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Pulse Position Access Codes

Pulse Position Access Codes. A.J. Han Vinck. content. 1. Motivation UWB, frequency hopping (M-FSK) 2. Synchronized 3. PPM word format 4. Unsynchronized permutation codes, M-ary FSK 5. Codes with low corelation. UWB signal emission spectrum mask ( 3.1-10.6 GHz )

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Pulse Position Access Codes

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  1. Pulse Position Access Codes A.J. Han Vinck

  2. content • 1. Motivation • UWB, frequency hopping (M-FSK) • 2. Synchronized • 3. PPM word format • 4. Unsynchronized • permutation codes, M-ary FSK • 5. Codes with low corelation A.J. Han Vinck

  3. UWB signal emission spectrum mask ( 3.1-10.6 GHz ) Signal bandwidth > 500 MHz A.J. Han Vinck

  4. Pulsed transmission UWB Example: On-Off keying binary 1 0 1 0 A.J. Han Vinck

  5. Pulsed transmission UWB 0 1 PPM < nS Nominal pulse position A.J. Han Vinck

  6. A.J. Han Vinck

  7. Time-Frequency /Code division Time-frequency inefficient, but easy  0 Code division efficient, but complex 1 signature A.J. Han Vinck

  8. Binary access model tr 1 rec 1 tr 2 rec 2 OR   rec T tr T We want: „Uncoordinated and Random Access“ A.J. Han Vinck

  9. (sync) Binary access model (cont‘d) In Out OR A.J. Han Vinck

  10. Maximum throughput Channel per user interference Maximum SUM throughput = 0.69 bits/channel use Compare ALOHA: 0.36 A.J. Han Vinck

  11. Superimposed codes  T code words should not produce a valid code word T words Valid word N ?  n  ? n Transmit signature := 1 Transmit no signature := 0 A.J. Han Vinck

  12. bounds Lower bound: # combinations for large N: superimposed signatures exist s.t. T log2 N < n < 3 T2 log2 N Obvious for T out of N items A.J. Han Vinck

  13. Example: T  2, n = 9, N = 12 User signature 1 001 001 010 2 001 010 100 3 001 100 001 4 010 001 100 5 010 010 001 6 010 100 010 7 100 001 001 8 100 010 010 9 100 100 100 10 000 000 111 11 000 111 000 12 111 000 000 R = 2/9 TDMA gives R = 2/12 Example: 011 101 101 = x OR y ? A.J. Han Vinck

  14. For PPM: make access model M-ary tr 1 rec 1 tr 2 rec 2 OR   rec T tr T A.J. Han Vinck

  15. M-ary Frequency hopping 0 1 f M frequencies t Symbol time Hopping period Different hopping patterns (signatures) A.J. Han Vinck

  16. A.J. Han Vinck

  17. Maximum throughput Normalized SUM throughput (M-1)/M 0.69 bits/channel use Hence: PPM does not reduce efficiency! -”On the Capacity of the Asynchronous T-User M-frequency noislesss Multiple Access Channel” IEEE Trans. on Information Theory, pp. 2235-2238, November 1996. (A.J. Han Vinck and Jeroen Keuning) A.J. Han Vinck

  18. Low density signaling - Note on ``On the Asymptotic Capacity of a Multiple-Access Channel'' by L. Wilhelmsson and K. Sh. Zigangirov, Probl. Peredachi Inf., 2000, vol. 36, no. 1, pp. 21--25, Gober, P. and Han Vinck, A.J.,[Probl. Inf. Trans. (Engl. Transl.), 2000, vol. 36, no. 1, pp. 19--22. A.J. Han Vinck

  19. Example 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

  20. M-ary Superimposed codes  T code words should not produce a valid code word M-1 words Valid word N n = 3 n Tlog2 N  nM  3T2 log2 N Transmit signature := 1 Transmit no signature := 0 A.J. Han Vinck

  21. 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 -“On Superimposed Codes,” in Numbers, Information and Complexity, Ingo Althöfer, NingCai, Gunter Dueck, Levon Khachatrian,Mark S. Pinsker, Andras Sarkozy, Ingo Wegener and Zhen Zhang (eds.), Kluwer Academic Publishers, February 2000, pp. 325-331. A.J. Han Vinck and Samwel Martirosyan. A.J. Han Vinck

  22. M-ary Error Correcting Codes minimum distance dmin = maximum number of agreements No „overlap“ if T ( n - dmin ) < n For M-ary RS codes (n,k,d = n-k+1 ) Rsuperimposed = T/nM RTDMA = T/Mk A.J. Han Vinck

  23. examples T = 3, M = 9; RS-code ( n, k, d ) = (7,3,5) N = 93 T ( n - dmin) = 3 (7 – 5) < 7 ! T = 3, M = 9; RS-code ( n, k, d ) = (4,2,3) N = 92 T ( n - dmin) = 3 (4 - 3) < 4 ! A.J. Han Vinck

  24. Condition: sufficient but not necessary Example: T = 2; n = 4; dmin = 2 0 0 0 0 0 1 1 0 0 2 2 1 1 1 2 2 1 2 0 1 1 0 1 0 2 2 1 1 2 0 2 1 2 1 0 1 2 2 2 0 0 0 1 2 2 2 0 2 T(n-d) = 2(4 – 2) = 4 = n ! A.J. Han Vinck

  25. Superimposed codes summary • - Construction hard • Must be in sync • More than T users give errors • can be used as protocol sequences in collision channels • better than TDMA for • N = 1024, T < 6 A.J. Han Vinck

  26. Permutation codes for access Properties: minimum distance dmin Signatures: length M M different symbols Examples: 0 1 2 0 1 2 1 0 2 1 2 0 dmin = 3 1 2 0 2 1 0 dmin = 2 2 0 1 2 0 1 0 2 1 A.J. Han Vinck

  27. properties Example: M = 3; dmin = 2; |C| = 6 In general cardinality: Reseach challenge: when equality? A.J. Han Vinck

  28. Interference property For minimum distance dmin = M-1 difference |C| = M(M-1) Maximum interference = M - dmin = 1 agreement CONCLUSION: up to M-1 users uniquely detectable always one unique position left A.J. Han Vinck

  29. Non-coherent detector structure Envelope detection 1 Threshold 1 > = 1 < = 0 Envelope detection 2 Threshold 2 in > = 1 < = 0  Envelope detection M Threshold M > = 1 < = 0 A.J. Han Vinck

  30. Coded Modulation for Power Line Communications”, AEÜ Journal, 2000, pp. 45-49, Jan 2000. A.J. Han Vinck

  31. A.J. Han Vinck

  32. M code words per user M code words  dmin = M n  M M-1 users; T active; dmin = M-1 A.J. Han Vinck

  33. Example: M = 3 1 2 0 1 0 2 2 1 0 2 0 1 0 1 2 0 2 1 6 users; <3 active; dmin = 2 n - dmin = 1 Rsuperimposed = 2/9 RTDMA = 2log23/18 User 1: 1 2 0 or 0 0 0 { ( (1,0), 2, (1,0) } = ? A.J. Han Vinck

  34. Example M = 5 0 1 2 0 2 4 0 3 1 0 4 3 1 2 3 1 3 0 1 4 2 1 0 4 2 3 4 2 4 1 2 0 3 2 1 0 3 4 0 3 0 2 3 1 4 3 2 1 4 0 1 4 1 3 4 2 0 4 3 2 4 users;  2 active; dmin = 2; n - dmin = 1 Rsuperimposed = 2log25/15 RTDMA = 2log25/20 Codewords for user 4 A.J. Han Vinck

  35. A.J. Han Vinck

  36. example A.J. Han Vinck

  37. A.J. Han Vinck

  38. A.J. Han Vinck

  39. 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

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

  41. Non-symbol-synchronized User A (Auto)-Correlation = 2 User B (Cross)-Correlation = 2 A.J. Han Vinck

  42. „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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

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

  49. 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

  50. Problem in construction • find good starting word • Find small value for blow up parameter -“A Construction for optical Orthogonal Codes with Correlation 1,” IEICE Trans. Fundamentals, Vol E85-A, No. 1, January 2002, pp. 269-272, Samwel Martirosyan and A.J. Han Vinck, A.J. Han Vinck

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