We generalize probabilistic amplitude shaping (PAS) with binary codes [1] to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.
We consider spatially coupled low-density paritycheck codes with finite smoothing parameters. A finite smoothing parameter is important for designing practical codes that are decoded using low-complexity windowed decoders. By optimizing the amount of coupling between spatial positions, we show that we can construct codes with excellent thresholds and small rate loss, even with the lowest possible smoothing parameter and large variable node degrees, which are required for low error floors. We also establish that the decoding convergence speed is faster with non-uniformly coupled codes, which we verify by density evolution of windowed decoding with a finite number of iterations. We also show that by only slightly increasing the smoothing parameter, practical codes with potentially low error floors and thresholds close to capacity can be constructed. Finally, we give some indications on protograph designs.
A class of circular 64-QAM that combines 'geometric' and 'probabilistic' shaping aspects is presented. It is compared to square 64-QAM in back-to-back, single-channel, and WDM transmission experiments. First, for the linear AWGN channel model, it permits to operate close to the Shannon limits for a wide range of signal-to-noise ratios. Second, WDM simulations over several hundreds of kilometers show that the obtained signal-to-noise ratios are equivalent to -or slightly exceed -those of probabilistic shaped 64-QAM. Third, for real-life validation purpose, an experimental comparison with unshaped 64-QAM is performed where 28% distance gains are recorded when using 19 channels at 54.2 GBd. This again is in line -or slightly exceeds -the gains generally obtained with probabilistic shaping. Depending upon implementation requirements (core forward-error correcting scheme for example), the investigated modulation schemes may be key alternatives for next-generation optical systems.
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