1. OCDMA Channel Coding Progress Report
UCLA Electrical Engineering Department-Communication Systems Laboratory
OCDMA Channel Coding Progress Report
Jun Shi
Andres I. Vila Casado
Miguel Griot
Richard D. Wesel

2. Main results this quarter
LDPC design for the Z Channel
Density Evolution
Design technique : Linear programming based on density evolution
Status
Simple codes for immediate hardware implementation

3. The OR Channel Bit synchronized
User 1
Output +
User 2
Bit synchronized
No multi-level detection (receiver detects only presence of light)
This means that the channel can be seen as an OR channel (1+ X = 1; 0 + X = X)

4. 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.

5. Successive Decoding: The Z-Channel
Successive decoding for n users:
User with lowest rate is decoded first
Other users are treated as noise
The decoded data of the first user is used in the decoding of the remaining users
First user sees a “Z-channel”
Where ai = 1-(1-p)n-i is the probability that at least one of the n-i remaining users transmits a 1

6. Successive Decoding: Z with erasures
Intermediate users see the following channel assuming perfect previous decoding
Where bi = 1-(1-p)i-1 is the probability that at least one of the I-1previously decoded users transmitted a 1 Xi
1 - a1 Yi bi
ai (1-bi) e bi 1 1
1-bi

7. Successive Decoding The last user sees a BEC channel a1 a1 X n Yna1 e a1 1 1
1 - a1

8. Successive Decoding For two users
User 1 sees a Z-channel
User 2 sees the following channel
When BER1 is small, the channel can be approximated to a BEC X2 Y2
1 - p
p(1-BER1)
p(BER1) e
p(1-BER1)+(1-p)BER1 1 1
(1-p)(1-BER1) + p(BER1)

9. A 3-user example ? 1 R1 1 User 1 User 2 User 3 1 1 R3 R2 User 1 1 11
User 1 1 1
User 2
User 3
Receiver

10. A 3-user example 1 ? e R1 1 User 1 User 2 User 3 1 1 R3 R2 User 1 1 11 ? e
User 1 1 1
User 2 e
User 3
Receiver

11. A 3-user example 1 ? e R1 1 User 1 User 2 User 3 1 1 R3 R2 User 1 1 11 ? e
User 1 1 1
User 2 e
User 3
Receiver

12. A 3-user example 1 R1 1 User 1 User 2 User 3 1 1 R3 R2 User 1 User 21
User 2
User 3
Receiver

13. LDPC codes
There are essentially two elements that must be designed when building and LDPC code :
Degree distributions
Edge positioning
This group has worked a lot on the edge positioning problem, thus we have all the necessary designing tools
Also, the degree distribution design problem for symmetric channels (AWGN, BSC, BEC) has been thoroughly studied by many authors.

14. LDPC codes for asymmetric channels
Density evolution is a concept developed by Richardson et al. that helps predict the LDPC code behavior in symmetric channels
This concept can be used in the design of good degree distribution in many different ways
Wang et al. recently generalized Richardson’s result for asymmetric channels.
We took all these concepts and tried several different linear programming based algorithms, and built a program that efficiently designs degree distributions for any binary memoryless channel.
Given a channel, the program tries to maximize the rate while maintaining an acceptable performance.

15. Degree Distribution Design for Z-channels
Target alpha =
Capacity
Rates
l(x) = x x x x11

16. Max variable node degree
Code Characteristics
Max variable node degree
Total number of edges
MD 12 12
7289
MD 13 13
7305
MD 12 v2
7413
Wang
7316
RCEV 10
7182

17. Simulation of Codes on the Z Channel

18. Simulation of Codes on the Z Channel

19. Degree Distribution Design for Z-channels
Capacity
Rates
Target alpha = 0.5

20. Another code on the Z channel

21. Simple codes
In order to have a hardware demo working for the May meeting, some very simple codes were produced.
This demo consists of two transmitter and two receivers
Both receivers decode the information independently

22. Simple Codes for Demo
Short codes have been designed for a simple demo for 2 users
These were chosen to be as simple to encode and decode as possible
Each bit is encoded separately
Bit synchronism is assumed, blocked asynchronism is allowed
Coordination is required
These codes are error free

23. Simple codes for Demo (2)
Source 1 1
Rate 1/4
Source 2
Rate 1/6
Receiver 1 looks for position of 0 (which always exists)
If 1 or 2, decide 1
If 3 or 4, decide 0
Sum Rate 5/12
Receiver 2 looks for
FIRST position of 0.
If 1, 3 or 5, decide 1
If 2, 4 or 6, decide 0
Worst Case :
block i
block i+1

24. OR Channel
Channel capacity for a two user OR channel

25. Density Transformer First approach
After the density transformer each user transmits a one with probability p

26. Density Transformer
The output bits of a binary linear code (such as LDPC codes) are equally likely 1 or 0 if the information bits are also equally likely
Thus we need to change this distribution in order to avoid such a large interference between users
Transmitter: Non-uniform mapper
1) Mapper
2) Huffman Decoder
Receiver: Use soft Huffman encoder

27. Future work in Successive Decoding
Design degree distributions that work well for the specific channels that the different users see
Develop the distribution transformer
Test this successive decoding idea with a simulation.

28. Joint decoding
Joint decoding will be done on graphs that look like this one
The formulas necessary in this joint decoding have already been found and simplified

29. Future work on Joint decoding
Develop a density evolution based design tool to construct good LDPC codes for joint decoding
Test and compare the performance of joint decoding and successive decoding
Also evaluate and compare the complexity of both techniques