On the Hilbert transform in the framework of generalized functions

In this paper, a definition of the Hilbert transform operating on Colombeau's temperated generalized functions is given. Similar results to some theorems that hold in the classical theory, or in certain subspaces of Schwartz distributions, have been obtained in this framework.

Hilbert transform

The Hilbert transform is an important tool in both pure and applied mathematics. It is largely used in the field of signal processing. Lately has been used in mathematical finance as the fast Hilbert transform method is an efficient and accurate algorithm for pricing discretely monitored barrier and Bermudan style options. The purpose of this report is to show the basic properties of the Hilbert transform and to check the domain of definition of this operator.

A simple but efficient voice activity detection algorithm through Hilbert transform and dynamic threshold for speech pathologies

A simple but efficient voice activity detector based on the Hilbert transform and a dynamic threshold is presented to be used on the pre-processing of audio signals -- The algorithm to define the dynamic threshold is a modification of a convex combination found in literature -- This scheme allows the detection of prosodic and silence segments on a speech in presence of non-ideal conditions like a spectral overlapped noise -- The present work shows preliminary results over a database built with some political speech -- The tests were performed adding artificial noise to natural noises over the audio signals, and some algorithms are compared -- Results will be extrapolated to the field of adaptive filtering on monophonic signals and the analysis of speech pathologies on futures works

The Riesz transform and simultaneous representations of phase, energy and orientation in spatial vision

To represent the local orientation and energy of a 1-D image signal, many models of early visual processing employ bandpass quadrature filters, formed by combining the original signal with its Hilbert transform. However, representations capable of estimating an image signal's 2-D phase have been largely ignored. Here, we consider 2-D phase representations using a method based upon the Riesz transform. For spatial images there exist two Riesz transformed signals and one original signal from which orientation, phase and energy may be represented as a vector in 3-D signal space. We show that these image properties may be represented by a Singular Value Decomposition (SVD) of the higher-order derivatives of the original and the Riesz transformed signals. We further show that the expected responses of even and odd symmetric filters from the Riesz transform may be represented by a single signal autocorrelation function, which is beneficial in simplifying Bayesian computations for spatial orientation. Importantly, the Riesz transform allows one to weight linearly across orientation using both symmetric and asymmetric filters to account for some perceptual phase distortions observed in image signals - notably one's perception of edge structure within plaid patterns whose component gratings are either equal or unequal in contrast. Finally, exploiting the benefits that arise from the Riesz definition of local energy as a scalar quantity, we demonstrate the utility of Riesz signal representations in estimating the spatial orientation of second-order image signals. We conclude that the Riesz transform may be employed as a general tool for 2-D visual pattern recognition by its virtue of representing phase, orientation and energy as orthogonal signal quantities.

Falling sheet envelope method for non-destructive testing time-dependent signals

In this work, an algorithm to compute the envelope of non-destructive testing (NDT) signals is proposed. This method allows increasing the speed and reducing the memory in extensive data processing. Also, this procedure presents advantage of preserving the data information for physical modeling applications of time-dependent measurements. The algorithm is conceived to be applied for analyze data from non-destructive testing. The comparison between different envelope methods and the proposed method, applied to Magnetic Bark Signal (MBN), is studied. (C) 2010 Elsevier Ltd. All rights reserved.

Calculating current densities and fields due to shielded bi-planar radio-frequency coils

A method for the accurate computation of the current densities produced in a wide-runged bi-planar radio-frequency coil is presented. The device has applications in magnetic resonance imaging. There is a set of opposing primary rungs, symmetrically placed on parallel planes and a similar arrangement of rungs on two parallel planes surrounding the primary serves as a shield. Current densities induced in these primary and shielding rungs are calculated to a high degree of accuracy using an integral-equation approach, combined with the inverse finite Hilbert transform. Once these densities are known, accurate electrical and magnetic fields are then computed without difficulty. Some test results are shown. The method is so rapid that it can be incorporated into optimization software. Some preliminary fields produced from optimized coils are presented.

Incorporating phase retardation effects into radiofrequency resonator models: application to magnetic resonance microscopy

A method is presented for including path propagation effects into models of radiofrequency resonators for use in magnetic resonance imaging. The method is based on the use of Helmholtz retarded potentials and extends our previous work on current density models of resonators based on novel inverse finite Hilbert transform solutions to the requisite integral equations. Radiofrequency phase retardation effects are most pronounced at high field strengths (frequencies) as are static field perturbations due to the magnetic materials in the resonators themselves. Both of these effects are investigated and a novel resonator structure presented for use in magnetic resonance microscopy.

Calculating current densities and fields produced by shielded magnetic resonance imaging probes

A method is presented for computing the fields produced by radio frequency probes of the type used in magnetic resonance imaging. The effects of surrounding the probe with a shielding coil, intended to eliminate stray fields produced outside the probe, are included. An essential feature of these devices is the fact that the conducting rungs of the probe are of finite width relative to the coil radius, and it is therefore necessary to find the distribution of current within the conductors as part of the solution process. This is done here using a numerical method based on the inverse finite Hilbert transform, applied iteratively to the entire structure including its shielding coils. It is observed that the fields are influenced substantially by the width of the conducting rungs of the probe, since induced eddy currents within the rungs become more pronounced as their width is increased. The shield is also shown to have a significant effect on both the primary current density and the resultant fields. Quality factors are computed for these probes and compared with values measured experimentally.

Comparative analysis of time-frequency methods estimating the time-varying microstructure of sleep EEG spindles

Proceedings of the Information Technology Applications in Biomedicine, Ioannina - Epirus, Greece, October 26-28, 2006

Cross-frequency coupling in EEG revealed by adaptive oscillation tracking.

Introduction: Neuronal oscillations have been the focus of increasing interest in the neuroscientific community, in part because they have been considered as a possible integrating mechanism through which internal states can influence stimulus processing in a top-down way (Engel et al., 2001). Moreover, increasing evidence indicates that oscillations in different frequency bands interact with one other through coupling mechanisms (Jensen and Colgin, 2007). The existence and the importance of these cross-frequency couplings during various tasks have been verified by recent studies (Canolty et al., 2006; Lakatos et al., 2007). In this study, we measure the strength and directionality of two types of couplings - phase-amplitude couplings and phase-phase couplings - between various bands in EEG data recorded during an illusory contour experiment that were identified using a recently-proposed adaptive frequency tracking algorithm (Van Zaen et al., 2010). Methods: The data used in this study have been taken from a previously published study examining the spatiotemporal mechanisms of illusory contour processing (Murray et al., 2002). The EEG in the present study were from a subset of nine subjects. Each stimulus was composed of 'pac-man' inducers presented in two orientations: IC, when an illusory contour was present, and NC, when no contour could be detected. The signals recorded by the electrodes P2, P4, P6, PO4 and PO6 were averaged, and filtered into the following bands: 4-8Hz, 8-12Hz, 15-25Hz, 35-45Hz, 45-55Hz, 55-65Hz and 65-75Hz. An adaptive frequency tracking algorithm (Van Zaen et al., 2010) was then applied in each band in order to extract the main oscillation and estimate its frequency. This additional step ensures that clean phase information is obtained when taking the Hilbert transform. The frequency estimated by the tracker was averaged over sliding windows and then used to compare the two conditions. Two types of cross-frequency couplings were considered: phase-amplitude couplings and phase-phase couplings. Both types were measured with the phase locking value (PLV, Lachaux et al., 1999) over sliding windows. The phase-amplitude couplings were computed with the phase of the low frequency oscillation and the phase of the amplitude of the high frequency one. Different coupling coefficients were used when measuring phase-phase couplings in order to estimate different m:n synchronizations (4:3, 3:2, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, 8:1 and 9:1) and to take into account the frequency differences across bands. Moreover, the direction of coupling was estimated with a directionality index (Bahraminasab et al., 2008). Finally, the two conditions IC and NC were compared with ANOVAs with 'subject' as a random effect and 'condition' as a fixed effect. Before computing the statistical tests, the PLV values were transformed into approximately normal variables (Penny et al., 2008). Results: When comparing the mean estimated frequency across conditions, a significant difference was found only in the 4-8Hz band, such that the frequency within this band was significantly higher for IC than NC stimuli starting at ~250ms post-stimulus onset (Fig. 1; solid line shows IC and dashed line NC). Significant differences in phase-amplitude couplings were obtained only when the 4-8 Hz band was taken as the low frequency band. Moreover, in all significant situations, the coupling strength is higher for the NC than IC condition. An example of significant difference between conditions is shown in Fig. 2 for the phase-amplitude coupling between the 4-8Hz and 55-65Hz bands (p-value in top panel and mean PLV values in the bottom panel). A decrease in coupling strength was observed shortly after stimulus onset for both conditions and was greater for the condition IC. This phenomenon was observed with all other frequency bands. The results obtained for the phase-phase couplings were more complex. As for the phase-amplitude couplings, all significant differences were obtained when the 4-8Hz band was considered as the low frequency band. The stimulus condition exhibiting the higher coupling strength depended on the ratio of the coupling coefficients. When this ratio was small, the IC condition exhibited the higher phase-phase coupling strength. When this ratio was large, the NC condition exhibited the higher coupling strength. Fig. 3 shows the phase-phase couplings between the 4-8Hz and 35-45Hz bands for the coupling coefficient 6:1, and the coupling strength was significantly higher for the IC than NC condition. By contrast, for the coupling coefficient 9:1 the NC condition gave the higher coupling strength (Fig. 4). Control analyses verified that it is not a consequence of the frequency difference between the two conditions in the 4-8Hz band. The directionality measures indicated a transfer of information from the low frequency components towards the high frequency ones. Conclusions: Adaptive tracking is a feasible method for EEG analyses, revealing information both about stimulus-related differences and coupling patterns across frequencies. Theta oscillations play a central role in illusory shape processing and more generally in visual processing. The presence vs. absence of illusory shapes was paralleled by faster theta oscillations. Phase-amplitude couplings were decreased more for IC than NC and might be due to a resetting mechanism. The complex patterns in phase-phase coupling between theta and beta/gamma suggest that the contribution of these oscillations to visual binding and stimulus processing are not as straightforward as conventionally held. Causality analyses further suggest that theta oscillations drive beta/gamma oscillations (see also Schroeder and Lakatos, 2009). The present findings highlight the need for applying more sophisticated signal analyses in order to establish a fuller understanding of the functional role of neural oscillations.

The response of a uniform horizontal temperature gradient to heating

The response of a uniform horizontal temperature gradient to prescribed fixed heating is calculated in the context of an extended version of surface quasigeostrophic dynamics. It is found that for zero mean surface flow and weak cross-gradient structure the prescribed heating induces a mean temperature anomaly proportional to the spatial Hilbert transform of the heating. The interior potential vorticity generated by the heating enhances this surface response. The time-varying part is independent of the heating and satisfies the usual linearized surface quasigeostrophic dynamics. It is shown that the surface temperature tendency is a spatial Hilbert transform of the temperature anomaly itself. It then follows that the temperature anomaly is periodically modulated with a frequency proportional to the vertical wind shear. A strong local bound on wave energy is also found. Reanalysis diagnostics are presented that indicate consistency with key findings from this theory.

Unimodality of wave amplitude in the northern Hemisphere

A novel statistic for local wave amplitude of the 500-hPa geopotential height field is introduced. The statistic uses a Hilbert transform to define a longitudinal wave envelope and dynamical latitude weighting to define the latitudes of interest. Here it is used to detect the existence, or otherwise, of multimodality in its distribution function. The empirical distribution function for the 1960-2000 period is close to a Weibull distribution with shape parameters between 2 and 3. There is substantial interdecadal variability but no apparent local multimodality or bimodality. The zonally averaged wave amplitude, akin to the more usual wave amplitude index, is close to being normally distributed. This is consistent with the central limit theorem, which applies to the construction of the wave amplitude index. For the period 1960-70 it is found that there is apparent bimodality in this index. However, the different amplitudes are realized at different longitudes, so there is no bimodality at any single longitude. As a corollary, it is found that many commonly used statistics to detect multimodality in atmospheric fields potentially satisfy the assumptions underlying the central limit theorem and therefore can only show approximately normal distributions. The author concludes that these techniques may therefore be suboptimal to detect any multimodality.

Uma arquitetura para detecção online de transientes em sinais de eletrocardiograma sobre o protocolo IEEE 802.3 com PM-AH

The occurrence of transients in electrocardiogram (ECG) signals indicates an electrical phenomenon outside the heart. Thus, the identification of transients has been the most-used methodology in medical analysis since the invention of the electrocardiograph (device responsible for benchmarking of electrocardiogram signals). There are few papers related to this subject, which compels the creation of an architecture to do the pre-processing of this signal in order to identify transients. This paper proposes a method based on the signal energy of the Hilbert transform of electrocardiogram, being an alternative to methods based on morphology of the signal. This information will determine the creation of frames of the MP-HA protocol responsible for transmitting the ECG signals through an IEEE 802.3 network to a computing device. That, in turn, may perform a process to automatically sort the signal, or to present it to a doctor so that he can do the sorting manually

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Avaliação e implementação de métodos de estimação de tempo de atraso de sinais de ultra-som

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)