Christian Häger

Design of APSK Constellations for Coherent Optical Channels with Nonlinear Phase Noise

We study the design of amplitude phase-shift keying (APSK) constellations for a coherent fiber-optical communication system where nonlinear phase noise (NLPN) is the main system impairment. APSK constellations can be regarded as a union of phase-shift keying (PSK) signal sets with different amplitude levels. A practical two-stage (TS) detection scheme is analyzed, which performs close to optimal detection for high enough input power. We optimize APSK constellations with 4, 8, and 16 points in terms of symbol error probability (SEP) under TS detection for several combinations of input power and fiber length. For 16 points, performance gains of 3.2 dB can be achieved at a SEP of 10-2 compared to 16-QAM by choosing an optimized APSK constellation. We also demonstrate that in the presence of severe nonlinear distortions, it may become beneficial to sacrifice a constellation point or an entire constellation ring to reduce the average SEP. Finally, we discuss the problem of selecting a good binary labeling for the found constellations.

Improving soft FEC performance for higher-order modulations via optimized bit channel mappings.

Soft forward error correction with higher-order modulations is often implemented in practice via the pragmatic bit-interleaved coded modulation paradigm, where a single binary code is mapped to a nonbinary modulation. In this paper, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed fiber-optical system without optical inline dispersion compensation. Our focus is on protograph-based low-density parity-check (LDPC) codes which allow for an efficient hardware implementation, suitable for high-speed optical communications. The optimization is applied to the AR4JA protograph family, and further extended to protograph-based spatially coupled LDPC codes assuming a windowed decoder. Full field simulations via the split-step Fourier method are used to verify the analysis. The results show performance gains of up to 0.25 dB, which translate into a possible extension of the transmission reach by roughly up to 8%, without significantly increasing the system complexity.

Optimized bit mappings for spatially coupled LDPC codes over parallel binary erasure channels

In many practical communication systems, one binary encoder/decoder pair is used to communicate over a set of parallel channels. Examples of this setup include multi-carrier transmission, rate-compatible puncturing of turbo-like codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly employed to determine how the coded bits are allocated to the channels. In this paper, we study spatially coupled low-density parity check codes over parallel channels and optimize the bit mapper using BICM as the driving example. For simplicity, the parallel bit channels that arise in BICM are replaced by independent binary erasure channels (BECs). For two parallel BECs modeled according to a 4-PAM constellation labeled by the binary reflected Gray code, the optimization results show that the decoding threshold can be improved over a uniform random bit mapper, or, alternatively, the spatial chain length of the code can be reduced for a given gap to capacity. It is also shown that for rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be initiated using only an optimized bit mapper.

Terminated and Tailbiting Spatially Coupled Codes With Optimized Bit Mappings for Spectrally Efficient Fiber-Optical Systems

We study the design of spectrally efficient fiber-optical communication systems based on different spatially coupled (SC) forward error correction (FEC) schemes. In particular, we optimize the allocation of the coded bits from the FEC encoder to the modulation bits of the signal constellation. Two SC code classes are considered. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes which are decoded using iterative hard-decision decoding. For both code classes, the bit allocation is optimized for the terminated and tailbiting SC cases based on a density evolution analysis. An optimized bit allocation can significantly improve the performance of tailbiting SC codes over the baseline sequential allocation, up to the point where they have a comparable gap to capacity as their terminated counterparts, at a lower FEC overhead. For the considered terminated SC codes, the optimization only results in marginal performance improvements, suggesting that in this case a sequential allocation is close to optimal.

Density Evolution for Deterministic Generalized Product Codes on the Binary Erasure Channel at High Rates

Generalized product codes (GPCs) are extensions of product codes (PCs), where code symbols are protected by two component codes but not necessarily arranged in a rectangular array. We consider a deterministic construction of GPCs (as opposed to randomized code ensembles) and analyze the asymptotic performance over the binary erasure channel under iterative decoding. Our code construction encompasses several classes of GPCs previously proposed in the literature, such as irregular PCs, blockwise braided codes, and staircase codes. It is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity. We show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph. Using a convergence result for these graphs, we derive the density evolution equations that characterize the asymptotic decoding performance. As an application, we discuss the design of irregular GPCs, employing a mixture of component codes with different erasure-correcting capabilities.

Deterministic and ensemble-based spatially-coupled product codes

Several authors have proposed spatially-coupled (or convolutional-like) variants of product codes (PCs). In this paper, we focus on a parametrized family of generalized PCs that recovers some of these codes (e.g., staircase and block-wise braided codes) as special cases and study the iterative decoding performance over the binary erasure channel. Even though our code construction is deterministic (and not based on a randomized ensemble), we show that it is still possible to rigorously derive the density evolution (DE) equations that govern the asymptotic performance. The obtained DE equations are then compared to those for a related spatially-coupled PC ensemble. In particular, we show that there exists a family of (deterministic) braided codes that follows the same DE equation as the ensemble, for any spatial length and coupling width.

Constellation optimization for coherent optical channels distorted by nonlinear phase noise

In order achieve beyond 400 Gb/s serial optical transmission using commercially available equipment, in this work, we propose the use of channel capacity achieving modulation formats. We present a method to determine the optimum signal constellation for an arbitrary dispersion map. This method can be described by the following steps. (i) Determine the probability density functions (pdfs) of the fiber-optics channel, which can be approximated by complex Gaussian function or estimated by evaluating histograms. (ii) Use the resulting pdfs to determine the optimum input distribution to achieve the channel capacity. A split-step method is presented to perform the optimization over the probabilities and mass points of input distributions. (iii) After quantizing the optimum input distribution the optimum signal constellation can be determined. We study two types of dispersion maps currently in use, which are suitable for metro and mediumhaul applications. For each of them we determine the optimum signal constellation. The numerical results demonstrate the efficiency of this method and show that the optimized signal constellation can increase the transmission distance at most by three spans against its counterpart.

Density evolution for deterministic generalized product codes with higher-order modulation

Generalized product codes (GPCs) are extensions of product codes (PCs) where coded bits are protected by two component codes but not necessarily arranged in a rectangular array. It has recently been shown that there exists a large class of deterministic GPCs (including, e.g., irregular PCs, half-product codes, staircase codes, and certain braided codes) for which the asymptotic performance under iterative bounded-distance decoding over the binary erasure channel (BEC) can be rigorously characterized in terms of a density evolution analysis. In this paper, the analysis is extended to the case where transmission takes place over parallel BECs with different erasure probabilities. We use this model to predict the code performance in a coded modulation setup with higher-order signal constellations. We also discuss the design of the bit mapper that determines the allocation of the coded bits to the modulation bits of the signal constellation.

On the Information Loss of the Max-Log Approximation in BICM Systems

We present a comprehensive study of the information rate loss of the max-log approximation for M-ary pulse-amplitude modulation (PAM) in a bit-interleaved coded modulation (BICM) system. It is widely assumed that the calculation of L-values using the max-log approximation leads to an information loss. We prove that this assumption is correct for all M-PAM constellations and labelings with the exception of a symmetric 4-PAM constellation labeled with a Gray code. We also show that for max-log L-values, the BICM generalized mutual information (GMI), which is an achievable rate for a standard BICM decoder, is too pessimistic. In particular, it is proved that the so-called harmonized GMI, which can be seen as the sum of bit-level GMIs, is achievable without any modifications to the decoder. We then study how bit-level channel symmetrization and mixing affect the MI and the GMI for max-log L-values. Our results show that these operations, which are often used when analyzing BICM systems, preserve the GMI. However, this is not necessarily the case when the MI is considered. Necessary and sufficient conditions under which these operations preserve the MI are provided.

Comparison of terminated and tailbiting spatially coupled LDPC codes with optimized bit mapping for PM-64-QAM

We optimize the allocation of coded bits to modulation bits for spatially coupled low-density parity-check (SC-LDPC) codes and PM-64-QAM. Tailbiting SC-LDPC codes are shown to offer a similar gap to capacity as their terminated counterparts, at a lower FEC overhead.