Instantaneous frequency and amplitude of orthocomplex modulated signals based on quaternion Fourier transform

• The paper extends the idea of the analytic signal, to the case of a complex, rather than real, signal.
• A quaternion Fourier transform is used to construct the analogue of the analytic signal.
• The idea of amplitude modulation is extended to the case where the modulating and modulated signals are both complex.
• The two complex signals are kept separate by treating them as orthogonal (in different complex planes, realised using quaternions).
• The paper presents examples and compares the approach taken with a specific signal model proposed by Lilly and Olhede.

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We explore the extension of these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We discuss its derivation and properties and show how to obtain a complex envelope and a real phase from it.A classical result in the case of real signals is that an amplitude modulated signal may be decomposed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Examples are presented to demonstrate these concepts.