Non-Linear Codes for Asymmetric Channels, applied to Optical Channels

1. Non-Linear Codes for Asymmetric Channels, applied to Optical Channels Miguel Griot

2. Outline • Motivation : Optical Channel, Uncoordinated Multiple Access. • Models and Capacity Calculation • Basic Model: the OR Channel • Treating other users as noise • Capacity loss vs. complexity reduction. • The Z channel • The need for non-linear codes • Optimal ones density • Non-linear Trellis Coded Modulation (NL-TCM) • Definition of distance in the Z-Channel • Characteristics of Trellis Codes • Design Technique • Results for 6-user OR-MAC & 100-user OR-MAC • Concatenation with High-Rate Block Codes • Results for 6-user OR-MAC • Conclusions • Future Work Oral Qualifying Exam - UCLA - Electrical Engineering

3. Motivation: Optical Channels, Multiple Access • Optical Channels: • provide very high data rates, up to tens to hundreds of gigabits per second. • Typically deliver a very low Bit Error Rate • Wavelength Division (WDMA) or Time Division (TDMA) are the most common forms of Multiple Access today. • However, they require considerable coordination. • Objective • Uncoordinated access to the channel. • Apply error correcting codes, in order to achieve the required BER. • Maximizing the rate at feasible complexity for optical speeds. Oral Qualifying Exam - UCLA - Electrical Engineering

4. User 1 User 2 User N Basic Model: The OR Multiple Access Channel (OR-MAC) • OR Channel model • Basic model that can describe the multiple-user optical channel with non-coherent combining • N users transmitting at the same time • If all users transmit a 0, then a 0 is received • If even one of them transmits a 1, a 1 is received • 0+X=X, 1+X=1 Receiver Oral Qualifying Exam - UCLA - Electrical Engineering

5. OR Channel: Theoretical characteristics • Achievable rate (Capacity): • The theoretical limits for the MAC, were given by Liao and Ahslwede. • In the case of the OR-MAC, the Theoretical Capacity is the triangle of all rate-pairs less than the maximum possible sum-rate, which is 1. • This sum-rate can be theoretically achieved by: • Joint Decoding. • Sequential decoding (requires coordination). • Time-Sharing or Wave-length sharing (requires coordination). Oral Qualifying Exam - UCLA - Electrical Engineering

6. Treating other users as noise: the Z-Channel • Joint Decoding and Successive Decoding are fully efficient in that one useful bit of information is transmitted per time-wavelength slot. • However, non of these are computationally feasible for optical speeds today. • A practical alternative is to treat all but a desired user as noise. • This alternative, while dramatically reducing the decoding complexity, looses up to 30% of full capacity, as we will see next. • When treating other users as noise in an OR-MAC, each user “sees” what is called the Z-Channel. • My research has been focused on the Z-Channel, resulting from the OR-MAC when treating other users as noise. Oral Qualifying Exam - UCLA - Electrical Engineering

7. 1 1 0 0 The Z-Channel • N users, all transmitting with the same ones density p: P(X=1)=p, P(X=0)=1-p. Focus on a desired user • If it transmits a 1, a 1 will be received. • If it transmits a 0, a 0 will be received only if all other N-1 users transmit a 0 Oral Qualifying Exam - UCLA - Electrical Engineering

8. Maximum achievable sum-rate, when treating other users as noise. • Information Theory tells us the optimal ones density to transmit for each user. • When the number of users tends to infinity, the optimal ones density tends to , which is also the optimal density for joint decoding. • In that case equal probabilities of 1 and 0 is perceived at the receiver. • Note that for a large number of users, the optimal ones density becomes very small. • Surprisingly, the maximum achievable sum-rate is always lower-bounded by ln(2)=0.6931 and tends to ln(2) when the number of users tends to infinity. Oral Qualifying Exam - UCLA - Electrical Engineering

9. Comparison of capacities Optimal ones densities: Oral Qualifying Exam - UCLA - Electrical Engineering

10. The need for non-linear codes • Linear codes provide equal density of ones and zeros in their output (p=0.5). • Most of the codes thoroughly studied in the literature are linear codes. • We observed in previous slide that, for linear codes, the achievable rate tends to zero as the number of users increase. • As the number of users increase, the optimal ones density tends to zero. • Non-linear codes with relatively low density of ones are required, to a achieve a good rate. • Only recently, there has been work on LDPC codes with arbitrary density of ones. Theoretical bounds are found to prove that these codes are capacity achieving under ML decoding. There is no design technique described for these codes. • Non-linear Trellis Coded Modulation • This work introduces a novel design technique for non-linear trellis codes with an arbitrary density of ones. • To my knowledge, it is the first work that addresses this task. Oral Qualifying Exam - UCLA - Electrical Engineering

11. Interleaver Division Multiple Access • One successful approach to uncoordinated multiple access is IDMA. • Every user has the same channel code, but each user’s code bits are interleaved by a randomly drawn interleaver, with very high probability of being unique. • The receiver is assumed to know the interleaver of the desired user. • With IDMA in the OR-MAC, a receiver should see the signal from a desired user, corrupted by a memory-less Z-Channel. • Performance obtained for a 6-user OR-MAC using IDMA, and for the corresponding Z-Channel were the same in my simulations. Oral Qualifying Exam - UCLA - Electrical Engineering

12. Trellis Codes Characteristics • Memory given by a state. In the trellis representation, for each state, and each possible input, an output value and the next state is given. • Generally next state and output given by generator polynomials. • Initial state: the all-zero state. • Zero Termination. • They are NOT capacity achieving • We are achieving around 30% of full capacity (around 43% of the achievable rate when treating other users as noise) • Low complexity compared to capacity achieving codes (Turbo-Codes, LDPC) • ML decoding: Viterbi Decoding State at time (t+1): State at time t: 0:010 1:100 Oral Qualifying Exam - UCLA - Electrical Engineering

13. Metric for the Z-Channel, for Maximum Likelihood decoding • Given a received word, the decoded codeword will be the one that maximizes , or given equally likely codewords . • For the Z-Channel, if one codeword has a 1 in a position where the received word r has a 0, then • Among the possible transmitted codewords (where there are no 1-to-0 transitions): where is the number of 0-to-1 transitions, and is the number of 0-to-0 transitions • Note that for the possible transmitted codewords is actually the number of zeros in the received word, which is the same for all possible codewords. • Now, , so the most likely codeword is the one that presents the less number of 0-to-1 transitions. Oral Qualifying Exam - UCLA - Electrical Engineering

14. Definition of distance for the Z-Channel • The distance between two codewords measures the likelihood that one transmitted codeword will be wrongly decoded as the other codeword. • In the Z-Channel, a transmitted 1 will always induce a received 1. • Define the directional Hamming distance as the number of ones that have to be added to a codeword so that all ones of codeword are present in the received word. • Example: • Now: • A Maximum-Likely (ML) decoder will always decode the codeword with larger Hamming weight. Oral Qualifying Exam - UCLA - Electrical Engineering

15. Definition of distance for the Z-Channel (2) • For two codewords with different Hamming weight, if the received word contains all ones from both codewords, the one with larger Hamming weight will be more likely than the codeword with smaller Hamming weight . • Only if is transmitted, an error will be produced in the decoder. • Then, the directional distance of interest is which is the larger of both directional Hamming distances. • For two codewords with equal Hamming weight, errors can be made in both directions, and both directional Hamming distances are equal, and equal to the maximum of both. • In any case, the proper pairwise deign metric is: • And the overall objective is to maximize: Oral Qualifying Exam - UCLA - Electrical Engineering

16. Greedy definition of distance • In a trellis code, the design is made branch-wise: for each state, and each input, we assign the next state, and the output. • Due to its non-linearity, last definition cannot be applied branch-wise. • It is impossible to tell from one branch, which codeword will have more Hamming weight. • Hence, we have to consider both branch-wise directional Hamming distances. • The safest branch-wise metric would be: • This is the definition used in our design of NL-TCM. Oral Qualifying Exam - UCLA - Electrical Engineering

17. State at time (t+1): State at time t: 0 1 Non-linear Trellis Coded Modulation • Desired density of ones p is given • Rate of the form: 1/n (1 input bit, n output bits). • states (represented by v bits) • 2S branches • Feed-forward encoder with 1 input: • Design: • Assign output values to the 2S branches of the trellis • Objective: Maximize the minimum distance (“greedy definition”) • Those outputs have to maintain the desired density of ones p. Oral Qualifying Exam - UCLA - Electrical Engineering

18. Assigning Hamming Weights • First step: assign Hamming weights to the output of each branch. • Using any of the definitions of distance given before, codewords with as equal Hamming weight between each other lead to better performance. • In the case of codewords with different Hamming weights, the worst-case performance will be driven by those codewords with smaller Hamming weight. • Criteria: assign as similar Hamming weights to the branches as possible, maintaining the density of ones as close to the desired density of ones as close to the desired p as possible. Oral Qualifying Exam - UCLA - Electrical Engineering

19. 0 1 0 1 Assigning Hamming Weights • Consider the following sub-graph: • There are S/2 of these sub-graphs. • Branches produced by an input bit equal to 0 for both states (or 1) go to the same state. • Define • In this subgroup of four branches, assign a Hamming weight of w+1 to i branches, and a Hamming weight of w to (4-i) branches. Oral Qualifying Exam - UCLA - Electrical Engineering

20. Assigning Hamming Weights, Examples: • 6-user OR-MAC, desired density of ones is . • n=20 : w=2, i=2 • 2 branches with Hw=2, 2 with Hw=3 (p=1/8). • n = 18 : w=2, i=1 • 3 branches with Hw=2, 1 with Hw=3 (p=1/8). • n = 17 : w=2, i=round(0.5) • 1 branch with Hw=3 and 3 with Hw=2 (p=0.132) • all with Hw=2 (p=2/17=0.118). • 100-user OR-MAC, • n = 400 : w=2, i=3 (p = 0.006875) • n = 360 : w=2, i=2 (p = 0.006944) Oral Qualifying Exam - UCLA - Electrical Engineering

21. Choosing all branches to have at least distance of 1 between each other • It would be desirable if possible, that all branches had at least distance of 1 between each other. • In the case where all branches have the same Hamming weight w then we can have up to different branches with a distance of at least 1. • If , it is possible. • In the case of branches with different Hamming weights, the computation is a little more complicated. • Two different codewords split at some point in their trellis paths, and their paths will not merge again until at least v+1 trellis sections after the split. • In case it is possible to have different output values for all branches, then the minimum distance of the code is lower-bounded by Oral Qualifying Exam - UCLA - Electrical Engineering

22. 0 A 1 0 B 1 Choosing all branches to have at least distance of 1 between each other • In case we need to repeat output values, we can allow the following branches to have same output value maintaining the bound • Again: consider the sub-graph • Branches in red (blue) can have same output value without affecting the minimum distance. • Consider two different paths, one traversing branch A, and the other traversing branch B at some trellis section. • They traverse at least v brancheswith different output values before that trellis section. • They traverse at least v brancheswith different output values after that trellis section. • Their distance is at least 2v. Oral Qualifying Exam - UCLA - Electrical Engineering

23. split merge merge split Ungerboeck’s rule • We have already assigned Hamming weights to the branches, and have enumerated all the possible output values in order to have different output values for all branches (allowing some branches to be equal according to previous slide) • Up to this point we have • We can further increase the minimum distance by applying Ungerboeck’s rule: maximize the distance between all splits and merges. • Remember that all output values had at least a Hamming distance of w. • For every two different codewords, their paths split and merge at least once, and there are at least v-1 branches between the split and the merge. • Hence: Oral Qualifying Exam - UCLA - Electrical Engineering

24. Extending Ungerboeck’s rule • One can extend Ungerboeck’s rule into the trellis. 0 1 Maximize split Oral Qualifying Exam - UCLA - Electrical Engineering

25. Extending Ungerboeck’s rule • One can extend Ungerboeck’s rule into the trellis. 0 0 1 1 0 1 Maximize Oral Qualifying Exam - UCLA - Electrical Engineering

26. Extending Ungerboeck’s rule • One can extend Ungerboeck’s rule into the trellis. Note that by maximizing the distance between the 8 branches, coming from a split 2 trellis section before, we are maximizing all groups of 4 branches coming from a split in the previous trellis section, and all splits. 0 0 1 1 Maximize 0 1 Oral Qualifying Exam - UCLA - Electrical Engineering

27. Extending Ungerboeck’s rule • The same idea applies for the merges, moving backwards in the trellis. • If we move h trellis sections forward from a split (including the split), and g sections backwards from a merge (including the merge), the new bound becomes: • Now, we have to compute the maximum possible values of h and g. Oral Qualifying Exam - UCLA - Electrical Engineering

28. Extending Ungerboeck’s rule • First, let’s compute the number of branches that need to have maximum distance between each otherto cover h sections from a split, and g sections backwards from a merge. • From a splitting point of view: • From the merging point of view: • Each branch, belongs to one group of and one group of • Thus, each branch has to have maximum distance with other branches. Oral Qualifying Exam - UCLA - Electrical Engineering

29. Extending Ungerboeck’s rule • Second, we can compute how many branches of maximum distance between each other we can have. Let’s denote this number T. • For branches with equal Hamming weight w, • In the general case • The constraints are: • Each branch has to belong to a group of and a group of • If we choose all constraints are satisfied Oral Qualifying Exam - UCLA - Electrical Engineering

30. Designing for a very low desired ones density • For a low enough desired ones density, all the branches can be chosen to have maximum distance. • The design becomes straight-forward. • Consider a NL-TCM code with S states, desired density p. • Denote M the sum of all the ones from the outputs of all 2S branches. • Then: • But if then • It is possible to choose all 2S branches so that there is at most 1 branch that has a 1 in a given position. • Straight-forward design: • Assign Hamming weights to branches • For each branch, add ones in positions that aren’t used in previous branches • Example: 100-user OR-MAC, Oral Qualifying Exam - UCLA - Electrical Engineering

31. Performance Results • For all implementations, states were used. • 6-user OR-MAC • n=20 : Sum-rate = 0.30 • 2 branches with Hw=2, 2 with Hw=3 (p=1/8). • h=3, g=2 : • n = 18 : Sum-rate = 1/3 • 3 branches with Hw=2, 1 with Hw=3 (p=1/8). • h=2,g=2 : • n = 17 : Sum-rate = 0.353 • all with Hw=2 (p=2/17=0.118). • h=2,g=2 : • 100-user OR-MAC, • n = 400 : w=2, i=3 (p = 0.006875) • n = 360 : w=2, i=2 (p = 0.006944) • for both cases Oral Qualifying Exam - UCLA - Electrical Engineering

32. Performance results • FPGA implementation: • In order to prove that NL-TCM codes are feasible today for optical speeds, a hardware simulation engine was built on the Xilinx Virtex2-Pro 2V20 FPGA. • Results for the rate-1/20 NL-TCM code are shown next. • Transfer Bound: • Wen-Yen Weng collaborated to this work, with the computation a Transfer Function Bound for NL-TCM codes. • It proved to be a very accurate bound, thus providing a fast estimation of the performance of the NL-TCM codes designed in this work. Oral Qualifying Exam - UCLA - Electrical Engineering

33. Performance Results : 6-user OR-MAC Oral Qualifying Exam - UCLA - Electrical Engineering

34. Performance Results : 6-user OR-MAC Oral Qualifying Exam - UCLA - Electrical Engineering

35. Results: observations • An error floor can observed for the FPGA rate-1/20 NL-TCM. • This is mainly due to the fact that, while theoretically a 1-to-0 transition means an infinite distance, for implementation constraints those transitions are given a value of 20. • Trace-back depth of 35. • The BER for all cases are not as low as required. Oral Qualifying Exam - UCLA - Electrical Engineering

36. Performance Results : 100-user OR-MAC Oral Qualifying Exam - UCLA - Electrical Engineering

37. Dramatically lowering the BER : Concatenation with Outer Block Code • Optical systems deliver a very low BER, in our work a was required. • Using only a NL-TCM, the rate would have to be very low. • A better solution is found using the fact that when the Viterbi decoding fails, with relatively high probability only a small number of bits are in error. • Thus, a high-rate block code that can correct a few errors can be attached as an outer code, dramatically lowering the BER. Block-Code Encoder NL-TCM Encoder Z-Channel Block-Code Decoder NL-TCM Decoder Oral Qualifying Exam - UCLA - Electrical Engineering

38. Reed-Solomon + NL-TCM : Results • A concatenation of the rate-1/20 NL-TCM code with (255 bytes,247 bytes) Reed-Solomon code has been tested for the 6-user OR-MAC scenario. • This RS-code corrects up to 8 erred bits. • The resulting rate for each user is (247/255).(1/20) • The results were obtained using a C program to apply the RS-code to the FPGA NL-TCM output. • Although we don’t have results for the 100-user case, it may be inferred that a similar BER would be achieved. Oral Qualifying Exam - UCLA - Electrical Engineering

39. Conclusions • I developed a novel design technique for non-linear trellis codes, that provide a wide range of ones density. • These codes have been designed for the Z-Channel, that arises in the optical multiple access channel. • A relatively low ones density is essential for the OR-MAC channel, and asymmetric channels in general. • An arbitrary number of users is supported, maintaining relatively the same efficiency (around 30%) • Although these codes are not capacity achieving,a good part of the capacity is achieved, with a suitable BER fr optical needs, and a complexity feasible for optical speeds with today’s technology. An FPGA implementation has been built to prove this fact. Oral Qualifying Exam - UCLA - Electrical Engineering

40. Future work (1): Capacity achieving codes • Capacity achieving codes. • Although they may not be feasible for optical speeds, with today’s technology, Turbo codes and LDPC codes will be feasible in the near future • Part of my immediate future’s work will be the design Turbo-Like codes, with an arbitrary ones density. • Most common Turbo-like codes are • Parallel concatenation of convolutional codes • Serially concatenated convolutional codes. • The convolutional codes will be replaced by properly designed NL-TCMs. Oral Qualifying Exam - UCLA - Electrical Engineering

41. Non-linear Turbo Like codes • Serial concatenation CC + NL-TCM: • Parallel concatenated NL-TCMs: CC Interleaver NL-TCM NL-TCM Interleaver NL-TCM Oral Qualifying Exam - UCLA - Electrical Engineering

42. Non-linear Turbo-like codes • The NL-TCM will not be a feed-forward encoder. • The design criteria changes. • However, the fundamental ideas hold. • The fact that the RS+NL-TCM concatenation (hard-decision transmitted from one decoder to the other) has such a good BER, makes the serial concatenation of CC+NL-TCM with soft-decoding look promising. Oral Qualifying Exam - UCLA - Electrical Engineering

43. 1 1 0 0 Future Work(2): More general Channel • Also, to be more general, I will study the Multiple access channel where the 1+1=0 case, has a positive (although very small) probability. • Treating other users as noise, one user “sees” an Binary Asymmetric Channel. • This will be change the metric in the Viterbi decoder, the definition of distance used, but shouldn’t change the design criteria Oral Qualifying Exam - UCLA - Electrical Engineering