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Institute for Experimental Mathematics Ellernstrasse 29 45326 Essen - Germany Data communication signatures A.J. Han Vinck July 29, 2004
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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
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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 3 3 2 1
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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)
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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
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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--
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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
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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: 123 312 231 132 321 213
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University Duisburg-Essendigital communications group A.J. Han Vinck Extension to M-ary Prime code construction: basis is permutation code with d min = 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!
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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
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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
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University Duisburg-Essendigital communications group A.J. Han Vinck Non-signature-synchronized User A # agreements = 2 (auto-correlation) User B # agreements = 2 (cross-correlation)
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University Duisburg-Essendigital communications group A.J. Han Vinck Other users noise OPTICAL matched filter TRANSMITTER/RECEIVER signature
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University Duisburg-Essendigital communications group A.J. Han Vinck 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
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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
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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
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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 auto correlation 2 Check! w = 3
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University Duisburg-Essendigital communications group A.J. Han Vinck Sketch of proof 1 1 1 1 * 1 – If * = 1, then interval A = B and auto correlation 2 A B
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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
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University Duisburg-Essendigital communications group A.J. Han Vinck Intervals between ones ? 1 0 1 1 0 0 0 1,5 2,3 4,6
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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
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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 1 ------1---------1----1 signature Y 1---------11----1-----1 1 ------1---------1----1 1---------11----1-----1
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University Duisburg-Essendigital communications group A.J. Han Vinck property 2: Cyclic shifted versions are not good as signature X1 ------1---------1----1 1 ------1---------1----1 X* --11 ------1---------1-- A shifted version of X* could give correlation 4
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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?
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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
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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 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
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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 802.11 network uses the length 11- Barker code
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University Duisburg-Essendigital communications group A.J. Han Vinck
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University Duisburg-Essendigital communications group A.J. Han Vinck –Application in 802.11b
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University Duisburg-Essendigital communications group A.J. Han Vinck
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University Duisburg-Essendigital communications group A.J. Han Vinck Application in Spread Spectrum
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