The hyperanalytic signal

TL;DR

This paper introduces the hyperanalytic signal, extending the analytic signal concept to complex signals using hypercomplex and quaternion frameworks, and explores its properties and applications.

Contribution

It presents the hyperanalytic signal concept, methods to derive it, and extends properness to deterministic complex signals, offering new tools for signal analysis.

Findings

Hyperanalytic signal can be obtained via Hilbert, Fourier, and quaternion Fourier transforms.

Allows derivation of complex envelope and phase from the hyperanalytic signal.

Generalizes modulation properties of classical analytic signals.

Abstract

The concept of the analytic signal is extended from the case of a real signal with a complex analytic signal to a complex signal with a hypercomplex analytic signal (which we call a hyperanalytic signal) The hyperanalytic signal may be interpreted as an ordered pair of complex signals or as a quaternion signal. The hyperanalytic signal contains a complex orthogonal signal and we show how to obtain this by three methods: a pair of classical Hilbert transforms; a complex Fourier transform; and a quaternion Fourier transform. It is shown how to derive from the hyperanalytic signal a complex envelope and phase using a polar quaternion representation previously introduced by the authors. The complex modulation of a real sinusoidal carrier is shown to generalize the modulation properties of the classical analytic signal. The paper extends the ideas of properness to deterministic complex signals using the hyperanalytic signal. A signal example is presented, with its orthogonal signal, and its complex envelope and phase.