A Hilbert transform product theorem

The usual product theorem for Hilbert transforms states that under certain conditions x(t)y(t), the Hilbert transform of a product of two signals x(t) and y(t), is equal to x(t)y^{^}(t), a result having repeated use in simplifying calculations involving modulated waveforms. Several sufficient conditions for the theorem to hold are well-known and appear in the literature. Here, using a time-domain approach, we derive a necessary and sufficient condition for the validity of the theorem and show as an example two signals which have the desired property but which do not satisfy any of the earlier sufficient conditions.