line sequence encoder + hard limiter power splitter optical correlator - optical correlator decoder A.J. Han Vinck

  24. +/- + Simple correlator and encoder balanced: equal weight signalling - Power splitting Cross correlation? A.J. Han Vinck

  25. 2 problems User 1: 1100000 or 0110000 11 User 2: 1000010 or 0100001 User 3: 1000100 or 0100010 0 1 0100001|1000010 correlation 2 ! A.J. Han Vinck

  26. Super Optical Orthogonal Codes AUTO CORRELATION CROSS CORRELATION SUPER-CROSS CORRELATION A.J. Han Vinck

  27. Super-cross correlation y y  1 x y y‘  1 x Y‘ could be shifted version A.J. Han Vinck

  28. Property shift sensitive 1100000 1010000 is a S-OOC 1001000 shifted code 1000001 1000010 is not a S-OOC 1000100 A.J. Han Vinck

  29. conclusions • Optical Orthogonal Codes • have nice correlation properties • Super Optical Orthogonal Codes • additional constraint: less code words A.J. Han Vinck

  30. Alternatives: M-ary Prime code pulse at position i Symbol i 1 i  M Example: 123 231 312 213 321 132 111 222 333 permutation code + extension A.J. Han Vinck

  31. Prime Code properties Permutation code has minimum distance M-1 i.e. Interference = 1 Cardinality permutation code  M (M-1) + extention M Cardinality PRIME code  M2 BAD AUTO- and CROSS-CORRELATION A.J. Han Vinck

  32. M-ary Superimposed codes  M-1 code words should not produce a valid code word M-1 words Valid word N M M-1 words Valid word N  2M2 A.J. Han Vinck

  33. Example: general construction 3 1 1 2 1 1 1 3 1 1 2 1 1 1 3 1 1 2 N N  M(M-1) M A.J. Han Vinck

  34. Difference: identification-decypherable Decypherable: 1 0 N = 3 0 1 Ex: { (01),(01) } covers ( 1 1 ) 1 1 but uniquely decodable  ( 1 0 )  ( 0 1) Identification: 1 0 N = 2 0 1 Ex: { (01),(01) } covers ( 1 1 ) 1 1 A.J. Han Vinck

  35. Example ( honest ? ) 2 users may transmit 1 bit of info at the same time User 1 112 or 222 User 2 121 or 222 User 3 211 or 222 User 4 122 or 222 Sum rate = 2/6 RTDMA = 2/8 Example: receive { (1), (1,2), 2 } =? A.J. Han Vinck

  36. conclusions We showed: - different optical signalling methods - problems with OOC code design Future: performance calculations A.J. Han Vinck

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