A Decoding Algorithm for Turbo Product Codes Using Optimality Test and Amplitude Clipping

This paper presents an iterative soft-input/soft-output (SISO) decoderfor product code using optimality test and amplitude clipping. A modifiedexpression for computing the soft-output of SISO decoder is proposed.The correlation discrepancy is employed to provide an optimality teston the decision codeword. The optimality test is performed in rowand column decoding to evaluate the reliability of row and columndecision codewords. Based on the optimality test, the variable reliabilityfactor is introduced for optimization of turbo decoding. A stoppingcriterion with very little performance degradation is also designedfor turbo decoding of product codes by using the optimality test.Besides, the amplitude clipping is employed to improve the performanceof turbo product code. Simulation results on the performance of theintroduced SISO decoder are presented.     

Symbol-by-symbol MAP decoding algorithm for high-rate convolutionalcodes that use reciprocal dual codes

A symbol-by-symbol maximum a posteriori (MAP) decoding algorithm for high-rate convolutional codes applying reciprocal dual convolutional codes is presented. The advantage of this approach is a reduction of the computational complexity since the number of codewords to consider is decreased. All requirements for iterative decoding schemes are fulfilled. Since tail-biting convolutional codes are equivalent to quasi-cyclic block codes, the decoding algorithm for truncated or terminated convolutional codes is modified to obtain a soft-in/soft-out decoder for high-rate quasi-cyclic block codes which also uses the dual code because of complexity reasons. Additionally, quasi-cyclic block codes are investigated as component codes for parallel concatenation. Simulation results obtained by iterative decoding are compared with union bounds for maximum likelihood decoding. The results of a search for high-rate quasi-cyclic block codes are given in the appendix     

Near-optimum decoding of product codes: block turbo codes

This paper describes an iterative decoding algorithm for any product code built using linear block codes. It is based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration. This soft-input/soft-output decoder is a Chase decoder which delivers soft outputs instead of binary decisions. The soft output of the decoder is an estimation of the log-likelihood ratio (LLR) of the binary decisions given by the Chase decoder. The theoretical justifications of this algorithm are developed and the method used for computing the soft output is fully described. The iterative decoding of product codes is also known as the block turbo code (BTC) because the concept is quite similar to turbo codes based on iterative decoding of concatenated recursive convolutional codes. The performance of different Bose-Chaudhuri-Hocquenghem (BCH)-BTCs are given for the Gaussian and the Rayleigh channel. Performance on the Gaussian channel indicates that data transmission at 0.8 dB of Shannon's limit or more than 98% (R/C>0.98) of channel capacity can be achieved with high-code-rate BTC using only four iterations. For the Rayleigh channel, the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel without using channel state information     

Improved Analysis of List Decoding and Its Application to Convolutional Codes and Turbo Codes

A list decoder generates a list of more than one codeword candidates, and decoding is erroneous if the transmitted codeword is not included in the list. This decoding strategy can be implemented in a system that employs an inner error correcting code and an outer error detecting code that is used to choose the correct codeword from the list. Probability of codeword error analysis for a linear block code with list decoding is typically based on the "worst case" lower bound on the effective weights of codewords for list decoding evaluated from the weight enumerating function of the code. In this paper, the concepts of generalized pairwise error event and effective weight enumerating function are proposed for evaluation of the probability of codeword error of linear block codes with list decoding. Geometrical analysis shows that the effective Euclidean distances are not necessarily as low as those predicted by the lower bound. An approach to evaluate the effective weight enumerating function of a particular code with list decoding is proposed. The effective Euclidean distances for decisions in each pairwise error event are evaluated taking into consideration the actual Hamming distance relationships between codewords, which relaxes the pessimistic assumptions upon which the traditional lower bound analysis is based. Using the effective weight enumerating function, a more accurate approximation is achieved for the probability of codeword error of the code with list decoding. The proposed approach is applied to codes of practical interest, including terminated convolutional codes and turbo codes with the parallel concatenation structure     

Complexity Reduction in SISO Decoding of Block Codes

SISO decoding for block codes can be carried out based on a trellis representation of the code. However, the complexity entailed by such decoding is most often prohibitive and thus prevents practical implementation. This paper examines a new decoding scheme based on the soft-output Viterbi algorithm (SOVA) applied to a sectionalized trellis for linear block codes. The computational complexities of the new SOVA decoder and of the conventional SOVA decoder, based on a bit-level trellis, are theoretically analyzed and derived for different linear block codes. These results are used to obtain optimum sectionalizations of a trellis for SOVA. For comparisons, the optimum sectionalizations for Maximum A Posteriori (MAP) and Maximum Logarithm MAP (Max-Log-MAP) algorithms, and their corresponding computational complexities are included. The results confirm that the new SOVA decoder is the most computationally efficient SISO decoder, in comparisons to MAP and Max-Log-MAP algorithms. The simulation results of the bit error rate (BER) performance, assuming binary phase -- shift keying (BPSK) and additive white Gaussian noise (AWGN) channel, demonstrate that the performance of the new decoding scheme is not degraded. The BER performance of iterative SOVA decoding of serially concatenated block codes shows no difference in the quality of the soft outputs of the new decoding scheme and of the conventional SOVA.     

Distance based adaptive scaling in suboptimal iterative decoding

This article develops an alternative adaptive iterative Chase (1972) based decoding algorithm for block turbo/product codes. The decoder considers only a small subset of codewords, so that estimates of the extrinsic information are required in some cases. This article develops such an estimate based on code distance properties     

CS-FSCL decoding algorithm of polar codes based on critical sets

In order to reduce the number of redundant candidate codewords generated by the fast successive cancellation list (FSCL) decoding algorithm for polar codes, a simplified FSCL decoding algorithm based on critical sets (CS-FSCL) of polar codes is proposed. The algorithm utilizes the number of information bits belonging to the CS in the special nodes, such as Rate-1 node, repetition (REP) node and single-parity-check (SPC) node, to constrain the number of the path splitting and avoid the generation of unnecessary candidate codewords, and thus the latency and computational complexity are reduced. Besides, the algorithm only flips the bits corresponding to the smaller log-likelihood ratio (LLR) values to generate the sub-maximum likelihood (sub-ML) decoding codewords and ensure the decoding performance. Simulation results show that for polar codes with the code length of 1 024, the code rates of 1/4, 1/2 and 3/4, the proposed CS-FSCL algorithm, compared with the conventional FSCL decoding algorithm, can achieve the same decoding performance, but reduce the latency and computational complexity at different list sizes. Specifically, under the list size of L=8, the code rates of R=1/2 and R=1/4, the latency is reduced by 33% and 13% and the computational complexity is reduced by 55% and 50%, respectively.     

An Efficient Stopping Criterion for Turbo Product Codes

In this letter, a stopping criterion using the error- detecting capability of linear block codes is proposed for the decoding of turbo product codes. The iterative decoding is stopped when the outputs from the Chase decoder are valid codewords for all rows and columns simultaneously. Simulation shows that the proposed method can reduce about one and half iterations compared with an existing stopping method, without noticeable BER performance loss. Some modification has also been discussed which may further reduce the decoding complexity.     

An efficient maximum-likelihood-decoding algorithm for linear blockcodes with algebraic decoder

A new soft decoding algorithm for linear block codes is proposed. The decoding algorithm works with any algebraic decoder and its performance is strictly the same as that of maximum-likelihood-decoding (MLD). Since our decoding algorithm generates sets of different candidate codewords corresponding to the received sequence, its decoding complexity depends on the received sequence. We compare our decoding algorithm with Chase (1972) algorithm 2 and the Tanaka-Kakigahara (1983) algorithm in which a similar method for generating candidate codewords is used. Computer simulation results indicate, for some signal-to-noise ratios (SNR), that our decoding algorithm requires less average complexity than those of the other two algorithms, but the performance of ours is always superior to those of the other two     

Iterative reduced-search decoding for coded partial-response channels

The full-complexity soft-input/soft-output (SISO) detector based on the BCJR algorithm for coded partial-response channels has a computational complexity growing exponentially with channel memory length. In this letter, we propose a low complexity soft-output channel detector based on the Chase decoding algorithm, which was previously applied to decode turbo product codes. At each iteration, the proposed detector forms a candidate list using all possible combinations of bit patterns in the weakest indices based on tentative hard estimates and a priori information fed back from the outer decoder. To demonstrate the performance/complexity tradeoff of the proposed detector, simulation results over rate-8/9 turbo-coded EPR4 and ME/sup 2/PR4 channels are presented, respectively. It is shown that the proposed detector can significantly reduce the computational complexity with only a small performance loss compared to the BCJR algorithm.     

Design and decoding of optimal high-rate convolutional codes

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.     

Performance of sphere decoding of block codes

A sphere decoder searches for the closest lattice point within a certain search radius. The search radius provides a tradeoff between performance and complexity. We focus on analyzing the performance of sphere decoding of linear block codes. We analyze the performance of soft-decision sphere decoding on AWGN channels and a variety of modulation schemes. A hard-decision sphere decoder is a bounded distance decoder with the corresponding decoding radius. We analyze the performance of hard-decision sphere decoding on binary and q-ary symmetric channels. An upper bound on the performance of maximum-likelihood decoding of linear codes defined over F_q (e.g. Reed- Solomon codes) and transmitted over q-ary symmetric channels is derived and used in the analysis.We then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes. The tradeoff between the performance and complexity of a sphere decoder is then discussed.     

Distributed decoding of cyclic block codes using a generalizationof majority-logic decoding

One-step majority-logic decoding is one of the simplest algorithms for decoding cyclic block codes. However, it is an effective decoding scheme for very few codes. This paper presents a generalization based on the “common-symbol decoding problem.” Suppose one is given M (possibly corrupted) codewords from M (possibly different) codes over the same field; suppose further that the codewords share a single symbol in common. The common-symbol decoding problem is that of estimating the symbol in the common position. This is equivalent to one-step majority logic decoding when each of the “constituent” codes is a simple parity check. This paper formulates conditions under which this decoding is possible and presents a simple algorithm that accomplishes the same. When applied to decoding cyclic block codes, this technique yields a decoder structure ideal for parallel implementation. Furthermore, this approach frequently results in a decoder capable of correcting more errors than one-step majority-logic decoding. To demonstrate the simplicity of the resulting decoders, an example is presented     

A soft-input soft-output APP module for iterative decoding ofconcatenated codes

Concatenated coding schemes consist of the combination of two or more simple constituent encoders and interleavers. The parallel concatenation known as “turbo code” has been shown to yield remarkable coding gains close to theoretical limits, yet admitting a relatively simple iterative decoding technique. The recently proposed serial concatenation of interleaved codes may offer superior performance to that of turbo codes. In both coding schemes, the core of the iterative decoding structure is a soft-input soft-output (SISO) a posteriori probability (APP) module. In this letter, we describe the SISO APP module that updates the APP's corresponding to the input and the output bits, of a code, and show how to embed it into an iterative decoder for a new hybrid concatenation of three codes, to fully exploit the benefits of the proposed SISO APP module     

Low complexity concatenated coding schemes for digital satellite communications

A study of reduced complexity concatenated coding schemes, for commercial digital satellite systems with low-cost earth terminals, is reported. The study explored trade-offs between coding gain, overall rate and decoder complexity, and compared concatenated schemes with single codes. It concentrated on short block and constraint length inner codes, with soft decision decoding, concatenated with a range of Reed-Solomon outer codes. The dimension of the inner code was matched to the outer code symbol size, and appropriate interleaving between the inner and outer codes was used. Very useful coding gains were achieved with relatively high-rate, low-complexity schemes. For example, concatenating the soft decision decoded (9,8) single parity check inner code with the CCSDS recommended standard Reed-Solomon outer code gives a coding gain of 4.8dB at a bit error probability of 10^?5, with an overall rate of 0-78.     

A low complexity block turbo decoder architecture - [transactions letters]

In this letter we present a low-complexity architecture designed for the decoding of block turbo codes. In particular we simplify the implementation of Pyndiah?s algorithm by not storing any of the concurrent codewords generated by the list decoder.     

BEAST decoding of block codes obtained via convolutional codes

BEAST is a bidirectional efficient algorithm for searching trees. In this correspondence, BEAST is extended to maximum-likelihood (ML) decoding of block codes obtained via convolutional codes. First it is shown by simulations that the decoding complexity of BEAST is significantly less than that of the Viterbi algorithm. Then asymptotic upper bounds on the BEAST decoding complexity for three important ensembles of codes are derived. They verify BEAST's high efficiency compared to other algorithms. For high rates, the new asymptotic bound for the best ensemble is in fact better than previously known bounds.     

New SISO decoding algorithms

A new maximum a posteriori (MAP)-equivalent soft-input soft-output (SISO) algorithm is derived together with its simplified versions. The proposed SISO algorithms provide a good compromise between complexity and performance. Our simplest SISO algorithm has lower complexity than the log-MAP, the max-log-MAP, and the soft-output Viterbi (1998) algorithm SISO algorithms, and it is an equivalent max-log-MAP algorithm. When this algorithm is used, turbo codes with block length as short as 150 bits will outperform convolutional codes when compared on the basis of equal decoder complexity.     

Parallel High Throughput Soft-Output Sphere Decoding Algorithm

Multiple-Input-Multiple-Output communication systems demand fast sphere decoding with high performance. To speed up the computation, we propose a scheme with multiple fixed complexity sphere decoders to construct a parallel soft-output fixed complexity sphere decoder (PFSD). The proposed decoder is highly parallel and has performance comparable to soft-output list fixed complexity sphere decoder (LFSD) and K-best sphere decoder. In addition, we propose a parallel QR decomposition algorithm to lower the preprocessing overhead, and a low complexity LLR algorithm to allow parallel update of LLR values. We demonstrate that the PFSD algorithm can increase the throughput and reduce bit error rate of a soft-output solution in a 4 × 4 16-QAM system, and has superior performance compared to other soft decoders with comparable throughput and computation complexity. The PFSD algorithm has been mapped onto Xilinx XC4VLX160 FPGA. The resulting PFSD decoder can achieve up to 75 Mbps throughput for 4 × 4 64-QAM configuration at 100MHz with low control overhead.     

A 40 Mb/s soft-output Viterbi decoder

Soft-output decoding has evolved as a key technology for new error correction approaches with unprecedented performance as well as for improvement of well established transmission techniques. In this paper, we present a high-speed VLSI implementation of the soft-output Viterbi algorithm, a low complexity soft-output algorithm, for a 16-state convolutional code. The 43 mm² standard cell chip achieves a simulated throughput of 40 Mb/s, while tested samples achieved a throughput of 50 Mb/s. The chip is roughly twice as big as a 16-state Viterbi decoder without soft outputs. It is thus shown with the design that transmission schemes using soft-output decoding can be considered practical even at very high throughput. Since such decoding systems are more complex to design than hard output systems, special emphasis is placed on the employed design methodology