UCLA Graduate School of Engineering Electrical Engineering Program

UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Joint Decoding on the OR Channel Communication System Laboratory

Joint Decoding Architecture Same Code Randomly picked (different with very high probability) Encoder 1 Interleaver 1 Encoder 2 Interleaver 2 Encoder N Interleaver N Decoder DEC-1 Decoder DEC-N De-Interl 1 Interleaver 1 De-Interl N Interleaver N Elementary Multi-User Decoder (Threshold) n Decoding is done by performing belief propagation over the receiver graph n Performs well at very high Sum. Rates n High decoding complexity for a large number of users n Requires either bit synchronism or timing knowledge of all the transmitters

Joint Decoding Results 6 users

UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Turbo Codes for the OR Channel Communication System Laboratory

Parallel Concatenated NL-TCs n Increases complexity and latency with respect to NLTC. n Capacity achieving. NL-TC Interleaver NL-TC n Design criteria: n An extension of Benedetto’s uniform interleaver analysis for parallel concatenated non-linear codes has been derived. n This analysis provides a good tool to design the constituent trellis codes.

Parallel Concatenated NL-TCs n 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. n Maximum-likelihood decoding is assumed. n However, this analysis doesn’t directly apply to our codes: n n n 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.

Results • Parallel concatenation of 8 -state, duo-binary NLTCs. • Sum-rate = 0. 6 • Block-length = 8192 • 12 iterations in message-passing algorithm 6 users

UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory 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

Theoretical answer n. Theoretically: YES.

Our Experience: NL-TCM n NL-TCM: looked like we don’t have a limit in the number of users. n Results for 100 user case: Rate Sum-rate p BER 1/360 0. 2778 0. 006944 0. 49837 1/400 0. 25 0. 006875 0. 49489 n And we were right in that case.

Comparison: Number of output bits n 0 & number of ones M vs number of users

Comparison: n 0(N) & M(N) Number of ones is increasing

Comparison: n 0(N) & M(N) Same number of ones. Ungerboeck’s extension: moving deeper into the trellis. increasing

Comparison: n 0(N) & M(N) increasing All branches different Best code at this point….

Comparison: n 0(N) & M(N) increasing is the best code at this point.

v=6 N n 0 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 150000 0. 3 0. 4908 We can support any number of users in the OR-MAC with basically same decoding complexity for each user, and practically same performance.

Moreover: Unused bits (Bunch of zeros)

Moreover: n Denote N* the minimum number of users for which n 0 > M. n For every N greater than N* we can use the same encoder and decoder n Design for N*. Encoder Add Zeros Decoder Delete unused bits Interleaver De-Interleaver

Limitation for Non-linear Turbo Codes n With 8 -state constituent non-linear trellis codes: n 16 -state constituent non-linear trellis codes should be used for more than 24 users.

Results • Parallel concatenation of 8 -state, duo-binary NLTCs. • Sum-rate = 0. 6 • Block-length = 8192 • 12 iterations in message-passing algorithm 6 users

With 16 -state constituent NL-TCs For 50 users: For 100 users: Around 50 users should be supported.

UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Code design for the Binary Asymmetric Channel Communication System Laboratory

Model for Optical MAC User 1 User 2 User N Receiver 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

Model n The can be chosen any way, depending on the actual model to be used. n Examples: n Coherent interference: threshold n n constant

Achievable sum-rates n n users with equal ones density p. n Joint Decoding n Treating other users as noise – Binary Asymmetric Channel: 1 1 0 0

Simulations

Simulations JD : Joint Decoding OUN: Other Users Noise

Simulations

Simulations

Simulations

Achievable sum-rates n n users with equal ones density p. n Joint Decoding n Treating other users as noise – Binary Asymmetric Channel: 1 1 0 0

Lower bounds for Sum-rate (1) n Joint Decoding: n n users, with equal ones density p. n Using n Then: n n n For the worst case ( constant) the bound is actually very tight. Note that for the case where Also note that if bound becomes 1 for (OR channel) , the lower.

Lower bounds for Sum-rate (2) n Treating other users as noise: n n users, with equal ones density p. n Using n Then: n n For the worst case ( constant) the bound is again very tight. Note that if (OR channel) , the lower bound becomes log(2) for.

Lower bound for different n This figure shows the lower bounds and the actual sum-rates for 200 users for the worst case ( JD : Joint Decoding OUN: Other Users Noise constant).

Lower bound for Sum-rate n For the Binary Asymmetric Channel, there is still a strictly positive achievable sum-rate for any number of users. n For the Coherent Interference Model, the lower bound for the achievable sum-rate is around 48% (vs. 70% for Z-Channel). n Our target sum-rate for Non-linear trellis codes is 20% (vs. 30% for Z-Channel). n For Parallel Concatenated NL-TCs, our goal will be to achieve a sum-rate of 40%. These codes are under design.

Metric for Z-Channel n We use a ‘greedy’ definition of distance (not the usual Hamming distance). n Directional distance between two codewords (denoted ) is the number of positions at which has a 0 and has a 1. n ‘Greedy’ definition of distance:

Design of NL-TCM for the BAC n The metric of the Viterbi decoder for the BAC is: n Where n n 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: n This is exactly the same criteria used for NL-TCM codes for the Z-Channel

Design of NL-TCM for the BAC n 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. n However, since the achievable rate is lower for the BAC, our target rate will be lower. n We have designed codes for the Coherent Interference Model. Nevertheless, this design technique applies to any model for the 1 -to-0 transition probabilies.

Design of NL-TCM for the BAC n Results (so far): n 6 -user MAC n 128 -state, rate 1/30 NLTC (Sum-rate = 0. 2) n Coherent interference model. n n n In order to achieve the same BER than in the OR Channel case: n The number of states had to be increased from 64 to 128 (Increase in complexity). n 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.