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

Abstract

A digital transmission system with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> possible transmitted symbols is considered. If the time between transmitted symbols is T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> seconds and the bandwidth is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</tex> , the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> possible symbols correspond to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> vectors in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2WT_{0}</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq 16</tex> 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.