On Optical Orthogonal Codes

1
On Optical Orthogonal Codes
or Cyclically Permutable Error-Correcting Codes
(Gilbert)

• A.J. Han Vinck

2
content

• 1. Optical Orthogonal codes
• properties
• 2. OOC transmission codes
• 3. Super OOCs
• 4. Alternatives

3
signature
Other users
noise
OPTICAL matched filter TRANSMITTER/RECEIVER
4
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
5

• We want
• weight w large high peak
• side peaks ? 1
• for other signatures cross correlation ? 1

6
Several possibilities
A
or
0
B
or
shifted
C
or
another
7
note
For situation A
or
0
A sequence might look like x 0 x x 0 0 ? ? ?
For situation C
or
another
A sequence might look like x y y x y ? ? ?
8
Optical Orthogonal Codes definition

• Property x, y ? 0, 1

AUTO CORRELATION
CROSS CORRELATION
x x y y
cross
x x
shifted
9
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
10
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 gt 1 impossible correlation ? 2
11
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
12
conclusion
Signature in sync peak of size w All other
situations contributions ? 1
What about code parameters?
13
Code size for code words of length n

• different intervals lt 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 0 0
• C(n,w,1) ? (n-1)/w(w-1) ( 6/6 1)

1, 2, 3, 4, 5, 6
14
Example C(7,2,1)
1000001 ? 1, 6 1000010 ? 2,
5 1000100 ? 3, 4
15
Construction (n,w,1)-OOC
IDEA starting word 110100000 w3, 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
16
Problem in construction

1. find good starting word
2. Find small value for blow up parameter

17
result
1. Code construction C(n,w,1) gt
2n/(w-1)w3 2. Using difference sets as starting
word Code construction C(n,w,1) gt
2n/(w-1)w2 problem existance of difference
sets Reference IEICE, January 2002
Upperbound C(n,w,1) ? (n-1)/w(w-1)
18
Difference set
A difference set is an ( n w(w-1) 1, w,
1 ) OOC with a single code vector X0
Example n 7 w 3 1 1 0 1 0 0 0
19
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)

20
application
All optical transmitter/ receiver is fast Use
signature of OOC to transmit information
21
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
22
Model for UWB ( EWO)
1 or 0

3 or -3
23
Transmitter / receiver(ref Tomoaki Ohtsuki)
data
Data selector
encoder
laser
Tunable optical delay line
sequence encoder

power splitter
hard limiter
optical correlator
-
optical correlator
decoder
24
/-
Simple correlator and encoder balanced equal
weight signalling - Power splitting Cross
correlation?
25
2 problems
User 1 1100000 or 0110000 11 User 2
1000010 or 0100001 User 3 1000100 or 0100010
0 1 01000011000010
correlation 2 !
26
Super Optical Orthogonal Codes
AUTO CORRELATION
CROSS CORRELATION
SUPER-CROSS CORRELATION
27
Super-cross correlation
y y
? 1
x
y y
? 1
x
Y could be shifted version
28
Property shift sensitive
1100000 1010000 is a S-OOC 1001000
shifted code 1000001 1000010 is not a
S-OOC 1000100
29
conclusions

• Optical Orthogonal Codes
• have nice correlation properties
• Super Optical Orthogonal Codes
• additional constraint less code words

30
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
31
Prime Code properties
Permutation code has minimum distance M-1 i.e.
Interference 1 Cardinality permutation code ?
M (M-1) extention M Cardinality PRIME code
? M2
BAD AUTO- and CROSS-CORRELATION
32
M-ary Superimposed codes
? M-1 code words should not produce a valid code
word
M-1 words Valid word
N
M
M-1 words Valid word
N ? 2M2
33
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
34
Difference identification-decypherable
Decypherable 1 0 N 3 0 1 Ex
(01),(01) covers ( 1 1 ) 1 1 but uniquely
decodable ? ( 1 0 ) ? ( 0 1)
Identification 1 0 N 2 0 1 Ex
(01),(01) covers ( 1 1 ) 1 1
35
Example ( honest ? )
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 ?
36
conclusions
We showed - different optical signalling
methods - problems with OOC code
design Future performance calculations