Probability Density Functions of Logarithmic Likelihood Ratios in Rectangular QAM

Closed-form expressions for the probability density function (PDF) of logarithmic likelihood ratios (LLR) in rectangular quadrature amplitude modulations are derived. Taking advantage of assumed Gray mapping, the problem is solved in one dimension corresponding to the real or imaginary part of the symbol. The results show that sought PDFs are linear combinations of truncated Gaussian functions. This simple result stands in contrast with often assumed Gaussian distribution for the LLRs. Histograms of LLRs obtained via simulations confirm our analysis