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

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

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

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

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

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

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

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“

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

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

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck Example: T  2, n = 9, N = 12 Usersignature R = 2/9 TDMA gives R = 2/12 Example: = x OR y ?

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

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

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

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 , November (A.J. Han Vinck and Jeroen Keuning)

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 , Gober, P. and Han Vinck, A.J.,[Probl. Inf. Trans. (Engl. Transl.), 2000, vol. 36, no. 1, pp

University Duisburg-Essendigital communications group A.J. Han Vinck Example 2 users may transmit 1 bit of info at the same time User or 222 User or 222 User or 222 User or 222 Sum rate = 2/6 R TDMA = 2/8 Example: receive { (1), (1,2), 2 } =?

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

University Duisburg-Essendigital communications group A.J. Han Vinck Example: general construction 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 A.J. Han Vinck and Samwel Martirosyan.

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

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 !

University Duisburg-Essendigital communications group A.J. Han Vinck Condition: sufficient but not necessary Example: T = 2; n = 4; d min = T(n-d) = 2(4 – 2) = 4 = n !

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

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: d min = d min =

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?

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

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     

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

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

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck Example M = 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

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

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

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck Alternatives: M-ary Prime code Symbol i 1  i  M pulse at position i Example: permutation code + extension

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

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

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

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 autocorrelation signature x side peak > 1 impossible correlation  2 w = 3

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 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 Example C(7,2,1)  = 1,  = 2,  = 3, 4

University Duisburg-Essendigital communications group A.J. Han Vinck Construction (n,w,1)-OOC IDEA: IDEA: starting word w=3, length n 0 =9 1 2 Blow up intervals *** 4 5 Parameter *** m = *** Proof OOC property: Proof OOC property: all intervals are different  correlation =1

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 , Samwel Martirosyan and A.J. Han Vinck,

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)

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 =

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)

University Duisburg-Essendigital communications group A.J. Han Vinck Transmission of 1 bit/user User 1: or User 2: or (OOO) User 3: or users can lead to wrong decision at sample moment +: simple transmitter -: not balanced

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!

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

University Duisburg-Essendigital communications group A.J. Han Vinck 2 problems User 1: or User 2: or correlation 2 ! User 3: or | correlation 2 !

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

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   

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

University Duisburg-Essendigital communications group A.J. Han Vinck Property shift sensitive is a S-OOC shifted code is not a S-OOC

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

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

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

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 why 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