Optimum receivers

In recent years there have been numerous studies of reception or communications systems for the demodulation of signals which are received with additive gaussian noise. Many of these studies have been based on the work of D.C. Youla ["The use of maximum likelihood in estimating continuously modulated intelligence which has been corrupted by nolse," IRE TRANS. ON INFORMATION THEORY, vol. PGIT-3, pp. 90-105, March, 1954]. In frequency modulation (FM) the signal is subjected to linear filtering (pre-emphasis) prior to modulation and it is not immediately apparent how to apply the work of Youla directly. Linear filtering prior to modulation is also used in many other situations which occur in practice (for example, SSB AM can be produced by a combination of linear operations on the signal), and it is therefore pertinent to extend Youla\´s work to this case. The purpose of this communication is to point out that Youla\´s work can be extended in a straightforward way to cover modulation schemes in which filtering prior to modulation is employed. The results have been known for some time but apparently are not widely available. We consider the case where one signal is sent over one communication channel. Extensions to more than one signal and more than one channel appear to be straightforward.