Ensemble Average Power Spectrum Research Articles

Enhancing the variability of simulated random processes using the spectral representation method

In this paper, the traditional spectral representation method for simulating stochastic processes is revisited. The main goal is to introduce various approaches to enhance the variability of the generated sample functions to better reflect variabilities observed in field measurements and experimental data, while keeping the ensemble-averaged power spectra unchanged. The general idea, gradually developed over the last decade, is to randomize the power spectral density function of the uncertain physical quantities by modeling certain parameters as random variables calibrated from experimental data. The potential of such approaches has not been fully explored yet. This work explores the pros and cons of using power spectral density functions with random parameters in the simulation of highly variable, stationary and nonstationary, Gaussian stochastic processes. The main characteristics of a stochastic process, such as distribution of maxima and variability in time and frequency, are examined. The use of a pass-band Butterworth filter to control the variability of the Fourier spectrum of the simulated samples is studied. The Butterworth filter is selected so that the ensemble-averaged power spectral density matches the target power spectrum, but each individual generated sample function possesses different amplitude and frequency distributions. Comparisons between samples generated using the traditional spectral representation method and those generated with enhanced variability are made through a series of illustrative examples. Results show that with the proposed enhanced-variability spectral representation method, certain characteristics of the stochastic process, such as the distribution of maxima, will be significantly different (while the ensemble-averaged power spectra will remain the same). The implications in reliability studies are obvious.

Numerical simulations to study nonlinear wave-based interaction and turbulent magnetic field amplification in the laboratory and astrophysical plasmas

A theoretical model is proposed to study the interaction of high-frequency oblique whistler wave (OWW) and low-frequency kinetic Alfvén waves (KAWs). In our previous investigation (Dewan et al 2020 Phys. Plasmas 27 032111), we established the existence of these two plasma modes (OWW and KAW) by virtue of parametric decay instability. For the present investigation, a numerical simulation technique is employed to solve the coupled system of equations. The ponderomotive force exerted by OWW excites the low-frequency KAW. This quasi-static force induces the genesis of density cavitation (regions of accumulation and depletion) in low-frequency KAW in the magnetized plasma. The simulation results give the localization of the OWW, which amplifies with time. At a later time, the structures become chaotic. The energy cascade is presented in terms of the ensemble-averaged power spectrum. We have also developed a semi-analytical model for this wave–wave interaction mechanism to understand the underlying physics of the field localization process.

Compression of the laser pulse in magnetized plasma having relativistic regime

A study of the interaction of laser with magnetized plasma in the relativistic regime has been presented. Using Maxwell equations, a modified nonlinear Schrodinger equation (MNLS) is derived for nonlinear propagation of the laser pulse in the presence of a magnetic field (where the propagation of laser pulse is along magnetic field). Using the pseudospectral and finite difference methods the (MNLS) equation has been solved numerically for typical laser plasma parameters. It is found that the compression of the ultra-short laser pulse is enhanced by application of static magnetic field and the laser pulse is gradually compressed from 30 to 12fs. We have also observed the splitting of the laser pulse inside the plasma and the generation of magnetic turbulence. The ensemble averaged power spectrum of magnetic turbulence has been presented. Our results follow the spectral index approximation scales, of the order of k−2 at larger scale and k−4 at a smaller scale. Such a nonlinear interaction is quite important in understanding turbulence in the astrophysical phenomenon.

Curvature perturbations from stochastic particle production during inflation

We calculate the curvature power spectrum sourced by spectator fields that are excited repeatedly and non-adiabatically during inflation. In the absence of detailed information of the nature of spectator field interactions, we consider an ensemble of models with intervals between the repeated interactions and interaction strengths drawn from simple probabilistic distributions. We show that the curvature power spectra of each member of the ensemble shows rich structure with many features, and there is a large variability between different realizations of the same ensemble. Such features can be probed by the cosmic microwave background (CMB) and large scale structure observations. They can also have implications for primordial black hole formation and CMB spectral distortions. The geometric random walk behavior of the spectator field allows us to calculate the ensemble-averaged power spectrum of curvature perturbations semi-analytically. For sufficiently large stochastic sourcing, the ensemble-averaged power spectrum shows a scale dependence arising from the time spent by modes outside the horizon during the period of particle production, in spite of there being no preferred scale in the underlying model. We find that the magnitude of the ensemble-averaged power spectrum overestimates the typical power spectra in the ensemble because the ensemble distribution of the power spectra is highly non-Gaussian with fat tails.

Cosmological Hydrodynamic Simulations with Suppressed Variance in the Lyα Forest Power Spectrum

Abstract We test a method to reduce unwanted sample variance when predicting Lyα forest power spectra from cosmological hydrodynamical simulations. Sample variance arises due to sparse sampling of modes on large scales and propagates to small scales through nonlinear gravitational evolution. To tackle this, we generate initial conditions in which the density perturbation amplitudes are fixed to the ensemble average power spectrum—and are generated in pairs with exactly opposite phases. We run 50 such simulations (25 pairs) and compare their performance against 50 standard simulations by measuring the Lyα 1D and 3D power spectra at redshifts z = 2, 3, and 4. Both ensembles use periodic boxes of 40 h − 1 Mpc containing 5123 particles each of dark matter and gas. As a typical example of improvement, for wavenumbers k = 0.25 h Mpc − 1 at z = 3, we find estimates of the 1D and 3D power spectra converge 34 and 12 times faster in a paired–fixed ensemble compared with a standard ensemble. We conclude that, by reducing the computational time required to achieve fixed accuracy on predicted power spectra, the method frees up resources for exploration of varying thermal and cosmological parameters—ultimately allowing the improved precision and accuracy of statistical inference.

Numerical simulation of turbulence and terahertz magnetosonic waves generation in collisionless plasmas

This paper presents numerical simulations of laser beam (x-mode) coupling with a magnetosonic wave (MSW) in a collisionless plasma. The coupling arises through ponderomotive non-linearity. The pump beam has been perturbed by a periodic perturbation that leads to the nonlinear evolution of the laser beam. It is observed that the frequency spectra of the MSW have peaks at terahertz frequencies. The simulation results show quite complex localized structures that grow with time. The ensemble averaged power spectrum has also been studied which indicates that the spectral index follows an approximate scaling of the order of ∼k−2.1 at large scales and scaling of the order of ∼k−3.6 at smaller scales. The results indicate considerable randomness in the spatial structure of the magnetic field profile which gives sufficient indication of turbulence.

Cosmological N-body simulations with suppressed variance

Abstract We present and test a method that dramatically reduces variance arising from the sparse sampling of wavemodes in cosmological simulations. The method uses two simulations which are fixed (the initial Fourier mode amplitudes are fixed to the ensemble average power spectrum) and paired (with initial modes exactly out of phase). We measure the power spectrum, monopole and quadrupole redshift-space correlation functions, halo mass function and reduced bispectrum at z = 1. By these measures, predictions from a fixed pair can be as precise on non-linear scales as an average over 50 traditional simulations. The fixing procedure introduces a non-Gaussian correction to the initial conditions; we give an analytic argument showing why the simulations are still able to predict the mean properties of the Gaussian ensemble. We anticipate that the method will drive down the computational time requirements for accurate large-scale explorations of galaxy bias and clustering statistics, and facilitating the use of numerical simulations in cosmological data interpretation.

Nonlinear magnetic field enhancement and turbulence in laboratory and astrophysical plasmas

The transient response of nonlinear coupling between relatively high frequency magnetosonic wave (HMSW) and low frequency magnetosonic wave (LMSW) in the overdense plasma has been presented in this paper. Along with this, the adiabatic response of HMSW also has been studied for low value of magnetic field in the underdense plasma. The ponderomotive force of relatively high frequency and high power pump magnetosonic wave is expected to excite the low frequency magnetosonic wave. The dynamical equation of LMSW in the presence of ponderomotive force of HMSW (pump wave) has been derived for this purpose. Using this coupled system of dynamical equations, the nonlinear behavior of the pump HMSW was studied, and the resulting turbulent spectrum has been presented. Numerical simulation has been carried out for dimensionless nonlinear coupled equations of HMSW and LMSW, and the results show quite complex localized structures that grow with time. The ensemble averaged power spectrum has also been studied which shows that the spectral index follows an approximate scaling of the order of ∼k−1.67 at larger scales and scaling of the order of ∼k−3.4 at smaller scales. The results indicate considerable randomness in the spatial structure of the magnetic field profile which gives sufficient indication of turbulence. In this context, filamentation of high frequency magnetosonic wave has been considered to be responsible for magnetic turbulence during the laser plasma interaction. The results of the present paper are found relevant with two different experimental observations. Such a nonlinear interaction is quite important in understanding turbulence in the astrophysical phenomenon.

Transformation of videocapsule images to detect small bowel mucosal differences in celiac versus control patients

BackgroundVideocapsule endoscopy can be useful to detect small intestinal pathology in celiac disease patients. However, presence of extraneous features including air bubbles and opaque fluids can complicate the analysis. A technique for quantitative analysis of videocapsule images is presented that is robust to presence of extraneous features. MethodVideocapsule clips were acquired from five small intestinal locations in 12 celiacs with villous atrophy and 11 control patients. Clips were 200 frames in length, their resolution was 576×576 pixels and 256 grayscale levels, with 2/s frame rate. The dominant period (DP), defined as the tallest peak in the ensemble average power spectrum, was computed over each clip without removal of extraneous features. Ensemble average basis images were constructed, and measurements were made of their frame-to-frame variation in brightness and texture. ResultsFrom pooled basis images, celiac images had greater texture than controls and exhibited more brightness variation (p<0.05 in mean and p<0.01 in standard deviation). In celiacs, correlation existed between greater textural alterations versus longer DP (r2=0.47), and between greater brightness variation and longer DP (r2=0.33). There was no significant correlation between quantitative features and DP in controls (r2<0.25). ConclusionsUsing this new method, celiac videoclips were quantitatively distinguishable from control videoclips without manual or computer-assisted detection, masking, and removal of extraneous image features. Furthermore, in celiac but not control basis images, larger textural and brightness alterations were correlated to longer DP. Greater textural and brightness alterations, and thus longer periodicities, are likely related to presence of villous atrophy.

Volume scattering of distributed microbubbles and its influence on blood flow estimation.

In recent years, microbubble contrast agents have become a potential adjunct in Doppler ultrasound diagnosis. In this paper, we show that volume scattering makes the effective band in Doppler spectrum shift downward after injection of microbubbles. Because the insonified volume comprises a collection of distributed microbubbles, the statistical properties such as the autocorrelation function and ensemble average power spectrum of the echoes from a collection of distributed microbubbles were derived first. It can be observed that, beyond a critical frequency, the theoretical volume backscattering cross section derived from the ensemble average power spectrum of microbubbles decreases with frequency. On the contrary, the volume backscattering cross section of red cells increases with frequency. Using two-dimensional (2-D) Fourier transform, the variation in Doppler spectrum caused by different volume backscattering cross section can be demonstrated, and the consequential downward shifts of the estimated Doppler parameters (e.g., the mean and maximum Doppler shifts, and the variance of Doppler power spectrum) after microbubble injection are shown. In addition, it can be observed that the variation gets larger as the transmitted bandwidth increases. And, the variations in Doppler parameters estimated with experimental data are presented to verify the theoretical deviations.

Scaling regimes of composite rainfall time series

The scaling behaviour of rainfall is analysed both for a range of scales in time and for a given scale in intensity using the statistics of the Fourier transform and the cumulative probability distribution. The analyses are applied to sets of long time series of daily rainfall (26 (8) files of 45 (90) years at 13 European stations) and sets of 5-min totals (13 single station summer seasons) thus covering a wide scaling range. The results of both analyses are interpreted in terms of their asymptotically hyperbolic (i.e. power law) behaviour: The ensemble averaged power spectra exhibit distinct scaling regimes with their associated power law behaviour, P ( f ) ∼ f –b : a regime of climatic variability (> 3 years: b ∼ 0.7), a spectral plateau (3 years to 1 month: b – 0) of general circulation fluctuations, a transition regime (1 month to 3 days: a dropping power spectrum without scaling), and a range governed by frontal systems (< 3 days: b ∼ 0.5). The transition region is interpreted as being generated by both of its neighbouring regimes whose scaling can therefore be expanded into the transition regime. Finally an apparent break in scaling (at 2.4 h) can possibly be attributed to the instruments inability to measure frequent weak signals. The tail-end of the hyperbolic distribution (characterizing the intermittency regime) is not approached smoothly but shows a break from the rest of the distribution. Finally, an outlook to multifractal scaling is given. DOI: 10.1034/j.1600-0870.1993.t01-3-00004.x

Flow characteristics of a motored spark ignition engine with a prechamber.

To understand the effect of a torch jet on the flow characteristics in a prechamber spark ignition engine, a laser Doppler anemometer (LDA) is used to measure the velocity in a main chamber under the motored condition. The engine is operated at a constant speed of 1000 rpm with a wide open throttle (WOT). The mean velocity, turbulence intensity and turbulence scales are calculated by a cycle-resolved analysis with a cut-off frequency of 650 Hz. The reasonable selection of the cut-off frequency is based on the ensemble-averaged power spectra. During the expansion stroke, the mean velocity, turbulence intensity and integral length scale are found to be strongly affected by the torch jet flow, which is dominated by the torch nozzle area size.

A Microwave Decoupled Electrode for the Electroencephalogram (Short Papers)

The recording of the electroencephalogram (EEG) in an amplitude-modulated microwave field presents two related but distinct problems when conventional electrodes are employed. The electrode and its associated conductors extract power from the incident radiation, resulting in increased local power deposition which confounds dosimetric arguments and imposes local thermalization; the electrode tissue interface is a nonlinear system that demodulates amplitude-modulated signals with the results that the demodulate is additively mixed with the EEG. The problems were studied in a series of bench tests with conventional and thin-film microwave integrated circuit (MIC) electrodes. The latter are decoupled from the magnetic component of the field by virtue of radically reduced dimensions of the thin-loop component of its geometry, and suppression of dipole (i.e., electric field interaction) currents by use of integrated Nichrome series resistance. The result is that the demodulation artifact is undetectable in the ensemble averaged-power spectrum from the in vitro electrode up to an S-band incident-power density of 100 mW/cm/sup 2/. Thermalization was studied in a dielectric brain phantom with high-resistance monofilament leads to the MIC with a result that 0.6/spl deg/C heating is attributable to the electrode with prolonged exposure to a 50-mW/cm/sup 2/ field.

Magnetohydrodynamic Turbulence Experiments in a Transverse Magnetic Field

Random fluctuations in a transverse magnetic field caused by a helium plasma produced in an arc discharge were measured with magnetic probes. Ensemble averaged power spectra of these fluctuations followed a power law over the frequency range from about 0.2 to 1 MHz, the exponents varying from −11/3 to −19/3. The time average of the fluctuation intensity parallel to the external magnetic field increased nearly linearly with the imposed field up to about 1000 G. Stronger magnetic fields inhibited these fluctuations and retarded the flow of the plasma. Integral and microscales were in the range of 40 to 100 mm and 20 to 40 mm, respectively. The random fluctuations in the magnetic field may have been due partly to random inhomogeneities in the conductivity of the plasma and only partly to vortical turbulence. Magnetic and conductivity probes indicated the presence of substantial electrical conductivity ahead of the luminous plasma. This conductivity was reduced by increasing magnetic field, interpreted as involving an electron flux.