On the Optimum Performance of N-ary Systems Having Two Degrees of Freedom

A digital transmission system with possible transmitted symbols is considered. If the time between transmitted symbols is T0seconds and the bandwidth is , the possible symbols correspond to vectors in a dimensional signal space. This paper considers the theoretical properties of a class of digital systems where the signal space is two-dimensional. Such systems are both amplitude-and phase-modulated. Approximate expressions are derived for the average probability of error for these systems as a function of the placement of the symbol vectors in the twodimensional signal space. Optimum placements are then given which minimize this probability of error for a given average or peak power SNR constraint. It is shown that the optimum channel structure is a function of the alphabet size , and the type of power constraint, as well as the SNR. In general the optimum system is a phase-modulated system for low SNR\´s and for alphabet sizes in the high SNR region. The performance of this optimum system in terms of channel capacity and probability of error is then compared with the performance of one-dimensional systems, AM-only and PM-only, in a complete set of curves for both peak and average power.