A-Posteriori Symbol Probabilities and Log-Likelihood Ratios for Coherently Detected \frac{π}{4}-DE-QPSK

In this letter, coherent detection of π/4-DE-QPSK is considered, but our analysis also holds for common DE-QPSK. It is shown that maximum a-posteriori (MAP) sequence detection can be regarded as an approximation, based on selecting dominant exponentials, of MAP symbol detection. A better approximation, relying on piecewise-linear fitting of the logarithm of the hyperbolic cosine, is proposed. This approximation results in a performance very close to optimal symbol detection. For the case where the symbols are produced by convolutional encoding and Gray mapping, the log-likelihood ratios are investigated. Again a simple approximation based on selecting dominant exponentials and an improved approximation relying on piecewise-linear fits, are discussed. As in the uncoded case the improved approximation gives a performance quite close to ideal. While the particular examples considered show modest gains in performance, this letter provides a way of improving performance when needed.