High-rate Convolutional Codes Research Articles

CRC-Aided High-Rate Convolutional Codes With Short Blocklengths for List Decoding

CRC-Aided High-Rate Convolutional Codes With Short Blocklengths for List Decoding

Construction of High-Rate Convolutional Codes Using Dual Codes

In this study, we consider techniques for searching high-rate convolutional code (CC) encoders using dual code encoders. A low-rate (R = 1/n) CC is a dual code to a high-rate (R = (n - 1)/n) CC. According to our past studies, if a CC encoder has a high performance, a dual code encoder to the CC also tends to have a good performance. However, it is not guaranteed to have the highest performance. We consider a method to obtain a high-rate CC encoder with a high performance using good dual code encoders, namely, high-performance low-rate CC encoders. We also present some CC encoders obtained by searches using our method.

Constraint length inflation in fixed high rate convolutional codes and their impact on performance of O-IDMA

<p>To cope up with the higher data rate existing in modern communication systems multiple access technologies has been investigated OIDMA (Optical Interleave Division Multiple Access Technology) is one of the prominent technology use to fulfill this demand. Convolutional codes used in OIDMA systems enhance the performances and reduces the bit error rate of the system. In the present paper, the convolutional codes with increasing number of memory elements are used, the constraint length which depends on number of memory elements are increased with a systematic manner and their combined effect on response of OIDMA system has been observed. Tree inter-leavers in taken into account for analysis purpose and BER with increasing number of users is plotted in graphical and tabular manner.</p>

Rotated Multi-D Constellations in Rayleigh Fading: Mutual Information Improvement and Pragmatic Approach for Near-Capacity Performance in High-Rate Regions

This paper studies the mutual information improvement attained by rotated multidimensional (multi-D) constellations via a unitary precoder G in Rayleigh fading. At first, based on the symmetric cut-off rate of the N-D signal space, we develop a design criterion with regard to the precoder G. It is then demonstrated that the use of rotated constellations in only a reasonably low dimensional signal space can significantly increase the mutual information in high-rate regimes. Based on parameterizations of unitary matrices, we then construct good unitary precoder G in 4-D signal space using a simple optimization problem, which involves only four real variables and it is applicable to any modulation scheme. To further illustrate the potential of multi-D constellation and to show the practical use of mutual information improvement, we propose a simple yet powerful bit-interleaved coded modulation (BICM) scheme in which a (multi-D) mapping technique employed in a multi-D rotated constellation is concatenated with a short-memory high-rate convolutional code. By using extrinsic information transfer (EXIT) charts, it is shown that the proposed technique provides an exceptionally good error performance. In particular, both EXIT chart analysis and simulation results indicate that a turbo pinch-off and a bit error rate around 10-6 happen at a signal-to-noise ratio that is well below the coded modulation and BICM capacities using traditional signal sets. For example, with code rates ranging from 2/3 to 7/8, the proposed system can operate 0.82 dB-2.93 dB lower than the BICM capacity with QPSK and Gray labeling. The mutual information gain offered by rotated constellations can be therefore utilized to design simple yet near Shannon limit systems in the high-rate regions.

An Efficient 4-D 8PSK TCM Decoder Architecture

This paper presents an efficient architecture for a 4-D eight-phase-shift-keying trellis-coded-modulation (TCM) decoder. First, a low-complexity architecture for the transition metric unit is proposed based on substructure sharing. This scheme significantly reduces the required computation without degrading the performance. Then, a new hybrid T -algorithm for a Viterbi decoder is developed by applying a T-algorithm on both branch metrics (BMs) and path metrics (PMs). TCM encoders usually employ high-rate convolutional codes that yield many more transition paths per state than low-rate codes do. This makes it feasible to purge unnecessary additions by applying the T -algorithm on BMs. Applying the T-algorithm on BMs instead of PMs allows one to move the ?search-for-the-optimal? operation out of the add-compare-select-unit (ACSU) loop. Hence, the clock speed will not be affected. In addition, by combining the T-algorithm on BMs and the T-algorithm on PMs, the hybrid T -algorithm can reduce the computations required with the conventional T-algorithm on PMs by as much as 50%.

Decoding of High Rate Convolutional Codes Using the Dual Trellis

This paper deals with a posteriori probability (APP) decoding of high-rate convolutional codes, using the dual code's trellis. After deriving the dual APP (DAPP) algorithm from the APP relation, its trellis-based implementation is addressed. The challenge involved in practical implementation of a DAPP decoder is then highlighted. Metric representation schemes similar to the log domain used for log-APP decoding are shown to be unattractive for DAPP decoding due to quantization requirements. After explaining the nature of the DAPP metrics, an arc hyperbolic tangent (AHT) scheme is proposed and its equivalent arithmetic operations derived. By using an efficient approximation, an addition is translated to an addition in the AHT domain. Efficient techniques for normalization and extrinsic log-likelihood ratio (LLR ) calculation are presented which reduce implementation complexity significantly. Simulation results with different high-rate codes are given to show that the AHT-DAPP decoder performs similarly to a log-APP decoder and at the same time performs better than a decoder for a punctured code. A fully fixed-point model of an AHT-DAPP decoder is shown to perform close to an optimum decoder. The decoding complexity of the log-APP and AHT-DAPP decoders are listed and compared for several rate-k/(k+1) codes. It is shown that an AHT-DAPP decoder starts to be less complex from a code rate of 7/8 . When compared against a max-log-APP decoder, the AHT-DAPP decoder is less complex at a code rate of 9/10 and above.

Symbol-by-symbol reception of signals corresponding to high-rate convolutional codes and turbo codes based on these convolutional codes

Computational procedures for symbol-by-symbol reception of signal ensembles corresponding to binary high-rate convolutional codes and to turbo codes formed by these convolutional codes are described. It is shown that the developed procedures are based on an optimum symbol-by-symbol reception algorithm that uses the fast Walsh-Hadamard transform algorithm.

On the performance of turbo equalization for precoded ISI channels

AbstractIn this paper, we present a semi‐analytical approach for analyzing the serial concatenation (SCC) of a high rate convolutional code and a precoded intersymbol interference (ISI) channel. The significance of the proposed approach is that it can be used to identify, for a given outer high rate code and fixed ISI channel, the precoder that results in the lowest error rate floor. In estimating the bit error performance in the floor region, we use analytical techniques developed for trellis codes to compute the overall system minimum squared Euclidean distance, and then use a semi‐analytical approach to find the corresponding multiplicity. We also demonstrate via simulations that the proposed technique may be extended to fading ISI channels with favorable results. We give examples that support our analysis. Copyright © 2006 John Wiley & Sons, Ltd.

High-Rate Punctured Convolutional Self-Doubly Orthogonal Codes for Iterative Threshold Decoding

The puncturing technique allows obtaining high-rate convolutional codes from low-rate convolutional codes used as mother codes. This technique has been successfully applied to generate good high-rate convolutional codes which are suitable for Viterbi and sequential decoding. In this paper, we investigate the puncturing technique for convolutional self-doubly orthogonal codes (CSO/sup 2/C) which are decoded using an iterative threshold-decoding algorithm. Based on an analysis of iterative threshold decoding of the rate-R=b/(b+1) punctured systematic CSO/sup 2/C, the required properties of the rate-R=1/2 systematic convolutional codes (SCCs) used as mother codes are derived. From this analysis, it is shown that there is no need for the punctured mother codes to respect all the required conditions, in order to maintain the double orthogonality at the second iteration step of the iterative threshold-decoding algorithm. The results of the search for the appropriate rate-R=1/2 SCCs used as mother codes to yield a large number of punctured codes of rates 2/3/spl les/R/spl les/6/7 are presented, and some of their error performances evaluated.

A modified Viterbi decoding algorithm for high-rate convolutional codes

It is well-known that the computational complexity of the Viterbi decoding algorithm for an (n,k,ν) convolutional code grows exponentially with k and the code memory ν, and thus it becomes quickly impractical as the code rate increases. A solution, so far, has been to use punctured convolutional codes, which strongly reduces the decoding complexity but leads to a slightly worse performance. Recently, it has been pointed out that every non-punctured high-rate convolutional code (with 2k > 2ν) can be viewed as the concatenation of a block code and a simpler convolutional code. In this paper, we exploit this property of high-rate codes to facilitate the implementation of the Viterbi algorithm. We propose a modification of the Viterbi decoding algorithm for non-punctured high-rate convolutional codes which results in very significant computational savings. Copyright © 2005 AEIT.

Sphere-Packing Bounds for Convolutional Codes

We introduce general sphere-packing bounds for convolutional codes. These improve upon the Heller (1968) bound for high-rate convolutional codes. For example, based on the Heller bound, McEliece (1998) suggested that for a rate (n - 1)/n convolutional code of free distance 5 with /spl nu/ memory elements in its minimal encoder it holds that n /spl les/ 2/sup (/spl nu/+1)/2/. A simple corollary of our bounds shows that in this case, n < 2/sup /spl nu//2/, an improvement by a factor of /spl radic/2. The bound can be further strengthened. Note that the resulting bounds are also highly useful for codes of limited bit-oriented trellis complexity. Moreover, the results can be used in a constructive way in the sense that they can be used to facilitate efficient computer search for codes.

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.

An Extensive Search for Good Punctured Rate-&amp;gt;tex&amp;lt;$ kover k+1$&amp;gt;/tex&amp;lt;Recursive Convolutional Codes for Serially Concatenated Convolutional Codes

In many practical applications requiring variable-rate coding and/or high-rate coding for spectral efficiency, there is a need to employ high-rate convolutional codes (CC), either by themselves or in a parallel or serially concatenated scheme. For such applications, in order to keep the trellis complexity of the code constant and to permit the use of a simplified decoder that can accommodate multiple rates, a mother CC is punctured to obtain codes with a variety of rates. This correspondence presents the results of extensive search for optimal puncturing patterns for recursive convolutional codes leading to codes of rate k/(k+1) (k an integer) to be used in serially concatenated convolutional codes (SCCC). The code optimization is in the sense of minimizing the required signal-to-noise ratio (SNR) for two target bit-error rate (BER) and two target frame-error rate (FER) values. We provide extensive sample simulation results for rate-k/(k+1) SCCC codes employing our optimized punctured CC.

A new approach to the construction of high-rate convolutional codes

This letter presents a new technique to construct high-rate convolutional codes using a structure formed by a high-rate block code and a simpler convolutional code. The goal is to obtain good convolutional codes in terms of free distance and number of nearest neighbors, with better performance than punctured codes. The obtained codes improve over the best known high-rate punctured codes with the same rate and memory in terms of both bit error probability and computational decoding complexity.

Fault tolerance in computing, compressing, and transmitting FFT data

Remote-sensing applications often calculate the discrete Fourier transform of sampled data and then compress and encode it for transmission to a destination. However, all these operations are executed on computing resources potentially affected by failures. Methods are presented for integrating various fault detection capabilities throughout the data flow path so that the momentary failure of any subsystem will not allow contaminated data to go undetected. New techniques for protecting complete source coding schemes are exemplified by examining a lossy compression system that truncates fast Fourier transform (FFT) coefficients to zero, then compresses the data further by using lossless arithmetic coding. Novel methods protect arithmetic coding computations by internal algorithm checks. The arithmetic encoding and decoding operations and the transmission path are further protected by inserting sparse parity symbols dictated by a high-rate convolutional symbol-based code. This powerful approach introduces limited redundancy at the beginning of the system but performs detection at later stages. While the parity symbols degrade efficiency slightly, the overall compression gain is significant because of the run-length coding. Well-known fault tolerance measures for FFT algorithms are extended to detect errors in the lossy truncation operations, maintaining end-to-end protection. Simulations verify that all single subsystem errors are detected and the overhead costs are reasonable.

Precoded PRML, serial concatenation, and iterative (turbo) decoding for digital magnetic recording

We show that a high rate serially concatenated code used in conjunction with a preceded partial response equalised magnetic recording channel and iterative decoding can provide similar performance to a turbo code on the same channel. The precoded partial response maximum-likelihood (PRML) read channel is incorporated as the inner code of the concatenated coding scheme and the outer code is a high rate convolutional code. Gains of 4.8 dB above conventional PRML at a bit error rate of 10/sup -5/ for a rate 13/14 code can be achieved.

MAP decoding of convolutional codes using reciprocal dual codes

A new symbol-by-symbol maximum a posteriori (MAP) decoding algorithm for high-rate convolutional codes using 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 for codes of rate greater than 1/2. The discussed algorithms fulfil all requirements for iterative ("turbo") decoding schemes. Simulation results are presented for high-rate parallel concatenated convolutional codes ("turbo" codes) using an AWGN channel or a perfectly interleaved Rayleigh fading channel. It is shown that iterative decoding of high-rate codes results in high-gain, moderate-complexity coding.

Symbol-by-symbol MAP decoding algorithm for high-rate convolutional codes 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.

Two-step trellis decoding of partial unit memory convolutional codes

We present a new soft-decision decoding method for high-rate convolutional codes. The decoding method is especially well-suited for PUM convolutional codes. The method exploits the linearity of the parallel transitions in the trellis associated with PUM codes. We provide bounds on the number of operations per decoded bit, and show that this number is dependent on the weight hierarchy of the linear block code associated with the parallel transitions. The complexity of the new decoding method for PUM codes is compared to the complexity of Viterbi decoding of comparable punctured convolutional codes. Examples from a special class of PUM codes show that the new decoding method compares favorably to Viterbi decoding of punctured codes.

The trellis complexity of convolutional codes

Convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm. Linear block codes also have a natural, though not in general a regular, "minimal" trellis structure, which allows them to be decoded with a Viterbi-like algorithm. In both cases, the complexity of an unenhanced Viterbi decoding algorithm can be accurately estimated by the number of trellis edge symbols per encoded bit. It would therefore appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes, are known to have trellis representations which are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the "minimal" trellis representation. Thus ironically, we seem to know more about the minimal trellis representation for block than for convolutional codes. We provide a remedy, by developing a theory of minimal trellises for convolutional codes. This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-canonical generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small.