A decoding rule is presented which minimizes the probability of symbol error over a time-discrete memory]ess channel for any linear error-correcting code when the codewords are equiprobable. The complexity of this rule varies inversely with code rate, making the technique particularly attractive for high rate codes. Examples are given for both block and convolutional codes.

/pdf/an-optimum-symbol-by-symbol-decoding-rule-for-linear-codes-745p591yvy.pdf

An optimum symbol-by-symbol decoding rule for linear codes

An equalization concept for the novel radio access scheme Enhanced Data rates for GSM Evolution (EDGE) is proposed by which high performance can be obtained at moderate computational complexity. Because high-level modulation is employed in EDGE, optimum equalization as usually performed in Global System for Mobile Communications (GSM) receivers is too complex and suboptimum schemes have to be considered. It is shown that delayed decision-feedback sequence estimation (DDFSE) and reduced-state sequence estimation (RSSE) are promising candidates. For various channel profiles, approximations for the bit error rate of these suboptimum equalization techniques are given and compared with simulation results for DDFSE. It turns out that a discrete-time prefilter creating a minimum-phase overall impulse response is indispensable for a favorable tradeoff between performance and complexity. Additionally, the influence of channel estimation and of the receiver input filter is investigated and the reasons for performance degradation compared to the additive white Gaussian noise channel are indicated. Finally, the overall system performance attainable with the proposed equalization concept is determined for transmission with channel coding.

Equalization concepts for EDGE

We propose a technique to reduce the number of trellis states in BCJR-type algorithms, i.e., algorithms. with a structure similar to that of the well-known algorithm by Bahl, Cocke, Jelinek, and Raviv (1974). This work is inspired by reduced-state sequence detection (RSSD). The key idea is the construction, during one of the recursions in the reduced-state trellis, of a "survivor map" to be used in the other recursion. In a more general setting, two distinct survivor maps could be determined in the two recursions and used jointly to approximate the a posteriori probabilities. Three examples of application to iterative decoding are shown: (1) coherent detection for intersymbol interference (ISI) channels; (2) noncoherent detection based on an algorithm previously proposed by the authors; and (3) detection based on linear prediction for Rayleigh fading channels. As in classical RSSD, the proposed algorithm allows significant state-complexity reduction with limited performance degradation.

/pdf/reduced-state-bcjr-type-algorithms-31ax6e5c7q.pdf

Reduced-state BCJR-type algorithms

Any symbol in a redundant code can be recovered when it belongs to certain erasure patterns. Several alternative expressions of a given symbol, to be referred to as its replicas, can therefore be computed in terms of other ones. Decoding is interpreted as decoding upon a received symbol, given itself and a number of such replicas, expressed in terms of other received symbols. For linear q-ary (n,k) block codes, soft-decision demodulation and memoryless channels, the maximum-likelihood decision rule on a given symbol is formulated in terms of r \leqn - k linearly independent replicas from the parity-check equations. All replicas deriving from the r selected replicas by linear combination are actually taken into account in this decision rule. Its implementation can be direct; use transformations or a sequential circuit implementing a trellis representation of the parity-check matrix. If r = n - k , decoding is optimum, in the sense of symbol-by-symbol maximum-likelihoed. Simplification results in the transformed and sequential implementations when r . If the selected replicas are disjoint, generalized ( q -ary, weighted) threshold decoding results. The decoding process can easily be modffied in order to provide word-by-word maximum-likelihood decoding. Convolutional codes are briefly considered. Two specific problems are discussed: the use of previous decisions, which leads to a weighted generalization of feedback decoding, and the extension of replication decoding to nonsystematic codes.

http://ieeexplore.ieee.org/iel5/18/22708/01056035.pdf

Replication decoding

This paper introduces a new approach for joint convolutional coding and decision feedback equalization (DPE). To minimize error propagation, the DFE uses a combination of soft decisions and delayed tentative decisions to cancel intersymbol interference (ISI). Soft decisions are obtained by passing the DFE output through a (soft) nonlinear device. This simple method is shown to perform almost as well as an optimum soft feedback approach on wireless channels with diversity. Tentative decisions from the Viterbi decoder are used to cancel ISI due to multipath with large delays, thus remedying the increasing effect of error propagation in channels with large delay spreads. We consider the use of this soft/delayed feedback DFE (S/D-DFE) technique in broadband wireless channels (with delay spreads up to several tens of the symbol period) typical in high-bitrate mobile data applications. Simulation results indicate that the proposed joint coding and S/D-DFE technique performs to within 1-2 dB [in required signal-to-noise ratio (SNR)] of an ideal coded DFE without error propagation. When combined with antenna diversity and a reduced-complexity DFE concept with adaptive feedforward tap assignment, it provides high packet throughput against Rayleigh fading, severe delay spreads, and high Doppler rates.

Joint coding and decision feedback equalization for broadband wireless channels

A recursive procedure is derived for decoding of rate R = 1/n binary convolutional codes which minimizes the probability of the individual decoding decisions for each information bit, subject to the constraint that the decoding delay be limited to Δ branches. This new decoding algorithm is similar to, but somewhat more complex than, the Viterbi decoding algorithm. A "real-time," i.e., fixed decoding delay, version of the Viterbi algorithm is also developed and used for comparison to the new algorithm on simulated channels. It is shown that the new algorithm offers advantages over Viterbi decoding in soft-decision applications, such as in the inner coding system for concatenated coding.

/pdf/real-time-minimal-bit-error-probability-decoding-of-2ltm8rciea.pdf

L. Lee

Papers

Real-Time Minimal-Bit-Error Probability Decoding of Convolutional Codes