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. 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

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. Communication Systems Laboratory

4. 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

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). Communication Systems Laboratory

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. Communication Systems Laboratory

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 Communication Systems Laboratory

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. Communication Systems Laboratory

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

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 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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