Institute for Experimental Mathematics Ellernstrasse Essen - Germany Data communication signatures A.J. Han Vinck July 29, 2004

University Duisburg-Essendigital communications group A.J. Han Vinck Content: 1. Optical transmission model 2. Prime codes constructed from permutation codes 3. Optical Orthogonal Codes optical matched filter receiver auto- and cross correlation bound on cardinality 4. Barker codes

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

University Duisburg-Essendigital communications group A.J. Han Vinck Synchronous Communication model –Transmit: –Transmit: 1:= signature; 0:= 0 sequence 1:= signature; 0:= 0 sequence – Overlap with other users –Detection –Detection: check presence of signature (yes or no)

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck (a)First signature is represented by placing a pulse at the 1 st, 10 th 13 th and 28 th chip positions. (b)Second signature is represented by placing a pulse at the 1 st, 5 th 12 th and 31 st chip positions. Two optical orthogonal signatures with length L = 32 –Both signatures interfere in only one position

University Duisburg-Essendigital communications group A.J. Han Vinck Example: permutation code signatures:  length M M symbols (positions) are different  minimum # of differences d min = M-1 i.e. maximum # of agreements = 1 Example: M = 3; M-1 = 2 Set of signatures:

University Duisburg-Essendigital communications group A.J. Han Vinck Extension to M-ary Prime code construction: basis is permutation code with d min = M permutation code + extension Property: any two signatures agree in at most 1 position! check!

University Duisburg-Essendigital communications group A.J. Han Vinck Prime Code properties - # of agreements between any 2 signatures  1 Cardinality permutation code  M (M-1) + extension M - Cardinality PRIME code  M 2

University Duisburg-Essendigital communications group A.J. Han Vinck performance – In the no-noise, signature synchronous situation – We can accept M-1 other users, since the „interference“ is  1

University Duisburg-Essendigital communications group A.J. Han Vinck Non-signature-synchronized User A # agreements = 2 (auto-correlation) User B # agreements = 2 (cross-correlation)

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

University Duisburg-Essendigital communications group A.J. Han Vinck What is the receiver doing? Collect all the ones in the signature: delay delay delay 3 weight w

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

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck Sketch of proof * 1 – If * = 1, then interval A = B and auto correlation  2 A B

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

University Duisburg-Essendigital communications group A.J. Han Vinck Intervals between ones ? ,5 2,3 4,6

University Duisburg-Essendigital communications group A.J. Han Vinck Important properties (for code construction) 1) All intervals  between two ones must be different  = 1,  = 2,  = 3, 4 C(7,2,1) cross 2) Cyclic shifts give cross correlation > 1 they are not in the OOC

University Duisburg-Essendigital communications group A.J. Han Vinck property 1: All intervals between ones are different, otherwise a shifted version of Y gives correlation 2 signature X signature Y

University Duisburg-Essendigital communications group A.J. Han Vinck property 2: Cyclic shifted versions are not good as signature X X* A shifted version of X* could give correlation 4

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?

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 |C(n,w,1)|  (n-1)/w(w-1) ( = 6/6 = 1) 1, 2, 3, 4, 5, 6

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

University Duisburg-Essendigital communications group A.J. Han Vinck 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 network uses the length 11- Barker code

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

University Duisburg-Essendigital communications group A.J. Han Vinck –Application in b

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

University Duisburg-Essendigital communications group A.J. Han Vinck Application in Spread Spectrum