Window Decoding Research Articles

Large Kernel Polar Codes With Efficient Window Decoding

Window decoding is useful for decoding polar codes defined by kernels that do not have Arikan's original form. We modify arbitrary polarization kernels of size $2^t \times 2^t$ to reduce the time complexity of window decoding. This modification is based on the permutation of the columns of the kernel. This method is applied to some of the kernels constructed in the literature of size 16 and 32, with different error exponents and scaling exponents such as eNBCH kernel. It is shown that this method reduces the complexity of the window decoding significantly without affecting the performance.

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding.

A Novel Concatenated Coding Scheme: RS-SC-LDPC Codes

A novel concatenated coding scheme is introduced, where the Reed-Solomon (RS) code is concatenated with the spatially coupled low-density parity-check (SC-LDPC) code, namely the RS-SC-LDPC code. We first propose the locally systematic encoding of the inner code and its termination when considering the concatenated code has a finite length. A decoding scheme is then proposed, in which the belief propagation (BP) based sliding window decoding (SWD) decodes the inner code with the assistance of the outer Berlekamp-Massey (BM) decoding feedback. This research shows the RS-SC-LDPC codes can achieve a high decoding performance without yielding an error floor until the bit error rate (BER) of 10−8. They also significantly outperform the incumbent RS-convolutional concatenated (RS-CC) codes, as well as the BCH-SC-LDPC codes.

Spatially Coupled Codes via Partial and Recursive Superposition for Industrial IoT With High Trustworthiness

For industrial Internet of Things (IIoT), data trustworthiness should be maintained both at the time of sensing and at the time of transmission. This article is concerned with trustworthiness during transmission, which is determined by transmission reliability. We present a low-complexity and flexible method via partial and recursive superposition to improve the transmission reliability of IIoT, resulting in an IIoT with high trustworthiness. In our method, a portion of the previously transmitted data are superimposed onto the current transmitted data to introduce memory among different transmissions, which are then exploited by the windowed decoder to obtain performance gain. The proposed method is referred to as partially recursive block Markov superposition transmission of low-density parity-check (PrBMST-LDPC) codes. This article is focused on the construction of low-complexity PrBMST-LDPC codes since IIoT is resource-limited in nature. The first construction is the memory-one PrBMST-LDPC code. We present a simplified density evolution algorithm to optimize the superposition ratio for memory-one PrBMST-LDPC code. Both the analytical and numerical results show that PrBMST with memory one can be used to reduce the packet loss ratio (PLR) of IIoT using LDPC codes. Particularly, around 1.0 dB performance gain is obtained by PrBMST. We then present a low-complexity construction for PrBMST-LDPC codes with encoding memory larger than one. Simulation results show that compared with memory-one PrBMST, a further PLR reduction of around one order of magnitude can be obtained.

Error Propagation Mitigation in Sliding Window Decoding of Braided Convolutional Codes

We investigate error propagation in sliding window decoding of braided convolutional codes (BCCs). Previous studies of BCCs have focused on iterative decoding thresholds, minimum distance properties, and their bit error rate (BER) performance at small to moderate frame length. Here, we consider a sliding window decoder in the context of large frame length or one that continuously outputs blocks in a streaming fashion. In this case, decoder error propagation, due to the feedback inherent in BCCs, can be a serious problem. To mitigate the effects of error propagation, we propose several schemes: a window extension algorithm where the decoder window size can be extended adaptively, a resynchronization mechanism where we reset the encoder to the initial state, and a retransmission strategy where erroneously decoded blocks are retransmitted. In addition, we introduce a soft BER stopping rule to reduce computational complexity, and the tradeoff between performance and complexity is examined. Simulation results show that, using the proposed window extension algorithm, resynchronization mechanism, and retransmission strategy, the BER performance of BCCs can be improved by up to four orders of magnitude in the signal-to-noise ratio operating range of interest, and the soft BER stopping rule can be employed to reduce computational complexity.

Designing Protograph-Based Quasi-Cyclic Spatially Coupled LDPC Codes With Large Girth

Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. In this paper, we introduce a systematic design to eliminate 4-cycles in a coupled protograph. Further using a quasi-cyclic (QC) lifting, we introduce a procedure for constructing QC-SC-LDPC codes of girth at least eight. This can be interpreted as a multi-stage graph lifting process that yields a greater flexibility in designing QC-SC-LDPC codes with a large girth than previous approaches. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random constructions. Finally, we determine the minimum coupling width required to eliminate 4-cycles in a coupled protograph.

Doubly-Recursive Block Markov Superposition Transmission: A Low-Complexity and Flexible Coding Scheme

In this paper, we introduce a novel method, called doubly-recursive block Markov superposition transmission (DrBMST), to construct high-performance spatially coupled codes. An important characteristic of DrBMST codes is that the degrees of the constraint nodes in their normal graphs are at most three. As a result, DrBMST codes can be decoded with low complexity. We first prove that the error probability of an enlarged DrBMST code ensemble can be made arbitrarily small under windowed maximum-likelihood decoding (MLD) by increasing the decoding window size. This result partially explains the superior performances of DrBMST codes. Then we propose to use the extrinsic information transfer (EXIT) chart analysis to estimate the iterative windowed decoding thresholds of DrBMST codes. The EXIT chart analyses show that, with such a simple structure, DrBMST codes are comparable to BMST codes with large encoding memories in terms of decoding thresholds. Finally, we carry out comparisons to validate the advantages of DrBMST codes in terms of error performances and decoding complexities. In particular, for a decoding latency of 20,000 bits, the DrBMST code performs better than the (4, 8)-regular spatially coupled low-density parity-check (SC-LDPC) code, but with lower computational complexity. Hence, DrBMST codes can be used in communication systems with limited computational resources. In addition, we show that DrBMST can be used to construct multiple-rate codes.

Protograph-Based Folded Spatially Coupled LDPC Codes for Burst Erasure Channels

In this letter, protograph-based folded spatially coupled (FSC) LDPC codes are proposed. The new folded-type structure is obtained by folding the spatial coupling chain of a conventional spatially coupled (SC) LDPC protograph and interlacing the nodes at staggered spatial positions. The proposed codes outperform the SC LDPC codes over single and multiple random-burst erasure channels. We extend the construction of the folded-type structure by connecting multiple one-sided SC LDPC chains for higher resilience to burst erasure channels. The FSC LDPC codes are also compatible with windowed decoder and outperform conventional SC LDPC codes.

Coupling Information Transmission With Window Decoding

We consider a coupling information transmission modulation format utilized for communication over a multiple-access channel. A multitude of packets is transmitted by a number of sources to a common receiver. Each packet is modulated via simple repetition, interleaving, and an application of a signature sequence. The packets are transmitted with time offsets that initiate coupling of the transmitted signals into a received signal that lends itself to an efficient window-based iterative estimation and interference cancellation decoding. We present the window decoding formulation, a theoretic analysis, and numerical results for the achievable sum-rate, gap to the channel capacity and the optimal window size.

Discrete Hand Motion Intention Decoding Based on Transient Myoelectric Signals

In the application of human-robot interaction (HRI) rehabilitation exercise controlled by Electromyogram (EMG), if the discrete motion intention decoded by EMG signals is within the range of electromechanical delay (EMD), through the rehabilitation training, it will largely enhance the user's feeling feedback and fully activate the brain plasticity. So the decoding of hand movement intention by transient EMG is investigated to reach ideal characteristics needed in the HRI. The high-density EMG signal (HDEMG) database CapgMyo was used to decode the motion intention based on the transient EMG signals within the EMD rather than the whole recorded signals. We investigate the impact of the different decoding window lengths (WL) and training sets constructing methods when using several machine learning algorithms. In addition, the transient EMG signals decoding performance of the sparse multi-electrode EMG signal database NinaPro performing the same hand movement was compared. The visual inspection of the EMG map was used to determine the onset of HDEMG. The proposed approach was tested on EMG decoding window length of 150 ms, demonstrating a mean±SD testing performance of 94.21%±4.84% after voting. However, it is worth noting that sparse EMG signal did not achieve the desired decoding accuracy. The result showed that the high-density EMG signal could be used to decode the motion intention within EMD by simple machine learning algorithms, and extending the window length of training set could improve the decoding accuracy.

Reduced Complexity Window Decoding of Spatially Coupled LDPC Codes for Magnetic Recording Systems

In channel coding theory, the performance of error correcting codes (ECCs) approaching the Shannon limit can be achieved through increasing code lengths. Unfortunately, the complexity of ECCs will be increased as the code length increases. Nowadays, the magnetic recording (MR) system takes advantage of powerful ECCs by using 4 Kbytes sector. Among the advanced ECCs, the spatially coupled LDPC (SC-LDPC) codes (also known as a LDPC convolutional code) [1]are shown to have the decoding latency and complexity lower than those of the underlying LDPC block codes (LDPC-BC). Moreover, the SC-LDPC codes with threshold decoding outperform the LDPC-BC codes [2]. Hence, the SC-LDPC codes are the strong candidate for the future MR systems, when the sector size is increased beyond 4 Kbytes. An SC-LDPC decoder can use sliding window decoding [3]whereby the received signals are decoded by sliding window along the bit sequence. The window decoder is called “uniform window decoding (U-WD),” when all variable nodes (VNs) within a window are updated. In order to reduce the complexity of window decoding, some researchers proposed the non-uniform window decoding (N-WD) [4], which do not update the VNs with no improvement in the bit error rate (BER). This approach provides about 35-50% reduction in complexity compared to U-WD. In this work, we consider the application of SC-LDPC codes in MR systems, whereby SC-LDPC decoder cooperates with BCJR detector to encounter inter-symbol interference (ISI). We propose the dynamic shifting of window decoding (DS-WD) to reduce the complexity of SC-LDPC codes. Herein, the number of shifted bits is defined according to their soft BERs which are estimated at each decoding position. In addition, we modify the N-WD [4]to reinforce our proposed algorithm called “dynamic-shifting non-uniform window decoding (DS-N-WD).” The DS-WD and DS-N-WD achieve the complexity reduction of 7% and 25% without any loss in performance compared to the N-WD algorithms.

Comparison of decoding performance between spike and local field potential signals during goal-directed decision-making task of pigeons

Both spike and local field potential (LFP) signals are two of the most important candidate signals for neural decoding. At present there are numerous studies on their decoding performance in mammals, but the decoding performance in birds is still not clear. We analyzed the decoding performance of both signals recorded from nidopallium caudolaterale area in six pigeons during the goal-directed decision-making task using the decoding algorithm combining leave-one-out and k-nearest neighbor (LOO- kNN). And the influence of the parameters, include the number of channels, the position and size of decoding window, and the nearest neighbor k value, on the decoding performance was also studied. The results in this study have shown that the two signals can effectively decode the movement intention of pigeons during the this task, but in contrast, the decoding performance of LFP signal is higher than that of spike signal and it is less affected by the number of channels. The best decoding window is in the second half of the goal-directed decision-making process, and the optimal decoding window size of LFP signal (0.3 s) is shorter than that of spike signal (1 s). For the LOO- kNN algorithm, the accuracy is inversely proportional to the k value. The smaller the k value is, the larger the accuracy of decoding is. The results in this study will help to parse the neural information processing mechanism of brain and also have reference value for brain-computer interface.

Partially Information-Coupled Turbo Codes for LTE Systems

We propose a new class of partially information-coupled (PIC) turbo codes to improve the transport block (TB) error rate performance for long-term evolution (LTE) systems, while keeping the hybrid automatic repeat request protocol and the turbo decoder for each code block (CB) unchanged. In the proposed codes, every two consecutive CBs in a TB are coupled together by sharing partial information bits. This coding structure introduces irregular variable node degree distributions in the Tanner graph of the PIC-turbo codes. We propose a feedforward and feedback decoding scheme and a windowed decoding scheme to decode the whole TB by exploiting the coupled information between CBs. We calculate the extrinsic information transfer (EXIT) functions for the PIC-turbo codes by assuming that the coupled information are perfectly decoded. An SNR gain upper bound of the PIC-turbo codes over the LTE turbo codes for various coupling ratios is derived by the calculated EXIT charts. Numerical results show that the proposed codes achieve an SNR gain of 0.26–0.73 dB for various code parameters at a TB error rate level of $10^{-2}$ , which complies with the derived SNR gain upper bound.

Reliability-Based Windowed Decoding for Spatially Coupled LDPC Codes

In this letter, we propose a reliability-based windowed decoding scheme for spatially coupled (SC) low-density parity-check (LDPC) codes. To mitigate the error propagation along the sliding windowed decoder of the SC LDPC codes, a partial message reservation method is proposed where only the reliable messages generated in the previous decoding window are reserved for the next decoding window. We also propose a partial syndrome check stopping rule for each decoding window, in which only the complete variable nodes are checked. Simulation results show that our proposed scheme significantly improves the error-floor performance compared to the sliding windowed decoder with the conventional weighted bit-flipping algorithm.

Recursive Block Markov Superposition Transmission of Short Codes: Construction, Analysis, and Applications

Extensive studies have demonstrated the effectiveness and the flexibility of constructing capacity-approaching codes by block Markov superposition transmission (BMST). However, to achieve high performance, BMST codes typically require large encoding memories and large decoding window sizes, which result in high decoding complexity and high decoding latency. To address these issues, we introduce the recursive BMST (rBMST), in which the block-oriented feedback convolutional code is used instead of the block-oriented feedforward convolutional code of BMST. We propose to use a modified extrinsic information transfer chart analysis, which relates the mutual information to the bit error rate, to study the convergence behaviors of rBMST codes. On one hand, rBMST code shares most merits of BMST code, including near-capacity performance, low-complexity encoding, and flexible construction. On the other hand, compared with BMST code, rBMST code requires a smaller encoding memory, hence a lower decoding complexity, to approach the capacity. In particular, both analytical and simulation results show that rBMST code with encoding memory three reveals a lower error floor than the BMST code with encoding memory twelve. Furthermore, we show by analysis and simulations that rBMST with fixed encoding memory ( $m = 3$ ) and fixed decoding delay ( $d = 12$ ) can be used to construct capacity-approaching multiple-rate codes. Finally, the comparison between rBMST codes and spatially coupled low-density parity-check codes is carried out, which shows the advantages of rBMST codes in terms of performances and decoding complexities.

Window-Interleaved Turbo Codes

We consider a class of turbo codes with a band structured interleaver, such that the interleaving (and thus deinterleaving) is done within a window. This ensures that the extrinsic information for each symbol is generated from symbols within a window, allowing the processing of large message blocks with a sliding window decoder. This provides flexibility in terms of decoding latency and also makes parallel decoding with high-throughput possible.

Improving Windowed Decoding of SC LDPC Codes by Effective Decoding Termination, Message Reuse, and Amplification

In this paper, we address a number of weaknesses of the windowed decoding of spatially coupled low-density parity-check (SC LDPC) codes and propose three modifications that simultaneously improve its performance, complexity, and latency. An effective termination method of the windowed decoding and the reuse of edge messages of previous target symbols provide a good performance-latency tradeoff when compared with the conventional windowed decoder. Also, we propose a scheme that lowers the error floor, in which the amplified edge messages of the previous window are used in the present window. The proposed windowed decoding, consisting of the three schemes, provides a significant performance gain with smaller latency. The validity of the new windowed decoding is verified by the evaluation with codes from different SC LDPC ensembles.

Partially Information Coupled Polar Codes

We propose a new class of partially information coupled (PIC) polar codes to improve the transmission efficiency of transport block (TB)-based communication standards. In the proposed PIC polar codes, every two consecutive systematic polar code blocks (CBs) in a TB are coupled by sharing a few systematic information bits. Dummy bits are inserted at the two ends of the TB to construct terminated PIC polar codes. We propose a CB decoding scheme which only uses the information associated with correctly decoded coupled bits to mitigate the serious error propagation problem in successive cancellation based polar code decoding algorithms. We also propose an inter-CB decoding scheme which realizes a windowed decoder with variable window size to achieve a flexible tradeoff between the decoding performance and complexity. We derive a closed form expression for the TB error rate (TBER) of the PIC polar codes. We further optimize the coupling scheme based on the derived TBER. Simulation results confirm the effectiveness of the TBER analysis and the coupling scheme optimization results. They also show that the PIC polar codes can significantly outperform the uncoupled polar codes for various code rates with a slightly increased decoding complexity.

Braided Convolutional Codes With Sliding Window Decoding

In this paper, we present a novel sliding window decoding scheme based on iterative Bahl–Cocke–Jelinek–Raviv decoding for braided convolutional codes, a class of turbo-like codes with short constraint length component convolutional codes. The tradeoff between performance and decoding latency is examined and, to reduce decoding complexity, both uniform and nonuniform message passing schedules within the decoding window, along with early stopping rules, are proposed. We also perform a density evolution analysis of sliding window decoding to guide the selection of the window size and message passing schedule. Periodic puncturing is employed to obtain rate-compatible code rates of 1/2 and 2/3 starting from a rate 1/3 mother code and a code rate of 3/4 starting from a rate 1/2 mother code. Simulation results show that, with nonuniform message passing and periodic puncturing, near capacity performance can be maintained throughout a wide range of rates with reasonable decoding complexity and no visible error floors.

Decoding Chinese stock market returns: Three-state hidden semi-Markov model

In this paper, we employ a three-state hidden semi-Markov model (HSMM) to explain the time-varying distribution of the Chinese stock market returns since 2005. Our results indicate that the time-varying distribution depends on the hidden states, which are represented by three market conditions, namely the bear, sidewalk, and bull markets. We find that the inflation, the PMI, and the exchange rate are significantly related to the market conditions in China. A simple trading strategy based on expanding window decoding shows profitability with a Sharpe ratio of 1.14.