The Theory of Analytic Band-Limited Signals Applied to Carrier Systems

The recently-developed concept of the "analytic signal," together with that of bandwidth-limited signals, is used to construct a general theory of amplitude modulation. The basic definition and properties of an "analytic signal" are reviewed with particular reference to band-limited signals. The "analytic signal" is a signal of the form
z(t) = x(t) + jy(t) = p(t)e^{j\varphi(t)}
where
x(t)
and
y(t)
have the same norms, are orthogonal and related to one another by the Hilbert transform, and where
p(t) = \sqrt{x^{2}+y^{2}}
defines an envelope function while
\varphi(t) = \tan^{-1}y(t)/x(t)
defines a phase function whose rate of change determines an instantaneous frequency. The Fourier transform
Z(\omega)
for
z(t)
is identically zero for
\omega < 0
. The theory of amplitude modulation based on the band-limited analytic signal is used to define clearly the properties of single-sideband transmission systems. The problem of amplifier overloading in multiplex single-sideband transmission is referred to and some comparisons are drawn between the amplitude probability distributions of the envelopes of double and single-sideband transmission of random signals. It is suggested that the concept of the analytic bandwidth-limited signal may facilitate the development of a general theory of frequency modulation.