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A new method to improve the bit error rate (BER) performance by optimising the bit energy is proposed in this study. This method is mainly based on a formula derived from the union bound that can estimate the BER for every bit in the codeword sequence. Then by mathematical optimisation and assisted with a BER distribution, the energy for every bit can be optimised so that the average BER achieves the minimum value. Compared with other methods, this method considering all the codewords with minimum Hamming weights and their multiplicity to each bit of the codewords. The simulation examples show that the average BER is improved not only at high signal-to-noise ratio (SNR), but also at low and moderate SNRs.

Based on the union bound, formulas to estimate the BER distribution of channel codes are derived. By using these formulas, the BER for every position in the information sequence can be estimated. Appling the formulas to Turbo codes, several examples were given, and the results are also compared with simulation results. The results show that the derived formulas can give out good estimations of the BER distributions for Turbo codes. Therefore this would be helpful for the BER analysis, especially the unequal error protection analysis of Turbo codes.

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