Download - Joint Decoding on the OR Channel
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Joint Decoding on the OR Channel
Communication System Laboratory
UCLA Graduate School of Engineering - Electrical Engineering ProgramUCLA Graduate School of Engineering - Electrical Engineering Program
Communication Systems Laboratory
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Joint Decoding Architecture
Decoding is done by performing belief propagation over the receiver graph
Performs well at very high Sum-Rates
High decoding complexity for a large number of users
Requires either bit synchronism or timing knowledge of all the transmitters
Encoder 2
Encoder 1
Encoder N
Interleaver 1
Interleaver 2
Interleaver N
Same CodeRandomly picked (different with very high probability)
ElementaryMulti-User
Decoder(Threshold)
Interleaver 1
Interleaver N
De-Interl 1
De-Interl N
DecoderDEC-1
DecoderDEC-N
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Joint Decoding Results
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.910
-6
10-5
10-4
10-3
10-2
Sum-Rate
BE
R
Joint DecodingJoint Decoding With Block Code
6 users
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Turbo Codes for the OR Channel
Communication System Laboratory
UCLA Graduate School of Engineering - Electrical Engineering ProgramUCLA Graduate School of Engineering - Electrical Engineering Program
Communication Systems Laboratory
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Parallel Concatenated NL-TCs Increases complexity and latency with respect
to NLTC. Capacity achieving.
Design criteria: An extension of Benedetto’s uniform
interleaver analysis for parallel concatenated non-linear codes has been derived.
This analysis provides a good tool to design the constituent trellis codes.
NL-TC
Interleaver NL-TC
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Parallel Concatenated NL-TCs The uniform interleaver analysis proposed by
Benedetto, evaluates the bit error probability of a parallel concatenated scheme averaged over all (equally likely) interleavers of a certain length.
Maximum-likelihood decoding is assumed. However, this analysis doesn’t directly apply to our
codes: It is applied to linear codes, the all-zero codeword is
assumed to be transmitted. The constituent NL-TCM codes are non-linear, hence all the possible codewords need to be considered.
In order to have a better control of the ones density, non-systematic trellis codes are used in our design. Benedetto’s analysis assumes systematic constituent codes.
An extension of the uniform interleaver analysis for non-linear constituent codes has been derived.
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Results
6 users
• Parallel concatenationof 8-state, duo-binaryNLTCs. • Sum-rate = 0.6• Block-length = 8192• 12 iterations in message-passing algorithm
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OR Channel when treating other users as noise:
Can we provide the same sum-rate and performance for any number of users?
Communication Systems Laboratory
UCLA Graduate School of Engineering - Electrical Engineering ProgramUCLA Graduate School of Engineering - Electrical Engineering Program
Communication Systems Laboratory
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Theoretical answer
Theoretically: YES.
2 3 4 5 6 7 8 9 10 11 120
0.2
0.4
0.6
0.8
1
Number of users
Cap
acity
Sum rate comparison
Other users as noise
Joint decodingp
1 = 0.5
ln(2)
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Our Experience: NL-TCM NL-TCM: looked like we don’t have a limit in
the number of users. Results for 100 user case:
And we were right in that case.
0.498370.0069440.27781/360
0.494890.0068750.251/400
BERpSum-rateRate 64.54 10
79.45 10
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Comparison: Number of output bits n0 & number of ones M vs number of users
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Comparison: n0(N) & M(N)
Number of ones is increasing
increasing
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Comparison: n0(N) & M(N)
Same number of ones.Ungerboeck’s extension: moving deeper into the trellis.
increasing
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Comparison: n0(N) & M(N)
Best code at this point….
All branches different
increasing
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Comparison: n0(N) & M(N)
is the best code at this point.
increasing
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v=6
N n0 SR BER
100 344 0.291 0.4777
300 1000 0.3 0.4901
900 3000 0.3 0.4906
1500 5000 0.3 0.4907
15000 50000 0.3 0.4908
51.1046 1051.2157 1051.2403 1051.2508 1051.2508 10
We can support any number of users in the OR-MAC with basically same decoding complexity for each user, and practically same performance.
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Moreover:
Unused bits(Bunch of zeros)
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Moreover: Denote N* the minimum number of users for
which n0 > M.
For every N greater than N* we can use the same encoder and decoder
Design for N* .
Encoder Add Zeros Interleaver
De-InterleaverDelete unused bits
Decoder
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Limitation for Non-linear Turbo Codes With 8-state constituent non-linear trellis
codes:
16-state constituent non-linear trellis codes should be used for more than 24 users.
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Results
6 users
• Parallel concatenationof 8-state, duo-binaryNLTCs. • Sum-rate = 0.6• Block-length = 8192• 12 iterations in message-passing algorithm
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With 16-state constituent NL-TCs
For 50 users:For 100 users:
0.4930 0.4965
Around 50 users should be supported.
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Code design for the Binary Asymmetric Channel
Communication System Laboratory
UCLA Graduate School of Engineering - Electrical Engineering ProgramUCLA Graduate School of Engineering - Electrical Engineering Program
Communication Systems Laboratory
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Model for Optical MAC
User 1 User 2 User N
Receiver
1X 2X NX
1 2( 0) ( , , , )NP Y f X X X
1,
( 0) 0,
,m
P Y
if all users transmit a 0
if one and only one user transmits a 1
if m users transmit a 1 and the rest a 0
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Model The can be chosen any way, depending on the
actual model to be used. Examples:
Coherent interference:
constant
'm s
m
2 , 0, 2m m
2 , 2mm m
( 2)( )(1 ), 2mm e m
2
1
, 2
0,2 ,
( 1/ 2)
i
mj
mi
i
P e m
U i
threshold
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Achievable sum-rates n users with equal ones density p. Joint Decoding
Treating other users as noise – Binary Asymmetric Channel:
0 2
0 1
( ) max 1 1
1, 0
n nn j n kj k
JD p j kj k
n nSR n H p p p p H
j k
0
1
0
1
YiX
p
1 p
1
1
11
0
11
11
11 1
11
nn kk
kk
nn kk
kk
np p
k
np p
k
( ) max 1 1 1BAC pSR n n H p p p H p H
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Simulations
0 20 40 60 80 100 120 140 160 180 2000.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95Sum-rate: Coherent interference
number of users
Sum
-rat
e
joint decoding
other users noise
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Simulations
JD : Joint DecodingOUN: Other Users Noise
0 20 40 60 80 100 120 140 160 180 2000.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Model: m
= + (-)(1-e-(m-2)) for m>=2
number of users
Sum
-rat
eJD = 0.23, = 0.001, = 0.05
JD = 0.16, = 0.0001, = 0.01
OUN = 0.23, = 0.001, = 0.05
OUN = 0.16, = 0.0001, = 0.01
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Simulations
0 20 40 60 80 100 120 140 160 180 2000.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Model: m
= ((m-2)) for m>=2
number of users
Sum
-rat
e
JD = 0.23, = 0.8
JD = 0.23, = 0.4
JD = 0.16, = 0.8
JD = 0.16, = 0.8
OUN = 0.23, = , 0.8
OUN = 0.23, = 0.4
OUN = 0.16, = 0.8
OUN = 0.16, = 0.4
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Simulations
0 20 40 60 80 100 120 140 160 180 2000.4
0.5
0.6
0.7
0.8
0.9
1
Model: m
= , constant
number of users
Sum
-rat
eJD = 0.3
JD = 0.23
JD = 0.15
OUN = 0.3
OUN = 0.23
OUN = 0.15
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Simulations
0 20 40 60 80 100 120 140 160 180 2000.4
0.5
0.6
0.7
0.8
0.9
1
Model: 2 = ,
m = 0 for m>2
number of users
Sum
-rat
eJD = 0.4
JD = 0.3
JD = 0.23
JD = 0.15
OUN = 0.4
OUN = 0.3
OUN = 0.23
OUN = 0.15
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Achievable sum-rates n users with equal ones density p. Joint Decoding
Treating other users as noise – Binary Asymmetric Channel:
0 2
0 1
( ) max 1 1
1, 0
n nn j n kj k
JD p j kj k
n nSR n H p p p p H
j k
0
1
0
1
YiX
p
1 p
1
1
11
0
11
11
11 1
11
nn kk
kk
nn kk
kk
np p
k
np p
k
( ) max 1 1 1BAC pSR n n H p p p H p H
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Lower bounds for Sum-rate (1) Joint Decoding:
n users, with equal ones density p. Using Then:
For the worst case ( constant) the bound is actually very tight.
Note that for the case where
Also note that if (OR channel) , the lower bound becomes 1 for .
1/1 , 1/ 2np
1/ 2,1, ( ) max ( ) max ( ) max ( )
( ) 1 (1 log( ))
m m m mn SR n H f f H
f
'm s
max 1/ 2 max ( ) (max ) ( )m m M m m m m MH H H
1/ 2,1, ( ) max ( ) ( ) ( )M Mn SR n H f f H 0M
1/ 2
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Lower bounds for Sum-rate (2) Treating other users as noise:
n users, with equal ones density p. Using Then:
For the worst case ( constant) the bound is again very tight.
Note that if (OR channel) , the lower bound becomes log(2) for .
1/1 , 1/ 2np
21/ 2,1
( )
1 ( )max log(1/ ) log(1/ ) 1 log
( )
log(1/ ) ( ) log(1/ ) (1 )
( ) 1 log( )
BAC
M
M
M
SR n
g
g
H g H
g
'm s
0M 1/ 2
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Lower bound for different This figure shows the lower bounds and the actual sum-rates for
200 users for the worst case ( constant) .
'M s
'm s
JD : Joint DecodingOUN: Other Users Noise
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.4
0.5
0.6
0.7
0.8
0.9
1
Sum-rate lower bound depending on maximum m
Maximum m
sum
rat
e bo
und
/ op
timum
JD optimum
JD sum-rate bound
JD sum-rate for 200 users
OUN optimum OUN sum-rate bound
OUN sum-rate for 200 users
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Lower bound for Sum-rate For the Binary Asymmetric Channel, there is still a
strictly positive achievable sum-rate for any number of users.
For the Coherent Interference Model, the lower bound for the achievable sum-rate is around 48% (vs. 70% for Z-Channel).
Our target sum-rate for Non-linear trellis codes is 20% (vs. 30% for Z-Channel).
For Parallel Concatenated NL-TCs, our goal will be to achieve a sum-rate of 40%. These codes are under design.
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Metric for Z-Channel We use a ‘greedy’ definition of distance (not
the usual Hamming distance). Directional distance between two codewords
(denoted ) is the number of positions at which has a 0 and has a 1.
‘Greedy’ definition of distance:
1 2( , )Dd c c
2c1c
, , min , , ,i j j i D i j D j id c c d c c d c c d c c
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Design of NL-TCM for the BAC The metric of the Viterbi decoder for the BAC is:
Where and are the number of 0-to-1 and 1-to-0 transitions from the codeword and the received word , respectively.
The decoded codeword is: The directional distance between two codewords
(denoted ) is the number of positions at which has a 0 and has a 1.
Both directional distances are relevant when computing the probability of error.
A good criteria is maximize the minimum of both directional distances:
This is exactly the same criteria used for NL-TCM codes for the Z-Channel
01 10
1 1| log logn nX Y N N
arg min |nn n n
XX X Y
1 2( , )Dd c c1c 2c
01N 10NnX
nY
, , min , , ,i j j i D i j D j id c c d c c d c c d c c
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Design of NL-TCM for the BAC Hence, although the metrics in the Viterbi
decoder are different on the Z-Channel and the BAC, we use the same design technique for both cases.
However, since the achievable rate is lower for the BAC, our target rate will be lower.
We have designed codes for the Coherent Interference Model. Nevertheless, this design technique applies to any model for the 1-to-0 transition probabilies.
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Design of NL-TCM for the BAC Results (so far):
6-user MAC 128-state, rate 1/30 NLTC (Sum-rate = 0.2) Coherent interference model.
In order to achieve the same BER than in the OR Channel case:
The number of states had to be increased from 64 to 128 (Increase in complexity).
The sum-rate was decreased from 0.3 to 0.2. Simulations for larger number of users are running. Parallel concatenated NL-TCs are being designed for
this channel.
5
0.283169
0.062156
1.044 10BER