Space-frequency Domain Research Articles

Comparison of topographic mapping and conventional EEG in acute stroke

In MEG source localization the estimated source parameters will be more reliable when the spatiotemporal covariance of the noise and background activity is taken into account. Since this covariance is in general too large to estimate based on the data and to invert efficiently, different parametrizations have been proposed in the literature. These models can be seen as special cases of the general decomposition of the covariance into a sum of Kronecker products of spatial matrices Xn and temporal matrices Tn (Van Loan, 2000).In this study we investigate the assumption of the matrices Tn being Toeplitz. If so, the covariance matrix in the space–frequency domain will have an approximate block-diagonal structure, facilitating inversion, which is a prerequisite for source localization. In this study we address the question whether the Toeplitz approximation is valid for data sets obtained in visual evoked field, auditory evoked field, somatosensory evoked field experiments and data sets containing spontaneous activity. It turns out that on average 87% is in the block-diagonal of the sample covariance, which is close to the values obtained for real Toeplitz matrices Tn. This implies that the space–frequency domain is very interesting for source localization since the major part of the entire covariance can be incorporated in that domain straightforwardly. Finally, the two major processes in the background activity are characterized in terms of their spatial and frequency patterns, yielding a focal and a non-focal pattern in 8 of 10 data sets analyzed in this study. The focal pattern represents the alpha frequency at parieto-occipital areas, whereas the non-focal pattern is more widespread both in space and in frequency.

Exact subband image decomposition and reconstruction in discrete space and discrete frequency domains

This work extends previously reported work on image multi-subband decomposition/reconstruction using DFT. The purpose of the method proposed here is to achieve exact decomposition and reconstruction using ideal band-pass filters implemented in the DFT domain without any excess data.

Coherence of Light at the Output of a Fibre Waveguide Analysed in the Space-Frequency Domain

Abstract Coherence of light in the space-frequency domain is investigated at the exit face of a weakly guiding fibre waveguide as well as in its far zone by using the coherent mode representation. Two limiting cases of fibre waveguide excitation by cross-spectrally pure, quasimonochromatic, spatially incoherent and coherent sources are considered in the evaluation of spectral changes occurring in the free space propagation of light in the paraxial regime from the exit face of the fibre waveguide to its far zone. In particular, the blue and the red shifts as well as the spectra modifications in the far zone of two-mode, weakly guiding, step-index fibre waveguide are presented for both spatially incoherent and coherent exciting sources. A frequency dependence of the modes is also considered which leads to a substantial reduction in spectral shifts. Moreover, it is revealed that a variable spectral modulation is present in the case of the fibre waveguide excitation by a spatially coherent source.

Relationships between the complex degrees of coherence in the space–time and in the space–frequency domains

A relationship is established between two correlation coefficients (complex degrees of coherence) frequently used in the theory of partially coherent optical fields. One of them characterizes correlation in the space-time domain, the other in the space-frequency domain. Some relations obtained previously by ad hoc methods are shown to readily follow from our general formula.

Generalization of the k-space formulation to elastodynamic scattering problems

A generalized k-space (GKS) formulation is presented for vectorial elastodynamic scattering problems. It represents a generalization of Bojarski’s scalar k-space formulation. From the basic second-order partial differential equation or its integral representation in the space-frequency (r-f) domain, a local equation is derived for the displacement field in the spectral-frequency (k-f) domain. This equation, together with the constitutive equation in the r-f domain, reduces the original scattering problem into two simultaneous local equations with two unknowns (displacement field and the induced source), which are then solved by the conjugate-gradient (CG) method. The connection between the k-f domain and r-f domain is obtained by the spatial fast Fourier transform (FFT) algorithm. The number of complex multiply-add operations per CG iteration is O(N log2 N), and the storage requirement is only O(N), where N is the number of spatial discretization points. This is much more efficient than the conventional method of moment combined with the CG procedures which requires O(N2) operations per iteration and O(N2) storage. In the spectral-time (k-t) domain, however, it is found that four simultaneous local equations have to be used to formulate the k-space algorithm because of the existence of two wave speeds. By virtue of the causality, a new local time-stepping algorithm is derived with the aid of two temporal propagators, i.e., the compressional and shear propagators. The connection between the r-t domain and k-t domain is again obtained by the spatial FFT algorithm. Therefore, in each time step, the number of complex multiply–add operations is O(N log2 N), and the storage requirement is O(N). Most importantly, for the same accuracy, N can be much smaller than that for the conventional finite-difference method.

Zero-phonon line as the foundation stone of high-resolution matrix spectroscopy, persistent spectral hole burning, single impurity molecule spectroscopy

The purely electronic zero-phonon line (ZPL) - the optical analog of the Mössbauer-γ-resonance line - is a remarkable feature of the low temperature impurity absorption and luminescence spectra of solids. The narrowness and high peak absorption cross section of ZPLs has opened new horizons for high resolution spectroscopy of impurity molecules (centers). The newest and most spectacular achievement is single impurity molecule spectroscopy. Persistent spectral hole burning (PSHB), serving as a tool to control by means of illumination the absorption coefficient, and bound to it, via the Kramers-Kronig relations, the refraction index, turns inhomogeneous broadening of ZPLs into a useful feature and opens a variety of applications including PSHB optical data storage and processing. Space-frequency and space-time domain holographies are developed. The latter comprises the causality relations between events subject to holography and opens new principal possibilities in optics.

Diffraction grating illuminated by partially coherent beam

A diffraction grating illuminated by partially coherent radiation has been investigated in the framework of scalar wave theory. The angular distribution of the light intensity behind the grating has been derived, using the formalism of correlation theory in the space-frequency domain. The source of radiation has been assumed to be quasi-homogeneous. In contrast to the coherent case, the angular dependence of the intensity differs, in general, from the spectrum of the radiation. However, in most cases of practical interest this effect is negligibly small.

Spectral modifications of partially coherent light passing through a periodic sequence of lenses

Changes in the spectrum of light passing through a periodic sequence of lenses separated by Gaussian amplitude filters are investigated theoretically on the basis of the modal expansion of the field within the framework of coherence theory in the space-frequency domain. The initial light launched into the structure is assumed to be polychromatic, spatially incoherent light with a single spectral line. By considering generally that the initially incoherent light becomes partially coherent after its propagation through the structure and by referring to the recent studies on the correlation-induced spectral changes of the Wolf effect, we satisfactorily manifest the trends in the spectral modifications that are due to the light propagation. Representative results of the analysis are illustrated by some numerical examples. As a result, the shift of the peak spectrum on the light propagation is found to depend strongly on the position of the observation. In addition, the contribution of each mode to the total spectral changes is also discussed, mainly from the viewpoint of the Wolf effect.

On the evaluation of explicit 2-D extrapolation operators

The 2-D extrapolation operator in the wavenumber-frequency domain is expanded in a series of Chebychev polynomials. Fourier-type coefficients, depending on the extrapolation step, the frequency and the current velocity, are derived in terms of standard functions of mathematical physics. The inverse Fourier transform to the space-frequency domain then gives an analytical solution of the explicit operator, which renders the calculation of filter coefficients a routine matter. Realizable operators are designed by application of suitable spatial window functions. Migration of synthetic zero-offset data is used to illustrate the merits of the analysis.

Higher-order Coherence Functions in the Space-frequency Domain

Abstract Some years ago a new coherence theory in the space-frequency domain was formulated. Subsequent research has revealed some difficulties regarding the representation, in this theory, of the higher-order correlation functions. In the present paper the origin of these difficulties is discussed and a new representation of the higher-order correlation functions is introduced, in terms of the coherent modes of the second-order theory.

Transparent switching node for optical frequency division multiplexed signals

A laboratory 4 × 4 transparent node for the space frequency domain has been set up in which optical frequency division multiplexed (OFDM) signals at different bit rates and modulation schemes can be switched simultaneously. The feasibility, signal path architecture, supervision, control, and performance of this node are reported.

Difference in the Definitions of Coherence in the Space-time Domain and in the Space-frequency Domain

Abstract Coherence in the space-time domain and in the space-frequency domain are compared and found to be non-equivalent. Coherence in the space-time domain is sufficient for coherence in the space-frequency domain. Coherence in the space-frequency domain is not sufficient for coherence in the space-time domain. The source spectrum as well as coherence in the space-frequency domain effect coherence in the space-time domain. The coherence in the radiation field from two very different localized sources is studied. It is found that the fields are coherent throughout the far field region only if the sources are monochromatic.

PLANE-WAVE CONSTRAINTS IN 2D FILTER DESIGN1

A method is presented for developing and/or evaluating 2D filters applied to seismic data. The approach used is to express linear 2D filtering operations in the space-frequency (x–ω) domain. Correction filters are then determined using plane-wave constraints. For example, requiring a vertically propagating plane wave to be unaffected by migration necessitates application of a half-derivative correction in Kirchhoff migration. The same approach allows determination of the region of time-offset space where half-derivative corrections are correct in x–t domain dip moveout. Finally, an x–ω domain dip filter is derived using the constraint that a plane wave be attenuated as its dip increases. This filter has the advantage that it is significantly faster than f–k domain dip filtering and can be used on irregularly spaced data. This latter property also allows the filter to be used for interpolation of irregular data onto a regular grid.

An approach to microwave imaging using a multiview moment method solution for a two-dimensional infinite cylinder

An approach based on a multiview solution to the inverse-scattering problem of a two-dimensional infinite cylinder is developed in a space-frequency domain. Microwave imaging is simulated by a computer algorithm using the moment method. To overcome ill-conditioning and solve nonsquare systems, a pseudoinverse transformation is employed. The equivalent current density and the complex conductivity are considered as object functions for image formation. The results of some numerical simulations in a noisy environment are reported. and a discussion of monoview and multiview imaging techniques for a space-frequency domain is presented. >

Spatial dealiasing of 3-D seismic reflection data

SUMMARY One of the main challenges of 3-D reflection seismology is providing the spatial sampling required to avoid aliasing. In practice, fine-scale and regular acquisition geometry is possible only in 2-D seismic surveys. 3-D surveys typically have sparse and irregular geometry that often results in spatial aliasing. In this paper we present a processing method to overcome this problem that is suitable for sparse and irregular acquisition geometry, and yields better results than does conventional processing such as dip moveout. The objective of the process is to obtain the hypothetical data that would be recorded in an adequately sampled zero-offset experiment, that best fits the data with a given velocity model. The process takes advantage of what seems to be redundancy in multifold seismic data, to overcome spatial aliasing which is a missing data problem. Our computer implementation involves pre-processing of gain, normal moveout removal, log-stretch transform and Fourier transform. After the log stretching, the relation between the data and the model is time invariant, making inversion practical in the space-frequency (X-F) domain, using a linear inversion, frequency by frequency. After the inversion, the data are inverse-Fourier and inverse-log transformed. The result is ready for post-stack migration.

Propagation invariance and self-imaging in variable-coherence optics

The concept of propagation invariance in partially coherent optics is introduced. Explicit expressions are given for the cross-spectral density and the angular correlation function (cross-angular spectrum) characterizing a class of fields that are propagation invariant in the sense that their correlation properties in the space-frequency domain are exactly the same in every transverse plane. The so-called diffraction-free beams are shown to be members of this new, wider class of wave fields, which itself is a subset of a generalized class of partially coherent self-imaging fields. The existence of partially coherent propagation-invariant fields with a sharp correlation peak is verified experimentally by considering radiation from a planar J0 Bessel-correlated source.

Limitations in recursive wavefield continuation with spatial wavelets (seismic reflection modeling)

Some problem that have to be avoided in wavefield continuation in the space-frequency domain are instability, inaccuracy, and poor steep-dip performance. The practical design in convolutional operators for one-way wavefield continuation in the space-frequency domain places constraints on the operator length and range of useful wavenumbers. These problems are inherent in the design of anti-aliased, finite-length continuation operators and their subsequent application in a recursive manner. The author discusses two major limiting factors anti-aliasing and aperture limiting. >

A spatial and spectral analysis of the electromagnetic sensitivity in a layered earth

Summary A sensitivity analysis of an electromagnetic (EM) technique can illustrate its theoretical resolution and is an essential step in using linearized inversion methods. Yet previous sensitivity studies in EM methods provide only a cursory examination of the complete, non-linear problem. We present a simplified analysis which illustrates the frequency characteristics and spatial variations of the EM sensitivity to buried layers in a stratified earth. The simplifications arise from ignoring the complicated geometrical effects of the sourcereceiver configuration and examining only the two fundamental response kernels which comprise the EM fields in a layered conductive earth. In this fashion, different EM prospecting methods can be compared and contrasted without dealing with specific aspects of survey methods. It is shown that although sensitivity is non-linearly related to conductivity, it is a qualitatively predictable function of frequency and source-receiver separation. Several general conclusions concerning electroprospecting can be derived in this manner. However, in all cases the realm (in space-frequency domain) of the maximum sensitivity in any layer is relatively small, suggesting that a diverse spatial and frequency sampling of the field is required in any high-resolution EM sounding survey.

Coherence theoretic algorithm to determine the transverse-mode structure of lasers

We introduce an algorithm to determine the relative modal weights of a laser beam with Hermite-Gaussian modes. The method is based on coherence theory of stable resonator modes in the space-frequency domain. Numerical simulations are presented that show that the algorithm is not sensitive to moderate levels of noise.

Optical coherence and Wolf’s theory for electromagnetic waves

Wolf’s theory [ J. Opt. Soc. Am.72, 343 ( 1982)] of coherence in the space–frequency domain is extended to the vector case, and propagation from sources in free space is described, using plane waves as basis states. The field is specified in terms of a correlation function given by the ensemble average of the basis-state amplitudes, and it is then shown how source radiation properties and a variety of traditional coherence functions may be calculated.