Space-time Correlation Function Research Articles

Space-time modeling of soil moisture

Abstract A physically derived space-time mathematical representation of the soil moisture field is carried out via the soil moisture balance equation driven by stochastic rainfall forcing. The model incorporates spatial diffusion and in its original version, it is shown to be unable to reproduce the relative fast decay in the spatial correlation functions observed in empirical data. This decay resulting from variations in local topography as well as in local soil and vegetation conditions is well reproduced via a jitter process acting multiplicatively over the space-time soil moisture field. The jitter is a multiplicative noise acting on the soil moisture dynamics with the objective to deflate its correlation structure at small spatial scales which are not embedded in the probabilistic structure of the rainfall process that drives the dynamics. These scales of order of several meters to several hundred meters are of great importance in ecohydrologic dynamics. Properties of space-time correlation functions and spectral densities of the model with jitter are explored analytically, and the influence of the jitter parameters, reflecting variabilities of soil moisture at different spatial and temporal scales, is investigated. A case study fitting the derived model to a soil moisture dataset is presented in detail.

Noise from a rotor ingesting a thick boundary layer and relation to measurements of ingested turbulence

Haystacked peaks in rotor noise spectra are produced by the correlated unsteady loading on neighboring blades. This noise can be predicted with knowledge of the turbulent inflow and blade response function, but this is not trivial, especially for complex turbulent inflows that are both inhomogeneous and anisotropic. This paper details the radiated noise and direct measurement of the unsteady upwash correlation in the rotating frame of a 10-bladed, 457mm diameter rotor immersed in planar wall boundary layers of different thickness at non-thrusting and thrusting advance ratios. At low thrust conditions, the measured upwash correlation can be predicted using the fixed frame space-time correlation function of the undisturbed inflow. However, as the advance ratio is lowered, predictions progressively deviate from measurements. This is shown to be due to both the distortion of the approaching turbulence and the formation of a separation region on the wall beneath the rotor. At these low advance ratios, haystacks at the blade passage frequency and harmonics are observed in the blade-to-blade upwash coherence spectra at spanwise locations near the blade tips. Also, a lateral contraction of the turbulence is not observed in measurement of the spanwise coherence with increasing thrust. Finally, increasing the boundary layer thickness increases spectral levels of the radiated noise at higher advance ratios near zero thrust by a factor similar to the increase in boundary layer thickness. At low advance ratios, peaks in the measured noise have the same approximate magnitude for similar rotational and freestream velocities regardless of the boundary layer thickness as these are likely produced by interaction with vortex structures in the separated region.

Midpoint Distribution of Directed Polymers in the Stationary Regime: Exact Result Through Linear Response

We obtain an exact result for the midpoint probability distribution function (pdf) of the stationary continuum directed polymer, when averaged over the disorder. It is obtained by relating that pdf to the linear response of the stochastic Burgers field to some perturbation. From the symmetries of the stochastic Burgers equation we derive a fluctuation-dissipation relation so that the pdf gets given by the stationary two space-time points correlation function of the Burgers field. An analytical expression for the latter was obtained by Imamura and Sasamoto [2013], thereby rendering our result explicit. In the large length limit that implies that the pdf is nothing but the scaling function $f_{{\rm KPZ}}(y)$ introduced by Pr\"ahofer and Spohn [2004]. Using the KPZ-universality paradigm, we find that this function can therefore also be interpreted as the pdf of the position y of the maximum of the Airy process minus a parabola and a two-sided Brownian motion. We provide a direct numerical test of the result through simulations of the Log-Gamma polymer.

A Floquet model for the many-body localization transition

The nature of the dynamical quantum phase transition between the many-body localized (MBL) phase and the thermal phase remains an open question, and one line of attack on this problem is to explore this transition numerically in finite-size systems. To maximize the contrast between the MBL phase and the thermal phase in such finite-size systems, we argue one should choose a Floquet model with no local conservation laws and rapid thermalization to "infinite temperature" in the thermal phase. Here we introduce and explore such a Floquet spin chain model, and show that standard diagnostics of the MBL-to-thermal transition behave well in this model even at modest sizes. We also introduce a physically motivated spacetime correlation function which peaks at the transition in the Floquet model, but is strongly affected by conservation laws in Hamiltonian models.

Performance Investigation of Spatial Modulation Systems Under Non-Stationary Wideband High-Speed Train Channel Models

In this paper, the bit error rate (BER) performance of a new multiple-input-multiple-output technique, named spatial modulation (SM), is studied under a novel non-stationary wideband high-speed train (HST) channel model in different scenarios. Time-varying parameters obtained from measurement results are used to configure the channel model to make all results more realistic. A novel statistic property called the stationary interval in terms of the space-time correlation function is proposed to describe the channel model’s time-varying behavior. The accurate theoretical BER expression of SM systems is derived under the time-varying wideband HST channel model with the non-ideal channel estimation assumption. The simulation results demonstrate that the BER performance of SM systems shows a time-varying behavior due to the non-stationary property of the employed HST channel model. The system performance can maintain a relative stationary status within the specified stationary interval. It can also be observed that the BER performance of SM systems under the HST channel model is mainly affected by the correlation between sub-channels, inter-symbol-interference, Doppler shift, and channel estimation errors.

Ensemble Space-Time Correlation of Plasma Turbulence in the Solar Wind.

Single point measurement turbulence cannot distinguish variations in space and time. We employ an ensemble of one- and two-point measurements in the solar wind to estimate the space-time correlation function in the comoving plasma frame. The method is illustrated using near Earth spacecraft observations, employing ACE, Geotail, IMP-8, and Wind data sets. New results include an evaluation of both correlation time and correlation length from a single method, and a new assessment of the accuracy of the familiar frozen-in flow approximation. This novel view of the space-time structure of turbulence may prove essential in exploratory space missions such as Solar Probe Plus and Solar Orbiter for which the frozen-in flow hypothesis may not be a useful approximation.

An efficient fully atomistic potential model for dense fluid methane

A fully atomistic model aimed to obtain a general purpose model for the dense fluid methane is presented. The new optimized potential for liquid simulation (OPLS) model is a rigid five site model which consists of five fixed point charges and five Lennard-Jones centers. The parameters in the potential model are determined by a fit of the experimental data of dense fluid methane using molecular dynamics simulation. The radial distribution function and the diffusion coefficient are successfully calculated for dense fluid methane at various state points. The simulated results are in good agreement with the available experimental data shown in literature. Moreover, the distribution of mean number hydrogen bonds and the distribution of pair-energy are analyzed, which are obtained from the new model and other five reference potential models. Furthermore, the space–time correlation functions for dense fluid methane are also discussed. All the numerical results demonstrate that the new OPLS model could be well utilized to investigate the dense fluid methane.

Spatiotemporal generalization of the Harris criterion and its application to diffusive disorder.

We investigate how a clean continuous phase transition is affected by spatiotemporal disorder, i.e., by an external perturbation that fluctuates in both space and time. We derive a generalization of the Harris criterion for the stability of the clean critical behavior in terms of the space-time correlation function of the external perturbation. As an application, we consider diffusive disorder, i.e., an external perturbation governed by diffusive dynamics, and its effects on a variety of equilibrium and nonequilibrium critical points. We also discuss the relation between diffusive disorder and diffusive dynamical degrees of freedom in the example of model C of the Hohenberg-Halperin classification and comment on Griffiths singularities.

Spatiotemporal correlation of high-performance military aircraft jet noise

Correlation analyses of pressure measurements on a ground-based array of microphones of noise from a tethered high-performance military aircraft provide insights into the sound field variation with position and engine conditions which are fundamental in the continued development of more complete jet noise models. Time-scaled, single-point (auto)correlation functions confirm that to the side of the nozzle exit, the temporal correlation envelope decays very rapidly, whereas the envelope decays more slowly in the maximum radiation region and farther downstream. Two-point space-time (cross) correlation functions confirm that noise from a single engine operating at intermediate power is more similar to that from heated, laboratory-scale jets, whereas additional features seen at military power and afterburner are unique, including a feature which is likely related to the dual directivity lobe observed in the far field of military aircraft. A complementary coherence analysis provides estimates of spatial coherence lengths as a function of frequency and location. Field coherence lengths are utilized in analyzing coherence lengths of equivalent source distributions obtained from applying DAMAS-C to the ground-based array data. The cumulative results of these investigations provide a full-scale military jet noise benchmark that should be considered when evaluating laboratory-scale jet studies and simulations of jet noise.

Strong Markov property of determinantal processes with extended kernels

Noncolliding Brownian motion (Dyson’s Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space–time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.

Cores of Dirichlet forms related to random matrix theory

We prove the sets of polynomials on configuration spaces are cores of Dirichlet forms describing interacting Brownian motion in infinite dimensions. Typical examples of these stochastic dynamics are Dyson's Brownian motion and Airy interacting Brownian motion. Both particle systems have logarithmic interaction potentials, and naturally arise from random matrix theory. The results of the present paper will be used in a forth coming paper to prove the identity of the infinite-dimensional stochastic dynamics related to the random matrix theories constructed by apparently different methods: the method of space-time correlation functions and that of stochastic analysis.

A ML-Based Framework for Joint TOA/AOA Estimation of UWB Pulses in Dense Multipath Environments

We present a joint estimator of the time of arrival (TOA) and angle of arrival (AOA) for impulse radio ultrawideband (UWB) systems in which an antenna array is employed at the receiver. The proposed method consists of two steps: 1) preliminary estimation of the TOA and the average power delay profile (APDP) using energy-based threshold crossing and log-domain least-squares fitting, respectively; and 2) joint TOA refinement and AOA estimation by local 2-D maximization of a log-likelihood function (LLF) that employs the preliminary estimates from the first step. The derivation of the LLF relies on an original formulation in which the superposition of images from secondary paths is modeled as a Gaussian random process, whose second-order statistical properties are characterized by a wideband space-time correlation function. In addition to the APDP, this function incorporates a special gating mechanism to represent the onset of the secondary paths, thereby leading to a novel form of the LLF. Closed-form expressions for the Cramer-Rao bound on the variance of the TOA and AOA estimators are also derived, which formally take into account pulse overlap through this gating mechanism. In simulation experiments based on multipath UWB channel models featuring both diffuse and directional image fields, our approach exhibits superior performance to that of a competing scheme from the recent literature.

Space-time correlation of acoustic signals in a turbulent atmosphere

Scattering by atmospheric turbulence diminishes the correlation, in both space and time, of acoustic signals. This decorrelation subsequently impacts beamforming, averaging, and other techniques for enhancing signal-to-noise ratio. Space-time correlation can be measured directly with a phased microphone array. In this paper, a general theory for the space-time correlation function is presented. The atmospheric turbulence is modeled using the von Karman spatial spectra of temperature and wind velocity fluctuations and locally frozen turbulence (i.e., the Taylor’s frozen turbulence hypothesis with convection velocity fluctuations). The theory developed is employed to calculate and analyze the spatial and temporal correlation of acoustic signals for typical regimes of an unstable atmospheric boundary layer, such as mostly cloudy or sunny conditions with light, moderate, or strong wind. The results obtained are compared with available experimental data.

Impact of Moving Scatterers on Vehicle-to-Vehicle Narrow-Band Channel Characteristics

A 3-D geometrical propagation model that includes both stationary and moving scatterers around the transmitter and the receiver for narrow-band multiple-input-multiple-output (MIMO) vehicle-to-vehicle (V-to-V) communications is proposed. Based on the geometrical model, a 3-D parametric reference model for narrow-band MIMO V-to-V multipath fading channels is developed. From the reference model, the corresponding space-time correlation function (ST-CF) and space-Doppler power spectral density (SD-PSD) are derived for a 3-D nonisotropic scattering environment. Finally, the theoretical results are compared with measured data in an urban surface street and Interstate highway road surfaces. The close agreement between the analytically and empirically obtained channel statistics confirms the utility of the proposed model and shows the importance of including moving scatterers in propagation models.

Space-time correlations in turbulent Rayleigh-Bénard convection

The recent development of the elliptic model (He, et al. Phy. Rev. E, 2006), which predicts that the space-time correlation function C u (r, τ) in a turbulent flow has a scaling form C u (r E , 0) with r E being a combined space-time separation involving spatial separation r and time delay τ, has stimulated considerable experimental efforts aimed at testing the model in various turbulent flows. In this paper, we review some recent experimental investigations of the space-time correlation function in turbulent Rayleigh-Benard convection. The experiments conducted at different representative locations in the convection cell confirmed the predictions of the elliptic model for the velocity field and passive scalar field, such as local temperature and shadowgraph images. The understanding of the functional form of C u (r, τ) has a wide variety of applications in the analysis of experimental and numerical data and in the study of the statistical properties of small-scale turbulence. A few examples are discussed in the review.

Simulating translation-induced laser speckle dynamics in photon Doppler velocimetry.

Historically, single-beam optical velocimetry has been limited to measuring only the component of velocity along the beam. However, theoretical work and recent experimental results have shown that laser speckle dynamics may be exploited to measure lateral motion, thereby gaining information about surface dynamics across an additional degree of freedom. In the use of photon Doppler velocimetry (PDV), this new information is considered "free" in that it is already contained within the PDV signal, needing only to be extracted and interpreted correctly. In this manuscript, we relate speckle dynamics to the lateral motion of a planar scattering surface in the PDV coordinate system via the space-time correlation function of the diffracted electric field. Next, we relate the characteristic time scale of speckle intensity fluctuations in the PDV signal to the rate of lateral surface translation and to parameters characterizing the optical probe. Analytical results are compared with a numerical simulation and found to be in close agreement.

Expressing entropy globally in terms of (4D) field-correlations

We have expressed the entropy of a scalar field directly in terms of its spacetime correlation function, assuming that the higher correlators are of "Gaussian" form. The resulting formula associates an entropy to any spacetime region, and when this region is globally hyperbolic with Cauchy surface then the entropy can be interpreted as the entropy of the reduced density-matrix belonging to the Cauchy surface. One acquires, in particular, a new expression for the entropy of entanglement across an event-horizon. Thanks to its spacetime character, this expression makes sense in a causal set as well as in a continuum spacetime.

Stochastic Analysis of Wideband Near-Field Emissions From Dipole Antennas and Integrated Circuits

A statistical method is developed for characterizing wideband emissions from planar antennas and circuits containing multiple radiating elements. It is shown how the space-time and space-frequency correlation functions for source currents can be deduced from near-field measurements. These correlations produce additional spectral distortion of the emitted near field, over and above the far-field spectra of individual elements and their combined spectrum for uncorrelated sources. For an arbitrary configuration and number of emitting dipoles, the contribution to this distortion by pairwise correlations is fully calculable from the second-order covariance theory. Simulation results for a 2 ×2 dipole array and near-field measurements on an L-shaped microstrip antenna with wide-band excitation demonstrate and validate the feasibility of the method and the theoretically predicted spectral distortion based on pairwise correlations only.

A Semiempirical MIMO Channel Model in Obstructed Viaduct Scenarios on High-Speed Railway

A semiempirical multiple-input multiple-output (MIMO) channel model is proposed for high-speed railway (HSR) viaduct scenarios. The proposed MIMO model is based on the combination of realistic single-input single-output (SISO) channel measurement results and a theoretical geometry-based stochastic model (GBSM). Temporal fading characteristics involvingK-factor and Doppler power spectral density (PSD) are derived from the wideband measurement under an obstructed viaduct on Zhengzhou-Xi’an HSR in China. The GBSM composed of a one-ring model and an elliptical model is employed to describe the entire propagation environment. Environment-related parameters in the GBSM are determined by the measured temporal fading properties. And a close agreement is achieved between the model results and measured data. Finally, a deterministic simulation model is established to perform the analysis of the space-time correlation function, the space-Doppler PSD, and the channel capacity for the measured scenario. This model is more realistic and particularly beneficial for the performance evaluation of MIMO systems in HSR environments.

Speckle suppression by doubly scattering systems

Speckle suppression in a two-diffuser system is examined. An analytical expression for the speckle space-time correlation function is derived, so that the speckle suppression mechanism can be investigated statistically. The grain size of the speckle field illuminating the second diffuser has a major impact on the speckle contrast after temporal averaging. It is shown that, when both the diffusers are rotating, the one with the lower rotating speed determines the period of the speckle correlation function. The coherent length of the averaged speckle intensity is shown to equal the mean speckle size of the individual speckle pattern before averaging. Numerical and experimental results are presented to verify our analysis in the context of speckle reduction.