Access Codes

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Access Codes

• A.J. Han Vinck

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content

• Permutation Codes
• Random Access bounds
• 3. Random Access Codes
• 4. Optical Orthogonal Codes

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PERMUTATION CODES

• Impulsive noise, broadcast, background
• Random access codes

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Typical noise situation
Narrow band (permanent) Broad band (single
impulse) Broad narrow band



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idea
Transmit messages as ? sequences (code words)
of length M where all M symbols are
different ? minimum distance ( of
differences) D
Example M 3 D 2 Code 123 312 231
132 321 213
f
time
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Communication structure example
( 3,2,1 )
encoder
modulator
message
t
M-FSK or PPM
f
Time and frequency diversity !
t

• gt T
• gt T
• gt T

3 Energy detectors
f
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Non-coherent detection (FFT)
Envelope detector
y1
filter matched to f1
1
Quantize gt Th 1 lt Th 0
Envelope detector
y2
filter matched to f2
0
X
???
???
Envelope detector
yM
0
filter matched to fM
sample
1 0 0 0
Detect Presence of code sequence
0 0 1 0
0 1 0 0
0 0 0 1
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Non-coherent detection Quadrature receiver
using correlators
()2
cos2?fit
?
sin2?fit
()2
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Non-coherent detection performance

• Decoder outputs sequence at minimum distance
• Error if noise generates valid sequence
• Advantage time and frequency diversity
• robusts against Broad- and narrowband noise

1 0 0 0
1 0 1 0
Detect Presence of code sequence
0 0 1 0
0 0 1 0
0 1 0 0
1 1 1 1
0 0 0 1
0 0 1 1
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2 Code words M 5 at D 3
C1 1,2,4,3,5 C2 1,2,3,5,4
frequency
time
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narrow band noise
C1 1,2,4,3,5
frequency
Narrow band
time
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broad band noise (impulse)
broad band
C1 1,2,4,3,5
frequency
time
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background noise
C1 1,2,4,3,- C2 1,2,-,5,4 Both code words 4
agreements!
frequency
insert delete
time
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Performance minimum distance decoding

• NOTE Sequences have minimum distance D
• Errors agree with sequences in only 1 positions
• gt D-1 errors are needed to create another
  sequence
• If E correct symbols disappear due to background
  noise
• gt D-1-E errors are needed to create decoding
  errors

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Upperbound on cardinality

• Order codewords specified by set of M-D symbols
• Set 1 1, 2, x, x, x 2, 1, x, x, x
• Set 2 x, 1, x, 2, x x, 2, x, 1, x
• Set 3 x. 1, 2, x, x x, 2, 1, x, x
• etc.
• For distance D, set constains ? (M-D)! D
  codewords
• There are different sets.
• Hence

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Code parameters (1)

• We showed that
• Q1 when do we achieve equality?
• Q2 if not, what is the upperbound
• References
• -Ian Blake, Permutation codes for discrete
  channels (1975, IT)
• -P. Frankl and M. Deza, On the max. of
  Permutations with
• given Max. Or Min. Distance (19977, Jrnl of
  Comb. Th.)

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Code parameters (2)

• Simple code constructions D 2, 3, M
• all cases M lt 7 solved
• Interesting cases left ?

D 2 3 4 5 6 7 6 x x x 18
x C 18 is the Klöve (2000) result 7 x x
? ? x x 8 x x ? ? x x
9 x x ? ? ? x 10 x x ? ?
? ?
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Simple codes D M

• Cyclic permutation of M integers has D M
• C M! / (M-1)! M
• Example 1 2 3
• 3 1 2
• 2 3 1

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Simple codes D 2

• The code with all M! permutation has D 2
• C M! / (2-1)! M!
• Proof
• All symbols are different
• Codewords differ in at least 2 positions
• x x a x x x b x
• x x b x x x a x

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Simple codes D M-1

• Construction for prime P example M 3
• - starting sets B 0 1 2
• 2B 0 2 1
• - add constant vector
• B 0 ? 0 1 2 2B 0 ? 0 2 1
• B 1 ? 1 2 0 2B 1 ? 1 0 2
• B 2 ? 2 0 1 2B 2 ? 2 1 0
• In general C M!/(M-2)! M(M-1)

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3
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Random access codes
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Optical 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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Time division central control
N users
????
inefficient, when small active
users synchronous easy
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Code division synchronized
0
Code division efficient, but complex
1
signature
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Several possibilities
A
or
0
B
or
shifted
C
or
another
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Superimposed codes
? T code words should not produce a valid code
word
T words Valid word
N
n
? ? N ? ?
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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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Example packet transmission
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(sync) Binary access model (contd)
In Out
OR
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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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Maximum throughput
Normalized SUM throughput 0.69
bits/channel use Note we have M-channels
available 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
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M-FSK multi-access (cont)

• Sender 1 1,2, ? ? ?, M
• Sender 2 1,2, ? ? ?, M
• ? ? ? Y 2M -1
• Sender N 1,2, ? ? ?, M SUM CAPACITY ? M-1
  bits/transm.
• Example for M 3 input 1, 2, 3 output
  (1), (2), (3), (1,3), (1,2), (2,3), (1,2,3)
• Simple time sharing gives R log2
  M bits/transm.

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M-FSK multi-access (cont)

• Capacity obtaining group time sharing!
• User M2 M 3 (2bits/tr) M
  4 (3 bits/tr.)
• I 0 1 0 1 0 1
• I1 0 2 0 2
• I2 0 3
• Output (0),(1) (0,1),(0,2),(1,2)
  (0,1),(0,2),(1,2)
• Y (0) (0), (1,2,3)
• (0,1,2),(0,1,3),(0,2,3)

Group I
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Frequency hopping 1 (many variations)
1 0
f
Symbol time
Hopping period
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Frequency hopping 2
1
0
f

t
Symbol time
Hopping period
Different hopping patterns
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Frequency hopping 3
f
Slow hopping
t
01 11 10 00
f
fast hopping
t
01 11 10 00
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advantages

• FH avoids tone jammers
• FH applies usually noncoherent modulation
• the hopping span can be very large
• FH is an avoidance system does not suffer on
  near-far effect!

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Non-coherent detection
Envelope detector
y1
filter matched to f1
1
Quantize gt Th 1 lt Th 0
Envelope detector
y2
filter matched to f2
0
X
???
???
Envelope detector
yM
0
filter matched to fM
sample
1 0 0 0
Detect Presence of signature
0 0 1 0
0 1 0 0
0 0 0 1
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Transmission model
Every user has a signature of length n Example n
4, M 3 3 2 2
1 1 send signature 0 send no pulses
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Channel model
tr 1
rec 1
tr 2
rec 2
????
OR
????
rec T
tr T
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M-ary random access codes
T words Valid word
T
N
n
OR of ? T signatures does not produce a non
transmitting valid signature
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Example
3 1 1 2 1 1 1 3 1 1 2 1 1 1 3 1 1 2
3 users produce (3,2,1), (1,3), (1) No
other signature covered Signatures uniquely
detectable!
N
M
N ? M(M-1)
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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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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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Error correcting codes as Multi-user access codes
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M-ary Error Correcting Codes
minimum number of differences dmin n -
maximum number of agreements
No covering for T users if T ( n - dmin )
lt n i.e. At least one position not covered!
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example Reed Solomon
Code property M 8, n 7, k 4, dmin 4
Then T ( 7 4 ) lt 7 or T ? 2 Conclusion
Total number of users N 84 T ? 2 active
users can be detected Data rate R ? 2/Mn 2/56

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Permutation codes
Transmit signatures as ? sequences (code words)
of length M where all M symbols are
different ? minimum distance ( of
differences) D
Example M 3 D 2 Code 123 312 231
132 321 213
f
time
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Interference property
For minimum distance D M-1
difference C M(M-1) Maximum
interference M - D 1 agreement CONCLUSIO
N lt M users uniquely detectable always one
unique position left
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Code shortening
Minimum distance M-1 length M Reducing word
length to n, reduced minimum distance to
n-1! CHECK!
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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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Code partitioning A lt n M users D n-1
n
n
n
???
M
???
???
???
User 1
User 2
User M-1
differences n-1 ? agreements (
interference ) 1 ? Pe 0! ? lt n active users
can never create a non participating
signature! Normalized sum rate same as single
user!
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Simple code using permutation codes(Titlebaum)

• M prime calculations modulo M
• M-1 Users 0 lt Ilt M
• Message 0 ? m lt M
• n ? M
• encode message m for user I gt o as
• CI(m) ( m , I ) 1 1 ? ? ? 1
• 0 1 ? ? ? n-1
• signature
• CI(m) is a permutation code word for a code with
  D n-1

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example

• M 5, n 3 user 2
• Decoding for user I check for I m

( 3, 2 ) 1 1 1 ( 3, 0, 2 )
0 1 2
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Error probability for n ? A ? M active users
Note R(TDMA)
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