The Study of Properties of n-D Analytic Signals and Their Spectra in Complex and Hypercomplex Domains

In the paper, two various representations of a n-dimensional (n-D) real signal u(x1,x2,…,xn) are investigated. The first one is the n-D complex analytic signal with a single-orthant spectrum defined by Hahn in 1992 as the extension of the 1-D Gabor’s analytic signal. It is compared with two hypercomplex approaches: the known n-D Clifford analytic signal and the Cayley-Dickson analytic signal defined by the Author in 2009. The signal-domain and frequency-domain definitions of these signals are presented and compared in 2-D and 3-D. Some new relations between the spectra in 2-D and 3-D hypercomplex domains are presented. The paper is illustrated with the example of a 2-D separable Cauchy pulse

The 2D Analytic Signal

This technical report covers a fundamental problem of 2D local phase based signal processing: the isotropic generalization of the analytic signal (D. Gabor) for two dimensional signals. The analytic signal extends a real valued 1D signal to a complex valued signal by means of the classical 1D Hilbert transform. This enables the complete analysis of local phase and amplitude information. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals. In case of 2D image signals the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n + 1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform (Riesz transform). In this technical report we present the 2D analytic signal as a novel generalization of the 2D monogenic signal which now extends the original 2D signal to a multivector valued signal in conformal space by means of higher order Hilbert transforms and by means of a hybrid matrix geometric algebra representation. The 2D analytic signal can be interpreted in conformal space which delivers a descriptive geometric interpretation of 2D signals. One of the main results of this work is, that all 2D signals exist per se in a 3D projective subspace of the conformal space and can be analyzed by means of geometric algebra. In case of 2D image signals the 2D analytic signal enables now the rotational invariant analysis of lines, edges, corners and junctions

A unique polar representation of the hyperanalytic signal

The hyperanalytic signal is the straight forward generalization of the classical analytic signal. It is defined by a complexification of two canonical complex signals, which can be considered as an inverse operation of the Cayley-Dickson form of the quaternion. Inspired by the polar form of an analytic signal where the real instantaneous envelope and phase can be determined, this paper presents a novel method to generate a polar representation of the hyperanalytic signal, in which the continuously complex envelope and phase can be uniquely defined. Comparing to other existing methods, the proposed polar representation does not have sign ambiguity between the envelope and the phase, which makes the definition of the instantaneous complex frequency possible.Comment: 2014 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP

Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

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