Data communication signatures

1. Data communicationsignatures A.J. Han Vinck July 29, 2004

2. Content:1. Optical transmission model2. Prime codes constructed from permutation codes 3. Optical Orthogonal Codes optical matched filter receiver auto- and cross correlation bound on cardinality4. Barker codes A.J. Han Vinck

3. Optical transmisison model • Consider Pulse Position Modulation (PPM) with optical „ON-OFF“ keying • Users transmit M-ary signatures Example: M = 3 (sub)slots for a signature of length 3 3 2 1 A.J. Han Vinck

4. Synchronous Communication model Transmit: 1:= signature; 0:= 0 sequence 1:= signature; 0:= 0 sequence Overlap with other users Detection: check presence of signature (yes or no) A.J. Han Vinck

5. How does it work as multi-access system? - Each user is assigned a unique signature ( length -L-) the unique signature is multiplied by each bit (1 or 0) the signature is only known to the receiver in order to recover the data. - The most important part for correct recovery is the set of signatures A.J. Han Vinck

6. Block Diagram Data Source # 1 Data Recovery Optical CDMA Encoder Optical CDMA Decoder Optical Star Coupler Data Recovery Data Source # N Optical CDMA Decoder Optical CDMA Encoder <----Transmitters-- <----Receivers-- A.J. Han Vinck

7. Two optical orthogonal signatures with length L = 32 • First signature is represented by placing a pulse at the 1st, 10th 13th and 28th chip positions. • Second signature is represented by placing a pulse at the 1st, 5th 12th and 31st chip positions. Both signatures interfere in only one position A.J. Han Vinck

8. Example: permutation code signatures:  length M M symbols (positions) are different  minimum # of differences dmin = M-1 i.e. maximum # of agreements = 1 Example: M = 3; M-1 = 2 Set of signatures: 123 312 231 132 321 213 A.J. Han Vinck

9. Extension to M-ary Prime code construction: basis is permutation code with dmin = M-1 123 231 312 213 321 132 111 222 333 permutation code + extension Property:any two signatures agree in at most 1 position! check! A.J. Han Vinck

10. Prime Code properties - # of agreements between any 2 signatures  1 Cardinality permutation code  M (M-1) + extension M - Cardinality PRIME code  M2 A.J. Han Vinck

11. performance • In the no-noise, signature synchronous situation • We can accept M-1 other users, since the „interference“ is  1 A.J. Han Vinck

12. Non-signature-synchronized User A # agreements = 2 (auto-correlation) User B # agreements = 2 (cross-correlation) A.J. Han Vinck

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

14. What is the receiver doing? 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

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

16. „Optical“ Orthogonal Codes (OOC) • Property: x, y  {0, 1} AUTO CORRELATION CROSS CORRELATION x x y y cross shifted x x A.J. Han Vinck

17. 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 auto correlation  2 Check! A.J. Han Vinck

18. Sketch of proof B A 1 1 1 1 * 1 1 1 1 1 * 1 If * = 1, then interval A = B and auto correlation  2 A.J. Han Vinck

19. 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

20. Intervals between ones ? 1,5 2,3 4,6 1 0 1 1 0 0 0 1 0 1 1 0 0 0 A.J. Han Vinck

21. 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

22. property 1: All intervals between ones are different, otherwise a shifted version of Y gives correlation 2 signature X 1 ------1---------1----1 signature Y 1---------11----1-----1 1 ------1---------1----1 1---------11----1-----1 A.J. Han Vinck

23. property 2: Cyclic shifted versions are not good as signature X 1 ------1---------1----1 1 ------1---------1----1 X* --11 ------1---------1-- A shifted version of X* could give correlation 4 A.J. Han Vinck

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

25. 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

26. Sequences with „good“ correlation properties Example: count # of agreements - # of disagreements agreements: 1-1 AND 0-0 Barker 7 1 1 1 0 0 1 0 1 1 1 0 0 1 0 7 - 1 1 1 0 0 1 0 shift one position to the right - - 1 1 1 0 0 -1 - - - 1 1 1 0 0 - - - - 1 1 1 -1 - - - - - 1 1 0 - - - - - - 1 -1 A.J. Han Vinck

27. Barker Codes examples Barker 11: [1,1,1,1,0,0,1,1,0,1,0] Barker 13: [1,1,1,1,1,0,0,1,1,0,1,0,1] The best we can do if „out of sync“: | # of agreements - # of disagreements |  1 Notes: Barker codes (Barker, 1950th) exist only for lengths: N = 2, 3, 4, 5, 7, 11, 13 IEEE 802.11 network uses the length 11- Barker code A.J. Han Vinck

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29. Application in 802.11b A.J. Han Vinck

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31. Application in Spread Spectrum A.J. Han Vinck