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When employing HD-FEC, the complex received symbol will be mapped to the nearest reference constellation point, converted to binary, and finally performed the error correction. Generally, for maximum likelihood (ML) detection, the threshold is at the middle point of any two constellation points which is not suitable for non-uniform signal like PS (and also HPGS) signal. MAP detection is considered as the optimal detection for PS signal in HDD. The MAP decision threshold between any 2 complex symbols x_i,x_j can be calculated from equation (1). Then, the MAP-BER can be calculated from equation (2)
,where P_X (X) is probability of each complex symbol, and h(x_i,x_j ) is the Hamming distance, and M is total bits in one N-QAM symbol. Note that, if the signal distribution is uniform, the second term in equation (1) will be zero and it turns into ML threshold and the BER in equation (2) is considered as ML-BER instead.
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