Analytic signals and product theorems for Hilbert transforms

Analytic signals are introduced as certain eigenfunctions of the Hilbert transform operator; that is,
z(\cdot)
is termed "analytic" if and only if
\hat{z}(t) = -jz(t)
for all
t
, where
\hat{z}(\cdot)
is the Hilbert transform of
z(\cdot)
. Similarly, "dual-analytic" signals are defined as solutions of the homogeneous equation
\hat{u} = ju
. Using this characterization of analytic signals (shown to be equivalent to the usual definition due to Ville [1]), simple proofs are obtained for all known product theorems of the form
\hat{f}g = f\hat{g}
, which are useful in the representation and analysis of modulated waveforms. In addition, parallel theorems for the class of dual-analytic and frequency-translated dual-analytic signals are proven.