Spatial Power Spectral Density Research Articles

Assessment of chemical scalars for heat release rate measurement in highly turbulent premixed combustion including experimental factors

This study uses direct numerical simulation (DNS) data from two turbulent premixed flames to explore the variables involved in experimental measurement of the heat release rate, h˙. Several commonly used markers of h˙ are compared, namely the CH, OH, and HCO radicals, as well as the spatial overlap of OH and CH2O and of H-atom and CH2O. The DNS data were also used to generate synthetic Planar Laser-Induced Fluorescence (PLIF) images that account for confounding experimental factors such as laser sheet thickness, optical blur, and shot noise. The overlap of OH and CH2O and the HCO radical appear to be the most promising markers of h˙, as they show excellent correlation with the high-intensity heat release. The synthetic-PLIF images show that increasing the laser sheet thickness and optical blur progressively de-correlates the measurement and the underlying DNS structures. However, for a sufficiently thin laser sheet (δsheet/δL ≲ 1), the correlation between the synthetic-PLIF and DNS remains strong even with significant optical blur. Similar results are observed in the measured layer thickness and the peak-to-peak distance between the PLIF structure and the heat release layers. The resolution of h˙ features was evaluated by measuring the average turbulent flame length, the spatial power spectral density, and the flame surface curvature. As the laser sheet thickness or optical blur is increased, the measured flame surface area progressively decreases, less fine-scale features are resolved, and regions of high curvature are eroded. Shot noise strongly affects spatial correlation with the DNS data and the layer thickness, but has relatively small impact on the turbulent flame length. Significantly greater deviations are observed for the overlap techniques requiring simultaneous measurement from two cameras, as optical blur and/or shot noise present in the two camera systems will both influence the resulting overlapped PLIF signal.

Processor dependent bias of spatial spectral estimates from coprime sensor arrays.

Coprime sensor arrays (CSAs) can estimate the directions of arrival of O(MN) narrowband plane wave sources using only O(M+N) sensors with the CSA product processor. Processing data from a finite aperture array effectively smears the true spatial power spectral density (PSD) with a kernel function determined by both the array geometry and the processing of the signals observed by the array. This paper examines the asymptotic behaviors of the kernel functions resulting from two different processors applied to a CSA sampling geometry in the limit of large aperture. The kernel functions of the product processed CSA and conventionally beamformed coprime sensor arrays (CBF CSA) are compared to the baseline of the kernel of a densely populated uniform line array (ULA) of similar aperture. At the limit of large aperture, the product processed CSA estimate is asymptotically unbiased like the ULA, while the CBF CSA estimate is not. The PSD estimates computed from the CSA processors are compared when spatially correlated Gaussian noise is an input to the array to highlight the bias issues.

Spatial power spectral density estimation using a multitapered coprime sensor array minimum processor.

A coprime sensor array (CSA) is a sparse array geometry that interleaves two spatially undersampled uniform linear arrays (ULAs) with coprime undersampling factors. The CSA Min processor achieves an asymptotically unbiased spatial power spectral density (PSD) estimate while approaching the variance of a ULA conventional beamformer. Nonstationary underwater sonar environments often preclude the number of snapshots required to achieve a desirable PSD variance. The multitaper method improves PSD variance by O(K) at the expense of resolution without additional snapshot cost by averaging uncorrelated PSD estimates obtained using a set of K orthogonal tapers. This paper proposes the multitapered Min processor to achieve unambiguous PSD estimates with desirable variance properties for passive beamforming scenarios. The probability density function and the first two moments of the MT-Min processor's PSD estimate are derived in closed-form for spatially white Gaussian processes. Simulations verify the variance reduction predicted by the analytical derivation for white processes and, by extension, for non-white processes. The multitaper method is then extended to an ad hoc mixture of Min and Product processors under constant noise plateau normalization that attenuates the spurious peaks occurring in the CSA PSD estimates in the presence of multiple planewave arrivals.

Gaussian Source Detection and Spatial Spectral Estimation Using a Coprime Sensor Array With the Min Processor

A coprime sensor array (CSA) interleaves two undersampled uniform linear arrays with coprime undersampling factors and has recently found broad applications in signal detection and estimation. CSAs commonly use the product processor by multiplying the scanned responses of two colinear subarrays to estimate the spatial power spectral density (PSD) of the received signal. This paper proposes a new CSA processor, the CSAmin processor, which chooses the minimum between the two CSA subarray periodograms at each bearing to estimate the spatial PSD. The proposed CSAmin processor resolves the CSA subarray spatial aliasing equally well as the product processor. For an extended aperture CSA, the CSAmin reduces the peak sidelobe height and total sidelobe area over the product processor for the same CSA geometry. Moreover, unlike the PSD estimate from the product processor, the PSD estimate from the CSAmin is guaranteed to be positive semidefinite. This paper derives the probability density function, the complementary cumulative distribution function (CCDF, or tail distribution), and the first two moments of the CSAmin PSD estimator in closed form for Gaussian sources in white Gaussian noise. Numerical simulations verify the derived CSAmin statistics and demonstrate that the CSAmin improves the performance over the product processor in detecting narrowband Gaussian sources in the presence of loud interferers and noise. The CSAmin spatial PSD estimate achieves lower variance than the product processor estimate, and keeps the PSD estimate unbiased for white Gaussian processes and asymptotically unbiased for nonwhite Gaussian processes.

Spatial Spectral Estimation with Product Processing of a Pair of Colinear Arrays

The product of the scanned responses of two colinear interleaved arrays estimates the spatial power spectral density of the received signal. This estimate has mathematical structure similar to the classical periodogram and hence is called the generalized periodogram. The generalized periodogram includes sparse arrays such as coprime arrays and two level nested arrays. This paper derives the analytical expected value of the generalized periodogram output for any signal. The generalized periodogram expected value is the convolution of the true spectral density with a windowing function. This result reveals that the expected product spectrum is not guaranteed to be positive definite. The paper also derives the covariance function of the generalized periodogram for a white Gaussian process. Finally, the paper illustrates the implicit equivalence between the product spectrum and estimation of the correlation function from the products of appropriately spaced pairs of sensors.

Investigation of the surface profile along the cutting trajectory and its correlation with cutting forces in single point diamond turning

Surface roughness is one of the key parameters affecting the surface finish quality in ultra-precision turning. As conventional surface roughness measurement is merely a statistical representation of the surface condition, it does not represent the dynamic relative movement between a workpiece and a tool. In this paper, an experimental investigation was conducted to correlate the roughness along the cutting trajectory profile and cutting forces along three different directions: the feed direction (FD), primary cutting direction (CD), and thrust direction (TD), respectively. It was found that the changes of cutting force along CD had a much higher impact on the surface generation than those along FD and TD. Based on the analysis of the spatial power spectral density (PSD), it was found that the surface generation is mostly affected by the tool-work vibration along the cutting trajectory. Cross-correlation of the spatial PSD and force PSD also revealed that when the depth of cut was greater than 5 μm, more similarities were found between these two PSDs. These findings will contribute to the development of online measurement technology of surface roughness in the future.

Estimating the spatial spectra of Gaussian processes with co-prime sensor arrays

A co-prime sensor array (CSA) is a nonuniform line array formed by interleaving two undersampled uniform line arrays. The CSA requires fewer sensors to span the same aperture as a densely sampled uniform line array (ULA), allowing the CSA to match the resolution of the ULA for direction of arrival estimation of narrowband planewaves. However, each CSA subarray suffers from aliasing, or grating lobes, due to the spatial undersampling. Vaidyanathan and Pal (2011) proved that if the subarray undersampling factors are co-prime, the aliasing can be unambiguously resolved by multiplying the spatial spectra of the subarrays. This product spatial spectra is the spatial cross-spectral density between the arrays, and is an estimate of the spatial power spectral density (PSD). In this talk, we extend the classic results of Jenkins and Watts (1968) on the periodogram PSD estimator for Gaussian processes to obtain the product processor's bias for spatially wide-sense stationary processes, and the processor's covariance for spatially white Gaussians. Additionally, we demonstrate that the CSA's product PSD estimate is not necessarily positive definite. Consequently, the CSA product spectrum may fail to detect weak signals in the presence of strong interferers. [Work supported by ONR BRC Program.]

Spatial power spectral estimation using coprime sensor array with the min processor

A coprime sensor array (CSA) commonly estimates the spatial power spectral density (PSD) of the observed signal by multiplying one conventionally beamformed subarray scanned response with the complex conjugate of the other [Vaidyanathan and Pal, 2011]. This product processor removes the CSA subarray spatial aliasing ambiguities, but has a peak sidelobe higher than the peak sidelobe of a fully populated uniform linear array (ULA) [Adhikari et al., 2014]. Moreover, the resulting spatial PSD estimate is not guaranteed positive semi-definite, and as a result, weak sources can be masked by the negative side lobes of strong interferers. This paper proposes choosing the minimum between the two CSA subarray periodograms at each bearing to resolve the spatial aliasing ambiguities. This min processor achieves lower peak sidelobe height and total sidelobe area than the product processor and preserves the PSD positive semi-definite characteristic. Closed-form expressions for the first two moments of the CSA min PSD estimator are available. Simulations and real data show that the min processor achieves a lower PSD estimate variance than the product processor while keeping the PSD estimate approximately unbiased. [Work supported by ONR BRC Program.]

Spatiotemporal Long-Range Persistence in Earth’s Temperature Field: Analysis of Stochastic–Diffusive Energy Balance Models

Abstract A two-dimensional stochastic–diffusive energy balance model (EBM) formulated on a sphere by G. R. North et al. is explored and generalized. Instantaneous and frequency-dependent spatial autocorrelation functions and local temporal power spectral densities are computed for local sites and for spatially averaged surface temperature signals up to the global scale. On time scales up to the relaxation time scale given by the effective heat capacities of the ocean mixed layer and land surface, respectively, scaling features are obtained that are reminiscent of what can be derived from the observed temperature field. On longer time scales, however, the EBM predicts a transition to a white-noise scaling, which is not reflected in the observed records. A fractional generalization, which can be considered as a spatial generalization of the zero-dimensional, long-memory EBM of M. Rypdal and K. Rypdal, is proposed and explored. It is demonstrated that this generalized model describes qualitatively the main correlation characteristics of the temperature field reported in the literature and those derived herein from 500-yr-long control simulations of the NorESM Earth system model. A further generalization of the model, to include long-term persistence in the stochastic forcing, is also discussed.

Degree of phase separation effects on the charge transfer properties of P3HT:Graphene nanocomposites

Graphene layers were introduced into the matrix of regioregular poly (3-hexylthiophene-2, 5-diyl) (RR-P3HT) via solution processing in the perspective of the development of organic nanocomposites with high P3HT/Graphene interfaces areas for efficient charge transfer process. P3HT and graphene act as electrons donor and electrons acceptor materials, respectively. Spatial Fourier Transforms (FFT) and power spectral density (PSD) analysis of the AFM images show that the phase separation decreases with increasing the graphene weight ratio in the P3HT matrix. The Raman spectra of the P3HT:Graphene nanocomposites shows that the G-band of graphene shifts to low frequencies with progressive addition of graphene which proves that there is an interaction between the nanowires of P3HT and the graphene layers. We suggest that the shift of the G-band is due to electrons transfer from P3HT to graphene. The quenching of the photoluminescence (PL) intensity of P3HT with addition of graphene proves also that an electrons transfer process occurred at the P3HT/Graphene interfaces.

Spatio-Temporal Scaling of Turbulent Photospheric Line-of-Sight Magnetic Field in Active Region NOAA 11158

We study structure and dynamics of the turbulent photospheric magnetic field in the active region NOAA 11158 by characterizing the spatial and temporal scaling properties of the line-of-sight (LOS) component. Using high-resolution high-cadence LOS magnetograms from SDO/HMI, we measured power-law exponents $\alpha$ and $\beta$ describing wavenumber- ($k$) and frequency-domain ($f$) Fourier power spectra, respectively, and investigated their evolution during the passage of the active region through the field of view of HMI. Flaring active region NOAA 11158 produces an average one-dimensional spatial power spectral density that follows approximately a $k^{-2}$ power law -- a spectrum that suggests parallel MHD fluctuations in an anisotropic turbulent medium. In addition, we found that values of $\alpha$ capture systematic changes in the configuration of LOS photospheric magnetic field during flaring activity in the corona. Position-dependent values of the temporal scaling exponent $\beta$ showed that, on average, the core of the active region scales with $\beta >$ 3 surrounded by a diffusive region with an approximately $f^{-2}$-type spectrum. Our results indicate that only about 1 - 3 \% of the studied LOS photospheric magnetic flux displays $\beta\approx\alpha$, implying that Taylor's hypothesis of frozen-in-flow turbulence is typically invalid for this scalar in presence of turbulent photospheric flows. In consequence, both spatial and temporal variations of the plasma and magnetic field must be included in a complete description of the turbulent evolution of active regions.

Gaussian mixture model approximation of total spatial power spectral density for multiple incoherently distributed sources

Practically, the spatial power spectral density (PSD) of single or multiple incoherently distributed (ID) sources is often unknown, and the total spatial PSD is suitable to model the spatial distribution characteristic of signals if the number of multiple ID sources is also unknown. In this study, the Gaussian mixture model (GMM) is employed to characterise the total spatial PSD of multiple ID sources, and two algorithms are proposed to estimate the parameters of the GMM. The first one is the covariance fitting method for multiple ID sources with Gaussian PSD, and the other is the iterative expectation maximisation (EM) algorithm. Simulation studies demonstrate that the EM algorithm outperforms other methods in approximating the shape of the total spatial PSD, especially for small spatial spread.

Experimental investigation on beam smoothing by combined spectral dispersion and lens array technology

The experimental performance of beam smoothing by combined one-dimensional (1D) spectral dispersion and lens array (LA) technology is presented, as applied in the ninth beam of SG-II. Using 3\omega spectral dispersion with a bandwidth of 270 GHz and a line dispersion that is 24.9 times the beam's diffraction-limited width decreases the focal spot non-uniformity of 80% energy concentration from 46% to 17%. The multiple-beam interference properties of the LA are theoretically and experimentally validated by spatial power spectral density analysis. Peak-spectra suppression ratios of 20 and 10 dB are achieved in the dispersion and orthogonal directions, respectively.

Simulation of Bus Ride Comfort under the Excitation of Road Irregularity

Bus vibration is studied by the finite element method (FEM) base on bus structure model. The bus mathematical model of vertical vibration is established and the vibration response variables were deduced with the modal analysis method. The finite element (FE) model is established and decoupled. The transformational relation between spatial frequency displacement power spectral density (PSD) and temporal frequency displacement PSD and the sampling characteristics of the road irregularity PSD in numerical computation are discussed. Road irregularity load is modeled in software. The FE model is solved using modal analysis method and the acceleration PSD of each keypoint can be gained. Finally, a road test experiment is carried on to verify the simulation results. The example indicated that study on vehicle ride comford by FEM has instructive meaning.

Similarities Between Simulated Spatial Spectra of Scalp EEG, MEG and Structural MRI

Electrical dipoles oriented perpendicular to the cortical surface are the primary source of the scalp EEGs and MEGs. Thus, in particular, gyri and sulci structures on the cortical surface have a definite possibility to influence the EEGs and MEGs. This was examined by comparing the spatial power spectral density (PSD) of the upper portion of the human cortex in MRI slices to that of simulated scalp EEGs and MEGs. The electrical activity was modeled with 2,650 dipolar sources oriented normal to the local cortical surface. The resulting scalp potentials were calculated with a finite element model of the head constructed from 51 segmented sagittal MR images. The PSD was computed after taking the fast Fourier transform of scalp potentials. The PSD of the cortical contour in each slice was also computed. The PSD was then averaged over all the slices. This was done for sagittal and coronal view both. The PSD of EEG and MEG showed two broad peaks, one from 0.05 to 0.22 cycles/cm (wavelength 20–4.545 cm) and the other from 0.22 to 1.2 cycles/cm (wavelength 4.545–0.834 cm). The PSD of the cortex showed a broad peak from 0.08 to 0.32 cycles/cm (wavelength 12.5–3.125 cm) and other two peaks within the range of 0.32 to 0.9 cycles/cm (wavelength 3.125–1.11 cm). These peaks are definitely due to the gyri structures and associated larger patterns on the cortical surface. Smaller peaks in the range of 1–3 cycles/cm were also observed which are possibly due to sulci structures. These results suggest that the spatial information was present in the EEG and MEG at the spatial frequencies of gyri. This also implies that the practical Nyquist frequency for sampling scalp EEGs should be 3.0 cycles/cm and an optimal interelectrode spacing of about 3 mm is needed for extraction of cortical patterns from scalp EEGs in humans.

Semi-active suspension system simulation using Simulink

This paper describes a simulation design procedure aimed to achieve improved performance of the vehicle semi-active suspension. The issues related to the design of vehicle models with skyhook control are discussed. Three basic models with linear parameters are explained: quarter-, half- and full-car. The road profile is generated from a spatial power spectral density (PSD) to represent a typical road (based on ISO 8608 classification). The normalized root-mean-square values of sprung mass acceleration and tyre load forces are used to assess the vehicle ride comfort and handling performance based on five benchmark road profiles employed in industrial tests.

A Physically Constrained Maximum-Likelihood Method for Snapshot-Deficient Adaptive Array Processing

This paper presents a physically constrained maximum-likelihood (PCML) method for spatial covariance matrix and power spectral density estimation as a reduced-rank adaptive array processing algorithm. The physical constraints of propagating energy imposed by the wave equation and the statistical nature of the snapshots are exploited to estimate the ldquotruerdquo maximum-likelihood covariance matrix that is full rank and physically realizable. The resultant matrix may then be used in adaptive processing for interference cancellation and improved power estimation in nonstationary environments where the amount of available data is limited. Minimum variance distortionless response (MVDR) power estimates are computed for a given environment at different levels of snapshot support using the PCML method and several other reduced-rank techniques. The MVDR power estimates from the PCML method are shown to have less bias and lower standard deviation at a given level of snapshot support than any of the other reduced-rank methods used. Furthermore, the estimated power spectral density from the PCML method is shown to offer better low-level source detection than the MVDR power estimates.

Origin, structure, and role of background EEG activity. Part 4: Neural frame simulation

Objective To develop a method for simulating background EEG based on the premise that the self-organized activity from synaptic interaction among populations of neurons creates sustained fluctuations that can be modeled with the filtered output of a random number generator. Methods The logarithm of the amplitude of activity was weighted in accordance with 1/ f, the log frequency in both temporal (PSD T) and spatial (PSD X) power spectral densities. The activity was spatially smoothed by volume conduction. Further deviation from full randomness was by sustained spatial coherence averaging 25% of total power. The departure from the background state to an active state, as seen in the awake EEG, was simulated by adding segments that were 90% correlated while attenuating by 50% the uncorrelated background activity in those segments. Spatial amplitude modulation was imposed on the correlated noise to create signals that simulated AM patterns. Results The statistical properties of the EEG that were replicated ( Freeman, 2004a,b, 2005) included the PSD T, PSD X, point spread function (PSF), partitioning of the variance with PCA, and the percentages of correct classification of AM patterns. Conclusions The origin of background EEG was traced to self-sustaining mutual excitation among pyramidal cells creating stable noise that was filtered by self-organized criticality to give 1/ f 2 PSD, by inhibitory feedback to give oscillations in the classic clinical bands, and by volume conduction to give smoothing. The essential change that identified a frame in EEG was transient synchrony by phase transition among cortical populations in beta and gamma bands of the PSD T. Significance This simulation can provide test data with which to optimize techniques for noninvasively extracting information from the EEG for diagnosis and treatment evaluation of neuropsychiatric disorders and for operation by paraplegics of prosthetic devices.

Reconstruction of ultrasonic fields by deconvolving the hydrophone aperture effects: I. Theory and simulation

In many ultrasound applications––especially in the area of medical diagnostics––conventional sensors and particularly piezoelectric hydrophones do not comply with the requirements for “point detectors”. It therefore suggests itself to develop methods, which allow ultrasonic fields to be reconstructed from the spatially averaged measurement values with the necessary spatial resolution. In this publication, this task will be treated as an inverse problem. It will be shown by simulations that it is possible with the aid of reconstruction methods to invert the aperture effects of the sensor. Three different reconstruction methods are tested: (1) Wiener's method furnishes the best approximation of the field to be reconstructed in the sense of the minimum mean square error. (2) The power spectral equalization method is based on equalization between the spatial power spectral densities of the field to be reconstructed and that of its estimated value. (3) With the maximum a-posteriori method, the maximum a-posteriori probability density function for the reconstructed field is searched. The results of this paper show that the quality of reconstruction depends on the signal-to-noise ratio (SNR) and the spatial frequency bandwidth of the sound field investigated. Good quality reconstructions were obtained at correlation coefficients larger than 0.995 and an SNR of at least 40 dB. In general, the quality of reconstruction decreases with decreasing SNR and is unacceptable at SNR=20 dB. The quality also decreases with decreasing axial distance from the source that is with increasing spectral frequency bandwidth.

2D spectral and turbulence length scale estimation with PIV

A new method to apply spatial two-dimensional power spectral density (2D PSD) analysis to the data measured with Particle Image Velocimetry (PIV) has been introduced. Applying the method to a set of the velocity vector fields characteristic turbulence length scales can be estimated. In this method the computation of 2D PSD has been performed to two kinds of pre-processed data. In the first set, the local average has been spatially subtracted (Spatial decomposition) and in the second set the time-average has been subtracted (Reynolds decomposition). In the computation of 2D PSD the 2D FFT with the variance scaling has been used. Besides 2D spectral analysis this paper uses the distribution analysis of the various turbulence quantities and a structure analysis method to estimate the dimensions of coherent structures in the flow. Another method to analyse turbulence length scales is the estimation of the spatial 2D Auto Correlation Coefficient Function (2D ACCF). All these methods applied side by side to the PIV data increase the understanding of the turbulence, its scales and the nature of the coherent structures.