We describe coding techniques that achieve the capacity of a discrete memoryless asymmetric channel. To do so, we discuss how recent advances in coding for symmetric channels yield more efficient solutions also for the asymmetric case. In more detail, we consider three basic approaches. The first one is Gallager's scheme that concatenates a linear code with a non-linear mapper, in order to bias the input distribution. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Further, we derive a scaling law between the gap to capacity, the cardinality of channel input and output alphabets, and the required size of the mapper. The second one is an integrated approach in which the coding scheme is used both for source coding, in order to create codewords with the capacity-achieving distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third approach is based on an idea due to Bocherer and Mathar and separates completely the two tasks of source coding and channel coding by “chaining” together several codewords. We prove that we can combine any suitable source code with any suitable channel code in order to provide optimal schemes for asymmetric channels. In particular, polar codes and spatially coupled codes fulfill the required conditions.

How to achieve the capacity of asymmetric channels

Polar codes for channels with memory are considered in this paper. Sasoglu proved that Arikan's polarization applies to finite-state memory channels but practical decoding algorithms of polar codes for such channels have been unknown. This paper proposes a generalization of successive cancellation algorithm for channels with memory where the complexity is polynomial in the number of states. In addition, two polar coding scheme are proposed to generate codewords following non-i.i.d. process required to achieve the capacity. Whereas one is a simple application of the polar coding scheme for asymmetric memoryless channels, the other one combines a polar code with a fixed-to-variable length homophonic coding scheme, which can realize the input distribution exactly equal to the target process.

Construction of polar codes for channels with memory

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. 
The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. 
The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. 
The third paradigm is based on an idea of Bocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager’s scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of Bocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy. Rather than focusing on the exact details of the schemes, the purpose of this tutorial is to present different coding techniques that can then be implemented with many variants. There is no absolute winner and, in order to understand the most suitable technique for a specific application scenario, we provide a detailed comparison that takes into account several performance metrics.

Fast polarization is crucial for the performance guarantees of polar codes. In the memoryless setting, the rate of polarization is known to be exponential in the square root of the block length. A complete characterization of the rate of polarization for models with memory has been missing. Namely, previous works have not addressed fast polarization of the high entropy set under memory. We consider polar codes for processes with memory that are characterized by an underlying ergodic finite-state Markov chain. We show that the rate of polarization for these processes is the same as in the memoryless setting, both for the high and for the low entropy sets.

Fast Polarization for Processes With Memory

Polar codes are a class of capacity-achieving codes for the binary-input discrete memoryless channels (B-DMCs). However, when applied in channels with intersymbol interference (ISI), the codes may perform poorly with BCJR equalization and conventional decoding methods. To deal with the ISI problem, in this paper a new joint successive cancellation (SC) decoding algorithm is proposed for polar codes in ISI channels, which combines the equalization and conventional decoding. The initialization information of the decoding method is the likelihood functions of ISI codeword symbols rather than the codeword symbols. The decoding adopts recursion formulas like conventional SC decoding and is without iterations. This is in contrast to the conventional iterative algorithm which performs iterations between the equalizer and decoder. In addition, the proposed SC trellis decoding can be easily extended to list decoding which can further improve the performance. Simulation shows that the proposed scheme significantly outperforms the conventional decoding schemes in ISI channels.

Joint Successive Cancellation Decoding of Polar Codes over Intersymbol Interference Channels

In this paper, we propose a parallel block-based Viterbi decoder (PBVD) on the graphic processing unit (GPU) platform for the decoding of convolutional codes. The decoding procedure is simplified and parallelized, and the characteristic of the trellis is exploited to reduce the metric computation. Based on the compute unified device architecture (CUDA), two kernels with different parallelism are designed to map two decoding phases. Moreover, the optimal design of data structures for several kinds of intermediate information are presented, to improve the efficiency of internal memory transactions. Experimental results demonstrate that the proposed decoder achieves high throughput of 598Mbps on NVIDIA GTX580 and 1802Mbps on GTX980 for the 64-state convolutional code, which are 1.5 times speedup compared to the existing fastest works on GPUs.

A Gb/s parallel block-based Viterbi decoder for convolutional codes on GPU

In this letter, we present a graphics processing unit (GPU)-based LDPC convolutional code (LDPC-CC) pipeline decoder with optimized parallelism. The proposed decoder exploits different granularities of decoding parallelism for both the compute unified device architecture (CUDA) kernel execution stage and the data transfer stage. Moreover, the parameter selection criteria for decoder implementation are designed to avoid exhaustive search of all the combinations of parameters. The experiments are carried out on Nvidia GTX460 and GTX580 platforms. The results demonstrate the proposed decoder achieves about 3 times speedup compared to the existing GPU-based work.

High Throughput Pipeline Decoder for LDPC Convolutional Codes on GPU

This paper tackles the problem of channel estimation and decoding in environments that exhibit time-varying block-memoryless impulsive noise. In order to perform the estimation in continuous value space and the decoding in discrete value space jointly, a novel message passing framework is proposed that combines the sampling-importance resampling (SIR) estimator and the LDPC sum-product decoder in an iterative manner. The resampling process within the framework is further improved based on the asymptotic performance analysis for mismatched decoding. The simulation results prove the validity of the enhanced SIR estimator in the proposed method and demonstrate our method can achieve better performance than the LDPC decoder with quantile estimator.

Yi Hou

Papers

Construction of polar codes for channels with memory

Joint Successive Cancellation Decoding of Polar Codes over Intersymbol Interference Channels

A Gb/s parallel block-based Viterbi decoder for convolutional codes on GPU

High Throughput Pipeline Decoder for LDPC Convolutional Codes on GPU

Joint Channel Estimation and LDPC Decoding Over Time-Varying Impulsive Noise Channels