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Information-Theoretic Study of Optical Multiple Access. Jun Shi and Richard D. Wesel UCLA 01/14/05. Princeton’s Scheme. Princeton uses a (4,101) 2D prime code. wavelength. 2 3 4 5 6 7 8 9 10 11 99 100 101 Time. User 1 User 2 User 3. - PowerPoint PPT Presentation
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Information-Theoretic Study of Optical Multiple Access
Jun Shi and Richard D. Wesel
UCLA
01/14/05
Princeton’s Scheme
• Princeton uses a (4,101) 2D prime code.
1 2 3 4 5 6 7 8 9 10 11 99 100 101Time
1
2
3
4
User 1
User 2
User 3
wav
elen
gth
The (4,101) code
• Asynchronous, coordinated.• Each bit takes 4 time/wavelength slots.• 1’s density per user (chip level) =4/808≈0.005• 1’s density per user (bit level) =1/2.• Upper bound on BER:
length codeword :
users of # :
code theof weight :
2 ,)1(
15.0
11
T
T
K
wi
iKi
N
K
w
N
wqqq
i
KPe
The Z-channel• All other users are treated as noise. • Each user sees a Z-channel
• Throughput (Sum-rate) :
0
1
0
1Pe
hs wavelengtof # :
length codeword:
channel-Z ofcapacity :
users of # :
N
N
C
K
NN
KC
T
z
T
z
• Due to asynchronism, in the worst case, a one from an interferer affects two bits of the desired users.
• Asynchronous channel is very complicated. The exactly capacity is still under investigation, but here is an approximation: synchronous double-interference
Double-Interference
User 1
User 2
2p
p
2p 2p
User 1
User 2 User 3 User 4
receiver
The Ideal case
• Under perfect synchronism and with joint decoding (other users are not noise but information), the throughput is a constant equal to 1bit/transmission.
• Let input 1’s density be 0.005, the chip density of Princeton’s scheme, we can plot throughput vs. # of users.
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
users
thro
ugh
pu
t, (
bit/
tran
smis
sion
)
Throughput Comparison
Ideal case, p=0.005double interference, p=0.005Princeton Scheme
Random Codes
• In Princeton’s approach, prime codes are assigned a priori, which requires coordination.
• We can assign the patterns randomly.
0 20 40 60 80 100 12010
-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1 Probability of Error
users
Bit
err
or
rate
Prime codeRandom Code
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2
0.25Throughput
users
thro
ugh
pu
t, (
bit/
tran
smis
sion
)
Prime coderandom code
Prime code constraint
• Princeton’s scheme is slightly better at small number of users while random code shows advantages with large number of users.
• This is due to the requirement that a prime codeword has at most one slot per wavelength, increasing the probability of collision.
Error Correcting Codes
• Prime codes do not correct errors. To achieve capacity, error-correcting codes are required.
• Encoding and decoding can be done in FPGA boards. This is an item for future work in Phase II.
User 1LDPCEncoder
User 2 LDPCEncoder
DecoderData 1
Data 2Decoder
Successive Decoding
• We can decode the first user by treating others as noise, then the first user’s ones become erasures for the other users. Proceed in this way until finish decoding all the users.
• This is called successive decoding. For binary OR channel, this process does not lose capacity as compared to joint decoding.
A 3-user example
1321 RRR
R1
R2R3
1
1
1
User 1
User 2
User 3
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
0 1 1 1 1 0 1 1
1
0
1
0
User 1
User 2
User 3
Receiver
A 3-user example
1321 RRR
R1
R2R3
1
1
1
User 1
User 2
User 3
0 1 0 0 1 0 0 1
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
0 e 1 1 e 0 0 e
1
0
1
0
e
User 1
User 2
User 3
Receiver
A 3-user example
1321 RRR
R1
R2R3
1
1
1
User 1
User 2
User 3
0 1 0 0 1 0 0 1
0 0 0 1 0 0 1 0
? ? ? ? ? ? ? ?
0 e 1 e e 0 e e
1
0
1
0
e
User 1
User 2
User 3
Receiver
A 3-user example
1321 RRR
R1
R2R3
1
1
1
User 1
User 2
User 3
0 1 0 0 1 0 0 1
0 0 0 1 0 0 1 0
0 1 1 0 0 0 0 0
0 1 1 1 1 0 1 1
User 1
User 2
User 3
Receiver
Density Transformer
• To achieve capacity and apply successive decoding, a key thing is to get the right one’s density. This is an item for future work under Phase II.
Source 1Density Transformer
LDPCEncoder
Source 2Density Transformer
LDPCEncoder
Source 3 Density Transformer
LDPCEncoder
½
½
½
p1
p2
½
½
½
p3
Synchronization
• In successive decoding, the receiver only needs to be synchronized to one user at a time.
Multiple looks
• To further increase the throughput, we should not treat other users as interference but as useful information.
• We want the receiver to align with each of the users, not just one user.
• This can be done in a star network where the receiver has all the information.
A 2-user example
User 1’s clock
User 2’s clock
Receiver
x11, x12, x13, …
x21, x22, x23, …
y11, y12, y13, …
y21, y22, y23, …
Receiver’s clocks
2 looks
Joint Decoding
User 1LDPCEncoder
User 2 LDPCEncoder
Joint Decoder
Data 1
Data 2
