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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 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.

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