Binary Convolutional Codes Research Articles

Feedback Decoding of Fixed-Point Arithmetic Convolutional Codes

Convolutional codes defined over the integers modulo a power of two, an arithmetic structure used for fixed-point arithmetic computations, employ well-known binary convolutional codes as their underlying generators. A recursive decoding technique that exploits binary expansion components of the code symbols uses any binary decoding algorithm valid for the underlying code.

The performance of incremental redundancy schemes based on convolutional codes in the block-fading Gaussian collision channel

The throughput performance of incremental redundancy (INR) schemes, based on short constraint length convolutional codes, is evaluated for the block-fading Gaussian collision channel. Results based on simulations and union bound computations are compared to estimates of the achievable throughput performance with random binary and Gaussian coding in the limit of large block lengths, obtained through information outage considerations. For low channel loads, it is observed that INR schemes with binary convolutional codes and limited block length may provide throughput close to the achievable performance for binary random coding. However, for these low loads, compared to binary random coding, Gaussian random coding may provide significantly better throughput performance, which prompts the use of larger modulation constellations. For high channel loads, a relatively large gap in throughput performance between binary convolutional codes and binary random codes indicates a potential for extensive performance improvement by alternative coding strategies. Only small improvements of the throughput have been observed by increasing the complexity through increased state convolutional coding.

Space-Time Bit-Interleaved Coded Modulation for OFDM Systems

Space-time coding techniques significantly improve transmission efficiency in radio channels by using multiple transmit and/or receive antennas and coordination of the signaling over these antennas. Bit-interleaved coded modulation gives good diversity gains with higher order modulation schemes using well-known binary convolutional codes on a single transmit and receive antenna link. By using orthogonal frequency division multiplexing (OFDM), wideband transmission can be achieved over frequency-selective fading radio channels without adaptive equalizers. In this correspondence, we combine these three ideas into a family of flexible space-time coding methods. The pairwise error probability is analyzed based on the correlated fading assumption. Near-optimum iterative decoders are evaluated by means of simulations for slowly varying wireless channels. Theoretical evaluation of the achievable degree of diversity is also presented. Significant performance gains over the wireless local area network (LAN) 802.11a standard system are reported.

Robust transmission of melp-compressed speech: an illustrative example of joint source-channel decoding

A mixed-excitation linear predictive (MELP) speech coder was selected as the US federal standard for 2400 b/s speech compression. This paper examines the quality of MELP-compressed speech when transmitted over noisy communication channels in conjunction with a variety of error-control schemes. The focus is on channel decoders that exploit the inherent in the MELP bitstream. This residual redundancy, which is manifested by the correlation in time and the nonuniform distribution of various MELP parameters, can be quantified by modeling the parameters as one-step Markov chains and computing the entropy rate of the Markov chains based on the relative frequencies of transitions. Moreover, this residual redundancy can be exploited by an appropriately tuned channel decoder to provide substantial coding gain when compared with decoders that do not exploit it. Channel coding schemes include conventional binary convolutional codes and iteratively-decoded parallel concatenated convolutional (turbo) codes.

Tailbiting codes obtained via convolutional codes with large active distance-slopes

The slope of the active distances is an important parameter when investigating the error-correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes. The slope of the active distances is equal to the minimum average weight cycle in the state-transition diagram of the encoder. A general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived. Moreover, a search technique, resulting in new tables of rate R=1/2 and rate R=1/3 convolutional encoders with high memories and large active distance-slopes is presented. Furthermore, we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances. Tables of rate R=1/2 and rate R=1/3 tailbiting codes with larger minimum distances than the best previously known quasi-cyclic codes are given. Two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length. One of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters.

On the complexity of bounded distance decoding for the AWGN channel

Earlier work has derived the storage complexity of the bounded distance decoder (BDD) for binary channel convolutional codes. We extend this work to the Gaussian noise channel and to partial-response codes. We show that the storage requirement /spl sim/(2/sup 1-R/ - 1)/sup -t/ paths for rate-R convolutional codes over the binary channel becomes /spl sim/2/sup 2Rt/ over the Gaussian channel, where the decoder must correct t errors. Thus, convolutional coding over the Gaussian channel is not only 3 dB more energy efficient, but its decoding is simpler as well. Next, we estimate the path storage for partial-response codes, i.e., real-number convolutional codes, over the Gaussian channel. The growth rate depends primarily on the bandwidth of the code. A new optimization procedure is devised to measure the maximum storage requirement in Gaussian noise for these two code types. An analysis based on difference equations predicts the asymptotic storage growth for partial response codes.

On (n, n-1) convolutional codes with low trellis complexity

We show that the state complexity profile of a convolutional code C is the same as that of the reciprocal of the dual code of C in case that minimal encoders for both codes are used. Then, we propose an optimum permutation for any given (n, n-1) binary convolutional code that will yield an equivalent code with the lowest state complexity. With this permutation, we are able to find many (n, n-1) binary convolutional codes which are better than punctured convolutional codes of the same code rate and memory size by either lower decoding complexity or better weight spectra.

Diversity Enhancement of Coded Spread Spectrum Video Watermarking

This paper analyses the effect of signal combination techniques in video watermark detection. A spatial spread spectrum based watermarking techique is used as embedding method, in combination with common error correction codes (BCH, Reed-Solomon with multilevel signaling, binary convolutional codes with Viterbi decoding). Besides an analytical evaluation of the bit error rate, the effectiveness of the channel coding and diversity techniques is also assessed experimentally under MPEG-2 video compression.

A maximum-likelihood soft-decision sequential decoding algorithm for binary convolutional codes

We present a trellis-based maximum-likelihood soft-decision sequential decoding algorithm (MLSDA) for binary convolutional codes. Simulation results show that, for (2, 1, 6) and (2, 1, 16) codes antipodally transmitted over the AWGN channel, the average computational effort required by the algorithm is several orders of magnitude less than that of the Viterbi algorithm. Also shown via simulations upon the same system models is that, under moderate SNR, the algorithm is about four times faster than the conventional sequential decoding algorithm (i.e., stack algorithm with Fano metric) having comparable bit-error probability.

Performance Limits of Compound Codes with Symbol-Based Iterative Decoding

Abstract We study the performance limit of concatenated codes based on binary constituent codes, under symbol-based iterative decoding. Although the results exposed herein are valid for any compound code built from both block and convolutional codes, we mainly focus on parallel turbo codes based on binary convolutional codes. The performance limit is found by evaluating the probability density function of the APP decoder output via a general propagation formula governing the density law through iterations. We propose also a second method to estimate the signal-to-noise ratio limit where the a priori density is assumed to be gaussian distributed. The results show that a gain in performance is achieved when bit nodes are combined into quaternary symbol nodes. For example, with a rate 1/2 Turbo code and a rate 2/3 binary RSC constituent, the minimal attainable signal-to-noise ratio per bit is 0.31 dB which is 0.13 dB away from the capacity of the AWGN channel with BPSK modulation.

On the error exponent for woven convolutional codes with inner warp

In this correspondence, the error exponents and decoding complexity of binary woven convolutional codes with outer and inner warp are studied. It is shown that for both constructions an error probability that is exponentially decreasing with the memory of the woven convolutional codes can be achieved with a nonexponentially increasing decoding complexity. Furthermore, the error exponent for woven convolutional codes with inner warp is larger than the one for woven convolutional codes with outer warp.

On the error exponent for woven convolutional codes with outer warp

In this correspondence the error exponent and the decoding complexity of binary woven convolutional codes with outer warp and with binary convolutional codes as outer and inner codes are studied. It is shown that an error probability that is exponentially decreasing with the product of the outer and inner code memories can be achieved with a nonexponentially increasing decoding complexity.

Binary multilevel convolutional codes with unequal error protection capabilities

Binary multilevel convolutional codes (CCs) with unequal error protection (UEP) capabilities are studied. These codes belong to the class of generalized concatenated (GC) codes. Binary CCs are used as outer codes. Binary linear block codes of short length, and selected subcodes in their two-way subcode partition chain, are used as inner codes. Multistage decodings are presented that use Viterbi decoders operating on trellises with similar structure to that of the constituent binary CCs. Simulation results of example binary two-level CC's are also reported.

Cosets of convolutional codes with least possible maximum zero- and one-run lengths

A communication or storage system may use a coset of a binary convolutional code for both symbol synchronization and error control. To facilitate symbol synchronization, the coset must have a short maximum zero-run length L/sub max/. General upper and lower bounds on L/sub max/ were given previously by Hole. In this correspondence we use these bounds to identify which convolutional codes have cosets with short L/sub max/. For such a code, we then show how to determine a coset with the least possible L/sub max/ among all cosets of the code. Exact expressions for the least possible L/sub max/ of convolutional code cosets are given, and examples of such cosets with large free distances are tabulated. Bounds on L/sub max/ for cosets of block codes are also provided. It is indicated how to tighten the bounds for block codes satisfying the one-way chain condition. We show that the cosets obtained from traditional high-rate block code constructions have larger L/sub max/ than cosets of convolutional codes with approximately the same rates. In some systems the convolutional code cosets must have short maximum one-run lengths as well as short maximum zero-run lengths to avoid loss of symbol synchronization. It is shown how to determine convolutional codes whose cosets with least possible maximum zero-run lengths also have least possible maximum one-run lengths.

On the Enumeration and Generation of Nonweight Equivalent Rate \frac{1}{2} Convolutional Codes

The weight equivalence of rate \frac{1}{2} binary convolutional codes and the problem of enumeration and generation of all nonweight equivalent codes is considered for a given memory order m. This is done during the generation process by first ordering the encoders into classes such that it becomes easy to recognize the weight equivalence as well as the catastrophic error propagation conditions. Subclasses of respectively self and nonself reciprocal as well as catastrophic and noncatastrophic encoders are introduced. The cardinal number of these classes and subsequently the number of ’’nonweight equivalent‘‘ codes are then computed recursively as a function of m. Finally, since all the relations amount to simple convolutions they are compactly represented by generating functions which are tabulated.

Multidimensional trellis coded phase modulation using a multilevel concatenation approach. I. Code design

The first part of this paper presents a simple and systematic technique for constructing multidimensional M-ary phase shift keying (MPSK) trellis coded modulation (TCM) codes. The construction is based on a multilevel concatenation approach. In which binary convolutional codes with good free branch distances are used as the outer codes and block MPSK modulation codes are used as the inner codes (or the signal spaces). Conditions on phase invariance of these codes are derived and a multistage decoding scheme for these codes is proposed. The proposed technique can be used to construct good codes for both the additive white Gaussian noise (AWGN) and fading channels as is shown in the second part of this paper.

A directed search approach for unit-memory convolutional codes

A set of heuristic algorithms to numerically search for binary unit-memory convolutional codes (UMC) are presented along with a large number of new codes for 2/spl les/k/spl les/8 and code rate 1/4/spl les/R<1. Combinatorial optimization is used which involves selecting and then pairwise-matching column vectors of the two (n,k) UMC tap weight matrices. The column selection problem is that of finding the best (2n,k) binary, linear block code (BC). In this correspondence, the best BC generator matrix G is found by successively refining G using directed local exhaustive searches. In particular, the set of minimum-weight codewords are used to find a subset of G to exhaustively search. The UMC search strategy (pairwise matching problem) uses a directed local exhaustive search similar to the BC directed search by using the concept of the terminated BC of the UMC. The heuristic algorithms developed in this correspondence are very robust and converge relatively quickly to the optimal or near-optimal UMC. In addition, although it is generally possible to achieve the block code upper bound for free distance, we give a class of UMCs which cannot achieve this bound.

Iterative decoding of binary block and convolutional codes

Iterative decoding of two-dimensional systematic convolutional codes has been termed "turbo" (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10/sup -4/ the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations.

Simplified trellis decoder

A new simplified trellis decoder (STD) Viterbi-type algorithm is proposed for fast trellis decoding of rate K/K+1 binary convolutional codes. Viterbi algorithm (VA) computation is dominated by add-compare-select (ACS) operations when k/spl ges/2. The STD can substantially reduce the number of ACS operations and allow for a trade-off between the computational load and the performance of the decoder, The STD is analyzed and simulated for a four-dimensional (4-D) rate 4/5 64-state convolutional encoder specified by the ITU-T V.34 modem recommendation.

DC- and Nyquist-free error correcting convolutional codes

A class of DC- and Nyquist-free codes with error correcting capability is proposed. These codes are minimum-bandwidth codes because they have spectral null at Nyquist frequency. They also have a good error correcting capability because they are constructed from binary convolutional codes.