Mapping strategies for short-length probabilistic shaping
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Tobias Fehenberger’s ECOC 2019 slides reveal the latest developments in the field of probabilistic amplitude shaping.
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Mapping strategies for short-length probabilistic shaping
- 1. ECOC 2019 Tobias Fehenberger1, David S. Millar2, Toshiaki Koike-Akino2, Keisuke Kojima2, and Kieran Parsons2 1ADVA, Munich, Germany 2Mitsubishi Electric Research Labs (MERL), Cambridge, MA, USA Mapping strategies for short- length probabilistic shaping
- 2. © 2019 ADVA Optical Networking. All rights reserved.22 Outline Introduction Numerical results Implementation aspects 1 2 3 4 Mapping strategies for PAS Conclusions5
- 3. © 2019 ADVA Optical Networking. All rights reserved.33 • First publications on PS in 2012, using trellis shaping [1] and shell mapping [2] • More work on PS has been published afterwards, such as [3, 4] • Hot topic in optics community since the proposal of probabilistic amplitude shaping (PAS) in 2014/2015 [5] • The first demonstrations [6–8] were published the same year • Since then hundreds of papers Literature review: probabilistic shaping in optics [1] B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber …,” JLT, 2012. [2] L. Beygi, E. Agrell, and M. Karlsson, “Adaptive coded modulation for nonlinear fiber-optical channels,” GLOBECOM, 2012 [3] M. P. Yankov, et al., “Constellation shaping for fiberoptic channels with QAM and high spectral efficiency,” PTL, 2014. [4] L. Beygi, E. Agrell, J. M. Kahn, and M. Karlsson, “Rate-adaptive coded modulation for fiber-optic communications,” JLT, 2014. [5] G. Böcherer, P. Schulte, F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check …,” TCOMM, 2015. [6] T. Fehenberger, G. Böcherer, A. Alvarado, N. Hanik, “LDPC coded modulation with probabilistic shaping for …,” OFC, 2015. [7] F. Buchali, G. Böcherer, W. Idler, L. Schmalen, P. Schulte, and F. Steiner, “Experimental demonstration of capacity increase and rate-adaptation by probabilistically shaped 64-QAM,” ECOC, 2015. [8] T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” PTL, 2016.
- 4. © 2019 ADVA Optical Networking. All rights reserved.44 • Up to 1 dB SNR improvement for high- order QAM • Achievable with various kinds of modifications to constellation • Different PS architectures possible • Could be also achieved with geometric shaping • Vary data rate while keeping modula- tion format, FEC, and symbol rate fixed • The throughput is controlled via the distribution or set explicitly, depending on the mapping • Many alternatives, such as time-hybrid modulation, vary FEC overhead, variable symbol rate Rate adaptivityShaping gain These benefits come at the expense of a new signal processing block Why the interest in PAS?
- 5. © 2019 ADVA Optical Networking. All rights reserved.55 Simplified block diagram of PAS Reverse concatenation principle: mapper outside FEC Mapping Inverse Mapping k uniform data bits n shaped amplitudes
- 6. © 2019 ADVA Optical Networking. All rights reserved.66 • Map k uniform data bits to n shaped amplitudes • At receiver: carry out inverse operation without errors • Amplitudes A, distribution PA, entropy H(A) • Key performance metric: rate loss • Differences between mapping schemes • Architecture properties of the output sequences • Algorithm how the mapping is carried out Common fundamentals of all PAS mappers Mapping k n
- 7. © 2019 ADVA Optical Networking. All rights reserved.77 • Explicitly set amplitude distribution PA • Transmission rate depends on the entropy H(A) and the rate loss • Examples: • Constant composition distribution matching (CCDM) [1] • Multiset partition DM (MPDM) [2] • Explicitly set the transmission rate • A shaped distribution is achieved implicitly • Examples: • Enumerative sphere shaping (ESS) [3] • Shell mapping [4] • Huffman coded sphere shaping [5] Indirect approachDirect approach This talk: CCDM, MPDM, and ESS Classification of mapping strategies [1] P. Schulte and G. Böcherer, “Constant composition distribution matching,” Trans IT, 2016. [2] T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Multiset-partition distribution matching,” TCOMM, 2019. [3] Y. C. Gültekin et al., “Enumerative sphere shaping for wireless communications with short packets,” arXiv, 2019. [4] R. Laroia, N. Farvardin, and S. A. Tretter, “On optimal shaping of multidimensional constellations,” Trans IT, 1994. [5] D. S. Millar, T. Fehenberger, T. Yoshida, T. Koike-Akino, K. Kojima, N. Suzuki, and K. Parsons, “Huffman coded sphere shaping with short length and reduced complexity,” ECOC, 2019
- 8. © 2019 ADVA Optical Networking. All rights reserved.88 • All shaped output sequen- ces are permutations of each other • Conventional algorithm: arithmetic coding • Optimal as block length goes to infinity • Use pairs of compositions such that their average gives target distribution • Layered extension of CCDM with better performance • CCDM algorithms in nodes • Sort output sequences lexicographically • Use them up to target rate • Precomputed trellis to obtain output sequence MPDM ESSCCDM Mapping strategies: (very) brief overview T. Fehenberger et al., “Multiset-partition distribution matching,” TCOMM, 2019. A. Amari et al., “Introducing Enumerative Sphere Shaping for Optical Communication Systems with Short Block Lengths,” arXiv, 2019. P. Schulte and G. Böcherer, “Constant composition distribution matching,” Trans IT, 2016.
- 9. © 2019 ADVA Optical Networking. All rights reserved.99 Energy considerations Binary input space Signal output space of size Mn Energyofoutputsequences ESS CCDM MPDM • Horizontal line through space on the right is a shell of some energy • By using a single composition, CCDM uses part of a shell • MPDM uses more compositions (thus more shells) but rarely a shell entirely • ESS uses entire sphere up to some maximum energy
- 10. © 2019 ADVA Optical Networking. All rights reserved.1010 • Target distribution for CCDM and MPDM: PA = [0.4378, 0.3212, 0.1728, 0.0682] • ESS set to same rate as MPDM Results: rate loss comparison From: Y. C. Gültekin, T. Fehenberger, A. Alvarado, and F. M. J. Willems, “Probabilistic Shaping for Finite Block Lengths: Distribution Matching and Sphere Shaping,” arXiv:1909.08886, Sept. 2019. Example: n=50, 64QAM, AIR = 5 bit/2D CCDM MPDM ESS Rate loss [bit/1D] 0.16 0.06 0.04 Rate loss [bit/2D] 0.32 0.12 0.08 Actual throughput [bit/2D] 4.68 4.88 4.92
- 11. © 2019 ADVA Optical Networking. All rights reserved.1111 Results: shaping gain (AWGN) Target: achieve approx. 50% of the maximum shaping gain 16QAM 64QAM 256QAM Max. gain [dB] 0.4 0.8 1 Target gain [dB] 0.2 0.5 0.6 Short-to-medium block lengths for some shaping gain
- 12. © 2019 ADVA Optical Networking. All rights reserved.1212 Results: rate adaptivity (AWGN) Target: match performance of uniform QAM (shaping gain equal to rate loss) Very short blocks are sufficient for rate adaptivity Required n (approx.) 16QAM @ 10 dB 64QAM @ 15 dB 256QAM @ 20 dB CCDM 85 90 155 MPDM 50 25 20 ESS 35 10 5
- 13. © 2019 ADVA Optical Networking. All rights reserved.1313 Results: FEC decoding (AWGN) 64QAM, 4.5 bit/2D, DVB-S2 LDPC codes (64800 bits), shaping block length n=180 0.5 dB0.4 dB From: Y. C. Gültekin, T. Fehenberger, A. Alvarado, and F. M. J. Willems, “Probabilistic Shaping for Finite Block Lengths: Distribution Matching and Sphere Shaping,” arXiv:1909.08886, Sept. 2019.
- 14. © 2019 ADVA Optical Networking. All rights reserved.1414 • Hard to analyze from high level • Many mapping schemes internally handle huge numbers • Is a bitwise comparison still easy in this case? • Unproblematic: In the kilobyte range (or less) for all mapping schemes • Lookup table (LUT) approaches seem to be feasible in some cases • ESS with n=5 and 256QAM needs at most 8^5=32k LUT entries, each 15-bit long • Number of loop iterations required for matching and dematching • CCDM & MPDM: • Conventional (arithmetic coding): k+n • Subset-ranking CCDM [1]: min(n1, n-n1)+1 where n1 is weight of a binary sequence • ESS: k+n Storage SerialismComputational complexity Which mapping strategy to choose? [1] T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Parallel-amplitude architecture and subset ranking for fast distribution matching," arXiv:1902.08556, Feb. 2019
- 15. © 2019 ADVA Optical Networking. All rights reserved.1515 • Typically, the scheme that achieves some fixed rate loss with the smallest block length is considered superior • Maybe true from (information-) theoretic point of view, but not when considering implementation aspects • Examples: • Binary DM transformations [1-3]: higher parallelization • Subset-ranking [1] for binary CCDM: less serialism than arithmetic coding • MPDM and HCSS [4]: small penalty for Huffman tree + CCDM per node • Hypothetical: Suppose we had a mapping algorithm that can be parallelized (cf. LDPC decoding with sum-product algorithm), long blocks would not be that bad Does the “shorter is better” paradigm make sense? [1] T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Parallel-amplitude architecture and subset ranking for fast distribution matching," arXiv:1902.08556, Feb. 2019 [2] M. Pikus and W. Xu, “Bit-level probabilistically shaped coded modulation,” IEEE Commun. Lett., 2017 [3] G. Böcherer, P. Schulte, F. Steiner, “High Throughput Probabilistic Shaping with Product Distribution Matching,” arXiv, 2017 [4] D. S. Millar, T. Fehenberger, T. Yoshida, T. Koike-Akino, K. Kojima, N. Suzuki, and K. Parsons, “Huffman coded sphere shaping with short length and reduced complexity,” ECOC, 2019
- 16. © 2019 ADVA Optical Networking. All rights reserved.1616 • Overview of mapping strategies for PAS • Different approaches: direct vs. indirect • Performance comparison • To achieve 50% of the available shaping gain: block lengths in the order of a few dozen amplitudes are sufficient • To get rate adaptivity: block lengths of a few amplitude symbols • Performance of MPDM and ESS very similar for moderate block lengths • Implementation aspects are key differentiator Conclusions
- 17. Thank you IMPORTANT NOTICE The content of this presentation is strictly confidential. ADVA Optical Networking is the exclusive owner or licensee of the content, material, and information in this presentation. Any reproduction, publication or reprint, in whole or in part, is strictly prohibited. The information in this presentation may not be accurate, complete or up to date, and is provided without warranties or representations of any kind, either express or implied. ADVA Optical Networking shall not be responsible for and disclaims any liability for any loss or damages, including without limitation, direct, indirect, incidental, consequential and special damages, alleged to have been caused by or in connection with using and/or relying on the information contained in this presentation. Copyright © for the entire content of this presentation: ADVA Optical Networking. tfehenberger@advaoptical.com