Comparison of M-ary Modulation Systems

T h e choice of waveforms for communicating over the Gaussian additive noise channel is a classic problem in communication theory. Orthogonal modulation systems (i.e., digital communications in which the alphabet consists of orthogonal waveforms) are known to result in good power efficiency at the expense of poor bandwidth utilization. 1 , 2 As the alphabet size M is increased, the energy-per-bit E required to achieve a given error probability P e diminishes, b u t the information rate to bandwidth ratio ( R / W ) diminishes even more rapidly. Biorthogonal and simplex modulation afford somewhat improved performance, b u t are likewise restricted to low values of R/W. There is considerable interest in finding large alphabet systems which have both good power efficiency and good bandwidth utilization. 843 844 T H E BELL SYSTEM TECHNICAL JOURNAL, M A Y - J U N E l f ) f i 7 Slepian 3 has given bounds on what can be achieved, but constructive techniques for approaching these bounds are generally unknown. Although computer evaluation is ultimately required for precise knowledge of error probability, it is of interest to consider simpler performance criteria which permit at least a qualitative comparison of various systems without extensive computation. It is the purpose of this paper to demonstrate the utility of the latter approach. After defining the problem more precisely in Section II, some wellknown bounds on the error probability are employed in Section III to obtain a simple analytic criterion for comparing systems in the limit of low P.