Uncoordinated Optical Multiple Access using IDMA and Nonlinear TCM

1. UCLA Electrical Engineering Department-Communication Systems Laboratory Uncoordinated Optical Multiple Access using IDMA and Nonlinear TCM PIs: Eli Yablanovitch, Rick Wesel, Ingrid Verbauwhede, Bahram Jalali, Ming Wu Students whose work is discussed here: Juthika Basak, Herwin Chan, Miguel Griot, Andres Vila Casado, Wen-Yen Weng

2. OCDMA Coding Architecture 1.2 Gbps 2 Gbps 60 Mbps Reed Solomon (255, 237) Trellis Code 1/20 int OR channel 93 Mbps 5 other tx Correct extra errors Separate different transmitters Asychronous Access code Communication Systems Laboratory

3. The system Reed Solomon (255, 237) sync Trellis Code 1/20 int 5 other tx For uncoor-dinated access To distinguish between users Initial synchroni-zation of tx-rx pair OR channel To bring final BER to 1e-9 BER Tester Reed Solomon (255, 237) Trellis Code 1/20 int Bit align sync Large feedback loop for rx synchronization Communication Systems Laboratory

4. Experimental Setup FPGA XMIT 1 AMP AMP Optical MOD AMP FPGA XMIT 2 AMP Optical MOD AMP AMP Optical MOD FPGA XMIT 3 Optical to Electrical AMP AMP Optical MOD FPGA XMIT 4 D Flip-Flop AMP AMP Optical MOD FPGA XMIT 5 AMP AMP Optical MOD FPGA XMIT 6 FPGA RCV 1 Communication Systems Laboratory

5. Six Users Communication Systems Laboratory

6. Communication Systems Laboratory

7. Probability of amplitudes for 6-users Communication Systems Laboratory

8. Asynchronous users Communication Systems Laboratory

9. Receiver Ones Densities for this code. Communication Systems Laboratory

10. 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 in 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. Communication Systems Laboratory

11. 6-user BER 10-5 C-Simulation Performance Results: 6-user OR-MAC Communication Systems Laboratory

12. 6-user BER 10-5 6-user OR-MAC:Simulation, Bound, FPGA (no optics) Communication Systems Laboratory

13. 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. • Additional coding required to lower BER to below 10-9. Communication Systems Laboratory

14. Dramatically lowering the BER : Concatenation with Outer Block Code • Optical systems deliver a very low BER, in our work a is required. • Using only a NL-TCM, the rate would have to be very low. • A better solution is found using the fact that Viterbi decoding fails gradually, 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 Communication Systems Laboratory

15. 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. Communication Systems Laboratory

16. C-Simulation Performance Results: NL-TCM only, 100-user OR-MAC Communication Systems Laboratory

17. Current Status • Decreased optical speed from 2 to 1.2 Gbps because FPGA can’t keep up at 2 Gbps. • Single Amplifier Results: • 2-Amplifier system in progress. • We need more amplifiers for six users. Last night, worked for 4 users, but two users need more power. Communication Systems Laboratory

18. Results • Demonstrated scalability to 100 users in a C simulation. • Working on our 6-user optical implementation. Communication Systems Laboratory

19. Outline of more detailed discussion • 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 • Design Technique • Conclusions • Future Work Communication Systems Laboratory

20. 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. Communication Systems Laboratory

21. 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 User 1 User 2 Receiver User N Communication Systems Laboratory

22. 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). Communication Systems Laboratory

23. 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. Communication Systems Laboratory

24. 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 Communication Systems Laboratory

25. 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. Communication Systems Laboratory

26. Comparison of capacities Optimal ones densities: Communication Systems Laboratory

27. 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 studied in the literature are linear codes. • 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. There is still no design technique described for these codes, and they can’t be decoded at optical speeds today. • This work introduces a novel design technique for non-linear trellis codes with an arbitrary density of ones. Communication Systems Laboratory

28. Interleaver Division Multiple Access (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 memoryless Z-Channel. • Performance obtained for a 6-user OR-MAC using IDMA, and for the corresponding Z-Channel were the same in our C simulations. Communication Systems Laboratory

29. 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. Communication Systems Laboratory

30. 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. Communication Systems Laboratory

31. 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. Communication Systems Laboratory

32. 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) Communication Systems Laboratory

33. split merge merge split Ungerboeck’s rule • We can 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 Ungerboeck’s rule delivers: Communication Systems Laboratory

34. Extending Ungerboeck’s rule • One can extend Ungerboeck’s rule into the trellis. 0 1 Maximize split Communication Systems Laboratory

35. Extending Ungerboeck’s rule • One can extend Ungerboeck’s rule into the trellis. 0 0 1 1 0 1 Maximize Communication Systems Laboratory

36. 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 Communication Systems Laboratory

37. 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. • It is possible to choose all 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, Communication Systems Laboratory

38. 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 Communication Systems Laboratory

39. Conclusions • 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 with IDMA. • 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. Communication Systems Laboratory

40. Future work: 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. Communication Systems Laboratory

41. Non-linear Turbo Like codes • Serial concatenation CC + NL-TCM: • Parallel concatenated NL-TCMs: CC Interleaver NL-TCM NL-TCM Interleaver NL-TCM Communication Systems Laboratory