Hypercomplex analytic signals : Extension of the analytic signal concept to complex signals

The analytic signal is a complex signal derived from a real signal such that its real part is identical to the original real signal, and its imaginary part is in quadrature (orthogonal) to the original signal. The analytic signal permits the envelope of the original signal to be computed, and it also admits the definition of an instantaneous frequency and phase. In this paper we present some initial results on extending this idea to the case of a complex signal using a hypercomplex analytic signal. We show that using the hypercomplex analytic signal it is possible to calculate a complex envelope of the original complex signal and that the modulus of this complex envelope is the envelope of the modulus of the original signal.