Power-law Power Spectral Density Research Articles

Simulation study on high spatio-temporal resolution acousto-electrophysiological neuroimaging.

Objective.Acousto-electrophysiological neuroimaging (AENI) is a technique hypothesized to record electrophysiological activity of the brain with millimeter spatial and sub-millisecond temporal resolution. This improvement is obtained by tagging areas with focused ultrasound (fUS). Due to mechanical vibration with respect to the measuring electrodes, the electrical activity of the marked region will be modulated onto the ultrasonic frequency. The region's electrical activity can subsequently be retrieved via demodulation of the measured signal. In this study, the feasibility of this hypothesized technique is tested.Approach.This is done by calculating the forward electroencephalography response under quasi-static assumptions. The head is simplified as a set of concentric spheres. Two sizes are evaluated representing human and mouse brains. Moreover, feasibility is assessed for wet and dry transcranial, and for cortically placed electrodes. The activity sources are modeled by dipoles, with their current intensity profile drawn from a power-law power spectral density.Results.It is shown that mechanical vibration modulates the endogenous activity onto the ultrasonic frequency. The signal strength depends non-linearly on the alignment between dipole orientation, vibration direction and recording point. The strongest signal is measured when these three dependencies are perfectly aligned. The signal strengths are in the pV-range for a dipole moment of 5 nAm and ultrasonic pressures within Food and Drug Administration (FDA)-limits. The endogenous activity can then be accurately reconstructed via demodulation. Two interference types are investigated: vibrational and static. Depending on the vibrational interference, it is shown that millimeter resolution signal detection is possible also for deep brain regions. Subsequently, successful demodulation depends on the static interference, that at MHz-range has to be sub-picovolt.Significance.Our results show that mechanical vibration is a possible underlying mechanism of acousto-electrophyisological neuroimaging. This paper is a first step towards improved understanding of the conditions under which AENI is feasible.

A search for planetary companions around 800 pulsars from the Jodrell Bank pulsar timing programme

ABSTRACT We have searched for planetary companions around 800 pulsars monitored at the Jodrell Bank Observatory, with both circular and eccentric orbits of periods between 20 d and 17 yr and inclination-dependent planetary masses from 10−4 to $100\, \mathrm{M}_{\oplus }$. Using a Bayesian framework, we simultaneously model pulsar timing parameters and a stationary noise process with a power-law power spectral density. We put limits on the projected masses of any planetary companions, which reach as low as 1/100th of the mass of the Moon ($\sim 10^{-4}\, \mathrm{M}_{\oplus }$). We find that two-thirds of our pulsars are highly unlikely to host any companions above $2-8\, \mathrm{M}_{\oplus }$. Our results imply that fewer than $0.5{{\ \rm per\ cent}}$ of pulsars could host terrestrial planets as large as those known to orbit PSR B1257+12 ($\sim 4\, \mathrm{M}_{\oplus }$); however, the smaller planet in this system ($\sim 0.02\, \mathrm{M}_{\oplus }$) would be undetectable in $95{{\ \rm per\ cent}}$ of our sample, hidden by both instrumental and intrinsic noise processes, although it is not clear whether such tiny planets could exist in isolation. We detect significant periodicities in 15 pulsars; however, we find that intrinsic quasi-periodic magnetospheric effects can mimic the influence of a planet, and for the majority of these cases we believe this to be the origin of the detected periodicity. Notably, we find that the highly periodic oscillations in PSR B0144+59 are correlated with changes in the pulse profile and therefore can be attributed to magnetospheric effects. We believe the most plausible candidate for planetary companions in our sample is PSR J2007+3120.

Generating fractal rough surfaces with the spectral representation method

The application of the spectral representation method in generating Gaussian and non-Gaussian fractal rough surfaces is studied in this work. The characteristics of fractal rough surfaces simulated by the spectral representation method and the conventional Fast Fourier transform filtering method are compared. Furthermore, the fractal rough surfaces simulated by these two methods are compared in the simulation of contact and lubrication problems. Next, the influence of low and high cutoff frequencies on the normality of the simulated Gaussian fractal rough surfaces is investigated with roll-off power spectral density and single power-law power spectral density. Finally, a simple approximation method to generate non-Gaussian fractal rough surfaces is proposed by combining the spectral representation method and the Johnson translator system. Based on the simulation results, the current work gives recommendations on using the spectral representation method and the Fast Fourier transform filtering method to generate fractal surfaces and suggestions on selecting the low cutoff frequency of the power-law power spectral density. Furthermore, the results show that the proposed approximation method can be a choice to generate non-Gaussian fractal surfaces when the accuracy requirements are not high. The MATLAB codes for generating Gaussian and non-Gaussian fractal rough surfaces are provided.

PKS 2155-304: A Case Study of Blazar Variability Power Spectrum at the Highest Energies and on the Longest Timescales

We present the results of our Power Spectral Density (PSD) analysis for the BL Lac object PKS 2155−304, utilizing the nightly-binned long-term light curve from the decade-long monitoring, as well as the minute-binned intra-night light curve from the High Energy Stereoscopic Survey (H.E.S.S.; >200 GeV). The source is unique for exhibiting the shortest flux-doubling timescale at Very High Energy (VHE) among its class and thus provides a rare opportunity to study the particle acceleration on the smallest spatial scales in blazar jets. The light curves are modeled in terms of the Continuous-Time Auto-Regressive Moving Average (CARMA) process. The combined long-term and intra-night PSD extends up to ∼6 decades in the temporal frequency range; unprecedented at the TeV energies for a blazar source. Our systematic approach reveals that PKS 2155−304 shows, on average, a complex shape of variability power spectrum, with more variability power on longer timescales. The long-term variability is best modeled by the CARMA(2,1) process, while the intra-night variability is modeled by a CARMA(1,0) process. We note that the CARMA(1,0) process refers to an Ornstein–Uhlenbeck process where the power-law PSD slope (PSD varies as a function of variability frequency to the power of the negative slope) changes from two to zero, above a certain “characteristic/relaxation” timescale. Even though the derived power spectrum of the intra-night light curve did not reveal a flattening, we speculate such relaxation must occur on timescales longer than a few hours for the source.

Measuring the Power Law Phase Noise of an RF Oscillator with a Novel Indirect Quantitative Scheme

In conventional phase noise metrology, the phase noise of an oscillator is measured by instruments equipped with specialized and sophisticated devices. Such hardware-based testing usually requires high-performance and costly apparatuses. In this paper, we carried out a novel phase noise measurement method based on a mathematical model. The relationship between the phase noise of a radio frequency oscillator and its power spectral density (PSD) was established, different components of the power law phase noise were analyzed in the frequency domain with their characteristic parameters. Based on the complete physical model of an oscillator, we fitted and extracted the parameters for the near-carrier Gaussian and the power law PSD with Levenberg-Marquardt optimization algorithm. The fitted parameters were used to restore the power law phase noise with considerable precision. Experimental validation showed an excellent agreement between the estimation from the proposed method and the data measured by a high-performance commercial instrument. This methodology can be potentially used to realize fast and simple phase noise measurement and reduce the overall cost of hardware.

Variability-selected Low-luminosity Active Galactic Nuclei Candidates in the 7 Ms Chandra Deep Field-South

Abstract In deep X-ray surveys, active galactic nuclei (AGNs) with a broad range of luminosities have been identified. However, cosmologically distant low-luminosity AGN (LLAGN, erg s−1) identification still poses a challenge because of significant contamination from host galaxies. Based on the 7 Ms Chandra Deep Field-South (CDF-S) survey, the longest-timescale (∼17 years) deep X-ray survey to date, we utilize an X-ray variability selection technique to search for LLAGNs that remain unidentified among the CDF-S X-ray sources. We find 13 variable sources from 110 unclassified CDF-S X-ray sources. Except for one source that could be an ultraluminous X-ray source, the variability of the remaining 12 sources is most likely due to accreting supermassive black holes. These 12 AGN candidates have low intrinsic X-ray luminosities, with a median value of 7 × 1040 erg s−1. They are generally not heavily obscured, with an average effective power-law photon index of 1.8. The fraction of variable AGNs in the CDF-S is independent of X-ray luminosity and is only restricted by the total number of observed net counts, confirming previous findings that X-ray variability is a near-ubiquitous property of AGNs over a wide range of luminosities. There is an anticorrelation between X-ray luminosity and variability amplitude for high-luminosity AGNs, but as the luminosity drops to ≲1042 erg s−1, the variability amplitude no longer appears dependent on the luminosity. The entire observed luminosity–variability trend can be roughly reproduced by an empirical AGN variability model based on a broken power-law power spectral density function.

Focus Meeting #3: Radio Galaxies – Resolving the AGN Phenomenon

In this presentation, we report the results of intraday variability in the optical (BVRI bands) and hard X-ray band (3-79 keV) in a number of blazars. In the optical microvariability studies of the blazars S5 0716+714 and BL Lac, we observed many interesting features such as rapid variability, large variability amplitude, presence of characteristic timescales, bluer-when-brighter achromatic behavior, and single power-law power spectral density. In {\it NuSTAR} observations of several blazars, using spectral and timing analysis, we found similar features consistent with the optical studies. In addition, in BL Lacs we estimated the Lorentz factor of the population of highest energy electrons emitting synchrotron emission, and whereas in flat-spectrum radio quasars, using external Compton models, we estimated the energy of the lower end of the injected electrons to be a few tens of Lorentz factors. In addition, we find that the low flux state exhibit more rapid variability in contrast to the previously reported results showing high flux states displaying rapid variability. In both the studies, the size of the emission regions estimated using variability timescales turn out to be an order magnitude smaller than the gravitational radius of a typical black-hole masses between $\sim 10^8-10^9$ solar masses, believed to be at the center of the radio-loud AGN. The results of the studies suggest that these low-amplitude rapid variability might originate as a result of magnetohydrodynamical instabilities near the base of the jets triggered by the processes modulated by the magnetic field at accretion disc.

Topological characterization of antireflective and hydrophobic rough surfaces: are random process theory and fractal modeling applicable?

The random process theory (RPT) has been widely applied to predict the joint probability distribution functions (PDFs) of asperity heights and curvatures of rough surfaces. A check of the predictions of RPT against the actual statistics of numerically generated random fractal surfaces and of real rough surfaces has been only partially undertaken. The present experimental and numerical study provides a deep critical comparison on this matter, providing some insight into the capabilities and limitations in applying RPT and fractal modeling to antireflective and hydrophobic rough surfaces, two important types of textured surfaces. A multi-resolution experimental campaign using a confocal profilometer with different lenses is carried out and a comprehensive software for the statistical description of rough surfaces is developed. It is found that the topology of the analyzed textured surfaces cannot be fully described according to RPT and fractal modeling. The following complexities emerge: (i) the presence of cut-offs or bi-fractality in the power-law power-spectral density (PSD) functions; (ii) a more pronounced shift of the PSD by changing resolution as compared to what was expected from fractal modeling; (iii) inaccuracy of the RPT in describing the joint PDFs of asperity heights and curvatures of textured surfaces; (iv) lack of resolution-invariance of joint PDFs of textured surfaces in case of special surface treatments, not accounted for by fractal modeling.

A method for the estimation of the significance of cross-correlations in unevenly sampled red-noise time series

We present a practical implementation of a Monte Carlo method to estimate the significance of cross-correlations in unevenly sampled time series of data, whose statistical properties are modelled with a simple power-law power spectral density. This implementation builds on published methods; we introduce a number of improvements in the normalization of the cross-correlation function estimate and a bootstrap method for estimating the significance of the cross-correlations. A closely related matter is the estimation of a model for the light curves, which is critical for the significance estimates. We present a graphical and quantitative demonstration that uses simulations to show how common it is to get high cross-correlations for unrelated light curves with steep power spectral densities. This demonstration highlights the dangers of interpreting them as signs of a physical connection. We show that by using interpolation and the Hanning sampling window function we are able to reduce the effects of red-noise leakage and to recover steep simple power-law power spectral densities. We also introduce the use of a Neyman construction for the estimation of the errors in the power-law index of the power spectral density. This method provides a consistent way to estimate the significance of cross-correlations in unevenly sampled time series of data.

The spectral energy distribution of compact jets powered by internal shocks

Internal shocks caused by fluctuations of the outflow velocity are likely to power the radio to IR emission of the compact jets of X-ray binaries. The dynamics of internal shocks and the resulting spectral energy distribution (SED) of the jet are very sensitive to the time-scales and amplitudes of the velocity fluctuations injected at the base of the jet. I present a new code designed to simulate the synchrotron emission of a compact jet powered by internal shocks. I also develop a semi-analytical formalism allowing one to estimate the observed SED of the jet as a function of the Power Spectral Density (PSD) of the assumed fluctuations of the Lorentz factor. I discuss the cases of a sine modulation of the Lorentz factor and Lorentz factor fluctuations with a power-law PSD shape. Independently of the details of the model, the observed nearly flat SEDs are obtained for PSDs of Lorentz factor fluctuations that are close to a flicker noise spectrum (i.e. P(f)~1/f). The model also presents a strong wavelength dependent variability that is similar to that observed in these sources.

Efficient estimation of generator instability characteristics in the time domain

This paper considers the problem of determining the true (nonsampling) variance of the Allan parameter-the main frequency instability characteristic of frequency generators, synthesizer, and standards in the time domain. The variables influencing the efficiency of estimating the parameter are determined and compared in magnitude. For frequency fluctuations with flicker noise power-law power spectral density, the parameter estimate accuracy is most significantly affected by the variance of the estimated average frequency—the quantity included in the Allan parameter and defining the quality of averaging over a finite time interval.

Stretched exponential relaxation and ac universality in disordered dielectrics

This paper is concerned with the connection between the properties of dielectric relaxation and alternating-current (ac) conduction in disordered dielectrics. The discussion is divided between the classical linear-response theory and a self-consistent dynamical modeling. The key issues are stretched exponential character of dielectric relaxation, power-law power spectral density, and anomalous dependence of ac conduction coefficient on frequency. We propose a self-consistent model of dielectric relaxation in which the relaxations are described by a stretched exponential decay function. Mathematically, our study refers to the expanding area of fractional calculus and we propose a systematic derivation of the fractional relaxation and fractional diffusion equations from the property of ac universality.

Mathematical Modeling and Computer-Aided Manufacturing of Rough Surfaces for Experimental Study of Seafloor Scattering

Diverse aspects of stochastic processes, time-series analysis, fractal geometry, and manufacturing technology are brought together to provide a unified theoretical framework for the mathematical characterization and physical fabrication of scale-model representations of the rough ocean bottom. These scale models can be used to validate the predictions of interface-scattering theories, particularly those relating to the dependence of scattering strength on roughness parameters. In acoustical oceanography, rough surfaces are conventionally described in terms of power-spectral density (PSD) or fractal dimension. Here, recently developed concepts describing modified power-law PSD and approximately self-affine stochastic fractals are used to account for issues of finite sample size and finite manufacturing resolution. The large- and small- bandwidth restrictions that these issues inherently impose are related to their effects on the properties of surfaces as characterized both mathematically and by visual observation. This development provides the groundwork for numerical generation of surfaces, using spectral methods, and their physical manufacture, using computer-aided manufacturing (CAM) techniques. Effects of the spectral method of numerical generation on the statistics of generated topography are detailed. A manufacturing technique using a computer numerically controlled (CNC) milling machine is discussed and topography-specific guidelines for accurate manufacture of rough surfaces are prescribed.

The high frequency power spectrum of Markarian 766

An analysis is presented of the power spectrum of X-ray variability of the bright Seyfert 1 galaxy Mrk 766 as observed by XMM–Newton. Over the 0.2–10 keV energy range the power spectral density (PSD) is well-represented by a power-law with a slope of αlow≈ 1 at low frequencies, breaking to a slope of αhi= 2.8+0.2−0.4 at a frequency fbr≈ 5 × 10−4 Hz. As has been noted before, this broken power-law PSD shape is similar to that observed in the Galactic black hole candidate Cygnus X-1. If it is assumed that Mrk 766 shows a power spectrum similar in form to that of Cyg X-1, and that the break time-scale scales linearly with black hole mass, then the mass of the black hole in Mrk 766 is inferred to be ≲ 5 × 105 M⊙. This rather low mass would mean Mrk 766 radiates above the Eddington limit. The coherence between different energy bands is significantly below unity implying that variations in the different energy bands are rather poorly correlated. The low coherence can be explained in the framework of standard Comptonization models if the properties of the Comptonizing medium are rapidly variable or if there are several distinct emission sites.

Fractals, wavelets, and stochastic interface modeling

The seafloor is one of many natural interfaces which, when viewed as a random process, exhibits a power law decay, i.e., the power spectral density (PSD) decays as A/fb, A>0, b>0 as f→∞. Power law spectra are usually taken as evidence that the stochastic process is multiscale or fractal. Multiscale interfaces exhibit features at all scales and their acoustical properties are distinct from those of single-scale interfaces, which contain features closely clustered about a mean size. Thus, determination of the scale structure, or size distribution of component features, is important in characterizing random interfaces for acoustical applications. However, because the PSD is only a second moment characterization, a power law PSD can allow both single-scale and multiscale processes. Only if the process is Gaussian is the PSD a complete description. Wavelet representations succeed where Fourier methods fail, as they are ideally suited for determining scale structure. In the case of fractal interfaces, the wavelet coefficients are Gaussian and independent across scale and space. Single-scale interfaces lead to a dependence of wavelet coefficients across scale for a given spatial location and vice-versa.