Pulse Position Access Codes

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Pulse Position Access Codes

• A.J. Han Vinck

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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

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UWB signal emission spectrum mask ( 3.1-10.6 GHz
) Signal bandwidth gt 500 MHz
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Pulsed transmission UWB
Example On-Off keying
binary
1 0 1 0
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Pulsed transmission UWB
0 1
PPM
lt nS
Nominal pulse position
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Time-Frequency /Code division
Time-frequency inefficient, but easy
????
0
Code division efficient, but complex
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signature
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Binary access model
tr 1
rec 1
tr 2
rec 2
????
OR
????
rec T
tr T
We want Uncoordinated and Random Access
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(sync) Binary access model (contd)
In Out
OR
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Maximum throughput
Channel per user
interference
Maximum SUM throughput 0.69 bits/channel use
Compare ALOHA 0.36
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Superimposed codes
? T code words should not produce a valid code
word
T words Valid word
N
? ? n ? ?
n
Transmit signature 1 Transmit no signature
0
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bounds
Lower bound
combinations
for large N superimposed signatures exist s.t.
T log2 N lt n lt 3 T2 log2 N
Obvious for T out of N items
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Example T ? 2, n 9, N 12 User signature 1
001 001 010 2 001 010 100 3 001 100
001 4 010 001 100 5 010 010 001 6 010 100
010 7 100 001 001 8 100 010 010 9 100 100
100 10 000 000 111 11 000 111 000 12 111
000 000
R 2/9 TDMA gives R 2/12 Example 011 101
101 x OR y ?
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For PPM make access model M-ary
tr 1
rec 1
tr 2
rec 2
????
OR
????
rec T
tr T
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M-ary Frequency hopping
1
0
f

M frequencies
t
Symbol time
Hopping period
Different hopping patterns (signatures)
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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)
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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.
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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 RTDMA 2/8
Example receive (1), (1,2), 2 ?
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M-ary Superimposed codes
? T code words should not produce a valid code
word
M-1 words Valid word
N
n 3
n
T log2 N ? nM ? 3T2 log2 N
Transmit signature 1 Transmit no signature
0
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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
-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.
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M-ary Error Correcting Codes
minimum distance dmin maximum number of
agreements
No overlap if T ( n - dmin ) lt n
For M-ary RS codes (n,k,d n-k1
) Rsuperimposed T/nM RTDMA T/Mk
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examples
T 3, M 9 RS-code ( n, k, d ) (7,3,5) N
93 T ( n - dmin) 3 (7 5) lt 7 ! T 3, M
9 RS-code ( n, k, d ) (4,2,3) N 92 T
( n - dmin) 3 (4 - 3) lt 4 !
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Condition sufficient but not necessary
Example T 2 n 4 dmin 2 0 0 0 0 0 1 1
0 0 2 2 1 1 1 2 2 1 2 0 1 1 0 1 0 2 2 1 1 2 0 2
1 2 1 0 1 2 2 2 0 0 0 1 2 2 2 0 2 T(n-d) 2(4
2) 4 n !
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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 lt 6

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Permutation codes for access
Properties minimum distance dmin Signatures
length M M different symbols Examples 0 1
2 0 1 2 1 0 2 1 2 0 dmin 3 1 2 0 2 1 0
dmin 2 2 0 1 2 0 1 0 2 1
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properties
Example M 3 dmin 2 C 6 In general
cardinality Reseach challenge when
equality?
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Interference property
For minimum distance dmin M-1
difference C M(M-1) Maximum
interference M - dmin 1
agreement CONCLUSION up to M-1 users uniquely
detectable always one unique position left
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Non-coherent detector structure
Envelope detection 1
Threshold 1
gt 1 lt 0
Envelope detection 2
Threshold 2
in
gt 1 lt 0
? ? ?
Envelope detection M
Threshold M
gt 1 lt 0
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Coded Modulation for Power Line Communications,
AEÜ Journal, 2000, pp. 45-49, Jan 2000.
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M code words per user
M code words
???
dmin M
n ? M
M-1 users T active dmin M-1
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Example M 3
1 2 0 1 0 2 2 1 0 2 0 1 0 1 2 0 2 1
6 users lt3 active dmin 2 n - dmin 1
Rsuperimposed 2/9 RTDMA 2log23/18
User 1 1 2 0 or 0 0 0
( (1,0), 2, (1,0) ?
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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 dmin 2 n - dmin
1 Rsuperimposed 2log25/15 RTDMA 2log25/20
Codewords for user 4
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example
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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
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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
? M2
BAD AUTO- and CROSS-CORRELATION
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Non-symbol-synchronized
User A (Auto)-Correlation 2
User B (Cross)-Correlation 2
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Optical Orthogonal Codes definition

• Property x, y ? 0, 1

AUTO CORRELATION
CROSS CORRELATION
x x y y
cross
x x
shifted
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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 gt 1
they are not in the OOC
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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
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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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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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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
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Example C(7,2,1)
1000001 ? 1, 6 1000010 ? 2,
5 1000100 ? 3, 4
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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
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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,
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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)
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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
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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)

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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
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Multi user Communication model for UWB
1 or 0
Signature 1
transmit
Signature 0

3 or -3
receive
Simple receiver structure!
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Transmitter / receiver(ref Tomoaki Ohtsuki)
data
Data selector
encoder
impulse
Tunable delay line
sequence encoder

hard limiter
correlator
splitter
-
correlator
decoder
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2 problems
User 1 1100000 or 0110000 11 User 2
1000010 or 0100001 correlation 2 ! User 3
1000100 or 0100010 0 1
01000011000010 correlation 2 !
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Super Optical Orthogonal Codes
AUTO CORRELATION
CROSS CORRELATION
SUPER-CROSS CORRELATION
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note
For situation A
or
0
A sequence might look like y 0 y y 0 0 ? ? ?
For situation C
or
another
A sequence might look like y y y y y ? ? ?
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Super-cross correlation
y y
? 1
x
y y
? 1
x
Y could be shifted version
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Property shift sensitive
1100000 1010000 is a S-OOC 1001000
shifted code 1000001 1000010 is not a
S-OOC 1000100
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conclusions

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

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conclusions
We showed - different signalling methods -
problems with OOC code design Future
performance calculations
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Application optical Multi-access
All optical transmitter/ receiver is fast Use
signature of OOC to transmit information
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signature
Other users
noise
OPTICAL matched filter TRANSMITTER/RECEIVER
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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
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• We want
• weight w large high peak
• side peaks ? 1
• for other signatures cross correlation ? 1