The properties of generalized offset linear canonical Hilbert transform and its applications

The Hilbert transform is tightly associated with the Fourier transform. As the offset linear canonical transform (OLCT) has been shown to be useful and powerful in signal processing and optics, the concept of generalized Hilbert transform associated with the OLCT has been proposed in the literature. However, some basic results for the generalized Hilbert transform still remain unknown. Therefore, in this paper, theories and properties of the generalized Hilbert transform have been considered. First, we introduce some basic properties of the generalized Hilbert transform. Then, an important theorem for the generalized analytic signal is presented. Subsequently, the generalized Bedrosian theorem for the generalized Hilbert transform is formulated. In addition, a generalized secure single-sideband (SSB) modulation system is also presented. Finally, the simulations are carried out to verify the validity and correctness of the proposed results.

Sampling in the Linear Canonical Transform Domain

This paper investigates the interpolation formulae and the sampling theorem for bandpass signals in the linear canonical transform (LCT) domain. Firstly, one of the important relationships between the bandpass signals in the Fourier domain and the bandpass signals in the LCT domain is derived. Secondly, two interpolation formulae from uniformly sampled points at half of the sampling rate associated with the bandpass signals and their generalized Hilbert transform or the derivatives in the LCT domain are obtained. Thirdly, the interpolation formulae from nonuniform samples are investigated. The simulation results are also proposed to verify the correctness of the derived results.

On Hankel conjugate functions

We consider some aspects of harmonic analysis of the differential operator . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform Hν. We present a fairly detailed analysis of the corresponding Poisson semigroup {Pt}t > 0: this is given in a weighted setting with Ap-weights involved. Then, we consider conjugate Poisson integrals of functions from Lp(w), w ∈ Ap, 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in Lp(w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform Hν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.

Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform

The extended Euler deconvolution algorithm is shown to be a generalization and unification of 2‐D Euler deconvolution and Werner deconvolution. After recasting the extended Euler algorithm in a way that suggests a natural generalization to three dimensions, we show that the 3‐D extension can be realized using generalized Hilbert transforms. The resulting algorithm is both a generalization of extended Euler deconvolution to three dimensions and a 3‐D extension of Werner deconvolution. At a practical level, the new algorithm helps stabilize the Euler algorithm by providing at each point three equations rather than one. We illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole.