This book explains the philosophy of the polar encoding and decoding technique. Polar codes are one of the most recently discovered capacity-achieving channel codes. What sets them apart from other channel codes is the fact that polar codes are designed mathematically and their performance is mathematically proven. The book develops related fundamental concepts from information theory, such as entropy, mutual information, and channel capacity. It then explains the successive cancellation decoding logic and provides the necessary formulas, moving on to demonstrate the successive cancellation decoding operation with a tree structure. It also demonstrates the calculation of split channel capacities when polar codes are employed for binary erasure channels, and explains the mathematical formulation of successive cancellation decoding for polar codes. In closing, the book presents and proves the channel polarization theorem, before mathematically analyzing the performance of polar codes.
The discovery of channel polarization and polar codes is universally recognized as an historic breakthrough in coding theory. Polar codes provably achieve the capacity of any memoryless symmetric channel, with low encoding and decoding complexity. Moreover, for short block lengths, polar codes under specific decoding algorithms are currently the best known coding scheme for binary-input Gaussian channels. Due to this and other considerations, 3GPP has recently decided to incorporate polar codes in the 5G wireless communications standard. Soon enough, a remarkably short time after their invention, we will be all using polar codes whenever we make a phone call or access the Internet on a mobile device. Our goal in this dissertation is to explore new frontiers in polar coding, thereby fundamentally advancing the current state-of-the-art in the field. Parts of the results are immediately relevant for successful deployment of polar codes in wireless systems, whereas other parts will focus on key theoretical problems in polar coding that have a longer time-horizon. We begin by studying the effect of the polarization kernels in the asymptotic behavior of polar codes. We show that replacing the conventional 2×2 kernel in the construction of polar codes with that of a larger size can reduce the gap to the capacity if the larger kernel is carefully selected. A heuristic algorithm is proposed that helps to find such kernels. Furthermore, we prove that a near-optimal scaling behavior is achievable if one is allowed to increase the kernel size as needed. We also study the computational complexity of decoding algorithms for polar codes with large kernels, which are viewed as their main implementation obstacle. Moving on to the decoding algorithms, we carefully analyze the performance of the successive cancellation decoder with access to the abstract concept of Arikan's genie. The CRC-aided successive-cancellation list decoding, the primary decoding method of polar codes, is commonly viewed as an implementation of the Arikan's genie. However, it comes short at completely simulating the genie since the auxiliary information (CRC) comes to the help only at the end of the decoding process. We overcome this problem by introducing the convolutional decoding algorithm of polar codes that is based on a high-rate convolutional pre-coder and utilizes Viterbi Algorithm to mimic the genie all the way through the SC decoding process. Lastly, we look into channels with deletions. A key assumption in the traditional polar coding is to transmit coded symbols over independent instances of the communication channel. Channels with memory and in particular, deletion channels, do not follow this rule. We introduce a modified polar coding scheme for these channels that depend on much less computational power for decoding than the existing solutions. We also extend the polarization theorems to provide theoretical guarantee and to prove the correctness of our algorithms.
Channel Coding Techniques for Wireless Communications
This book discusses the latest channel coding techniques, MIMO systems, and 5G channel coding evolution. It provides a comprehensive overview of channel coding, covering modern techniques such as turbo codes, low-density parity-check (LDPC) codes, space–time coding, polar codes, LT codes, and Raptor codes as well as the traditional codes such as cyclic codes, BCH, RS codes, and convolutional codes. It also explores MIMO communications, which is an effective method for high-speed or high-reliability wireless communications. It also examines the evolution of 5G channel coding techniques. Each of the 13 chapters features numerous illustrative examples for easy understanding of the coding techniques, and MATLAB-based programs are integrated in the text to enhance readers’ grasp of the underlying theories. Further, PC-based MATLAB m-files for illustrative examples are included for students and researchers involved in advanced and current concepts of coding theory.
Providing in-depth treatment of error correction Error Correction Coding: Mathematical Methods and Algorithms, 2nd Edition provides a comprehensive introduction to classical and modern methods of error correction. The presentation provides a clear, practical introduction to using a lab-oriented approach. Readers are encouraged to implement the encoding and decoding algorithms with explicit algorithm statements and the mathematics used in error correction, balanced with an algorithmic development on how to actually do the encoding and decoding. Both block and stream (convolutional) codes are discussed, and the mathematics required to understand them are introduced on a “just-in-time” basis as the reader progresses through the book. The second edition increases the impact and reach of the book, updating it to discuss recent important technological advances. New material includes: Extensive coverage of LDPC codes, including a variety of decoding algorithms. A comprehensive introduction to polar codes, including systematic encoding/decoding and list decoding. An introduction to fountain codes. Modern applications to systems such as HDTV, DVBT2, and cell phones Error Correction Coding includes extensive program files (for example, C++ code for all LDPC decoders and polar code decoders), laboratory materials for students to implement algorithms, and an updated solutions manual, all of which are perfect to help the reader understand and retain the content. The book covers classical BCH, Reed Solomon, Golay, Reed Muller, Hamming, and convolutional codes which are still component codes in virtually every modern communication system. There are also fulsome discussions of recently developed polar codes and fountain codes that serve to educate the reader on the newest developments in error correction.
This book covers the key technologies associated with the physical transmission of data on fifth generation (5G) mobile systems. Following an overview of these technologies, a high-level description of 3GPP's mobile communications standard (5G NR) is given and it is shown how the key technologies presented earlier facilitate the transmission of control data and very high-speed user data. In the final chapter, an overview and the physical layer aspects of 5G NR enabled Fixed Wireless Access (FWA) networks is presented. This book is intended for those practicing engineers and graduate and upper undergraduate engineering students who have an interest in 3GPP's 5G enabled mobile and or FWA networks and want to acquire, where missing, the necessary technology background in order to understand 3GPP's physical layer specifications and operation. Provides a comprehensive covering of key 3GPP 5G NR physical layer technologies, presented in a clear, tractable fashion, with sufficient mathematics to make it technically coherent; Addresses all key 5G NR technologies, including digital modulation, LDPC and Polar coding, multicarrier based multiple access techniques, and multiple antenna techniques including MIMO and beamforming; Presents an overview of 5G NR Radio Access Network (RAN) architecture and a detailed understanding of how user and control data is transported in the physical layer by the application of the technologies presented; Provides an overview and addresses physical layer aspects of 5G NR enabled Fixed Wireless Access networks.
Broadband Coding Modulation and Transmission Engineering
This book brings together papers from the 2018 International Conference on Communications, Signal Processing, and Systems, which was held in Dalian, China on July 14–16, 2018. Presenting the latest developments and discussing the interactions and links between these multidisciplinary fields, the book spans topics ranging from communications, signal processing and systems. It is aimed at undergraduate and graduate electrical engineering, computer science and mathematics students, researchers and engineers from academia and industry as well as government employees.
"In 2008, a new class of block error correction codes, known as polar codes, were provenby Erdal Arıkan to be able to achieve the Shannon limit. Through inventive new de-coding algorithms and fast code constructions, polar codes have become an attractivehigh-performance error correction code for practical use. These innovations have resultedin adoption of polar codes in the upcoming 3GPP 5 th generation standard for New Ra-dio. Still, polar codes are hindered by certain inflexible characteristics. Arıkan's originalpolar code definition limits block lengths to powers of two, due to a recursive Kroneckerproduct of the 2 × 2 polarizing kernel. This constraint presents a considerable obstacle,as many realistic scenarios call for all code lengths to be readily available. Rate-matchingtechniques, known as puncturing and shortening, allow for flexible polar code lengths,albeit with inefficient decoding complexity. Multi-kernel polar codes produce native codelengths that are powers of two and/or three with the addition of a 3 × 3 ternary kernel,although they necessitate specialized decoders and code design. This thesis will exploreand propose techniques that are intended for maximizing the flexibility and efficiencyof polar codes, as well as analyze any trade-offs affecting error correction performance.An in-depth study is presented that compares state-of-the-art length-flexible polar codeswith the 3GPP standardized polar codes. This inquiry finds that the 5G standard offersa highly simplified polar code construction with minimal loss to error correction per-formance. Further, multi-kernel polar codes were found to have a negative correlationbetween error correction performance and the quantity of ternary Kronecker constituents.This thesis also proposes a new fast successive cancellation decoder that is compliant withmulti-kernel polar codes. The ternary kernel is further investigated by testing its rate-matching and systematic properties. Finally, this thesis proposes a new scheme calledasymmetric polar codes. We present details on generator matrix definition, informa-tion set design, and decoding schedules, as well as perform comparisons with competingschemes using simulations and a comprehensive analysis. Asymmetric polar codes offerflexible block lengths with decoding complexity lower than equivalent length-compatiblepolar codes under successive cancellation. The enclosed findings indicate that asymmetricpolar codes afford comparable error correction performance to the competing schemes,while dividing the number of successive cancellation decoding operations by up to a fac-tor of two. The thesis is then concluded by recommending appropriate extensions of thiswork for future research." --
Channel Coding Techniques for 5G Using Polar Codes
Coding and modulation in the crown known as the communications technology, embodies a national basic theory of the overall strength of communication science. Channel coding is a way of encoding data in a communication channel that adds patterns of redundancy into the transmission path in order to lower the error rate. Such methods are widely used in wireless communications. 5G is the coming fifth-generation wireless broadband technology based on the IEEE 802.11ac standard. 5G will provide better speeds and coverage than the current 4G. It operates with a 5 Ghz signal and is set to offer speeds of up to 1 Gb/s for tens of connections or tens of Mb/s for tens of thousands of connections. Commonly accepted use cases for 5G networks are eMBB (Enhanced Mobile Broadband), Massive IoT (Internet of Things) and URLLC (UltraReliable and Low Latency Communications). eMBB covers Internet access with high data rates to enable rich media applications, cloud storage and applications, and augmented reality for entertainment. All these demanding scenarios make use of many 5G standards of which polar codes is used as the channel coding scheme for eMBB scenario as short codes for control channel. A new class of codes, polar codes, recently made a breakthrough in coding theory. In 2008, Erdal Arıkan at Bilkent University invented polar codes, providing a new mathematical framework to solve this problem. The construction itself was first described by Stolte, and later independently by Erdal Arıkan in 2008 This thesis focuses on study of the key technology of polar code including the construction encoding and decoding. In this work, we analyze a method, known as channel polarization, to construct block codes that achieve the symmetric capacity of any binary-input discrete memoryless channel (B-DMC). The proof of their capacity achieving property is also given. In particular, we show that the algorithm can find almost all the “good” channels with computing complexity which is essentially linear in block-length. This thesis explores the structure and features of polar codes to improve their performance using Gaussian approximation-based construction of polar codes. Several schemes of polar codes are compared with each other like successive cancellation decoding(SC), list decoding(LS), list decoding with CRC(LS+CRC) and finally the existing adaptive decoder is shown to outperform all the schemes.