Selection Method of Test Patterns in Soft-Input and Output Iterative Bounded Distance Decoding Algorithm

Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.

In high density patterned media storage system, the “multiple islands per read head” model is taken into account for inter-track interference (ITI) in bit-patterned media (BPM). It has been proposed that low-density parity-check (LDPC) codes could improve the symbol error rate (SER) performance significantly when only considering AWGN noise alone over the storage channel. LDPC decoding algorithm could be applied in LDPC joint decoder of patterned media storage system. In this paper, we proposed a decoding algorithm of serial belief propagation (SBP) with a stop updating criterion for LDPC decoder. Simulation results show that the proposed decoding algorithm could reduce the times of generating and propagating the message in decoding with negligible error rate performance loss over AWGN channels when appropriately choosing the parameter that determines which variable node has been converged. By taking advantage of extrinsic information transfer (EXIT) charts, it is demonstrated that the proposed decoding algorithm could maintain superior SER performance and convergence rate compared with the traditional SBP decoding algorithm. To further explore the performance of the proposed decoding algorithm in “multiple islands per read head” model, we performed a large number of simulations over this channel model with different distribution percentages between the AWGN and the island position jitter noise. Simulation results show that the LDPC codes could improve the SER performance when considering jitter noise while the proposed decoding algorithm behaves excellently over the patterned media storage channel and over the AWGN channel.

Iterative decoding structure is the main feature of Turbo code decoding. It can improve the performance of Turbo decoding, but also increase the decoding complexity, time delay and power consumption. This paper proposes a parallel decoding structure of Turbo code based on double prediction control, i.e., using double prediction control module instead of two iterative decoding algorithms. Since the computation quantity and computing time of the Double Prediction Control Algorithm (DPCA) is smaller than the traditional decoding algorithm, the complexity and delay of the decoding algorithm can be reduced by the new decoding scheme presented in this paper. Exterior information simulations show that comparing to the traditional iterative decoding, time delay can be reduced by at least three times by using prediction scheme each time, while the BER performance simulation results show that the error-rate performance of the new decoding structure based on DPCA is only lower by 0.1 dB than the traditional Turbo decoding structure, but has a good approximation BER curves to the traditional one. When reasonably choose the two prediction intervals, it can achieve the coincident BER curve.

This paper proposes an enhanced guessing codeword decoding (GCD) algorithm, termed GCD with most reliable cyclic basis (MRCB-GCD), specifically tailored for BCH codes. Unlike original GCD, the proposed method selects the k consecutive bits with the highest aggregate reliability as the re-encoding basis, leveraging the cyclic structure of BCH codes. To further enhance search efficiency, a local constraint (LC) mechanism is introduced, where the extended reliable bits of length k+δ enable the algorithm to skip numerous unnecessary test error patterns (TEPs). To stop the search process, we introduce two termination criteria and prove that the proposed approach with the trivial termination criterion forms a non-exhaustive-search maximum likelihood (ML) decoding algorithm. A saddlepoint-based numerical method is developed to approximately calculate the upper bound on the performance gap to the ML decoding and estimate the average number of searches, showing the advantages of the proposed MRCB-GCD over the original GCD in terms of both decoding performance and search efficiency. Simulation results demonstrate that: 1) MRCB- GCD outperforms original GCD in both frame error rate (FER) and number of searches, 2) the LC mechanism and termination criteria further reduce the number of searches, and 3) the proposed approach achieves finite-length capacity across various code rates but without requiring Gaussian elimination.

Turbo codes have a wide range of applications in 3G mobile communications, deep-sea communications, satellite communications and other power constrained fields. In the paper, the Turbo Code Decoding Principle and several major decoding methods are introduced. Simulations of Turbo code performance under different parameters of AWGN channel are made and the effects of the different interleaving length, the number of iterations, and the decoding algorithm to Turbo code performance are also discussed in AWGN channel. Simulation results show that under the same signal-to-noise ratio, the more the number of iterations is, the longer the sequence of information is, and the more excellent decoding algorithm is, the better the performance of Turbo codes is.

Reed-Solomon convolutional concatenated (RSCC) code has been widely applied in wireless and space communications. However, iterative soft-decision decoding of the concatenated code is yet to be developed. This paper proposes a novel iterative soft decoding algorithm for the concatenated coding scheme. The maximum a posteriori (MAP) algorithm is used to decode the inner convolutional code. Its soft output will be deinterleaved and then passed to the soft-in-soft-out (SISO) decoding algorithm for the outer Reed-Solomon (RS) code. The outer SISO decoder integrates the adaptive belief propagation (ABP) algorithm and the Koetter-Vardy (KV) list decoding algorithm, attempting to find out the transmitted message. If it is found, the deterministic probabilities of the corresponding RS coded bits will be fed back. Otherwise, the extrinsic probabilities that are yielded by the ABP algorithm will be given as the feedback. With the proposed soft information exchange decoding mechanism, error-correction potential of the concatenated code can be better exploited. Our simulation results show that significant performance improvement can be achieved over the existing decoding algorithms.

A BP-based algorithm with 2-dimensional classified normalized correction is developed to reduce the complexity and improve the performance of decoding algorithm for the low density parity check (LDPC) codes. The algorithm first utilizes classification according to the absolute values of incoming messages in check nodes. Then it uses 2-dimensional normalization to correct the minimum and sub-minimum values. The 2-dimensional normalized factors can be calculated respectively by using probability and statistic theory in the initialization step. Simulation results illustrate that the proposed algorithm achieves better performance in bit error ratio (BER) and average iteration number than normalized BP-based algorithm in the high signal noise ratio (SNR) region, i.e., it can achieve 0.4 dB SNR gain and reduce 20% number of iterations at BER= 5 10  , whose complexity is also much less than that of belief propagation (BP) algorithm. It is concluded that the proposed algorithm offers better tradeoff between performance and complexity.  Index Terms—channel coding theory, density evolution, Iterative decoding algorithm

In this paper, we investigate the decoding of short Non-Binary (NB)-LDPC codes of rate 1/2 using a non-iterative approach based on the Maximum-Likelihood (ML) principle. The traditional decoding algorithms used to decode the NB-LDPC codes are by nature iterative where the Variables Nodes (VN) and Check Nodes (CN) exchange data iteratively during, at least, eight iterations which imposes a long decoding time to achieve good performance in terms of Frame Error Rate (FER). In this paper we propose a decoding algorithm based on the Maximum Likelihood (ML) search named Near ML approach where the number of tested words considered as potential codewords is highly reduced. Simulation of codes of lengths 16 and 48 are presented and the results show that the proposed algorithm achieves the performance offered by the EMS algorithm. The NB-LDPC of length 16 is shown to outperform the EMS algorithm.

Turbo codes are well known to be one of the error correction techniques which achieve closer results to the Shannon limit. Nevertheless, the specific performance of the code highly depends on the particular decoding algorithm used at the receiver. In this sense, the election of the decoding algorithm involves a trade off between the gain introduced by the code and the complexity of the decoding process. In this work we perform a thorough analysis of the different iterative decoding techniques and analyze their suitability for being implemented in the user terminals of new cellular and broadcast systems which are based on orthogonal frequency division multiplexing (OFDM). The analyzed iterative decoding algorithms are the max-log-MAP and the soft output Viterbi algorithm (SOVA), since both of them have a relative low computational complexity, simplifying their implementation in cost efficient terminals. Simulation results have been obtained for different encoder structures, block sizes and considering realistic channel conditions (an OFDM transmission over a wireless channel).

For binary linear block codes, we introduce a multiple generalized minimum distance (GMD) decoding algorithm, where GMD-like decoding is iterated around a few appropriately selected search centers. Compared with the original GMD decoding by Forney (1966), this decoding algorithm provides better error performance by increasing the number of iterations of erasure and error correction moderately. To reduce the number of iterations, we derive new effective sufficient conditions on the optimality of decoded codewords.

An iterative decoding algorithm of rate-compatible punctured low-density parity-check (RCP-LDPC) codes of high coding rates is developed. This algorithm performs a predetermined recovering process of punctured bits sums at the beginning of each iteration of the standard belief-propagation (BP) decoding algorithm. By propagating messages of two punctured bits sum, this algorithm can recover much more punctured bits than the standard BP decoding algorithm. It is shown that the proposed algorithm is applicable for RCP-LDPC codes of higher coding rates with little increase of decoding complexity.

The key challenges in real time voice communication in long term evolution mobile are reduction in complexity and latency. Efficient encoding and decoding algorithms can cater to these. The implementation of such polar code based efficient algorithms is proposed in this paper. The overall latency of 3.8 ms is needed to process 8 bit block length. The novel sub-matrix near to identity matrix is presented. This resulted into minimization of loops among least reliable bits due to iterated parity check matrix. Look-up table based memory mapping is used in encoder to reduce latency while Euclidian decoding technique is used in decoder. The number of iterations is reduced by 50%. The experimentation is performed with additive white Gaussian noise and QPSK modulation. The proposed modified iterative decoding algorithm requires SNR of 5.5 dB and 192 computations for targeted bit error rate of 10−4. The second proposed method needs 9 dB, 2 iterations for 384 computations. The penalty paid is quantization error of 0.63% due to restricting computations to fourth order series of hyperbolic function with same 8 bit block length.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

In this contribution, we propose a signal constellation for the the phase noise channel which does not require pilots thus increasing the effective information rate in a communication system. This constellation does not present rotational symmetry thus enabling decoding algorithms such as SPA, to converge without the use of pilots. We will provide Bit Error Rate (BER) simulations which show the superiority of this constellation over standard MPSK with pilots. Moreover, we will provide a method to analyze any arbitrary signal constellation and provide a figure of merit for its performance when iterative decoding algorithms are used.

A multiple-input multiple-output interleave division multiple access (MIMO-IDMA) system with Triple Polarized Division Multiplexing (TPDM) is presented in this paper. The present methodology replaces three independent linearly polarized antennas with a single triple polarized antenna at both the transmitter and receiver. The users in the communication link are accommodated and separated using a user-specific interleaver combined with low rate spreading sequence. To eliminate the effects of multi-stream interference (MSI), minimum mean square error (MMSE) algorithm based on successive interference cancellation (SIC) Multi-user detection (MUD) technique is employed at the receiver. Furthermore, log-maximum a posteriori probability (MAPP) decoding algorithm is implemented at the mobile stations (MSs) to overcome the effects of multi-user interference (MUI) effects. The paper also evaluates the effects of coded MIMO-IDMA in the downlink communication by adopting the Stanford University Interim (SUI) and Long-term Evolution (LTE)channel model specifications. In comparison with the traditional uncoded system, the present solution considering turbo coded triple-polarized MIMO-IDMA system with iterative decoding algorithm provides better bit error rate (BER) with reduced signal to noise ratio (SNR). The simulation results also show that though the SNR requirement is higher for the proposed technique compared to the conventional uni-polarized antenna based MIMO-IDMA system, it gives the advantages of achieving higher data rate with reduced cost and space requirements in the context of a downlink (DL).