Information-Theoretic Study of Optical Multiple Access

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Information-Theoretic Study of Optical Multiple
Access
Jun Shi and Richard D. Wesel UCLA 01/14/05
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Princetons 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
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The (4,101) code

• Asynchronous, coordinated.
• Each bit takes 4 time/wavelength slots.
• 1s density per user (chip level) 4/8080.005
• 1s density per user (bit level) 1/2.
• Upper bound on BER

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The Z-channel

• All other users are treated as noise.
• Each user sees a Z-channel
• Throughput (Sum-rate)

1
1
Pe
0
0
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Double-Interference

• 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

p
User 1
receiver
2p
2p
2p
User 2
User 3
User 4
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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 1s density be 0.005, the chip density
  of Princetons scheme, we can plot throughput vs.
  of users.

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Random Codes

• In Princetons approach, prime codes are assigned
  a priori, which requires coordination.
• We can assign the patterns randomly.

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Prime code constraint

• Princetons 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.

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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.

Decoder
LDPC Encoder
Data 1
User 1
Data 2
Decoder
LDPC Encoder
User 2
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Successive Decoding

• We can decode the first user by treating others
  as noise, then the first users 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.

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A 3-user example
R1
1
User 1
User 2
1
User 3
1
R3
R2
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
0 1 1 1 1 0 1 1
User 1
1
1
User 2
User 3
0
0
Receiver
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A 3-user example
R1
1
User 1
User 2
1
User 3
1
R3
R2
0 1 0 0 1 0 0 1
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
0 e 1 1 e 0 0 e
User 1
1
1
User 2
e
User 3
0
0
Receiver
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A 3-user example
R1
1
User 1
User 2
1
User 3
1
R3
R2
0 1 0 0 1 0 0 1
0 0 0 1 0 0 1 0
? ? ? ? ? ? ? ?
0 e 1 e e 0 e e
User 1
1
1
User 2
e
User 3
0
0
Receiver
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A 3-user example
R1
1
User 1
User 2
1
User 3
1
R3
R2
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
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Density Transformer

• To achieve capacity and apply successive
  decoding, a key thing is to get the right ones
  density. This is an item for future work under
  Phase II.

Density Transformer
LDPC Encoder
½
½
Source 1
p1
½
p2
Density Transformer
LDPC Encoder
½
Source 2
p3
½
LDPC Encoder
Density Transformer
½
Source 3
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Synchronization

• In successive decoding, the receiver only needs
  to be synchronized to one user at a time.

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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.

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A 2-user example
x11, x12, x13,
2 looks
User 1s clock
y11, y12, y13,
Receiver
x21, x22, x23,
y21, y22, y23,
User 2s clock
Receivers clocks
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Joint Decoding
LDPC Encoder
Data 1
User 1
Joint Decoder
Data 2
LDPC Encoder
User 2