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

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

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)

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.

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

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

Successive Decoding The last user sees a BEC channel a1 a1 X n Yn a1 e a1 1 1 1 - a1

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)

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

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

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

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

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.

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.

Degree Distribution Design for Z-channels Target alpha = 0.2731 0.371989 0.380872 0.389073 0.396646 0.403669 0.410180 0.416236 0.421862 0.427106 0.431998 0.436565 0.440834 0.444828 0.448566 0.452070 0.455357 0.458442 0.461342 0.464062 0.466625 0.469032 0.471299 0.473436 0.475448 0.477345 0.479132 0.480820 0.482412 0.483915 0.485333 0.486675 0.487941 0.489139 0.490271 0.491342 0.492354 0.493311 0.494217 0.495076 0.495891 0.496662 0.497393 0.498086 0.498743 0.499366 0.499957 0.500516 0.501047 0.501550 0.502027 Capacity 0.532439 Rates l(x) = 0.27571047 x + 0.15042832 x2 + 0.18575028 x3 + 0.38811080 x11

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

Simulation of Codes on the Z Channel

Simulation of Codes on the Z Channel

Degree Distribution Design for Z-channels -0.051744 -0.026240 -0.003107 0.017941 0.037158 0.054751 0.070872 0.085686 0.099326 0.111926 0.123588 0.134406 0.144449 0.153785 0.162473 0.170558 0.178103 0.185163 0.191761 0.197937 0.203717 0.209134 0.214218 0.218996 0.223478 0.227695 0.231663 0.235397 0.238912 0.242225 0.245348 0.248085 0.250882 0.253517 0.256000 0.258348 0.260559 0.262657 0.264637 0.266504 0.268278 0.269951 0.271536 0.273036 0.274453 0.275797 0.277072 0.278278 0.279422 0.280506 0.281533 0.282507 0.283431 0.284306 0.285137 0.285925 0.286673 0.287384 0.288059 0.288703 0.289321 0.289910 0.290461 0.290986 0.291486 0.291964 Capacity 0.321928 Rates Target alpha = 0.5

Another code on the Z channel

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

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

Simple codes for Demo (2) Source 1 1 2 3 4 1 Rate 1/4 Source 2 1 2 3 4 5 6 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

OR Channel Channel capacity for a two user OR channel

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

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

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.

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

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