Binary Convolutional Codes Research Articles

IC design of an adaptive Viterbi decoder

We implement an integrated circuit (IC) design for a decoder of a 64-state binary convolutional code. The decoder is based on a reduced-state adaptive Viterbi algorithm (VA) for which the decoding speed is faster than the standard VA while the error performance remains almost the same. With the adaptive VA, less bits are needed to store and to calculate the metrics of the decoding trellis and less power dissipation is needed, as compared to the standard VA. The IC design is based on a 0.8 /spl mu/m CMOS technology. The number of quantization levels in the decoding is Q=8.

Simplified Viterbi decoding of geometrically uniform TCM codes

We present a procedure to design maximum likelihood (ML) decoders for the new class of geometrically uniform (GU) trellis coded modulation (TCM) codes, exploiting the algebraic properties of such codes. The proposed design has a very efficient VLSI implementation. The design of the decoders for the GUTCM codes is more complicated in comparison to the standard convolutional codes because between any pair of states in the trellis diagram of a GUTCM code, there is usually a large number of parallel transitions, and the trellis diagram of the code has a much higher degree of connectivity in comparison to binary convolutional codes. We present a novel technique for solving the parallel transitions using the algebraic structure of the GUTCM codes, which represents a significant reduction in complexity in comparison to the direct approach. The proposed technique is applied to the design of a simplified Viterbi decoder (VD) for a 64-state nonbinary GUTCM code defined over (Z/sub 8/)/sup 4/. For this example, we obtain a 58 fold reduction in complexity for the parallel transition solver in comparison to a direct implementation.

Variable-complexity trellis decoding of binary convolutional codes

Considers trellis decoding of convolutional codes with selectable effort, as measured by decoder complexity. Decoding is described for single parent codes with a variety of complexities, with performance near that of the optimal fixed receiver complexity coding system. Effective free distance is examined. Criteria are proposed for ranking parent codes, and some codes found to be best according to the criteria are tabulated, Several codes with effective free distance better than the best code of comparable complexity were found. Asymptotic (high SNR) performance analysis and error propagation are discussed. Simulation results are also provided.

Coded QAM using a binary convolutional code

As an alternative to trellis coding, a binary convolutional code is considered for use with such nonbinary modulation schemes as quadrature amplitude modulation (QAM). A Gray code is used to map the encoder output to the M-ary QAM constellation. The focus is on the design of 16-ary coded QAM with a rate 3/4 punctured convolutional code of a constraint length 7. A quantized binary metric generation method is proposed and shown to be suboptimum as compared to the direct use of a M-ary unquantized metric. Impressive coding gains and bandwidth efficiency are shown in comparison with uncoded systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Soft syndrome decoding of binary convolutional codes

We present an efficient recursive algorithm for accomplishing maximum likelihood (ML) soft syndrome decoding of binary convolutional codes. The algorithm consists of signal-by-signal hard decoding followed by a search for the most likely error sequence. The number of error sequences to be considered is substantially larger than in hard decoding, since the metric applied to the error bits is the magnitude of the log likelihood ratio rather than the Hamming weight. An error-trellis (alternatively, a decoding table) is employed for describing the recursion equations of the decoding procedure. The number of its states is determined by the states indicator, which is a modified version of the constraint length of the check matrix. Methods devised for eliminating error patterns and degenerating error-trellis sections enable accelerated ML decoding. In comparison with the Viterbi algorithm, the syndrome decoding algorithm achieves substantial reduction in the average computational complexity, particularly for moderately noisy channels.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A single-chip Viterbi decoder for a binary convolutional code using an adaptive algorithm

By using an adaptive algorithm, we design a single-chip Viterbi decoder for a rate 1/2 binary convolutional code. The adaptive algorithm is realized by threshold checking at each stage. The survivor paths that are less likely are ignored and hence the number of states at each decoding stage to be computed is reduced significantly in comparison with the conventional Viterbi decoder. Therefore, the decoding speed can be increased. A single-chip hard-decision decoder for a rate 1/2 convolutional code with constraint length K=6 (64 states) has been fabricated in 1.2 /spl mu/m CMOS technology. Experimental results show that this improved algorithm can achieve a data throughput rate about 3 times faster than that using a conventional algorithm without sacrificing the decoding reliability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Coded continuous phase modulation using ring convolutional codes

Rate 1/2 convolutional codes over the ring of integers modulo M are combined with M-ary continuous phase modulation (CPM) schemes whose modulation indices are of the form h=1/M. An M-ary CPM scheme with h=1/M can be modeled by a continuous-phase encoder (CPE) followed by a memoryless modulator (MM), where the CPE is linear over the ring of integers modulo M. The fact that the convolutional code and the CPE are over the same algebra allows the state of the CPE to be fed back and used by the convolutional encoder. A modified Euclidean distance function that substantially simplifies the search for good codes has been derived and used to find new codes. Numerical results show that this approach consistently improves the performance as compared to coded schemes using binary convolutional codes with the same decoding complexity.

Simulated error performance of encoded MSK signalsin Gaussian and Rician fading channels

New trellis codes are constructed that are based on binary concordant convolutional codes and minimum shift keying. The codes are investigated in various communication channels by means of computer simulation.

Reduced-state sequence detection with convolutional codes

Reduced-state sequence detection (RSSD) reduces the state trellis of a channel code by forming the states into classes. States within a class are such that paths into the states lie further than a distance parameter d from each other. An RSSD decoder retains only one survivor per class at each trellis level. The authors apply RSSD to ordinary binary convolutional codes. They first give a class-forming algorithm that finds the greatest reduction. It turns out that no commonly tabulated good code benefits from RSSD. However, RSSD is an effective way to repair weaker codes, such as quick look-in and RCPC codes. Finally, the authors show that RSSD cannot be more efficient than the M-algorithm. >

Coding for DPSK noisy phase channels

Low to moderate complexity coding schemes are suggested and examined for binary DPSK modulation. The model of the communication channel consists of both a Brownian motion phase noise and an additive white Gaussian noise. Decoding utilizes a mismatched soft-decision metric, which comprises the prefiltering of the phase noise impaired signals followed by differential demodulation. The resulting equivalent, binary-input, output-symmetric, discrete-time, coding channel is stationary and memoryless. It accounts as well for an underlying time-diversity, thus providing an effective simple, means of boosting the capabilities of less powerful codes to cope with the existing phase noise levels. An analytical upper bound on the coded error probability is derived, which strictly addresses the Brownian motion phase model. It features a convenient decoupling of the code structure from the coding channel. The latter is completely specified via a single, scalar, generic parameter which relies on the univariate moments admitted by certain exponential functionals of the Brownian motion sample path, the exact statistics of which is intractable. The theory is illustrated while studying the performance of low constraint length, rate 1/n, binary convolutional codes, which are optimally combined with time-diversity. The advantages obtained by less trivial forms of code concatenation are examined. The approach treated is to use a two level concatenation scheme for which the binary inner code dimension matches the alphabet size of an outer non-binary code.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Repeated convolutional codes for high-error-rate channels

An error-correction scheme for an M-ary symmetric channel (MSC) characterized by a large error probability p/sub e/ is considered. The value of p/sub e/ can be near, but smaller than, 1-1/M, for which the channel capacity is zero, such as may occur in a jamming environment. The coding scheme consists of an outer convolutional code and an inner repetition code of length m that is used for each convolutional code symbol. At the receiving end, the m inner code symbols are used to form a soft-decision metric, which is passed to a soft-decision decoder for the convolutional code. The effect of finite quantization and methods to generate binary metrics for M>2 are investigated. Monte Carlo simulation results are presented. For the binary symmetric channel (BSC), it is shown that the overall code rate is larger than 0.6R/sub 0/, where R/sub 0/ is the cutoff rate of the channel. New union bounds on the bit error probability for systems with a binary convolutional code on 4-ary and 8-ary orthogonal channels are presented. For a BSC and a large m, a method is presented for BER approximation based on the central limit theorem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A time-varying convolutional encoder better than the best time-invariant encoder

A periodically-time-varying binary (3, 2) convolutional code with period T=2 and memory M=1 having larger free distance than the best time-invariant (3, 2) convolutional code with M=1 is presented. >

Geometrically uniform partitions of L*MPSK constellations and related binary trellis codes

The theory of geometrically uniform trellis codes is applied to the case of multidimensional PSK (phase shift keying) constellations. The symmetry group of an L*MPSK (M-ary PSK) constellation is completely characterized. Conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given. Through suitable algorithms, geometrically uniform partitions of L*MPSK (M=4,8,16 and L=1,2,3,4) constellations are found, which present good characteristics in terms of the set of distances at a given partition level, the maximum obtainable rotational invariance, and the isomorphism of the quotient group associated with the partition. These partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Performance of robust metrics with convolutional coding and diversity in FHSS systems under partial-band noise jamming

The performance of robust metrics (metrics that can be computed from the outputs of the matched filters only) with convolutional coding and diversity under worst-case partial-band noise jamming is analyzed. Both binary and dual-k convolutional codes employing these metrics with diversity are compared via Union-Chernoff bounds. The performances of metrics considered in the literature that assume perfect side-information are given for comparison purposes. It is found that there exist very good robust metrics that provide performance comparable to metrics using perfect side-information. Among the robust metrics considered, the self-normalized metric offers the best performance and achieves performance practically identical to that of the square-law-combining metric with perfect side-information for M=8. >

Multilevel trellis coded modulations for the Rayleigh fading channel

The authors present coded 8-phase-shift-keyed (8-PSK) modulations for the Rayleigh fading channel. The schemes are based on multilevel trellis-coded-modulation constructions and utilize maximum free Hamming distance binary convolutional codes as building blocks. A suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated. Examples are constructed to show that the proposed schemes outperform the best modified codes of the Ungerboeck type due to significantly higher implicit time diversity, yielding seven branches of built-in time diversity, whereas the Ungerboeck code yields four branches of time diversity for a 64-state system. The transmission delay is higher, however. The new schemes can provide three levels of unequal error protection when 8-PSK or 8-differential-phase-shift-keying (8-DPSK) modulations are used. They provide 10-14-dB channel signal-to-noise ratio gain over uncoded 4-DPSK at a bit error rate of 10/sup -3/ for a modest decoding complexity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Convolutional coding for BPSK with pilot tone over Rayleigh fading channels

To improve the performance of digital communication systems over fading channels, several pilot signal techniques have been proposed in the literature. The Letter investigates the use of binary convolutional coding for such systems and derives an upper bound on the bit error probability (BEP), the optimum power split ratio between data and pilot signals, and the channel cutoff rate.

A survey of array error control codes

AbstractArray error control codes are linear block or convolutional codes which are constructed from several single parity check or other component codes, assembled in two or more geometrical dimensions or directions, with emphasis on simple component codes and low complexity methods of decoding. This survey attempts to introduce, relate and compare all known types of array codes and their applications. After an introduction to the basic properties and uses of array codes, the second section of the paper describes binary array code constructions for random, burst and cluster error control. The third section describes a number of binary convolutional array codes, for both random and burst error control. Non‐binary and byte‐oriented array codes, both block and convolutional, are covered in the fourth section, which mentions some quite powerful and yet practical constructions. The fifth section discusses various enhancements which can be applied to array codes. The paper ends with conclusions, open problems, and an extensive set of references.

Forward error correction for an atmospheric noise channel

Models for predicting the performance of error correcting codes on an atmospheric noise channel are presented and verified. Two Markov chains are used to model the memory of the atmospheric noise channel. The transition probabilities for these chains are derived from atmospheric noise error processes which were recorded at 306 kHz. The models are used to estimate the probability of codeword error, and these estimates are compared to codeword error rates which are obtained directly from the recorded error processes. The comparisons are made for the Golay code with a variety of bit interleaving depths, and a Reed-Solomon code with a variety of symbol interleaving depths. Both Markov channel models predict the actual performance of the codes with much greater accuracy than the binary symmetric channel. Results for binary convolutional codes are presented. >

Selforthogonal convolutional codes derived from linear congruences

A systematic procedure is presented for the construction of class, of binary (n, k, m) selforthogonal convolutional codes (SOCC) of minimum distance d = n−k+1 = J+1, where 1&lt;J&lt; p, k&lt;p, m+1&lt;p, and p is a prime number. An extension of the basic construction technique is provided for the case where a power of a prime, i.e. q = pe, is used instead of p. Unequal error protection SOCCs are obtained through a minor modification in the parity check matrix.

Hybrid trellis-coded 8/4-PSK modulation systems

By using codes of rate 2/3 followed by 8-PSK modulation, a gain of 3-6 dB is obtained over uncoded 4-PSK, for an ideal coherent transmission on the white Gaussian channel. In the presence of carrier phase offset, it has been shown that trellis-coded 8-PSK systems are more sensitive than uncoded 4-PSK. A more robust performance can be achieved by using rate 2/3 trellis-coded 8-PSK signals and 4-PSK signals in a time-varying manner. Only the mapper from the output of the binary convolutional code to the signal point number to be transmitted has to be periodically time-varying. In its simplest form, trellis-coded 8-PSK and 4-PSK signals alternate in time. An examination has also been made of systems where the mixture of 8-PSK and 4-PSK signals varies, with a short periodic sequence of time-varying mapping rules. The distance spectra and error probability are evaluated with and without phase offset. Simulation results of bit error rate (BER) and performance of the recovery loop (S curve) are presented. It is concluded that systems which are more robust against jitter can be achieved by means of time-varying hybrid trellis-coded 8/4-PSK systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>