On the Performance of Digital Modulation Systems That Expand Bandwidth

It is well known that protection against additive gaussian noise can be obtained in m-ary digital modulation systems by expanding bandwidth or by increasing the channel signal-to-noise ratio. It is also well known that arbitrarily small error probabilities can be attained in digital systems by using long and complex encoding and decoding procedures. Based on the results of Shannon and Slepian, we derive for an optimal system lower bounds to the channel signal-to-noise ratio for various probabilities of error, for various bandwidth expansions, and for a processing interval not greater than the signaling interval of the source. It is assumed that all m characters have equal a priori probabilities and that maximum likelihood detection is used in the receiver. For a bandwidth expansion of two, and for equal energy code words, we also show that the performance of a coherent phase-shift keyed system is as good as that of the optimal system.