Stephen J. Sangwine

Nicolas Le Bihan, Stephen J. Sangwine, Todd A. Ell

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 extend 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 present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed 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. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts.

Nicolas Le Bihan, Stephen J. Sangwine

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.

Stephen J. Sangwine

A vector-valued signal in N dimensions is a signal whose value at any time instant is an N-dimensional vector, that is, an element of $\mathbb{R}^N$. The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N-dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N-dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in $\mathbb{R}^N$. Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions.

Salem Said, Nicolas Le Bihan, Stephen J. Sangwine

A discrete complexified quaternion Fourier transform is introduced. This is a generalization of the discrete quaternion Fourier transform to the case where either or both of the signal/image and the transform kernel are complex quaternion-valued. It is shown how to compute the transform using four standard complex Fourier transforms and the properties of the transform are briefly discussed.

Stephen J. Sangwine, Nicolas Le Bihan

We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation of the svd of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from the product of the Householder transformations and the real singular vectors of B. We show in the paper that left and right quaternionic Householder transformations are different because of the noncommutative multiplication of quaternions and we present formulae for computing the Householder vector and matrix in each case.