A digital transmission system with n possible transmitted symbols is considered. If the time between transmitted symbols is T 0 seconds and the bandwidth is W , the n possible symbols correspond to n vectors in a 2WT_{0} 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 n 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 n , 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 n \leq 16 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.

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