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Analytic signals and product theorems for Hilbert transforms

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Analytic signals are introduced as certain eigenfunctions of the Hilbert transform operator; that is,z(cdot)is termed "analytic" if and only ifhat{z}(t) = -jz(t)for allt, wherehat{z}(cdot)is the Hilbert transform ofz(cdot). Similarly, "dual-analytic" signals are defined as solutions of the homogeneous equationhat{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 formhat{f}g = fhat{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.

Published in:

Circuits and Systems, IEEE Transactions on  (Volume:21 ,  Issue: 6 )

Date of Publication:

Nov 1974

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