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 New Insight into the Theory of 2-D Complex and Quaternion Analytic Signals

The paper presents the overview of the theory of 2-D complex and quaternion analytic signals with the 1st-quadrant spectrum support. Both signals are expressed as complex/hypercomplex sums of partial and total 2-D Hilbert transforms. Moreover, starting with the definition of 2-D complex and quaternion Fourier transforms, the 2-D Hilbert transforms are derived in the form of sums of parts of different parity with respect to signal-domain variables. Some new relations for Hilbert quaternion spectra have been derived. The paper is illustrated with the example of the 2-D separable Cauchy signal

Quasi-analytic multidimensional signals

In a recent paper, the authors have presented the unified theory of n-dimensional (n-D) complex and hypercomplex analytic signals with single-orthant spectra. This paper describes a specific form of these signals called quasi-analytic. A quasi-analytic signal is a product of a n-D low-pass (base-band) real (in general non-separable) signal and a n-D complex or hypercomplex carrier. By a suitable choice of the carrier frequency, the spectrum of a low-pass signal is shifted into a single orthant of the Fourier frequency space with a negligible leakage into other orthants. A measure of this leakage is defined. Properties of quasi-analytic signals are studied. Problems of polar representation of quasi-analytic signals and of its lower rank representation are discussed

Plasma-based X-ray laser at 21 nm for multidisciplinary applications

An overview of recent advances in applications of currently the most energetic X-ray laser at 21 nm is given. The unique parameters of this half-cavity based X-ray laser such as record output energy of 10 mJ, highly symmetric beam, robustness and reproducibility, have made it possible to carry out a number of multidisciplinary scientific projects featuring novel applications of intense coherent X-ray radiation. Selected results obtained in these experiments are reviewed, including X-ray laser probing of dense plasmas, measurements of transmission of focused soft X-ray radiation at intensities of up to 1012 W cm-2, measurements of infrared laser ablation rates of thin foils, and ablative microstructuring of solids