An Analytical Method for Performance Evaluation of Binary Linear Block Codes

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An analytical method for performance evaluation of binary linear block codes using an Additive White Gaussian Noise (AWGN) channel model with Binary Phase Shift Keying (BPSK) modulation is presented. We focus on the probability distribution function (pdf) of the bit LogLikelihood Ratio (LLR) which is expressed in terms of the Gram-Charlier series expansion. This expansion requires knowledge of the statistical moments of the bit LLR. We introduce an analytical method for calculating these moments. This is based on some straight-forward recursive calculations involving certain weight enumerating functions of the code. It is shown that the estimate of the bit error probability provided by the proposed method will asymptotically converge to the true bit error performance. Numerical results are provided for the (15,11) Cyclic code which demonstrate close agreement with the simulation results.

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  • Biometrika 06/1963; 50(3). DOI:10.2307/2333784 · 1.42 Impact Factor
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    ABSTRACT: The maximum a posterioriprobability (MAP) algorithm is a trellis-based MAP decoding algorithm. It is the heart of turbo (or iterative) decoding that achieves an error performance near the Shannon limit. Unfortunately, the implementation of this algorithm requires large computation and storage. Furthermore, its forward and backward recursions result in a long decoding delay. For practical applications, this decoding algorithm must be simplifled and its decoding complexity and delay must be reduced. In this paper, the MAP algorithm and its variation's, such as log-MAP and max-log-MAP algorithms, are first applied to sectionalized trellises for linear block codes and carried out as two-stage decodings. Using the structural properties of properly sectionalized trellises, the decoding complexity and delay of the MAP algorithms can be reduced. Computation-wise optimum sectionalizations of a trellis for MAP algorithms are investigated. Also presented in this paper are bidirectional and parallel MAP decodings
    IEEE Transactions on Communications 05/2000; 48(4-48):577 - 587. DOI:10.1109/26.843125 · 1.99 Impact Factor
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    ABSTRACT: A simple bound on the probability of decoding error for block codes is derived in closed form. This bound is based on the bounding techniques developed by Gallager. We obtained an upper bound both on the word-error probability and the bit-error probability of block codes. The bound is simple, since it does not require any integration or optimization in its flnal version. The bound is tight since it works for signal-to-noise ratios (SNRs) very close to the Shannon capacity limit. The bound uses only the weight distribution of the code. The bound for nonrandom codes is tighter than the original Gallager bound and its new versions derived by Sason and Shamai and by Viterbi and Viterbi. It also is tighter than the recent simpler bound by Viterbi and Viterbi and simpler than the bound by Duman and Salehi, which requires two-parameter optimization. For long blocks, it competes well with more complex bounds that involve integration and parameter optimization, such as the tangential sphere bound by Poltyrev, elaborated by Sason and Shamai, and investigated by Viterbi and Viterbi, and the geometry bound by Dolinar, Ekroot, and Pollara. We also obtained a closed-form expression for the minimum SNR threshold that can serve as a tight upper bound on maximum-likelihood capacity of nonrandom codes. We also have shown that this minimum SNR threshold of our bound is the same as for the tangential sphere bound of Poltyrev. We applied this simple bound to turbo-like codes.
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