We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, $L$ decoding paths are considered concurrently at each decoding stage, where $L$ is an integer parameter. At the end of the decoding process, the most likely among the $L$ paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of $L$ . Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the $L$ most likely paths. However, straightforward implementation of this algorithm requires $\Omega (L n^{2})$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only $O(L n \log n)$ time and $O(L n)$ space.

List Decoding of Polar Codes

We show that successive cancellation list decoding can be formulated exclusively using log-likelihood ratios. In addition to numerical stability, the log-likelihood ratio based formulation has useful properties that simplify the sorting step involved in successive cancellation list decoding. We propose a hardware architecture of the successive cancellation list decoder in the log-likelihood ratio domain which, compared with a log-likelihood domain implementation, requires less irregular and smaller memories. This simplification, together with the gains in the metric sorter, lead to $ 56\%$ to $137\%$ higher throughput per unit area than other recently proposed architectures. We then evaluate the empirical performance of the CRC-aided successive cancellation list decoder at different list sizes using different CRCs and conclude that it is important to adapt the CRC length to the list size in order to achieve the best error-rate performance of concatenated polar codes. Finally, we synthesize conventional successive cancellation decoders at large block-lengths with the same block-error probability as our proposed CRC-aided successive cancellation list decoders to demonstrate that, while our decoders have slightly lower throughput and larger area, they have a significantly smaller decoding latency.

LLR-Based Successive Cancellation List Decoding of Polar Codes

Methods and apparatus in a fifth-generation wireless communications, including an example method, in a wireless device, that includes receiving a downlink signal comprising an uplink access configuration index, using the uplink access configuration index to identify an uplink access configuration from among a predetermined plurality of uplink access configurations, and transmitting to the wireless communications network according to the identified uplink access configuration. The example method further includes, in the same wireless device, receiving, in a first subframe, a first Orthogonal Frequency-Division Multiplexing (OFDM) transmission formatted according to a first numerology and receiving, in a second subframe, a second OFDM transmission formatted according to a second numerology, the second numerology differing from the first numerology. Variants of this method, corresponding apparatuses, and corresponding network-side methods and apparatuses are also disclosed.

Network architecture, methods, and devices for a wireless communications network

Since the early 1990s, a large number of chaos-based communication systems have been proposed exploiting the properties of chaotic waveforms. The motivation lies in the significant advantages provided by this class of non-linear signals. For this aim, many communication schemes and applications have been specially designed for chaos-based communication systems where energy, data rate, and synchronization awareness are considered in most designs. Recently, the major focus, however, has been given to the non-coherent chaos-based systems to benefit from the advantages of chaotic signals and non-coherent detection and to avoid the use of chaotic synchronization, which suffers from weak performance in the presence of additive noise. This paper presents a comprehensive survey of the entire wireless radio frequency chaos-based communication systems. First, it outlines the challenges of chaos implementations and synchronization methods, followed by comprehensive literature review and analysis of chaos-based coherent techniques and their applications. In the second part of the survey, we offer a taxonomy of the current literature by focusing on non-coherent detection methods. For each modulation class, this paper categorizes different transmission techniques by elaborating on its modulation, receiver type, data rate, complexity, energy efficiency, multiple access scheme, and performance. In addition, this survey reports on the analysis of tradeoff between different chaos-based communication systems. Finally, several concluding remarks are discussed.

Wireless Chaos-Based Communication Systems: A Comprehensive Survey

A new Multi-Carrier Differential Chaos Shift Keying (MC-DCSK) modulation is presented in this paper. The system endeavors to provide a good trade-off between robustness, energy efficiency and high data rate, while still being simple compared to conventional multi-carrier spread spectrum systems. This system can be seen as a parallel extension of the DCSK modulation where one chaotic reference sequence is transmitted over a predefined subcarrier frequency. Multiple modulated data streams are transmitted over the remaining subcarriers. This transmitter structure increases the spectral efficiency of the conventional DCSK system and uses less energy. The receiver design makes this system easy to implement where no radio frequency (RF) delay circuit is needed to demodulate received data. Various system design parameters are discussed throughout the paper, including the number of subcarriers, the spreading factor, and the transmitted energy. Once the design is explained, the bit error rate performance of the MC-DCSK system is computed and compared to the conventional DCSK system under multipath Rayleigh fading and an additive white Gaussian noise (AWGN) channels. Simulation results confirm the advantages of this new hybrid design.

Design and Analysis of a Multi-Carrier Differential Chaos Shift Keying Communication System

Polar codes provably achieve the symmetric capacity of a memoryless channel while having an explicit construction. The adoption of polar codes however, has been hampered by the low throughput of their decoding algorithm. This work aims to increase the throughput of polar decoding hardware by an order of magnitude relative to successive-cancellation decoders and is more than 8 times faster than the current fastest polar decoder. We present an algorithm, architecture, and FPGA implementation of a flexible, gigabit-per-second polar decoder.

/pdf/fast-polar-decoders-algorithm-and-implementation-564rc1553x.pdf

Fast Polar Decoders: Algorithm and Implementation

Polar codes asymptotically achieve the symmetric capacity of memoryless channels, yet their error-correcting performance under successive-cancellation (SC) decoding for short and moderate length codes is worse than that of other modern codes such as low-density parity-check (LDPC) codes. Of the many methods to improve the error-correction performance of polar codes, list decoding yields the best results, especially when the polar code is concatenated with a cyclic redundancy check (CRC). List decoding involves exploring several decoding paths with SC decoding, and therefore tends to be slower than SC decoding itself, by an order of magnitude in practical implementations. In this paper, we present a new algorithm based on unrolling the decoding tree of the code that improves the speed of list decoding by an order of magnitude when implemented in software. Furthermore, we show that for software-defined radio applications, our proposed algorithm is faster than the fastest software implementations of LDPC decoders in the literature while offering comparable error-correction performance at similar or shorter code lengths.

/pdf/fast-list-decoders-for-polar-codes-41nhedfzl8.pdf

Fast List Decoders for Polar Codes

In this paper, we present hardware and software implementations of flexible polar systematic encoders and decoders. The proposed implementations operate on polar codes of any length less than a maximum and of any rate. We describe the low-complexity, highly parallel, and flexible systematic-encoding algorithm that we use and prove its correctness. Our hardware implementation results show that the overhead of adding code rate and length flexibility is little, and the impact on operation latency minor compared with code-specific versions. Finally, the flexible software encoder and decoder implementations are also shown to be able to maintain high throughput and low latency.

/pdf/flexible-and-low-complexity-encoding-and-decoding-of-2fe703mgfo.pdf

Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes

Modern communication systems must cope with varying channel conditions and differing throughput constraints. Polar codes despite being the first error-correcting codes with an explicit construction to achieve the symmetric capacity of memoryless channels are not currently employed against other older coding protocols such as low-density parity check (LDPC) codes as their performance at short/moderate lengths has been inferior and their decoding algorithm is serial leading to low decoding throughput. Accordingly techniques to address these issues are identified and disclosed including decoders that decode constituent codes without recursion and/or recognize classes of constituent directly decodable codes thereby increasing the decoder throughput. Flexible encoders and decoders supporting polar codes of any length up to a design maximum allow adaptive polar code systems responsive to communication link characteristics, performance, etc. whilst maximizing throughput. Further, designers are provided flexibility in implementing either hardware or software implementations.

Flexible polar encoders and decoders

Polar codes asymptotically achieve the symmetric capacity of memoryless channels, yet their error-correcting performance under successive-cancellation (SC) decoding for short and moderate length codes is worse than that of other modern codes such as low-density parity-check (LDPC) codes. Of the many methods to improve the error-correction performance of polar codes, list decoding yields the best results, especially when the polar code is concatenated with a cyclic redundancy check (CRC). List decoding involves exploring several decoding paths with SC decoding, and therefore tends to be slower than SC decoding itself, by an order of magnitude in practical implementations. In this paper, we present a new algorithm based on unrolling the decoding tree of the code that improves the speed of list decoding by an order of magnitude when implemented in software. Furthermore, we show that for software-defined radio applications, our proposed algorithm is faster than the fastest software implementations of LDPC decoders in the literature while offering comparable error-correction performance at similar or shorter code lengths.

/pdf/fast-list-decoders-for-polar-codes-s7pycf8oka.pdf

Pascal Giard

Papers

Fast Polar Decoders: Algorithm and Implementation

Fast List Decoders for Polar Codes

Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes

Flexible polar encoders and decoders

Fast List Decoders for Polar Codes