Power Spectrum Estimation Research Articles

Robust statistics towards detection of the 21 cm signal from the Epoch of Reionization

We explore methods for robust estimation of the 21 cm signal from the Epoch of Reionisation (EoR). A Kernel Density Estimator (KDE) is introduced for measuring the spatial temperature fluctuation power spectrum from the EoR. The KDE estimates the underlying probability distribution function of fluctuations as a function of spatial scale, and contains different systematic biases and errors to the typical approach to estimating the fluctuation power spectrum. Extraction of histograms of visibilities allows moments analysis to be used to discriminate foregrounds from 21 cm signal and thermal noise. We use the information available in the histograms, along with the statistical dis-similarity of foregrounds from two independent observing fields, to robustly separate foregrounds from cosmological signal, while making no assumptions about the Gaussianity of the signal. Using two independent observing fields to robustly discriminate signal from foregrounds is crucial for the analysis presented in this paper. We apply the techniques to 13 hours of Murchison Widefield Array (MWA) EoR data over two observing fields. We compare the output to that obtained with a comparative power spectrum estimation method, and demonstrate the reduced foreground contamination using this approach. Using the second moment obtained directly from the KDE distribution functions yields a factor of 2-3 improvement in power for k < 0.3hMpc^{-1} compared with a matched delay space power estimator, while weighting data by additional statistics does not offer significant improvement beyond that available for thermal noise-only weights.

Full-sky beam convolution for cosmic microwave background applications

We introduce a publicly available full-sky beam convolution code library intended to inform the design of future cosmic microwave background (CMB) instruments and help current experiments probe potential systematic effects. The code can be used to assess the impact of optical systematics on all stages of data reduction for a realistic experiment, including analyses beyond power spectrum estimation, by generating signal timelines that may serve as input to full analysis pipelines. The design and mathematical framework of the Python code is discussed along with a few simple benchmarking results. We present a simple two-lens refracting telescope design and use it together with the code to simulate a year-long dataset for 400 detectors scanning the sky on a satellite instrument. The simulation results identify a number of sub-leading optical non-idealities and demonstrate significant B-mode residuals caused by extended sidelobes that are sensitive to polarized radiation from the Galaxy. For the proposed design and satellite scanning strategy, we show that a full physical optics beam model generates B-mode systematics that differ significantly from the simpler elliptical Gaussian model. The code is available at https://github.com/adrijd/beamconv

An advanced bispectrum features for EEG-based motor imagery classification

Motor imagery (MI)-related brain activities can be effectively described by frequency analysis. Bispectrum is developed to overcome the drawback of power spectrum that the estimation of power spectrum discards the phase relationship among frequency components. However, the widely used bispectral features extraction method adds up all bispectral values as one feature, which could lead to the loss of effective information and increase of the sensitivity to non-linear and non-Gaussian noises. Thus, the representative bispectral features extraction method may be inefficient for MI classification. In addition, recent research suggests that the variations of EEG signals could provide more useful underlying information of event-related brain responses. This paper presents an advanced variations based bispectral feature extraction method to improve the performance of MI classification. The proposed method calculates the variations of MI-related EEG signals as input to bispectrum estimation. Besides, a new segmented bispectral sum features are developed to reduce the influence of non-linear and non-Gaussian noises and emphasize the valuable information for MI classification. The dataset collected in our laboratory and BCI Competition IV dataset 2b were adopted to validate the proposed method. The results indicate that the proposed method outperforms the power spectrum based methods and the representative bispectral features based methods. Moreover, compared to other state-of-the-art works, our approach also achieves the greater performance for MI classification.

Estimating statistics of sky brightness using radio interferometric observations

Radio interferometers are used to construct high resolution images of the sky at radio frequencies and are the key instruments for accessing the statistical properties of the evolution of neutral hydrogen over cosmic time. Here we use simulated observations of the model sky to assess the efficacy of different estimators of the large-scale structure and power spectrum of the sky brightness distribution. We find that while the large-scale distribution can be reasonably estimated using the reconstructed image from interferometric data, estimates of the power spectrum of the intensity fluctuations calculated from the image are generally biased. This bias is found to be more pronounced for diffuse emission. The visibility based power spectrum estimator, however, gives an unbiased estimate of the true power spectrum. This work demonstrates that for an observation with diffuse emission the reconstructed image can be used to estimate the large-scale distribution of the intensity, while to estimate the power spectrum, visibility based methods should be preferred. With the upcoming experiments aimed at measuring the evolution of the power spectrum of the neutral hydrogen distribution, this is a very important result.

Supercritical and subcritical Hopf-bifurcations in a two-delayed prey–predator system with density-dependent mortality of predator and strong Allee effect in prey

One of the possible ways to visualize the effect of intra- and inter-species synergistic and antagonistic interactions in a natural ecosystem is the detailed analysis of the underlying prey-predator model, and the subsequent analytical findings may provide a definite justification towards the species coexistence, which often maintains biodiversity in nature. Here, our central motivation is to understand the combined effect of the Allee threshold and intra-species competition on the evolution of interacting species, which often experience delays in evolution due to its complex ecological and physiological processes. Therefore, in the present paper, we extensively analyze the stability of a two-delayed prey-predator system in the presence of strong Allee effects in prey and intra-species competition in predator. In addition, we capture the reality of time difference between the lifespan of the prey and predator through proper nondimensionalization. The two delays in the proposed retarded system correspond to the intra-specific competition-induced feedback time lag to the prey and predator gestation period. In the absence of intra-predator competition, the present dynamics unveils supercritical Hopf-bifurcation around the interior point of coexistence which is in-line with the existing literature. It is noteworthy to mention that the proposed Allee system exhibits subcritical Hopf-bifurcation in the presence of intra-species competition in predator. We confirm the occurrence of both supercritical and subcritical Hopf-bifurcations via calculating the direction and stability of Hopf-bifurcating periodic solutions using the normal form method and the center manifold theory. Moreover, the suggested delayed schema presents supercritical Hopf-bifurcation at the boundary steady-state, where the population density of prey exists at its maximum carrying capacity. We recognize the bistability between extinction and coexistence, and the proposed model also exhibits the ‘chaotic concurrence between prey and predator,’ which happens through the period-doubling bifurcation. The existence of chaos is validated using the estimated power spectrum and the spectrum of Lyapunov exponents. The primary finding of this paper is that Allee threshold induces the capability to the density-dependent death rate of predator towards changing the stability of ‘oscillatory coexistence between prey and predator.’

Spatial power spectrum of the photospheric magnetic field during solar minimum

Context. During solar minima the spatial power spectrum of the photospheric magnetic field is dominated by the low-degree zonal (axisymmetric; m = 0) harmonic components, reflecting the large polar coronal holes of unipolar magnetic field. However, measuring polar fields is difficult because of the unequal visibility of the two poles during most of the year and the small line-of-sight component of the roughly radial field at high solar latitudes. Aims. In this paper we derive the spatial power spectrum of the photospheric magnetic field in terms of the harmonic coefficients of the radial component (Br) as well as in terms of the harmonic coefficients of the internal potential (known as Gauss coefficients). We calculate the zonal spatial power spectrum using Mount Wilson Observatory synoptic maps from 1995–1996, during the solar minimum between solar cycles 22 and 23, and investigate how filling or not filling the polar data gaps affects the zonal harmonic coefficients. Methods. We eliminated the vantage point effect by removing the highest 5° of the measured magnetic field and calculating the latitudinal profile of the zonal median field over the two years, which ensured equal latitudinal data coverage of both solar hemispheres. We then derived the zonal harmonic coefficients using this latitudinal profile of Br. Results. We find that when the polar data gaps are left unfilled, a strong artificial power above l = 8 is produced. Only the first five zonal harmonic coefficients can be considered reliable in this case. Therefore polar filling is essential to obtain a realistic spatial power spectrum. Filling the polar gap with a constant (non-zero) value yields zonal harmonics that are reliable up to l = 9. We find that the zonal octupole component contributes most to the total spatial power, more than the zonal dipole, even during the solar minimum conditions. This difference is seen more clearly in the case of polar filling. We also prove that the asymmetry of the polar fields during this solar minimum is statistically significant. Conclusions. Our results emphasize the importance of filling the polar data gaps in order to obtain a correct estimate of the spatial power spectrum of the photospheric field. This helps in estimating the reliability of polar fields and the large-scale structure in synoptic maps of different origin. Our results also verify the asymmetric nature of the polar fields, which is important for the heliospheric magnetic field and for solar dynamo modeling.

Bayesian power spectrum estimation at the Epoch of Reionization

We introduce a new method for performing robust Bayesian estimation of the three-dimensional spatial power spectrum at the Epoch of Reionization (EoR), from interferometric observations. The versatility of this technique allows us to present two approaches. First, when the observations span only a small number of independent spatial frequencies ($k$-modes) we sample directly from the spherical power spectrum coefficients that describe the EoR signal realisation. Second, when the number of $k$-modes to be included in the model becomes large, we sample from the joint probability density of the spherical power spectrum and the signal coefficients, using Hamiltonian Monte Carlo methods to explore this high dimensional ($\sim$ 20000) space efficiently. This approach has been successfully applied to simulated observations that include astrophysically realistic foregrounds in a companion publication (Sims et al. 2016). Here we focus on explaining the methodology in detail, and use simple foreground models to both demonstrate its efficacy, and highlight salient features. In particular, we show that including an arbitrary flat spectrum continuum foreground that is $10^8$ times greater in power than the EoR signal has no detectable impact on our parameter estimates of the EoR power spectrum recovered from the data.

Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage

This paper investigates the windowing design and performance assessment for mitigation of spectral leakage. A pretreatment method to reduce the spectral leakage is developed. In addition to selecting appropriate window functions, the Welch method is introduced. Windowing is implemented by multiplying the input signal with a windowing function. The periodogram technique based on Welch method is capable of providing good resolution if data length samples are selected optimally. Windowing amplitude modulates the input signal so that the spectral leakage is evened out. Thus, windowing reduces the amplitude of the samples at the beginning and end of the window, altering leakage. The influence of various window functions on the Fourier transform spectrum of the signals was discussed, and the characteristics and functions of various window functions were explained. In addition, we compared the differences in the influence of different data lengths on spectral resolution and noise levels caused by the traditional power spectrum estimation and various window-function-based Welch power spectrum estimations.

Covariance Matrix Reconstruction via Residual Noise Elimination and Interference Powers Estimation for Robust Adaptive Beamforming

Recently, a number of robust adaptive beamforming (RAB) methods based on Capon power spectrum estimator integrated over a specific region for covariance matrix reconstruction have been proposed. However, all of these methods ignore the residual noise existing in the Capon spectrum estimator, which results in reconstruction errors. In this paper, we propose a RAB algorithm via residual noise elimination and interference powers estimation to reconstruct covariance matrix. First, the proposed algorithm demonstrates the existence of residual noise and analyze its relationship to actual noise. Then, after eliminating the residual noise, the modified Capon power spectrum estimator is utilized to reconstruct the covariance matrix and desired signal SV. Moreover, to reduce the influence of the desired signal on interference powers estimation, we project the snapshots onto the complementary subspace of the desired signal and estimated interference powers are derived according to the theoretical formulation of the interference covariance matrix (ICM). The simulation results demonstrate that the proposed method is robust against various mismatches and can achieve superior performance.

Gridded and direct Epoch of Reionisation bispectrum estimates using the Murchison Widefield Array

AbstractWe apply two methods to estimate the 21-cm bispectrum from data taken within the Epoch of Reionisation (EoR) project of the Murchison Widefield Array (MWA). Using data acquired with the Phase II compact array allows a direct bispectrum estimate to be undertaken on the multiple redundantly spaced triangles of antenna tiles, as well as an estimate based on data gridded to the uv-plane. The direct and gridded bispectrum estimators are applied to 21 h of high-band (167–197 MHz; z = 6.2–7.5) data from the 2016 and 2017 observing seasons. Analytic predictions for the bispectrum bias and variance for point-source foregrounds are derived. We compare the output of these approaches, the foreground contribution to the signal, and future prospects for measuring the bispectra with redundant and non-redundant arrays. We find that some triangle configurations yield bispectrum estimates that are consistent with the expected noise level after 10 h, while equilateral configurations are strongly foreground-dominated. Careful choice of triangle configurations may be made to reduce foreground bias that hinders power spectrum estimators, and the 21-cm bispectrum may be accessible in less time than the 21-cm power spectrum for some wave modes, with detections in hundreds of hours.

Correlation Between Power Line Harmonic Radiation and Magnetospheric Line Radiation Over China

Power line harmonic radiation (PLHR) and magnetospheric line radiation (MLR) are recognized as electromagnetic pollutants in the near-earth space, and their correlation has been a controversial issue. An identification procedure based on short-time discrete Fourier transform and Welch power spectrum estimation was applied to detect 114 PLHR events and 328 MLR events that occurred as DEMETER satellite flew over China from 2008 to 2010. PLHR events feature parallel horizontal lines with a frequency interval of 50/100 Hz in spectrogram, whereas MLR events feature multi-parallel spectral lines with frequency drift as a baseline for possible triggered emissions as well as triggered emissions with no baseline. We statistically compared the temporal and geographical distribution, Kp and Dst indices, and propagation characteristics between PLHR and MLR events. PLHR events showed more obvious variations annually than MLR events, and the former had a close relationship with the development of Chinese power system. Both PLHR and MLR presented similar diurnal and seasonal differences, owing likely to the ionospheric state; moreover, they appeared to have no significant connection with geomagnetic activity level. The geographical distributions of PLHR and MLR differed significantly. PLHR events occurred in the range of (20° N–45° N), whereas MLR events were concentrated in southwestern low-latitude regions of (18° N–28°N). Furthermore, PLHR waves were right-hand circularly polarized, whereas MLR waves were approximately linearly polarized. In summary, MLR is more likely caused by natural radiation that is not directly related to PLHR, whereas PLHR is closely related to terrestrial power systems.

Virtual prediction and constraint of contour errors induced by cutting force disturbances on multi-axis CNC machine tools

This paper presents a strategy to virtually predict and constrain the contouring errors contributed by cutting force disturbances on feed drives. The tracking errors on each feed drive are predicted as a linear function of tangential feed by evaluating the product of estimated power spectrum of cutting forces and disturbance frequency response function along the tool path in virtual CAM environment. The corresponding tool tip contouring and tool axis orientation errors are estimated and constrained by scaling the feed along the tool path. The algorithm is experimentally illustrated to improve the machining accuracy on a 5 Axis CNC machine tool.

Ionospheric Attenuation of Polarized Foregrounds in 21 cm Epoch of Reionization Measurements: A Demonstration for the HERA Experiment

Foregrounds with polarization states that are not smooth functions of frequency present a challenge to HI Epoch of Reionization (EoR) power spectrum measurements if they are not cleanly separated from the desired Stokes I signal. The intrinsic polarization impurity of an antenna's electromagnetic response limits the degree to which components of the polarization state on the sky can be separated from one another, leading to the possibility that this frequency structure could be confused for HI emission. We investigate the potential of Faraday rotation by the Earth's ionosphere to provide a mechanism for both mitigation of, and systematic tests for, this contamination. Specifically, we consider the delay power spectrum estimator, which relies on the expectation that foregrounds will be separated from the cosmological signal by a clearly demarcated boundary in Fourier space, and is being used by the Hydrogen Epoch of Reionization Array (HERA) experiment. Through simulations of visibility measurements which include the ionospheric Faraday rotation calculated from real historical ionospheric plasma density data, we find that the incoherent averaging of the polarization state over repeated observations of the sky may attenuate polarization leakage in the power spectrum by a factor of 10 or more. Additionally, this effect provides a way to test for the presence of polarized foreground contamination in the EoR power spectrum estimate.

Optimizing primordial non-Gaussianity measurements from galaxy surveys

Galaxy clustering data from current and upcoming large scale structure surveys can provide strong constraints on primordial non-Gaussianity through the scale-dependent halo bias. To fully exploit the information from galaxy surveys, optimal analysis methods need to be developed and applied to the data. Since the halo bias is sensitive to local non-Gaussianity predominately at large scales, the volume of a given survey is crucial. Consequently, for such analyses we do not want to split into redshift bins, which would lead to information loss due to edge effects, but instead analyse the full sample. We present an optimal technique to directly constrain local non-Gaussianity parametrised by $f_\mathrm{NL}^\mathrm{loc}$, from galaxy clustering by applying redshift weights to the galaxies. We derive a set of weights to optimally measure the amplitude of local non-Gaussianity, $f_\mathrm{NL}^\mathrm{loc}$, discuss the redshift weighted power spectrum estimators, outline the implementation procedure and test our weighting scheme against Lognormal catalogs for two different surveys: the quasar sample of the Extended Baryon Oscillation Spectroscopic Survey (eBOSS) and the emission line galaxy sample of the Dark Energy Spectroscopic Instrument (DESI) survey. We find an improvement of 30 percent for eBOSS and 6 percent for DESI compared to the standard Feldman, Kaiser $\&$ Peacock weights, although these predictions are sensitive to the bias model assumed.

Power spectrum estimation in innovation models

Abstract In this paper, a regularized nuclear norm minimization-based subspace identification algorithm is proposed to estimate discrete-time power spectra of multi-input/multi-out systems in innovation models from output-only time-domain measurements. Parametrization and implementation issues are carefully addressed. The new identification algorithm and three popular subspace algorithms are tested on a numerical example and three real-life applications drawn from automotive engineering, acoustics, and renewable energy.

Characterizing Signal Loss in the 21 cm Reionization Power Spectrum: A Revised Study of PAPER-64

Abstract The Epoch of Reionization (EoR) is an uncharted era in our universe’s history during which the birth of the first stars and galaxies led to the ionization of neutral hydrogen in the intergalactic medium. There are many experiments investigating the EoR by tracing the 21 cm line of neutral hydrogen. Because this signal is very faint and difficult to isolate, it is crucial to develop analysis techniques that maximize sensitivity and suppress contaminants in data. It is also imperative to understand the trade-offs between different analysis methods and their effects on power spectrum estimates. Specifically, with a statistical power spectrum detection in HERA’s foreseeable future, it has become increasingly important to understand how certain analysis choices can lead to the loss of the EoR signal. In this paper, we focus on signal loss associated with power spectrum estimation. We describe the origin of this loss using both toy models and data taken by the 64-element configuration of the Donald C. Backer Precision Array for Probing the Epoch of Reionization (PAPER). In particular, we highlight how detailed investigations of signal loss have led to a revised, higher 21 cm power spectrum upper limit from PAPER-64. Additionally, we summarize errors associated with power spectrum error estimation that were previously unaccounted for. We focus on a subset of PAPER-64 data in this paper; revised power spectrum limits from the PAPER experiment are presented in a forthcoming paper by Kolopanis et al. and supersede results from previously published PAPER analyses.

A Bayesian method for combining theoretical and simulated covariance matrices for large-scale structure surveys

Accurate and precise covariance matrices will be important in enabling planned cosmological surveys to detect new physics. Standard methods imply either the need for many N-body simulations in order to obtain an accurate estimate, or a precise theoretical model. We combine these approaches by constructing a likelihood function conditioned on simulated and theoretical covariances, consistently propagating noise from the finite number of simulations and uncertainty in the theoretical model itself using an informative Inverse-Wishart prior. Unlike standard methods, our approach allows the required number of simulations to be less than the number of summary statistics. We recover the linear ‘shrinkage’ covariance estimator in the context of a Bayesian data model, and test our marginal likelihood on simulated mock power spectrum estimates. We conduct a thorough investigation into the impact of prior confidence in different choices of covariance models on the quality of model fits and parameter variances. In a simplified setting, we find that the number of simulations required can be reduced if one is willing to accept a mild degradation in the quality of model fits, finding that even weakly informative priors can help to reduce the simulation requirements. We identify the correlation matrix of the summary statistics as a key quantity requiring careful modelling. Our approach can be easily generalized to any covariance model or set of summary statistics, and elucidates the role of hybrid estimators in cosmological inference.

Events detection and recognition by the fiber vibration system based on power spectrum estimation

One of the important successes of optical fiber sensor established for the security system is the detection and the recognition of any type of events. The performance parameters (event recognition, event detection position, and time of detection) are unavoidable and describe the validity of any perimeter detection system. An event recognition is any signal detected within the protected area, and it is related to a non-intrusion event and an intrusion event. To achieve the detection and the recognition events at the real time, an effective two-level vibration recognition method and a technique are proposed and presented in this article. The signal characteristics (short-term energy and short-time over-threshold) have been used and compared to the dynamic threshold to judge the type of event. Then the extraction of the power distribution features on the frequency domain through power spectral estimation on the suspected intrusion signal samples is carried out and finally combined with the time-domain characteristics as feature vector through Support Vector Machine to determine the efficiency and effectiveness of the proposed vibration recognition method. The experimental simulation results show that the proposed method is effective and reliable. With collected data, it can detect and recognize the type of event in real time.

Resolving Gender Difference in Problem Solving Based On the Analysis of Electroencephalogram (EEG) Signals

Problem solving is regarded as one of the core work-related abilities and skills, which are highly demanded by the workplace and industry. Current literature suggests that problem solving abilities might differ from one individual to another due to biological factors such as brain activationa, cognitive functions and hormones, as well as due to socio-cultural and socio-economic factors like gender roles, self-perceptions and stereotyping. Hence, this study used electroencephalogram (EEG) signals to investigate the differences in problem solving skills among the Malaysian undergraduates based on their gender differences. 29 undergraduate students from the Faculty of Electrical Engineering, Universiti Teknologi Malaysia (UTM) served as the subjects of the experiments in this research. Specifically, 16 female and 13 male subjects engaged in two main problem-solving tasks: mental arithmetic task and Tower of Hanoi (TOH) task. The EEG data were analysed using partial directed coherence (PDC) and power spectrum estimation (PSE). Based on the results, female subjects achieved only 1% higher performance in mental arithmetic task, while male subjects achieved about 13% higher performance in TOH task. The differences in terms of the functional connectivity between brain regions, i.e. in PDC, as well as the power distribution of 6 EEG waveforms, i.e. delta, theta, alpha, beta, gamma and high gamma bands are also highlighted and represented graphically in this paper.

Effect of prenatal stress on neural oscillations in developing hippocampal formation

The effect of prenatal stress (PS) on offspring's behavior was reported previously. Several studies attempted to reveal the mechanisms of PS on synaptic and molecular levels. However, the influences of PS on neural oscillations and their interaction in hippocampus are still unknown. In the present study, a PS rat model was established by using restraint stress. The local-field potentials (LFPs) were simultaneously recorded from the hippocampal CA3 and CA1 regions in young, adolescent and early-adult offspring rats. After that, LFPs were analyzed by analytic algorithms for estimating power spectrum, coherence, phase synchronization and cross-frequency coupling. The results showed that there was a significant influence of PS on power distribution from 1 to 100 Hz during different developmental stages. The identical-frequency synchronizations between CA3 and CA1 regions, including coherence and phase synchronization, were significantly reduced in PS rats compared to that in normal rats. Meanwhile, PS significantly impaired the cross-frequency coupling strength between theta and gamma rhythms. These data show that PS alters the neural oscillations and their interaction on the hippocampal CA3-CA1 pathway, which may be associated with the behavior outcomes and synaptic impairments previously reported in PS offspring rats. Moreover, the significant PS × age interactions between the effects of PS and age have been only found in the cross-frequency coupling, implying that the cross-frequency coupling more appropriately reflects the differences of the behavioral effects of PS in different postnatal ages.