Polar Coded Modulation Research Articles

Probabilistic Shaped Joint Source-Channel Polar-Coded Modulation

Probabilistic Shaped Joint Source-Channel Polar-Coded Modulation

Performance of Adaptive Bit-Interleaved Polar Coded Modulation in FSOC System

This paper proposes an adaptive bit-interleaved polar coded modulation (A-BIPCM) method based on minimum logarithmic upper bound weight (MLUW). It is designed to reduce the fading effects and long string of bit error interference caused by atmospheric turbulence in free-space optical communications (FSOC). To assess the effectiveness of this method across turbulent channels of varying intensities, we conducted an evaluation of the bit error rate (BER) performance of polar codes in turbulent channels. The results demonstrate significant performance improvements provided by the A-BIPCM method compared to conventional polar code encoding and decoding. Specifically, under weak, moderate, and strong turbulence conditions, the A-BIPCM method achieves performance gains of 0.96 dB, 1.66 dB, and 1.35 dB, respectively. Additionally, an experimental verification platform for FSOC employing intensity modulation direct detection (IM/DD) with an atmospheric turbulence simulation channel, is established in this work. When the optical power of the detector is −16 dBm, the traditional polar code encoding and decoding performance at BER = 2.36 × 10−5, whereas the A-BIPCM scheme exhibits a significantly higher performance at BER = 2.11 × 10−6. The BER has been improved by representing an order of magnitude.

Polar-Coded Cross-Packet HARQ Based On Polarizing Matrix Extension

In this letter, polar-coded cross-packet hybrid automatic repeat request (XP-HARQ) using multilevel coded modulation (MLCM) to improve throughput performance is investigated. Firstly, a novel polar-coded XP-HARQ framework combining polar codes, MLCM, and level-independent XP-HARQ by using polarizing matrix extension is proposed. Then, a two-step throughput-based code design method is developed to generate optimal code rates and corresponding bit protect pairs and new information bit sets in extended codes. Simulation results demonstrate that the proposed scheme outperforms level-independent Turbo-coded XP-HARQ and state-of-art hybrid automatic repeat request (HARQ) using polar-coded modulation.

Conditional Entropy-Based Construction of Multilevel Polar-Coded Modulation With Probabilistic Shaping

Code construction is a critical issue for polar-coded systems to ensure competitive performance. In this letter, we propose a conditional entropy-based method to construct multilevel polar-coded modulation (MLC-PCM) with probabilistic shaping. More specifically, we allocate the bit positions in accordance with their conditional entropies induced by channel and source polarization. We show that a surrogate channel can be used to determine the frozen set in each component polar code efficiently. The remaining positions are assigned to the information and shaping bits by the source Bhattacharyya parameter, where the shaping bits are used to achieve non-uniform channel inputs. Approximate bounds are also derived to evaluate the decoding error probability of the probabilistically shaped MLC-PCM.

Sliding Window Soft-SCL Decoders for Asynchronous Polar-Coded Modulation

Asynchronous polar-coded modulation (A-PCM) outperforms multilevel polar-coded modulation by taking advantage of spatial coupling among several coded blocks. To further improve the performance of A-PCM, a cyclic-redundancy-check (CRC) aided sliding window soft successive cancellation list (CA-SWSCL) decoder is proposed to exploit the benefits of spatial coupling and CRC-aided iterative message passing. This approach includes both soft-output and hard-output iteration processes, where the CRC results are not only used for early termination but also served as an execution flag of the hard-output iteration process. In the case of rate half polar-coded modulation with frame length 128, A-PCM with the proposed CA-SWSCL achieves the best block-error-rate (BLER) performance among the state-of-art polar-coded modulation schemes.

Short-Block Length Polar-Coded Modulation for the Relay Channel

Short-block length communication is becoming increasingly important for applications such as machine-to-machine communication, among others. This paper studies polar-coded modulation in the short-block length regime for decode-forward, amplify-forward and compress-forward relays. Our work seamlessly combines multi-level signaling, polar coding/decoding, and the requirements of relaying, into encoders and decoders that exhibit excellent performance. Compared with the state of the art in decode-forward, our work demonstrates 2.5 dB improvement (at block length 512). For shorter block lengths 256 and 128, this work is the first reported implementation for any coded modulation and any relaying protocol. In the category of polar-coded full-duplex relaying, this work presents the first implementation at any block length and for any relaying protocol. For decode-forward, we propose and analyze joint iterative belief propagation decoding with polar codes, and successive list decoding with polarization-adjusted convolutional (PAC) codes. For amplify-forward, we utilize PAC codes and successive list decoding. For the three relaying protocols, the respective dispersion bounds are presented for comparison, and the error exponent of the multi-level relaying coded modulations are analyzed to shed light on the results. Extensive simulations verify the performance of the proposed coding schemes.

Block Polarization HARQ for Polar-Coded Modulation

In this paper, we propose a block polarization (BP) scheme as a framework to study polar-coded hybrid automatic repeat request (Polar-HARQ) under bit-interleaved polar-coded modulation (BIPCM). The BP scheme of BIPCM can combine multiple independent polar subcodes to form a longer polar code under the block polarization between polar subcodes. The BP scheme of HARQ further carries out block polarization based on the BP scheme of BIPCM, which can combine multiple transmitted BP schemes of BIPCM together to form a longer polar code. When the underlying channels are non-uniform, such as the polarization effect of high-order modulation will lead to non-uniform reliabilities, it is not suitable to use the fixed sequence (e.g., the polarization weight (PW) sequence or Polar sequence in the 5G standard) to construct polar codes. The underlying channels on each polar subcode have uniform reliability by dividing different blocks according to the reliabilities of the underlying channel. Based on this characteristic of BP scheme, we propose a rate-allocation (RA) method to study fast code construction of BP scheme. The key idea of the proposed RA method is finding out the equivalent channel whose average symmetric capacity equals the target transmission rate. The allocated rates are computed according to the symmetric capacities of split channels under the block polarization between polar subcodes. Then the ranking of bit indices within each polar subcode can be obtained by fixed sequence. In this way, the RA code construction is concise and robust for diverse configurations which are the desired features for practical implementation. Simulation results show that the proposed BP scheme under RA construction can achieve almost identical performance as the Gaussian approximation construction with much lower complexity.

Performance Research of MDM System based on Bit Interleaved Polar Coded Modulation

Performance Research of MDM System based on Bit Interleaved Polar Coded Modulation

Analysis and Design of Polar-Coded Modulation

Conventional design methods of polar-coded modulation schemes aim to minimize the block error rate (BLER) under successive cancellation (SC) decoding. However, codes designed by conventional methods are not competitive under successive cancellation list (SCL) decoding. This paper presents a new design method based on the BLER upper bound under maximum-likelihood (ML) decoding (ML-BLER upper bound). The ML-BLER upper bound depends on the weight enumerating function (WEF) of the polar-coded modulation scheme over the squared Euclidean distance. In this paper, the polar-coded modulation is randomized by the concept of interleaved polar (i-polar) codes, and the WEF averaged over the ensemble of the polar-coded modulation schemes can be derived. Three polar-coded modulation schemes are considered, i.e., the bit-interleaved polar-coded modulation with a single interleaver (BIPCM-SI), the bit-interleaved polar-coded modulation with multiple interleavers (BIPCM-MI), and the multi-level polar-coded modulation (MLPCM). A new bit channel selection algorithm for polar-coded modulation schemes is proposed, which takes the polarization effect and the ML-BLER upper bound as design criteria. Design examples show that, under SCL decoding, the polar-coded modulation schemes (without CRC) with the proposed channel selection algorithm outperform those with conventional algorithms and are competitive as compared to the state-of-the-art 5G LDPC codes.

Partially Information Coupled Bit-Interleaved Polar Coded Modulation

In this paper, we propose partially information coupled bit-interleaved polar coded modulation (PIC-BIPCM), which is a class of spatially coupled polar coding schemes designed for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{Q}$ </tex-math></inline-formula> -ary modulations. Specifically, we consider PIC-BIPCM schemes respectively constructed with three BIPCM schemes: direct BIPCM, punctured BIPCM, and multi-kernel BIPCM. We analyze the error performance of the proposed PIC-BIPCM over the binary erasure channel (BEC) via density evolution. With the analysis as a guideline, we jointly design the positions of coupled bits and the modulation bit-mapper by taking into account the partial polarization of finite length polar codes as well as the unequal error protection of high order modulations. Simulation results demonstrate significant performance improvement of the proposed PIC-BIPCM over the uncoupled BIPCM on both BEC and AWGN channels.

Systematic Polar Coded Modulation for Informed Receivers

The problem of coding for informed receivers (IR) is considered, where a transmitter multicasts a bunch of messages to its associated receivers, each of which already has a subset of messages as side information. Which receiver has what side information is unknown to the transmitter. A family of polar codes for IR was proposed for binary-input channels in (Huang and Shieh, 2018), which was shown to provide excellent side information gain regardless of side information configuration and content. A trial extension of this scheme to high-order modulations, however, was unsatisfactory. In this paper, two families of systematic polar coded modulations for IR are proposed. In the first family, by building a connection between the message bits and modulated symbols under systematic polar codes, we leverage the bit assignments and labeling techniques such that receiver side information can be directly translated to a minimum distance gain. In the second family, a permutation operation is added before modulation such that the message bits can be properly reordered to further advance and balance the gains among different side information configurations. Simulations show that the proposed schemes provide large and balanced side information gains and significantly outperform the system directly adapted from (Huang and Shieh, 2018) onto high-order modulations.

Progressive Rate-Filling: A Framework for Agile Construction of Multilevel Polar-Coded Modulation

We propose the progressive rate-filling method as a framework to study agile construction of multilevel polar-coded modulation. We show that the bit indices within each component polar code can follow a fixed, precomputed ranking sequence (e.g., the Polar sequence in the 5G standard) while their allocated rates (i.e., the number of information bits of each component polar code) can be fast computed by exploiting the target sum-rate approximation and proper rate-filling methods. In particular, we propose two rate-filling methods based on the capacity and the rate considering the finite blocklength effect. The proposed construction methods can be performed independently of the actual channel condition with ${O(m)}$ ( ${m}$ denotes the modulation order) complexity and robust to diverse modulation and coding schemes (MCSs) in the 5G standard, which is a desired feature for practical systems.

Convolutional Polar Coded Modulation

$2^m$-ary modulation creates $m$ bit channels which are neither independent nor identical, and this causes problems when applying polar coding because polar codes are designed for independent identical channels. Different from the existing multi-level coding (MLC) and bit-interleaved coded modulation (BICM) schemes, this paper provides a convolutional polar coded modulation (CPCM) method that preserves the low-complexity nature of BICM while offering improved spectral efficiency. Numerical results are given to show the good performance of the proposed method.

Performance Enhanced 256-QAM BIPCM-DMT System Enabled by CAZAC Precoding

In the paper, bit-interleaved polar-coded modulation discrete multi-tone (DMT) (BIPCM-DMT) with 256-QAM enabled by constant amplitude zero auto-correlation (CAZAC) precoding is proposed and experimentally demonstrated in a low-cost intensity-modulation and direct-detection (IM-DD) system. In order to conquer the fading effect on mapped bit-levels, two types of interleaver including quadratic polynomial permutation (QPP) interleaver and random interleaver are compared to optimize the system performance. However, for 256-QAM BIPCM-DMT system, the signal-to-noise ratio (SNR) fades dramatically as the data subcarrier index increases, it is difficult to match the well-designed operating point of Monte Carlo in the process of polar code construction. Thus, the precoding schemes including CAZAC and orthogonal circulant transform (OCT) are proposed to equalize the uneven SNRs in BIPCM-DMT system. Compared with the bit / power loading and pre-equalization schemes, it will greatly reduce the cost and complexity of the system because channel information obtained by the transmitter via round-trip feedback is not considered. In addition, CAZAC precoding is selected to optimize the BIPCM-DMT system because it is superior to OCT precoding in reducing PAPR. The experimental results show that aided by CAZAC precoding, 20.63 Gb/s BIPCM-DMT signal can be received with an error-free over 50-km standard single-mode fiber (SSMF). Furthermore, compared with BIPCM-free, only CAZAC precoding, and only random interleaver schemes, the proposed scheme can achieve Q-factor enhancement of 4.82 dB, 2.46 and 1.78 dB, respectively, at the received optical power (ROP) of −5 dBm.

An Experimental Demonstration of Rate-Adaptation Using Shaped Polar Codes for Flexible Optical Networks

In this paper, we propose a polar-coded transmission system with adjustable data rate for coherent optical transmission employing quadrature amplitude modulation (QAM). The proposed system is based on a many-to-one constellation shaping method for achieving a range of data rates with arbitrarily small rate steps. An implicitly punctured polar-coded modulation system is then designed by combining polar coded bit-interleaved coded modulation system with many-to-one labelings. The scheme is experimentally evaluated in a wavelength division multiplexed (WDM) system with five carriers modulated at 16 GBaud with polarization multiplexed (PM) 256QAM. Data rates ranging from 121 to 182 Gb/s per carrier are experimentally demonstrated and the system can be directly extended to achieve higher data rates. Synchronization and equalization of PM-256QAM received symbols are performed with a pilot rate of 5%. The experimental results show 1.2 dB of shaping gain over the conventionally punctured polar codes in the optical back-to-back scenario. The maximum transmission system reach is increased by 200 to 400 km w.r.t the conventionally punctured polar codes. It is shown that the rate adaptation does not require a change of modulation format and/or underlying forward error-correction code. Finally, the performance of all five WDM channels is validated for the optimal input power.

Throughput-Based Design for Polar-Coded Modulation

Typically, forward error correction (FEC) codes are designed based on the minimization of the error rate for a given code rate. However, for applications that incorporate hybrid automatic repeat request (HARQ) protocol and adaptive modulation and coding, the throughput is a more important performance metric than the error rate. Polar codes, a new class of FEC codes with simple rate matching, can be optimized efficiently for maximization of the throughput. In this paper, we aim to design HARQ schemes using multilevel polar-coded modulation (MLPCM). Thus, we first develop a method to determine a set-partitioning-based bit-to-symbol mapping for high-order quadrature amplitude modulation (QAM) constellations. We simplify the log-likelihood ratio estimation of set-partitioned QAM constellations for a multistage decoder, and we introduce a set of algorithms to design throughput-maximizing MLPCM for the successive cancellation decoding (SCD). These codes are specifically useful for non-combining, chase-combining, and incremental redundancy HARQ protocols. Furthermore, since optimized codes for SCD are not optimal for SC list decoders (SCLDs), we propose a rate matching algorithm to find the best rate for SCLD while using the polar codes optimized for SCD. The resulting codes provide throughput close to the capacity with low decoding complexity when used with HARQ protocols.

Scaling Analysis of Multilevel Polar Coded Modulation

In this paper, we analyze and prove the capacity-achieving property of the equivalent asymmetric channels of multilevel polar-coded modulation in view of the scaling assuming regime, which employ some techniques of symmetric channels. Based on the previous framework of the proof of multilevel coding modulation, we apply the scaling exponent to the equivalent asymmetric channels of multilevel polar-coded modulation under the constraint of the finite code length of component polar codes. Then, we prove that the overall capacity is still achievable under the capacity rule of the design concepts of multilevel coding. In order to accomplish reliable communication of the proposed scheme, we combine Gallager’s mapping conception and source-channel coding theorems to generate the component polar codes with the optimal input distribution for each equivalent asymmetric channel. The optimal property refers to asymptotic optimization under a total variation distance measure. Finally, we describe the detailed design process of multilevel polar-coded modulation versus multistage decoding applying polar codes as the component codes analytically.

Mapping Design for $2^{M}$ -Ary Bit-Interleaved Polar Coded Modulation

This paper proposes a mapping design for bit-interleaved polar coded modulation (BIPCM) systems with belief propagation (BP) decoding. We first introduce a two-layer bipartite graph to represent BIPCM, where a new mapping graph linking polar graph to modulator is added to the conventional factor graph. Then, a mapping design is proposed and the design paradigm is to separate sub-channels with lower reliability to different stopping trees of polar codes, aiming to make sure that each stopping tree receives reliable extrinsic information from demodulator. The proposed mapping algorithm is employed for BIPCM with traditional polar codes over 16-quadrature amplitude modulation (QAM) and 256-QAM. Numerical results show that our scheme can improve the error-correcting performance compared to the conventional scheme with a random mapping. Furthermore, to meet code-length requirement of different modulation orders, we propose an efficient method to construct flexible-length polar code (FLPC) by coupling several short length polar codes with a repeat-accumulate (RA) code. Also, the proposed FLPC is employed in the BIPCM system, with the designed mapping algorithm, simulation result also reveals that the block error rate performance of proposed BIPCM scheme with BP decoding outperforms the one with successive cancellation decoding by providing a gain up to 1 dB.

Symbol Mapping Design for Bit-Interleaved Polar-Coded Modulation With Iterative Decoding

In this letter, we design a bit-interleaved polar-coded modulation system with iterative decoding subject to constraints on complexity, error floor, and waterfall region performance. We show that the hill-like trajectory of polar code extrinsic information transfer curves allows us to predict the gap of the bottleneck region that a polar code achieves at a given received signal-to-noise ratio for a given labeling rule. Using the gap, we propose a 3-D linear objective function to optimize for error floor, waterfall region performance, and complexity. The objective function requires Monte Carlo simulations for only three points on the extrinsic information transfer curve of the demapper for a given labeling rule. Our simulations show that the proposed method designs labeling rule and polar code duos that trade off error-floor and waterfall region performance with complexity and latency.

Bit-interleaved polar coded modulation scheme for high-order QAM encoded DDO-OFDM systems

Abstract In the paper, a simple generation scheme of the twin-SSB-OFDM signal based on one OFDM transmitter is proposed in the direct-detection optical orthogonal frequency-division multiplexing (DDO-OFDM) system. Compared with traditional generation scheme that requires two OFDM transmitters, it has lower implementation complexity and lower cost. In addition, to enhance the spectral efficiency (SE) for DDO-OFDM system, an efficient bit-interleaved polar coded modulation (BIPCM) scheme adopting 256 quadrature-amplitude modulation (256-QAM) is proposed. Moreover, a quadratic polynomial permutation (QPP) interleaver is adopted for polar coded modulation (PCM) to further improve the overall system performance. The simulation results exhibit that the proposed scheme can achieve more than 4 dB and 5 dB Q-factor improvement compared to the conventional schemes with PCM and without BIPCM, respectively. At the same time, aided by the efficient 256-QAM BIPCM scheme, the receiver sensitivity is improved and the laser linewidth requirement is less strict compared to the conventional schemes.