Based on pruning techniques that interpret polar codes as low-density parity-check-like (LDPC-like) codes, this paper develops novel belief propagation (BP)-based decoders utilizing the concept of schedule diversity. By analyzing the impact of hidden variable nodes (VNHs), scheduling algorithms that guarantee fast convergence speeds and superior error performance are devised for the LDPC-like sparse factor graph (SPF), incorporating both standard sequential scheduling (SSS) and informed dynamic scheduling (IDS) algorithms. To further enhance error performance, cyclic redundancy check (CRC) bits are integrated into the LDPC-like SPF, and random update sequences are introduced to reduce computational complexity while implementing schedule diversity. Inspired by belief propagation list decoding, the proposed approaches eventually update messages in the LDPC-like SPF according to various schedules, thereby producing diverse decoding outcomes. Compared to the CRC-aided successive cancellation list (CA-SCL) decoder and benchmark soft-output decoders in the literature, the proposed schemes achieve comparable or improved error performance while reducing computational complexity in the high SNR region. These results serve as preliminary findings on the BP algorithm over LDPC-like SFGs and provide useful insights for the design and practical improvement of polar decoders.

This paper presents a method for blind carrier frequency offset (CFO) estimation for quadrature amplitude modulation (QAM). Our proposed algorithm has three major steps, including two-stage full modulation removal, symbol aggregation, and iterative parabolic interpolation. In the first step, a preliminary CFO estimate will be evaluated using the QPSK sub-constellations. The modulation values of the remaining constellation points are partially removed based on the preliminary CFO estimate, and the residual modulation values are quantized to achieve the full modulation removal. In the second step, modulation-removed symbols are paired and weighted based on their reliability levels. This symbol aggregation step improves the estimation accuracy and reduces the FFT size of the following FFT-based CFO estimation. A trade-off between performance and complexity can be achieved by adjusting the aggregation length. In the final step, an iterative parabolic interpolation technique is introduced to mitigate the estimation bias. Performance evaluation shows that the proposed method provides better mean square error (MSE) performance and comparable complexity compared to the existing CFO methods for QAM.

This paper presents two methods for blind carrier frequency offset (CFO) estimation for amplitude phase shift keying (APSK) signals. The first approach carries out modulation removal using the outermost symbols of APSK and then utilizes an iterative parabolic interpolation technique to mitigate the bias of FFT-based CFO estimation. The second method conducts two-stage full modulation removal and aggregates symbols to reduce the FFT size for CFO estimation. In the symbol aggregation process, modulation-removed symbols are paired and weighted based on their reliability to achieve accurate estimation. The first method demonstrates better performance, particularly for high-precision requirements, while the second method offers a more adaptable trade-off between performance and complexity by adjusting the aggregation length. Both methods are more efficient and provide better mean square error (MSE) performance compared to existing ones. Moreover, numerical complexity analyses show the efficiency of the proposed algorithms.

In order to minimize the decoding period and the response time for Polar Codes, an early termination (ET) scheme based on additional check points (ACPs) is proposed in this work. For conventional ET schemes based on distributed parity-check (PC) bits, ET can only be triggered when the decoding process reaches the PC bits. The ACPs are selected from the information bits, and extend the feature of the PC bits, where ET can also be triggered at the ACPs, meaning that a more rapid ET is available that does not impact the error-rate performance. With sophisticated method of selecting ACPs based on the channel-independent polarization weight (PW), the proposed ET scheme is able to reduce the response time by about 5.

This paper presents hardware-friendly LDPC decoding schedules for 5G hybrid automatic repeat request (HARQ) applications. Since there are built-in punctured blocks in the parity check matrix (PCM), a scheduling technique is proposed that allows the punctured nodes to be efficiently recovered. For HARQ using the Chase combining (CC), the previous decoding results corresponding to the punctured part are retained, and the proposed layered decoding is arranged according to the row weight. For HARQ using the incremental redundancy (IR) approach, the parity bits corresponding to the pure-row-orthogonal part of the PCM are transmitted first. The hardware implementation shows that the throughput can be increased by 21.37% for the first decoding attempt, 56.9% for CC-HARQ and 14.51% for IR-HARQ when the code rate reaches 0.303.

We construct polar codes for OFDM systems over the frequency selective channel. The channel coefficients are assumed to be available to the transmitter such that the encoder of polar codes can correctly distinguish the input reliabilities and make proper arrangement of frozen bits and message bits. Instead of using the well-known Gaussian approximation method to construct polar codes, we propose a curve fitting method which not only closely approximate the corresponding channel capacity function but also significantly reduce the computational complexity. With the proposed method, the polar coded OFDM systems can achieve good error performances for systems using high- rate polar codes.

This paper proposes an iterative soft-decision decoding algorithm for Reed-Solomon (RS) codes. The proposed decoding algorithm combines the concepts of adapting the parity-check matrix and informed dynamic scheduling decoding. The parity-check matrix is re-arranged before each iteration, where the systematic part is mapped to the least reliable bits, consequently reducing their influence on the other bits. Using dynamic scheduling, the more important decoding messages are updated to these least reliable bits, meaning that the majority of the error bits with low reliability can be corrected. When the proposed integrated decoding is applied to (255, 239) RS code, the difference between its frame error rate performance (FER) and the maximum-likelihood (ML) bound can be reduced to 0.8 dB, and a gain of about 0.1 dB is achieved compared to all the previously recorded soft-decision decoding for RS codes.

In this paper, a coded modulation scheme using extremely sparse low-density parity-check (LDPC) codes is proposed for the stored signal of triple-level-cell (TLC) NAND flash, where both the encoding and the decoding complexity can be significantly reduced with the advantage of the extremely sparse code graph. In order to enhance the performance of decoder, iterative detection decoding (IDD) is introduced to extract the extrinsic information from the symbol detector, and the cooperative non-Gray mapping is also designed. In addition, for error floor lowering, an interleaver is inserted to ensure the cascaded degree-2 variable nodes are separated to individual symbols. The simulation results show that the proposed coded modulation scheme can provide a practical error floor performance with a low decoding complexity.

Low-density parity-check convolutional codes (LDPC-CCs) are generally decoded using sliding- window based message passing decoding. Based on the sliding-window decoding, an informed dynamic scheduling (IDS) for LDPC-CC is proposed in this work, where the decoding convergence can be significantly accelerated. Since the number of processors required for a satisfactory performance can be reduced, the decoder can be simplified. A set of rate-compatible (RC) puncturing patterns is also proposed and is used to construct RC LDPC-CCs for the performance evaluation of the proposed IDS. Although the proposed puncturing pattern cannot be optimized for individual code rate owing to the rate-compatible constraint, they can still provide a comparable error rate performance compared to the codes defined in the IEEE 802.16m standard. It is worth noting that these standard codes are not rate-compatible.

This paper shows that the decoding convergence of the punctured low-density parity-check (LDPC) codes can be significantly increased using edge wise scheduling belief-propagation. If rate- compatible (RC)-LDPC codes constructed based on puncturing are considered, the maximum mutual information increase (M^2I^2)-based algorithm can be used to arrange fixed schedules for incremental decoding, and further reduce the required number of iterations. With the assistance of the proposed decoding schedules, the puncture-based RC-LDPC codes can be a potential solution for delay- sensitive HARQ (Hybrid-Automatic Repeat reQuest) applications.

In this paper, an iterative soft-decision decoding algorithm is proposed for Reed-Solomon (RS) codes. The proposed decoding algorithm combines the concepts of adapting the parity-check matrix and informed dynamic scheduling. Before each iteration, the parity-check matrix is re-arranged according to the reliability of the codeword bits, meaning that the influence of the least reliable variable nodes on the decoding process can be reduced. Consequently, the important decoding messages can be scheduled to be updated first, and the reliability of the least reliable bits can be enhanced. The simulation results show that the proposed decoding algorithm can provide significant improvement in the error-rate performance. By using the proposed algorithm, a gain of 0.5 dB can be achieved compared to the conventional adapting belief propagation algorithm.

This paper presents a maximum mutual information increase (M2I2)-based algorithm that can be used to arrange low-density parity-check (LDPC) decoding schedules for faster convergence, where the increase is used to guide the arrangement of the fixed decoding schedule. The predicted mutual information for the messages to be updated is used in the calculation of the increase. By looking ahead for several decoding stages, a high-order prediction can be realized, which can then be used to devise a schedule with an even faster convergence. For a single received frame, different decoding results can be obtained using different schedules, and, hence, schedule diversity, that lowers the error floor resultant from the dominant trapping sets, is proposed. By adopting the M2I2-based schedule together with the schedule diversity, both the convergence speed in the waterfall region and the error-rate in the floor region can be improved.

After the publication of the residual belief-propagation (RBP) algorithm, many low-density parity-check (LDPC) decoders using informed dynamic scheduling (IDS) have been investigated. In this paper, we propose the twofold-RBP (T-RBP) decoder that combines two residuals. Using T-RBP, significant improvement can be achieved in both convergence speed and convergence error-rate performance. In addition to T-RBP, the simplified-RBP (S-RBP) decoder is also proposed in order to reduce the complexity. Instead of comparing all the residuals of all edges in the code graph, as with previous IDS decoders, S-RBP only compares the residuals of the edges connected to a single check node, thus dramatically reduces the complexity without any significant degradation in performance. The proposed T-RBP and S-RBP can improve not only the performance of dedicated codes, but also that of punctured codes, especially the convergence speed. The improvement can make punctured LDPC codes more practical and becoming a competitive candidate for the rate compatible applications.

In this paper, we focus on the design of differential amplitude/phase modulation schemes by a look-up table for Rayleigh fading channels. In the first part of this paper, we first propose the noncoherent distance measure for each two signal vectors based on the maximum-likelihood (ML) signal detection under Rayleigh fading channels. The design criterion/procedure of the look-up table is then investigated. In the second part of this paper, linear dispersion codes are used to construct a multi-dimensional differential amplitude/phase modulation scheme, with the aid of the genetic algorithm. Both schemes are shown to provide improvements in error performance, when comparing to the conventional scheme provided in the literature.

In this paper, a full-duplex compress-and-forward (CF) protocol for cooperative communication systems using Raptor codes is first investigated. It is verified that the throughput of the full-duplex CF protocol is superior to that of the full-duplex decode-and-forward (DF) protocol when the relay-to- destination channel has a much higher quality compared to the source-to-relay channel, while the full-duplex DF protocol works more effectively than the full-duplex CF protocol in the reverse situation. A full-duplex adaptive relaying protocol that is devised based on both CF and DF protocols is then proposed. In the proposed protocol, the CF and DF protocols are adaptively switched, depending on the channel conditions of the wireless links. In addition, the throughput of the overall system is enhanced using an outer Raptor code. Simulation results show that the proposed scheme smoothly switches between the two conventional protocols to cope with different channel conditions.

In the encoding of Luby transform (LT) codes, information bits (variable nodes) are randomly connected to the check nodes. Although the check-node degrees have been optimized for error-rate performance, the arbitrary connections in the code graph cannot guarantee efficient propagation of the channel information (intrinsic messages), and may slow down the convergence speed of the iterative decoding. This paper presents a design for a generator matrix such that the intrinsic messages can be efficiently propagated through the arranged connections. In addition, the convergence speed of rateless codes can also be accelerated using the proposed degree-oriented scheduling (DOS), where the intrinsic messages are propagated based on the order of check node degrees. An improvement in the convergence speed can be achieved without sacrificing the error-rate performance. In the case of Raptor codes, the BER performance can be significantly improved.

When residual belief-propagation (RBP), which is a kind of informed dynamic scheduling (IDS), is applied to low-density parity-check (LDPC) codes, the convergence speed in error-rate performance can be significantly improved. However, the RBP decoders presented in previous literature suffer from poor convergence error-rate performance due to the two phenomena explored in this paper. The first is the greedy-group phenomenon, which results in a small part of the decoding graph occupying most of the decoding resources. By limiting the number of updates for each edge message in the decoding graph, the proposed Quota-based RBP (Q-RBP) schedule can reduce the probability of greedy groups forming. The other phenomenon is the silent-variable-nodes issue, which is a condition where some variable nodes have no chance of contributing their intrinsic messages to the decoding process. As a result, we propose the Silent-Variable-Node-Free RBP (SVNF-RBP) schedule, which can force all variable nodes to contribute their intrinsic messages to the decoding process equally. Both the Q-RBP and the SVNF-RBP provide appealing convergence speed and convergence error-rate performance compared to previous IDS decoders for both dedicated and punctured LDPC codes.

As part of the tradeoff between error performance, decoding complexity and the overhead of rateless codes, the combination of incremental decoding (ID) and informed dynamic scheduling (IDS), known as IDIDS (incremental decoding with informed dynamic scheduling), is an attractive solution in binary symmetric channels (BSC). However, applying IDIDS in the AWGN (additive white Gaussian noise) channel may cause a degradation in BER performance, since the incrementally received symbols from the AWGN channel may not be immediately utilized in the decoding process directed by IDS. In addition, a stopping criterion combined with IDS may cause the current decoding attempt to terminate too early, and the channel information contained in the originally received codeword will not be used efficiently. In this paper, a dynamic decoding schedule strategy is proposed. In the proposed decoder, the new received symbols can be immediately utilized in the decoding process. For Luby transform (LT) codes and Raptor codes over the AWGN channel, the proposed algorithm provides a more balanced tradeoff. In the case of Raptor codes, the BER performance is obviously improved.