Two-Stage Coded Modulation for Hurwitz Constellations in Fiber-Optical Communications

Abstract

Four-dimensional signal constellations based on the checkerboard lattice D 4 offer a packing gain over conventional QAM constellations per polarization. Due to the increased number of nearest neighbors such power-efficient 4D formats cannot be Gray-labeled and bit-interleaved coded modulation results in a considerable performance loss. Instead, a suited coded-modulation scheme must be tailored to the properties of the underlying signal lattice. We apply a low-complexity two-stage coded-modulation scheme for constellations based on the set of Hurwitz integers, an isomorphic representation of the D 4 lattice. The proposed signaling scheme is evaluated in numerical simulations and fiber-optical system experiments. The performance is compared to a reference implementation using square QAM formats as proposed in the current 400ZR standardization activities. We demonstrate a 0.8 dB gain in required OSNR of the 512-ary Hurwitz constellation over dual-polarization 16-QAM (with 256 signal points in 4D) at the same symbol and bit rate. Additional numerical and experimental results show the potential for applications in the long-haul as well as the short-reach regime. Index Terms-Two-stage coded modulation, Hurwitz constellations, 4D modulation techniques, D 4 lattice, packing gain, bit-interleaved coded modulation, OIF 400ZR. I. INTRODUCTION T HE application of four-dimensional (4D) and higherdimensional modulation has been intensively investigated over the last decade ever since it was first introduced to the fiber-optical communications community in [1], [2]. The four orthogonal components of the electromagnetic field constitute the natural basis for 4D signal sets. It is well known that the density of signal points per unit volume can be doubled when Manuscript

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

“…Then, the transmit symbols are consistently drawn from a quaternion-valued system constellation A ⊂ H with the (4D) cardinality M q = |A|. The signal points can, e.g., be chosen as a subset of the Lipschitz or the Hurwitz integers [1], [67]. The related variance reads σ 2 x,q .…”

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

“…Then, the transmit symbols are consistently drawn from a quaternion-valued signal constellation A ⊂ H with the (4D) cardinality M q = |A|. The signal points can, e.g., be chosen as a subset of the Lipschitz or the Hurwitz integers [1], [68].…”

“…In complex transmission, most often QAM constellations are employed that form (shifted) subsets of G. Besides, especially in optical communications, dual-polarized QAM [31] is popular which corresponds to (shifted) subsets of L if interpreted over quaternion space. Alternatively, lattice-based constellations can be defined as A ⊂ E [2], [28], [41], [42] and A ⊂ H [1], [31], [68], [74]- [76], respectively. Since these rings constitute denser packings than their related Z Dr -based ones, the number of signal points within a certain hypervolume is increased, i.e., a packing gain is achieved.…”

Geometric Constellation Shaping for Concatenated Two-Level Multi-Level Codes

“…It has been demonstrated in [2]- [6] that, for this role, twolevel multi level codes (MLCs) [7], [8] that encode only the least reliable bit (LRB) of a mapper input by an inner FEC code achieve better performance-complexity trade-offs than the conventional bit-interleaved coded modulation (BICM) [9], [10] that protects all the bit levels equally by an inner code. Recently, similar two-level coded modulation approach has been also proposed in [11], [12] for four-dimensional lattices, termed Hurwitz constellations.…”

“…In this paper, we propose a neural network-based design of GS for a concatenated FEC scheme with inner two-level MLC. Note that there exist several works that apply power-efficient lattice constellations to MLC, e.g., [11], [12], [46], which may be seen as GS. However, to the best of our knowledge, this is the first attempt to optimize GS for MLC by means of an autoencoder, which takes fiber nonlinearities into account unlike the conventional works.…”