Asymptotic Extraction Research Articles

On the robustness of Singular Spectrum Analysis for long signals

The paper is devoted to theoretical investigation of the robustness for Singular Spectrum Analysis (briefly, SSA) if the length N of time series tends to infinity. The last condition discerns the paper from many other works on this theme. Note that here the general theoretical approach to the asymptotic extraction of the signal series from the additively perturbed signal with the help of SSA was used. Therefore, taking the series corresponding to the outliers as the noise series, we can check whether the so-called reconstruction errors tend uniformly as N to infinity. If this convergence takes place then the method is robust. Several examples of such an approach for concrete signals and outlies are considered, some of them are illustrated by computational experiments.

Post-processing of a lavender flowers solvent extract using supercritical CO2 fractionation

BackgroundSupercritical Fluid Extraction (SFE) was performed using an unconventional material, the solid extract of lavender flowers obtained by liquid solvent extraction and subsequent solvent elimination. MethodsSystematic extraction experiments were carried out at 8 MPa and 40 °C, and the extract was fractionated in semi-continuous mode in the SFE plant. To understand mass transfer phenomena driving the process, CO2 mass flow rates ranging between 0.60 and 1.50 kg/h were used, and yield vs. time curves were obtained. Significant findingsFractionation produced a cuticular waxes selective precipitation in the first separator and the floral fragrance in the second separator. The most abundant species in the extract were τ-cadinol (13%), lavandulol (10.5%), β-caryophyllene (10%), viridiflorene (8.5%), isocaryophyllene (6%), cedrenalol (4.5%), linalool (4%) and 1,8-cineol (4%). The fragrance contained no waxes, indicating that the fractionation was successful. At higher mass flow rates (from 0.90 to 1.50 kg/h), an asymptotic extraction yield of 5.2% w/w was obtained; whereas, at lower mass flow rates, the extraction yield was lower (2.3% w/w) since the vegetable bed was not completely wetted by the extraction fluid. The overall results indicated that an external mass transfer resistance controlled the process.

Efficient Semi-Analytical Method for the Analysis of Large Finite Connected Slot Arrays

We present an efficient method for the analysis of finite connected slot arrays in the presence of stratified media. The formulation is based on a spectral method of moments, where only one basis function is considered for each array element and one for each slot edge. An expression for the mutual impedance is derived in terms of a double spectral integral. Asymptotic extraction techniques are employed to largely reduce the computation time of one of the spectral integrals. For the other integral, when a guided wave contribution dominates the mutual coupling between two array elements, the result can be approximated as the residue of the spectral polar singularity, providing a closed-form solution of the coupling for elements at electrically large distances. The complete method enables simulations of entire finite arrays with hundreds or even thousands of elements in minutes. The same structure would require impractical computation time when analyzed with general-purpose commercial software. The method allows estimating the performance of finite connected arrays. This is particularly relevant because wideband connected arrays are known to exhibit higher edge effects compared to narrowband arrays, due to the high interelement mutual coupling.

On the asymptotical separation of linear signals from harmonics by singular spectrum analysis

The general theoretical approach to the asymptotic extraction of the signal series from the additively perturbed signal with the help of singular spectrum analysis (briefly, SSA) was already outlined in Nekrutkin (2010), SII, vol. 3, 297–319. In this paper we consider the example of such an analysis applied to the linear signal and the additive sinusoidal noise. It is proved that in this case the so-called reconstruction errors r_i(N) of SSA uniformly tend to zero as the series length N tends to infinity. More precisely, we demonstrate that max_i |r_i(N)| = O(N^(−1)) if N → ∞ and the “window length” L equals (N + 1)/2. It is important to mention, that the completely different result is valid for the increasing exponential signal and the same noise. As it is proved in Ivanova, Nekrutkin (2019), SII, vol. 12, 1, 49–59, in this case any finite number of last terms of the error series does not tend to any finite or infinite values.

Towards fundamentally based heap leaching scale-up

A procedure is proposed for the extrapolation of laboratory column leaching kinetics for commercial heap leaching design. It employs previously published models based on dual-porosity hydrology and chemical reactions occurring during diffusion via the regions holding immobile solution. Case studies were analysed for which both the column leaching and heap leaching performance had been recorded on acid-soluble copper, sulphide-copper and gold ores. Theoretical relationships were developed for the manner in which bulk density and dripper spacing were expected to affect the diffusional path length via the immobile solution of the column/heap. The resulting correlations were fitted to the case study data by means of a single empirical constant, p, to which an average value of 4.5 (a dimensionless value) could be assigned. The magnitude of p suggests that densification, i.e. compaction of the ore particle bed, results in a slight increase in the spacing of solution flow channels but a large increase of the tortuosity of diffusion pathways. In the case of narrow dripper spacing, the decrease in leaching kinetics upon scale-up can be attributed to the increase in bulk density. The proposed method identifies the maximum dripper spacing that can be used before the width of the dripper spacing has a stronger effect on the diffusional path length than the increase in bulk density. The effect of dripper spacing on the asymptotic extractions (extrapolated to time infinity) of the case studies was quantified. Confidence in the reliability of the method can be further improved if more case study data can be accessed.

A parallel‐plate waveguide antenna radiating through a perfectly conducting wedge

Parallel-plate waveguide antennas hosted in an electrically perfectly conducting wedge are studied. The analysis relies on an exact singular integral equation formulation. Asymptotic extraction techniques are employed to recast the kernel functions as the sum of two terms: one singular in closed form and another having the form of either a rapidly converging eigenfunctions expansion or a computationally efficient diffraction integral. The obtained integral equation is discretised in the framework of the Nystrom method using rapidly converging quadratures that take into account both the singularities of the kernels and the geometrical singularities at the edges. Numerical examples and case studies demonstrate the high accuracy and efficiency of the proposed algorithms. The analysis covers the canonical problem of a parallel-plate waveguide antenna backed on a perfectly conducting wedge as a special case.

The Asymptotic Spectrum of LOCC Transformations

We study the exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability, known as the converse error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterization of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1] , where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent, the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case, we prove that this bound holds with equality.

Two asymptotic approaches for the exponential signal and harmonic noise in Singular Spectrum Analysis

The general theoretical approach to the asymptotic extraction of the signal series from the perturbed signal with the help of Singular Spectrum Analysis (briefly, SSA) was already outlined in Nekrutkin 2010, SII, v. 3, 297--319. In this paper we consider the example of such an analysis applied to the increasing exponential signal and the sinusoidal noise. It is proved that if the signal rapidly tends to infinity, then the so-called reconstruction errors of SSA do not uniformly tend to zero as the series length tends to infinity. More precisely, in this case any finite number of last terms of the error series do not tend to any finite or infinite values. On the contrary, for the scheme with the bounded from above exponential signal, all elements of the error series tend to zero. This effect shows that the discretization model can be an effective tool in the theoretical SSA considerations with increasing signals.

Evaluation of Efficient Green’s Functions for Spherically Stratified Media

An efficient procedure for computing Green’s function for an arbitrary directed magnetic point source radiating inside a spherically stratified media is presented. The singularity related to the spatial co-location of the field and source points is effectively treated without asymptotic extraction. In addition, the source-free resonant frequency of the antenna modes can also be efficiently computed from the formulation. The incorporation of a new source component or the addition of a new layer can be readily accommodated in the formulation without the recomputation of Green’s function unknowns. The technique is applied to the analysis of several spherical multilayer antenna configurations.

Asymptotic extraction of common signal subspaces from perturbed signals

Asymptotic extraction of common signal subspaces from perturbed signals

A novel model for multicomponent supercritical fluid extraction and its application to Ruta graveolens

In this work, a multicomponent model for understanding the relationship between the chemical compositions of an extract and extracted plant is developed and presented for the first time. Its robustness and efficiency were validated on the example of furanocoumarins and waxes extracted from aerial parts of common rue with supercritical CO2 at pressures and temperatures (12–30)MPa and (313–333)K. For the purpose of modeling, the waxes are represented by n-tritriacontane and furanocoumarins – by xanthotoxin, the solubility of which in CO2 was measured using the dynamic method. A rigorous thermodynamic modeling framework was applied to calculate the composition-dependent solubilities of extracted compounds, which were incorporated in the kinetics model in order to calculate the extraction curves. For individual extraction curves, the average absolute deviation between the calculated and experimental extraction yields, expressed as percentage of the asymptotic extraction yield, was in the range (0.2–5.8) %.

Convergent Expressions for Periodic Potentials in Stratified Media Using Asymptotic Extractions

The efficient computation of periodic Green’s functions is discussed here for an arbitrarily directed array of point sources in layered media. These Green’s functions are necessary to formulate boundary integral equations for arrays of scatterers inside a general layered medium, solved with the method of moments in the spatial domain. For this reason, mixed-potential Green’s functions—having a mild spatial singularity—are selected. The case of horizontally oriented dipoles is rather simple and has been previously solved. On the other hand, the case of vertically oriented dipoles (i.e., aligned perpendicular to the layers) is more intricate, since the extracted terms cannot be transformed into well-known Green’s functions. Previous works dealt with arrays of line and point sources, but did not address the critical task of computing the curl of the dyadic potentials, required to treat slot arrays and dielectric inclusions, whose available Floquet series expressions do not converge if the source and observation points lie in the same transverse plane.

Efficient Computation of 1-D Periodic Layered Mixed Potentials for the Analysis of Leaky-Wave Antennas With Vertical Elements

An efficient mixed-potential integral equation formulation is proposed for the analysis of one-dimensional (1-D) periodic leaky-wave antennas (LWAs) based on planar stratified configurations with inclusions of arbitrarily oriented metallic or dielectric perturbations. Both the transverse and vertical components of the mixed-potential Green’s functions due to a 1-D phased array of dipoles in a layered medium are accelerated using suitable homogeneous-medium asymptotic extractions from the standard spectral series of Floquet harmonics. A novel acceleration procedure is applied for the computation of the vertical potentials whose extracted terms can be expressed as potentials from a 1-D phased array of half-line sources in a homogeneous medium. Their numerical calculation requires a suitable modification of the Ewald method, thus resulting in new modified spectral and spatial series, having Gaussian convergence even in the case of complex modes and improper harmonics. Numerical comparisons for the 1-D periodic potentials, both in the case of bounded and unbounded (e.g., leaky) harmonics, validate the efficiency and accuracy of the proposed acceleration technique. The method is illustrated and verified by determining the dispersion behavior of both bound and leaky modes for several LWA test cases.

Calculation of the Impedance of a Rail Track With Earth Return for the High-Speed Railway Signal Circuit Using Finite-Element Method

The impedance of the rail track is one of basic parameters to analyze the high-speed railway signal circuit. In this paper, an accurate and fast approach to calculate the impedance of the rail track with the earth return is presented. Two imagined current-carrying sheaths going in opposite directions are placed to encase the rail track. With these imaginary currents, the impedance can actually be divided into imagined internal and external impedance, respectively. The imagined internal impedance can be calculated using the finite-element method without numerical difficulties, and the imagined external one can be evaluated using the analytical formula with the help of the asymptotic extraction technique. The numerical examples validate that the proposed approach and show the proposed approach has a satisfactory accuracy and reasonable computation time.

Moment Method Analysis of a Conformal Curl Antenna Printed Within Layered Dielectric Cylindrical Media

A curl antenna printed within a grounded cylindrical dielectric substrate is analyzed efficiently using the method of moments (MoM). The antenna consists of a single Archimedean spiral arm printed on the cylindrical surface which is fed through the substrate using a radial probe, and a rigorous model is developed for this feeding probe. The cylindrical media is modeled using dyadic Green's functions (DGF), where a novel asymptotic extraction approach and the generalized pencil of functions (GPOF) method are employed to expedite the analysis. Computations are compared with those of antennas in equivalent planar structures, as well as with the results of commercial software, and good agreement is achieved in all cases.

Quasi 3-D Mixed Potential Integral Equation Formulation for Periodic Structures

This paper introduces the use of the quasi 3-D mixed potential integral equation formulation for structures with a periodicity in two dimensions. The quasi 3-D formulation allows to use both horizontal and vertical conductors, and dielectric inclusions. The key innovation is the use of an effective algorithm to calculate Periodic Green's Functions (PGFs) for vertical currents in the planar multilayered background medium. This algorithm retains the same scheme as for a single source Green's Function (GF). In both cases, the GFs are calculated via an inverse Fourier transform of the GFs in the spectral domain, which are known in closed form. In order to remedy the poor convergence for small source—observation distances, an asymptotic extraction is used, yielding a very compact algorithm, and acceleration techniques are applied. Simple guidelines are proposed to construct asymptotes having the required leading terms and a good convergence. It is shown that many acceleration algorithms widely used in literature are actually able to perform well only in specific circumstances. As far as the authors can see, there is still no general and robust technique, yielding highly accurate results under all circumstances. Several examples illustrate the efficiency of the technique.

Full-Wave Analysis of Microstrip Antennas in Three-Layered Spherical Media

A model of three-layered spherical microstrip antenna has been analyzed based on Rao-Wilton-Glisson (RWG) triangular basis functions using mixed potential integral equation (MPIE). Firstly, the model of antenna and the dyadic Green’s function in spherical microstrip antennas are given at the beginning of this paper. Then, due to the infinite series convergence problem, asymptotic extraction approach is presented to accelerate the Green’s functions convergence speed when source and field points are located in the same layer and different layers. The convergence speed can be accelerated observably by using this method. Finally, in order to simplify impedance matrix elements calculation at the junction of the probe and patch, a novel division fashion of pair of triangles is adopted in this paper. The input impedance result obtained shows the validity and effectiveness of the analysis method comparing with published data.

An enhanced integral-equation formulation for accurate analysis of frequency-selective structures

In this work, a very efficient mixed-potential integral-equation formulation is implemented for the rigorous analysis of multilayered structures with arbitrarily shaped two-dimensional periodic metallic and/or dielectric inclusions. Original acceleration techniques have been developed for the computation of the components of the scalar and dyadic Green's functions, based on different types of asymptotic extractions according to the potential considered. The theoretical approach and its computational convenience have been validated through different full-wave analyses concerning both scattering problems and complex-mode dispersive behaviors in various frequency-selective structures for microwave applications.

Measurements and Analysis of a Probe-Fed Circularly Polarized Loop Antenna Printed on a Layered Dielectric Sphere

Radiation characteristics of a probe-fed open circular loop antenna are presented when the antenna is printed on a layered dielectric sphere. A smaller parasitic circular loop has been incorporated in the configuration to enhance the circular polarization (CP) bandwidth. The structure has been analyzed rigorously using the method of moments (MoM), where the sphere has been represented using the spherical dyadic Green's function (DGF). The computation efficiency has been enhanced using asymptotic extraction as well as closed form integrations to compute the MoM impedance matrix elements. The computed results are in close agreement with those obtained from measurements.

Calculation of the Magnetic Field in Air Produced by the Underground Conductor Using Pollaczek Integral

A fast approach for evaluating the magnetic field in air generated by an underground conductor is proposed in this paper. By means of the asymptotic extraction technique, the magnetic field can be decomposed into two parts. One is the contribution from the asymptotic integrand, which can be calculated analytically; and the other can be expressed by a sum of the definite integral and the exponential integrals. Due to the fact that there is no improper integral involved, the approach proposed here is highly efficient. Meanwhile, the impact of the eddy current to the magnetic field is quantitatively analyzed. The proposed approach could be used in the electromagnetic environment assessment for underground power cable systems.