Theory of Analytic Modulation Systems

A general theory of analytic modulation systems is developed where the transmitted signal is of the form σ(t) = Re {eiwctf(z(t))}. Here f(z) is an analytic function (modulation law), and z(t) = x(t) + iy(t) is the analytic baseband signal whose real part x(t) is a bounded bandlimited signal of spectral support [–Ω,Ω] which is assumed to have a bounded Hilbert transform y(t). It is shown for a large class of {z(t)} and modulation laws that z(t) may be recovered using a receiver incorporating the inverse function of f as a detector with appropriate pre- and post-detection filtering. The theory also shows that in the procedure for factoring certain positive bandlimited signals, an approximate Hilbert transform operator (bandlimited) may be used. A related result is that signals subjected to logarithmic companding (one-sided) and filtering may be recovered by a non-iterative method.