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

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.Comment: This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-22-12-1454

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. 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 with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. 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 with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

On Signal Constellations and Coding for Long-Haul Fiber-Optical Systems

Motivated by the realization that even the enormous bandwidth available in an optical fiber is finite and valuable, the design of spectrally efficient long-haul fiber-optical communication systems has become an important research topic. Compared to other wireline technologies, e.g., transmission over coaxial cables, the main challenge comes from the inherent nonlinearity of the underlying communication channel caused by the relatively high signal intensities. In this thesis, we study the design of spectrally efficient fiber-optical systems for both uncoded and coded transmission scenarios.We consider the problem of designing higher-order signal constellations for a system that is severly impaired by nonlinear phase noise. By optimizing amplitude phase-shift keying constellations, which can be seen as the union of phase-shift keying constellation with different amplitude levels, gains of up to 3.2 dB at a symbol error probability of 10^(−2) are shown to be achievable compared to conventional constellations. We also illustrate a somewhat counterintuitive behavior of optimized constellations for very high input powers and nonlinear distortions. In particular, sacrificing a constellation point or ring may be beneficial in terms of the overall performance of the constellation.Furthermore, we study polarization-multiplexed transmission, where spectral efficiency is increased by encoding data onto both polarizations of the light. For a memoryless fiber-optical channel, we introduce a low-complexity detector which is based on an amplitude- dependent phase rotation and subsequent threshold detection. The complexity compared to the four-dimensional maximum likelihood detector is considerably reduced, albeit at the expense of some performance loss.Lastly, we consider the design of a coded fiber-optical system operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed fiber-optical system that is based on the bit-interleaved coded modulation paradigm. This technique, which we refer to as bit mapper optimization, is extended to the class of spatially coupled low-density parity- check codes, which have shown outstanding performance over memoryless binary-input channels. For a transmission scenario without optical inline dispersion compensation, the results show that the transmission reach can be extended by roughly up to 8%, without significantly increasing the system complexity

On Signal Constellations and Coding for Long-Haul Fiber-Optical Systems

Motivated by the realization that even the enormous bandwidth available in an optical fiber is finite and valuable, the design of spectrally efficient long-haul fiber-optical communication systems has become an important research topic. Compared to other wireline technologies, e.g., transmission over coaxial cables, the main challenge comes from the inherent nonlinearity of the underlying communication channel caused by the relatively high signal intensities. In this thesis, we study the design of spectrally efficient fiber-optical systems for both uncoded and coded transmission scenarios.We consider the problem of designing higher-order signal constellations for a system that is severly impaired by nonlinear phase noise. By optimizing amplitude phase-shift keying constellations, which can be seen as the union of phase-shift keying constellation with different amplitude levels, gains of up to 3.2 dB at a symbol error probability of 10^(−2) are shown to be achievable compared to conventional constellations. We also illustrate a somewhat counterintuitive behavior of optimized constellations for very high input powers and nonlinear distortions. In particular, sacrificing a constellation point or ring may be beneficial in terms of the overall performance of the constellation.Furthermore, we study polarization-multiplexed transmission, where spectral efficiency is increased by encoding data onto both polarizations of the light. For a memoryless fiber-optical channel, we introduce a low-complexity detector which is based on an amplitude- dependent phase rotation and subsequent threshold detection. The complexity compared to the four-dimensional maximum likelihood detector is considerably reduced, albeit at the expense of some performance loss.Lastly, we consider the design of a coded fiber-optical system operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed fiber-optical system that is based on the bit-interleaved coded modulation paradigm. This technique, which we refer to as bit mapper optimization, is extended to the class of spatially coupled low-density parity- check codes, which have shown outstanding performance over memoryless binary-input channels. For a transmission scenario without optical inline dispersion compensation, the results show that the transmission reach can be extended by roughly up to 8%, without significantly increasing the system complexity