Signal selection for non-symmetric sources in two dimensions

An attempt is made to determine which two-dimensional signal constellation minimizes the probability of error under an average power constraint for nonsymmetric sources. Two different approaches to the problem are presented. First, a numerical method with the help of Lagrange multipliers is described. Then, a normalized conjugate gradient method and a normalized gradient method based on the approach given by G.J. Foschini et al. (1974) are provided. The gradient-based methods are powerful tools that can be applied to signal sets of any size and of any statistics. The normalized conjugate gradient method is a faster than the normalized gradient method