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

Improved Polarization Tracking in the Presence of PDL

We propose a novel tracking algorithm for optical channels suffering from fast state of polarization (SOP) rotations and polarization-dependent loss (PDL). Unlike gradient descent-based algorithms that require step size adjustment when the channel conditions change, our algorithm performs similarly or better without parameter tuning

Decoding Reed-Muller Codes Using Redundant Code Constraints

The recursive projection-aggregation (RPA) decoding algorithm for Reed-Muller (RM) codes was recently introduced by Ye and Abbe. We show that the RPA algorithm is closely related to (weighted) belief-propagation (BP) decoding by interpreting it as a message-passing algorithm on a factor graph with redundant code constraints. We use this observation to introduce a novel decoder tailored to high-rate RM codes. The new algorithm relies on puncturing rather than projections and is called recursive puncturing-aggregation (RXA). We also investigate collapsed (i.e., non-recursive) versions of RPA and RXA and show some examples where they achieve similar performance with lower decoding complexity

Wideband Time-Domain Digital Backpropagation via Subband Processing and Deep learning

We propose a low-complexity sub-banded DSP architecture for digital backpropagation where the walk-off effect is compensated using simple delay elements. For a simulated 96-Gbaud signal and 2500 km optical link, our method achieves a 2.8 dB SNR improvement over linear equalization

Approaching Miscorrection-Free Performance of Product Codes with Anchor Decoding

Product codes (PCs) protect a 2-D array of bits using short component codes. Assuming transmission over the binary symmetric channel, the decoding is commonly performed by iteratively applying bounded-distance decoding to the component codes. For this coding scheme, undetected errors in the component decoding - also known as miscorrections - significantly degrade the performance. In this paper, we propose a novel iterative decoding algorithm for PCs which can detect and avoid most miscorrections. The algorithm can also be used to decode many recently proposed classes of generalized PCs, such as staircase, braided, and half-product codes. Depending on the component code parameters, our algorithm significantly outperforms the conventional iterative decoding method. As an example, for double-error-correcting Bose-Chaudhuri-Hocquenghem component codes, the net coding gain can be increased by up to 0.4 dB. Moreover, the error floor can be lowered by orders of magnitude, up to the point where the decoder performs virtually identical to a genie-aided decoder that avoids all miscorrections. We also discuss post-processing techniques that can be used to reduce the error floor even further

Nonlinear Interference Mitigation via Deep Neural Networks

A neural-network-based approach is presented to efficiently implement digital backpropagation (DBP). For a 32x100km fiber-optic link, the resulting "learned" DBP significantly reduces the complexity compared to conventional DBP implementations