  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.
