1. Institute for Experimental Mathematics Ellernstrasse 29 45326 Essen - Germany Pulse Position Access Codes A.J. Han Vinck

2. University Duisburg-Essendigital communications group 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

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

4. University Duisburg-Essendigital communications group A.J. Han Vinck Pulsed transmission UWB 1 0 binary Example: On-Off keying

5. University Duisburg-Essendigital communications group A.J. Han Vinck Pulsed transmission UWB 0 1 Nominal pulse position < nS PPM

6. University Duisburg-Essendigital communications group A.J. Han Vinck

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

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

9. University Duisburg-Essendigital communications group A.J. Han Vinck (sync) Binary access model (cont‘d) In Out OR

10. University Duisburg-Essendigital communications group A.J. Han Vinck Maximum throughput Channel per user Maximum SUM throughput = 0.69 bits/channel use Compare ALOHA: 0.36 interference

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

12. University Duisburg-Essendigital communications group A.J. Han Vinck bounds Lower bound: for large N: superimposed signatures exist s.t. T log 2 N < n < 3 T 2 log 2 N Obvious for T out of N items # combinations

13. University Duisburg-Essendigital communications group A.J. Han Vinck Example: T  2, n = 9, N = 12 Usersignature 1001 001 010 2001 010 100 3001 100 001 4010 001 100 5010 010 001 6010 100 010 7100 001 001 8100 010 010 9100 100 100 10000 000 111 11000 111 000 12111 000 000 R = 2/9 TDMA gives R = 2/12 Example: 011 101 101 = x OR y ?

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

15. University Duisburg-Essendigital communications group A.J. Han Vinck M-ary Frequency hopping f t Symbol timeHopping period Different hopping patterns (signatures) 10 M frequencies

16. University Duisburg-Essendigital communications group A.J. Han Vinck

17. University Duisburg-Essendigital communications group A.J. Han Vinck 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)

18. University Duisburg-Essendigital communications group A.J. Han Vinck 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.

19. University Duisburg-Essendigital communications group A.J. Han Vinck 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 R TDMA = 2/8 Example: receive { (1), (1,2), 2 } =?

20. University Duisburg-Essendigital communications group A.J. Han Vinck M-ary Superimposed codes  T code words should not produce a valid code word n N M-1 words Valid word T log 2 N  nM  3T 2 log 2 N n = 3 Transmit signature:= 1 Transmit no signature := 0

21. University Duisburg-Essendigital communications group A.J. Han Vinck Example: general construction 3 1 1 2 1 1 1 3 1 1 2 1 1 1 3 1 1 2 N M N  M(M-1) -“On Superimposed Codes,” in Numbers, Information and Complexity, Ingo Althöfer, Ning Cai, 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.

22. University Duisburg-Essendigital communications group A.J. Han Vinck M-ary Error Correcting Codes minimum distance d min = maximum number of agreements No „overlap“ if T ( n - d min ) < n For M-ary RS codes (n,k,d = n-k+1 ) R superimposed = T/nM R TDMA = T/M k

23. University Duisburg-Essendigital communications group A.J. Han Vinck examples T = 3, M = 9; RS-code ( n, k, d ) = (7,3,5) N = 9 3 T ( n - d min ) = 3 (7 – 5) < 7 ! T = 3, M = 9; RS-code ( n, k, d ) = (4,2,3) N = 9 2 T ( n - d min ) = 3 (4 - 3) < 4 !

24. University Duisburg-Essendigital communications group A.J. Han Vinck Condition: sufficient but not necessary Example: T = 2; n = 4; d min = 2 0 0 0 00 1 1 00 2 2 11 1 2 2 1 2 0 11 0 1 02 2 1 12 0 2 1 2 1 0 12 2 2 00 0 1 22 2 0 2 T(n-d) = 2(4 – 2) = 4 = n !

25. University Duisburg-Essendigital communications group A.J. Han Vinck 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

26. University Duisburg-Essendigital communications group A.J. Han Vinck Permutation codes for access Properties:minimum distance d min Signatures: length M M different symbols Examples: 0 1 20 1 21 0 2 1 2 0 d min = 31 2 02 1 0 d min = 2 2 0 12 0 10 2 1

27. University Duisburg-Essendigital communications group A.J. Han Vinck properties Example: M = 3; d min = 2; |C| = 6 In general cardinality: Reseach challenge: when equality?

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

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

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

31. University Duisburg-Essendigital communications group A.J. Han Vinck

32. University Duisburg-Essendigital communications group A.J. Han Vinck M code words per user M code words M-1 users; T active; d min = M-1 d min = M  n  M

33. University Duisburg-Essendigital communications group A.J. Han Vinck Example: M = 3 1 2 01 0 2 2 1 02 0 1 0 1 20 2 1 6 users; <3 active; d min = 2 n - d min = 1 R superimposed = 2/9 R TDMA = 2log 2 3/18 User 1: 1 2 0 or 0 0 0 { ( (1,0), 2, (1,0) } = ?

34. University Duisburg-Essendigital communications group A.J. Han Vinck 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; d min = 2; n - d min = 1 R superimposed = 2log 2 5/15 R TDMA = 2log 2 5/20 Codewords for user 4

35. University Duisburg-Essendigital communications group A.J. Han Vinck

36. University Duisburg-Essendigital communications group A.J. Han Vinck example

37. University Duisburg-Essendigital communications group A.J. Han Vinck

38. University Duisburg-Essendigital communications group A.J. Han Vinck

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

40. University Duisburg-Essendigital communications group A.J. Han Vinck 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  M 2 BAD AUTO- and CROSS-CORRELATION

41. University Duisburg-Essendigital communications group A.J. Han Vinck Non-symbol-synchronized User A (Auto)-Correlation = 2 User B (Cross)-Correlation = 2

42. University Duisburg-Essendigital communications group A.J. Han Vinck „Optical“ Orthogonal Codes: definition Property: x, y  {0, 1} AUTO CORRELATION CROSS CORRELATION x x y y x xshifted cross

43. University Duisburg-Essendigital communications group A.J. Han Vinck 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) cross 2) Cyclic shifts give cross correlation > 1 they are not in the OOC

44. University Duisburg-Essendigital communications group A.J. Han Vinck autocorrelation 0 0 0 1 0 1 1 signature x 0 0 0 1 0 1 1 1 1 1 3 1 1 1 side peak > 1 impossible correlation  2 w = 3

45. University Duisburg-Essendigital communications group A.J. Han Vinck Cross correlation 0 0 0 1 0 1 1 signature x * * * 1 * * * signature y * * * 1 * * * * * * 1 * * ? Suppose that ? = 1 then cross correlation with x = 2 y contains same interval as x  impossible

46. University Duisburg-Essendigital communications group A.J. Han Vinck conclusion Signature in sync: peak of size w w must be large All other situations contributions  1 What about code parameters?

47. University Duisburg-Essendigital communications group A.J. Han Vinck 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 |C(n,w,1)|  (n-1)/w(w-1) ( = 6/6 = 1) 1, 2, 3, 4, 5, 6

48. University Duisburg-Essendigital communications group A.J. Han Vinck Example C(7,2,1) 1000001  = 1, 6 1000010  = 2, 5 1000100  = 3, 4

49. University Duisburg-Essendigital communications group A.J. Han Vinck Construction (n,w,1)-OOC IDEA: IDEA: starting word 110100000 w=3, length n 0 =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: Proof OOC property: all intervals are different  correlation =1

50. University Duisburg-Essendigital communications group A.J. Han Vinck Problem in construction 1.find good starting word 2.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,

51. University Duisburg-Essendigital communications group A.J. Han Vinck result 1. Code construction: |C(n,w,1)| > 2n/(w-1)w 3 2. Using difference sets as starting word: Code construction|C(n,w,1)| > 2n/(w-1)w 2 problem: existance of difference sets Reference: IEICE, January 2002 Upperbound: |C(n,w,1)|  (n-1)/w(w-1)

52. University Duisburg-Essendigital communications group A.J. Han Vinck Difference set A difference set is : an ( n = w(w-1) + 1, w, 1 ) – OOC with a single code vector X 0 Example: n = 7; w = 3 1 1 0 1 0 0 0

53. University Duisburg-Essendigital communications group A.J. Han Vinck 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)

54. University Duisburg-Essendigital communications group A.J. Han Vinck 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

55. University Duisburg-Essendigital communications group A.J. Han Vinck Multi user Communication model for UWB 1 or 0 * +3 = or -3 transmit receive Signature 1 Signature 0 Simple receiver structure!

56. University Duisburg-Essendigital communications group A.J. Han Vinck Transmitter / receiver (ref: Tomoaki Ohtsuki) Data selector data Tunable delay line impulse sequence encoder splitter hard limiter correlator - + encoder decoder

57. University Duisburg-Essendigital communications group A.J. Han Vinck 2 problems User 1: 1100000 or 0110000 11 User 2: 1000010 or 0100001 correlation 2 ! User 3: 1000100 or 0100010 0 1 0100001|1000010 correlation 2 !

58. University Duisburg-Essendigital communications group A.J. Han Vinck Super Optical Orthogonal Codes AUTO CORRELATION CROSS CORRELATION SUPER-CROSS CORRELATION

59. University Duisburg-Essendigital communications group A.J. Han Vinck note or 0 A sequence might look like: y 0 y y 0 0    For situation A: or another For situation C: A sequence might look like: y y‘ y‘ y‘ y   

60. University Duisburg-Essendigital communications group A.J. Han Vinck Super-cross correlation y y x y y‘ x  1 Y‘ could be shifted version

61. University Duisburg-Essendigital communications group A.J. Han Vinck Property shift sensitive 1100000 1010000 is a S-OOC 1001000 shifted code 1000001 1000010 is not a S-OOC 1000100

62. University Duisburg-Essendigital communications group A.J. Han Vinck conclusions Optical Orthogonal Codes have nice correlation properties Super Optical Orthogonal Codes additional constraint: less code words

63. University Duisburg-Essendigital communications group A.J. Han Vinck conclusions We showed: - different signalling methods - problems with OOC code design Future: performance calculations

64. University Duisburg-Essendigital communications group A.J. Han Vinck Application optical Multi-access All optical transmitter/ receiver is fast Use signature of OOC to transmit information

65. University Duisburg-Essendigital communications group A.J. Han Vinck Other users noise OPTICAL matched filter TRANSMITTER/RECEIVER signature

66. University Duisburg-Essendigital communications group A.J. Han Vinck 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

67. University Duisburg-Essendigital communications group A.J. Han Vinck We want: 1.weight w large high peak 2.side peaks  1 for other signatures cross correlation  1