Fast List Decoding of High-Rate Polar Codes

Due to the ability to provide superior error-correction performance, the successive cancellation list (SCL) algorithm is widely regarded as one of the most promising decoding algorithms for polar codes with short-to-moderate code lengths. However, the application of SCL decoding in low-latency communication scenarios is limited due to its sequential nature. To reduce the decoding latency, developing tailored fast and efficient list decoding algorithms of specific polar substituent codes (special nodes) is a promising solution. Recently, fast list decoding algorithms are proposed by considering special nodes with low code rates. Aiming to further speedup the SCL decoding, this paper presents fast list decoding algorithms for two types of high-rate special nodes, namely single-parity-check (SPC) nodes and sequence rate one or single-parity-check (SR1/SPC) nodes. In particular, we develop two classes of fast list decoding algorithms for these nodes, where the first class uses a sequential decoding procedure to yield decoding latency that is linear with the list size, and the second further parallelizes the decoding process by pre-determining the redundant candidate paths offline. Simulation results show that the proposed list decoding algorithms are able to achieve up to 70.7\% lower decoding latency than state-of-the-art fast SCL decoders, while exhibiting the same error-correction performance.Comment: 13 pages, 8 figure

Low Power, Area Efficient Architecture for Successive Cancellation Decoder

Polar codes have recently emerged as an error-correcting code and have become popular owing to their capacity-achieving nature. Polar code based communication system primarily consists of two parts, including Polar Encoder and Decoder. Successive Cancellation Decoder is one of the methods used in the decoding process. The Successive Cancellation Decoder is a recursive structure built with the building block called Processing Element. This article proposes a low power, area-efficient architecture for the Successive Cancellation Decoder for polar codes. Successive Cancellation Decoder with code length 1024 and code rate 0.5 was designed in Verilog HDL and implemented using 45-nm CMOS technology. The proposed work focuses on developing an area-efficient Successive Cancellation Decoder architecture by presenting a new Processing Element architecture. The proposed architecture has produced about 35% lesser area with a 12% reduced gate count. Moreover, power is also reduced by 50%. A substantial reduction in the latency and improvement in the Technology Scaled Normalized Throughput value was observed

Polar-Coded OFDM with Index Modulation

Polar codes, as the first error-correcting codes with an explicit construction to provably achieve thesymmetric capacity of memoryless channels, which are constructed based on channel polarization, have recently become a primary contender in communication networks for achieving tighter requirements with relatively low complexity. As one of the contributions in this thesis, three modified polar decoding schemes are proposed. These schemes include enhanced versions of successive cancellation-flip (SC-F), belief propagation (BP), and sphere decoding (SD). The proposed SC-F utilizes novel potential incorrect bits selection criteria and stack to improve its error correction performance. Next, to make the decoding performance of BP better, permutation and feedback structure are utilized. Then, in order to reduce the complexity without compromising performance, a SD by using novel decoding strategies according to modified path metric (PM) and radius extension is proposed. Additionally, to solve the problem that BP has redundant iterations, a new stopping criterion based on bit different ratio (BDR) is proposed. According to the simulation results and mathematical proof, all proposed schemes can achieve corresponding performance improvement or complexity reduction compared with existing works. Beside applying polar coding, to achieve a reliable and flexible transmission in a wireless communication system, a modified version of orthogonal frequency division multiplexing (OFDM) modulation based on index modulation, called OFDM-in-phase/quadrature-IM (OFDM-I/Q-IM), is applied. This modulation scheme can simultaneously improve spectral efficiency and bit-error rate (BER) performance with great flexibility in design and implementation. Hence, OFDM-I/Q-IM is considered as a potential candidate in the new generation of cellular networks. As the main contribution in this work, a polar-coded OFDM-I/Q-IM system is proposed. The general design guidelines for overcoming the difficulties associated with the application of polar codes in OFDM-I/Q-IM are presented. In the proposed system, at the transmitter, we employ a random frozen bits appending scheme which not only makes the polar code compatible with OFDM-I/Q-IM but also improves the BER performance of the system. Furthermore, at the receiver, it is shown that the \textit{a posteriori} information for each index provided by the index detector is essential for the iterative decoding of polar codes by the BP algorithm. Simulation results show that the proposed polar-coded OFDM-I/Q-IM system outperforms its OFDM counterpart in terms of BER performance

Hardware implementation aspects of polar decoders and ultra high-speed LDPC decoders

The goal of channel coding is to detect and correct errors that appear during the transmission of information. In the past few decades, channel coding has become an integral part of most communications standards as it improves the energy-efficiency of transceivers manyfold while only requiring a modest investment in terms of the required digital signal processing capabilities. The most commonly used channel codes in modern standards are low-density parity-check (LDPC) codes and Turbo codes, which were the first two types of codes to approach the capacity of several channels while still being practically implementable in hardware. The decoding algorithms for LDPC codes, in particular, are highly parallelizable and suitable for high-throughput applications. A new class of channel codes, called polar codes, was introduced recently. Polar codes have an explicit construction and low-complexity encoding and successive cancellation (SC) decoding algorithms. Moreover, polar codes are provably capacity achieving over a wide range of channels, making them very attractive from a theoretical perspective. Unfortunately, polar codes under standard SC decoding cannot compete with the LDPC and Turbo codes that are used in current standards in terms of their error-correcting performance. For this reason, several improved SC-based decoding algorithms have been introduced. The most prominent SC-based decoding algorithm is the successive cancellation list (SCL) decoding algorithm, which is powerful enough to approach the error-correcting performance of LDPC codes. The original SCL decoding algorithm was described in an arithmetic domain that is not well-suited for hardware implementations and is not clear how an efficient SCL decoder architecture can be implemented. To this end, in this thesis, we re-formulate the SCL decoding algorithm in two distinct arithmetic domains, we describe efficient hardware architectures to implement the resulting SCL decoders, and we compare the decoders with existing LDPC and Turbo decoders in terms of their error-correcting performance and their implementation efficiency. Due to the ongoing technology scaling, the feature sizes of integrated circuits keep shrinking at a remarkable pace. As transistors and memory cells keep shrinking, it becomes increasingly difficult and costly (in terms of both area and power) to ensure that the implemented digital circuits always operate correctly. Thus, manufactured digital signal processing circuits, including channel decoder circuits, may not always operate correctly. Instead of discarding these faulty dies or using costly circuit-level fault mitigation mechanisms, an alternative approach is to try to live with certain malfunctions, provided that the algorithm implemented by the circuit is sufficiently fault-tolerant. In this spirit, in this thesis we examine decoding of polar codes and LDPC codes under the assumption that the memories that are used within the decoders are not fully reliable. We show that, in both cases, there is inherent fault-tolerance and we also propose some methods to reduce the effect of memory faults on the error-correcting performance of the considered decoders

Digital VLSI Architectures for Advanced Channel Decoders

Error-correcting codes are strongly adopted in almost every modern digital communication and storage system, such as wireless communications, optical communications, Flash memories, computer hard drives, sensor networks, and deep-space probes. New and emerging applications demand codes with better error-correcting capability. On the other hand, the design and implementation of those high-gain error-correcting codes pose many challenges. They usually involve complex mathematical computations, and mapping them directly to hardware often leads to very high complexity. This work aims to focus on Polar codes, which are a recent class of channel codes with the proven ability to reduce decoding error probability arbitrarily small as the block-length is increased, provided that the code rate is less than the capacity of the channel. This property and the recursive code-construction of this algorithms attracted wide interest from the communications community. Hardware architectures with reduced complexity can efficiently implement a polar codes decoder using either successive cancellation approximation or belief propagation algorithms. The latter offers higher throughput at high signal-to-noise ratio thanks to the inherently parallel decision-making capability of such decoder type. A new analysis on belief propagation scheduling algorithms for polar codes and on interconnection structure of the decoding trellis not covered in literature is also presented. It allowed to achieve an hardware implementation that increase the maximum information throughput under belief propagation decoding while also minimizing the implementation complexity

Polar codes combined with physical layer security on impulsive noise channels

Ph. D. ThesisThe need for secure communications is becoming more and more impor- tant in modern society as wired and wireless connectivity becomes more ubiquitous. Currently, security is achieved by using well established encryption techniques in the upper layers that rely on computational complexity to ensure security. However, processing power is continu- ally increasing and well-known encryption schemes are more likely to be cracked. An alternative approach to achieving secure communication is to exploit the properties of the communication channel. This is known as physical layer security and is mathematically proven to be secure. Phys- ical layer security is an active research area, with a significant amount of literature covering many different aspects. However, one issue that does not appear to have been investigated in the literature is the effect on physical layer security when the noise in the communication channel is impulsive. Impulsive noise adds large spikes to the transmitted signal for very short durations that can significantly degrade the signal. The main source of impulsive noise in wireless communications is electromag- netic interference generated by machinery. Therefore, this project will investigate the effect of impulsive noise on physical layer security. To ensure a high level of performance, advanced error-correcting codes are needed to correct the multiple errors due to this harsh channel. Turbo and Low-Density Parity-Check (LDPC) codes are capacity-approaching codes commonly used in current wireless communication standards, but their complexity and latency can be quite high and can be a limiting fac- tor when required very high data rates. An alternative error-correcting code is the polar code, which can actually achieve the Shannon capacity on any symmetric binary input discrete memoryless channel (B-DMC). Furthermore, the complexity of polar codes is low and this makes them an attractive error-correcting code for high data rate wireless commu- nications. In this project, polar codes are combined with physical layer security and the performance and security of the system is evaluated on impulsive noise channels for the first time. This project has three contributions: Polar codes designed for impulsive noise channels using density evo- lution are combined with physical layer security on a wire-tap chan- nel experiencing impulsive noise. The secrecy rate of polar codes is maximised. In the decoding of polar codes, the frozen bits play an important part. The posi- tions of the frozen bits has a significant impact on performance and therefore, the selection of optimal frozen bits is presented to opti- mise the performance while maintaining secure communications on impulsive noise wire-tap channels. Optimal puncturing patterns are investigated to obtain polar codes with arbitrary block lengths and can be applied to different modu- lation schemes, such as binary phase shift keying (BPSK) and M- ary Quadrature Amplitude Modulation (QAM), that can be rate compatible with practical communication systems. The punctured polar codes are combined with physical layer security, allowing the construction of a variety of different code rates while maintaining good performance and security on impulsive noise wire-tap chan- nels. The results from this work have demonstrated that polar codes are ro- bust to the effects of impulsive noise channel and can achieve secure communications. The work also addresses the issue of security on im- pulsive noise channels and has provided important insight into scenarios where the main channel between authorised users has varying levels of impulsiveness compared with the eavesdropper's channel. One of the most interesting results from this thesis is the observation that polar codes combined with physical layer security can achieve good perfor- mance and security even when the main channel is more impulsive than the eavesdropper's channel, which was unexpected. Therefore, this thesis concludes that the low-complexity polar codes are an excellent candidate for the error-correcting codes when combined with physical layer security in more harsh impulsive wireless communication channels

Polar Coding Schemes for Cooperative Transmission Systems

: In this thesis, a serially-concatenated coding scheme with a polar code as the outer code and a low density generator matrix (LDGM) code as the inner code is firstly proposed. It is shown that that the proposed scheme provides a method to improve significantly the low convergence of polar codes and the high error floor of LDGM codes while keeping the advantages of both such as the low encoding and decoding complexity. The bit error rate results show that the proposed scheme by reasonable design have the potential to approach a performance close to the capacity limit and avoid error floor effectively. Secondly, a novel transmission protocol based on polar coding is proposed for the degraded half-duplex relay channel. In the proposed protocol, the relay only needs to forward a part of the decoded source message that the destination needs according to the exquisite nested structure of polar codes. It is proved that the scheme can achieve the capacity of the half-duplex relay channel while enjoying low encoding/decoding complexity. By modeling the practical system, we verify that the proposed scheme outperforms the conventional scheme designed by low-density parity-check codes by simulations. Finally, a generalized partial information relaying protocol is proposed for degraded multiple-relay networks with orthogonal receiver components (MRN-ORCs). In such a protocol, each relay node decodes the received source message with the help of partial information from previous nodes and re-encodes part of the decoded message for transmission to satisfy the decoding requirements for the following relay node or the destination node. For the design of polar codes, the nested structures are constructed based on this protocol and the information sets corresponding to the partial messages forwarded are also calculated. It is proved that the proposed scheme achieves the theoretical capacity of the degraded MRN-ORCs while still retains the low-complexity feature of polar codes

Polar Codes: Finite Length Implementation, Error Correlations and Multilevel Modulation

Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arıkan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arıkan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length