statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

analysis of subspace fitting and ml techniques for parameter estimation from sensor array data it is shown that the multidimensional signal subspace method termed weighted subspace fitting wsf is asymptotically efficient this results in a novel compact matrix expression for the cramer rao bound crb on the estimation error variance the asymptotic analysis of the maximum likelihood ml and wsf methods is extended to deterministic emitter signals the asymptotic properties of the estimates for this case are shown to be identical to the gaussian emitter signal case i e independent of the actual signal waveforms conclusions concerning the modeling aspect of the sensor array problem are drawn

statistics of intense turbulent vorticity events we investigate statistical properties of vorticity fluctuations in fully developed turbulence which are known to exhibit a strong intermittent behavior taking as the starting point the navier stokes equations with a random force term correlated at large scales we obtain in the high reynolds number regime a closed analytical expression for the probability distribution function of an arbitrary component of the vorticity field the central idea underlying the analysis consists in the restriction of the velocity configurational phase space to a particular sector where the rate of strain and the rotation tensors can be locally regarded as slow and fast degrees of freedom respectively this prescription is implemented along the martin siggia rose functional framework whereby instantons and perturbations around them are taken into account within a steepest descent approach

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

on gauss quadrature formulas with equal weights it is well known that the chebyshev weight function 1 x2 1 2 is the only weight function up to a linear transformation for which then point gauss quadrature formula has equal weights for alln in this paper we describe all weight functions for which thenm point gauss quadrature formula has equal weights for alln wherem is fixed

singularities and the distribution of density in the burgers adhesion model abstract we are interested in the tail behavior of the pdf of mass density within the one and d dimensional burgers adhesion model used e g to model the formation of large scale structures in the universe after baryon photon decoupling we show that large densities are localized near kurtoparabolic singularities residing on space time manifolds of codimension two d 2 or higher d 3 for smooth initial conditions such singularities are obtained from the convex hull of the lagrangian potential the initial velocity potential minus a parabolic term the singularities contribute universal power law tails to the density pdf when the initial conditions are random in one dimension the singularities are preshocks nascent shocks whereas in two and three dimensions they persist in time and correspond to boundaries of shocks in all cases the corresponding density pdf has the exponent 7 2 originally proposed by e et al phys rev lett 78 1997 1904 for the pdf of velocity gradients in one dimensional forced burgers turbulence we also briefly consider models permitting particle crossings and thus multi stream solutions such as the zel dovich approximation and the jeans vlasov poisson equation with single stream initial data they have singularities of codimension one yielding power law tails with exponent 3

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a preliminary study of the burgers equation with symbolic computation a novel approach based on recursive symbolic computation is introduced for the approximate analytic solution of the burgers equation once obtained appropriate numerical values can be inserted into the symbolic solution to explore parametric variations the solution is valid for both inviscid and viscous cases covering the range of reynolds number from 500 to infinity whereas current direct numerical simulation dns methods are limited to reynolds numbers no greater than 4000 what further distinguishes the symbolic approach from numerical and traditional analytic techniques is the ability to reveal and examine direct nonlinear interactions between waves including the interplay between inertia and viscosity thus preliminary efforts suggest that symbolic computation may be quite effective in unveiling the anatomy of the myriad interactions that underlie turbulent behavior however due to the tendency of nonlinear symbolic operations to produce combinatorial explosion future efforts will require the development of improved filtering processes to select and eliminate computations leading to negligible high order terms indeed the initial symbolic computations present the character of turbulence as a problem in combinatorics at present results are limited in time evolution but reveal the beginnings of the well known saw tooth waveform that occurs in the inviscid case i e re future efforts will explore more fully developed 1 d flows and investigate the potential to extend symbolic computations to 2 d and 3 d potential applications include the development of improved subgrid scale sgs parameterizations for large eddy simulation les models and studies that complement dns in exploring fundamental aspects of turbulent flow behavior

tracking of signal subspace projectors a new subspace tracking approach that directly operates on the projection matrix onto the signal subspace instead of tracking its unitary eigenbasis is proposed therefore the difficulties arising from introducing a calculus on the manifold of projection matrices are overcome by a proper parameterization of the projectors simulation results are presented from the field of angular frequency retrieval in array signal processing

on the number of signals resolvable by a uniform linear array an algebraic limitation on the maximum number of directions of arrival of plane waves that can be resolved by a uniform linear sensor array is studied achievable lower and upper bounds are derived on that number as a function of the number of elements in the array number of snapshots and the rank of the source sample correlation matrix the signals are assumed narrow band and of identical and known center frequency the results are also applicable in the coherent signal case and when directions of arrival are estimated from few snapshots while in the multiple snapshot case the lower bounds coincide with known asymptotic results the upper bounds indicate the potential for resolving more signals than by present methods of array processing

instanton theory of burgers shocks and intermittency a lagrangian approach to burgers turbulence is carried out along the lines of the field theoretical martin siggia rose formalism of stochastic hydrodynamics we derive from an analysis based on the hypothesis of unbroken galilean invariance the asymptotic form of the probability distribution function of negative velocity differences the origin of burgers intermittency is found to rely on the dynamical coupling between shocks identified to instantons and noncoherent background fluctuations which char22 then char22 cannot be discarded in a consistent statistical description of the flow

phase diagram of optimal paths we show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media such as directed percolation invasion percolation and directed and nondirected spanning polymers we also introduce a simple and efficient algorithm which solves the d dimensional model numerically in o n 1 df d steps where df is the fractal dimension of the path using extensive simulations in two dimensions we identify the phase boundaries of the directed polymer universality class a new strong disorder phase occurs where the optimum paths are self affine with parameter dependent scaling exponents furthermore the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines

universality classes for self similarity of noiseless multidimensional burgers turbulence and interface growth the present work is devoted to the evolution of random solutions of the unforced burgers and kpz equations in d dimensions in the limit of vanishing viscosity we consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell we show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function three classes of initial distribution leading to self similar evolution are identified a distributions with a power law tail b compactly supported potential c stretched exponential tails in class c we find that the mean potential mean height of the surface increases logarithmically with time and the turbulence energy e t mean square gradient of the surface decays as 1 t times a logarithmic correction in classes a and b we find that the changes in the mean potential and energy have a power law time dependence in class c the roughness of the surface measured by its mean square gradient may either decrease or increase with time we discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the burgers turbulence

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

calibrating nested sensor arrays with model errors we consider the problem of direction of arrival doa estimation based on a nonuniform linear nested array which is known to provide o n 2 degrees of freedom dofs using only n sensors both subspace based and sparsity based algorithms require certain modeling assumptions e g exactly known array geometry including sensor gain and phase in practice however the actual sensor gain and phase are often perturbed from their nominal values which disrupts the existing doa estimation algorithms in this paper we investigate the self calibration problem for perturbed nested arrays proposing corresponding robust algorithms to estimate both the model errors and the doas the partial toeplitz structure of the covariance matrix is employed to estimate the gain errors and the sparse total least squares stls is used to deal with the phase error issue in addition we provide the cramer rao bound crb to analyze the robustness of the estimation performance of the proposed approaches furthermore we extend the calibration strategies to general nonuniform linear arrays numerical examples are provided to verify the effectiveness of the proposed strategies

doa estimation and detection in colored noise using additional noise only data in a typical array processing scenario noise acting on the array can not be assumed spatially white it is in many cases necessary to use quiet periods when only noise is received to estimate the noise covariance if estimation of the signal parameters and noise covariance is performed jointly performance can be improved this is especially true when stationarity considerations limit the amount of available valid noise only data an asymptotically valid approximative maximum likelihood method aml for the estimation problem is derived in this work the resulting criterion is when concentrated with respect to the signal parameters relatively simple in numerical experiments aml shows promising small sample performance compared to earlier methods the criterion function is also well suited for numerical optimization the new criterion function allows for the development of a novel mode like non iterative estimation procedure if the array belongs to the important class of uniform linear arrays the resulting procedure retains the asymptotic properties of maximum likelihood and numerical simulations indicate superior threshold performance when compared to an optimally weighted subspace fitting wsf formulation of mode for the detection problem no method has been presented that takes the unknown noise covariance into account here a well known detection scheme for wsf is extended to work in this scenario as well the derivations of this scheme further stress the importance of using the correct weighting in wsf when the noise covariance is unknown it is also shown that the minimum value of the criterion function associated with aml can be used for the detection purpose numerical experiments indicate very promising performance for the aml detection scheme

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

waves weighted average of signal subspaces for robust wideband direction finding existing algorithms for wideband direction finding are mainly based on local approximations of the gaussian log likelihood around the true directions of arrival doas assuming negligible array calibration errors suboptimal and costly algorithms such as classical or sequential beamforming are required to initialize a local search that eventually furnishes doa estimates this multistage process may be nonrobust in the presence of even small errors in prior guesses about angles and number of sources generated by inherent limitations of the preprocessing and may lead to catastrophic errors in practical applications a new approach to wideband direction finding is introduced and described the proposed strategy combines a robust near optimal data adaptive statistic called the weighted average of signal subspaces waves with an enhanced design of focusing matrices to ensure a statistically robust preprocessing of wideband data the overall sensitivity of waves to various error sources such as imperfect array focusing is also reduced with respect to traditional cssm algorithms as demonstrated by extensive monte carlo simulations

analysis of the asymptotic relative efficiency of the music algorithm an analytical performance evaluation of the errors of the direction of arrival estimates obtained by the music algorithm for uncorrelated sources is provided explicit asymptotic formulas are derived for the means and the covariance of the estimates the covariances are then compared to the cramer rao lower bound it is shown that for a single course the music algorithm is asymptotically efficient for multiple sources the algorithm is not efficient in general however it approaches asymptotic efficiency when the snrs signal to noise ratios of all sources tend to infinity it is illustrated by several test cases that the relative efficiency of the music algorithm is nearly one under a wide range of parameter variations the analytic performance evaluation thus confirms empirical evidence to the excellent performance of the music algorithm for narrowband signals

on unique localization of multiple sources by passive sensor arrays necessary and sufficient conditions are derived for the unique localization of narrowband sources having the same known center frequency by passive sensor arrays the conditions specify the maximum number of sources that can be uniquely localized by a general array that satisfies some mild geometric constraints the conditions are expressed in terms of the number of sensors and the rank of the correlation matrix of the sources two different conditions are presented the first guarantees uniqueness for every batch of sampled data the second which is weaker guarantees uniqueness for almost every batch of sampled data with the exception of a set of batches of measure zero it is shown that a condition that is slightly weaker than the second one is also necessary

music for multidimensional spectral estimation stability and super resolution this paper presents a performance analysis of the multiple signal classification music algorithm applied on d dimensional single snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain inspired by the matrix pencil form we construct a d fold hankel matrix from the measurements and exploit its vandermonde decomposition in the noiseless case music amounts to identifying a noise subspace evaluating a noise space correlation function and localizing frequencies by searching the s smallest local minima of the noise space correlation function in the noiseless case 2s d measurements guarantee an exact reconstruction by music as the noise space correlation function vanishes exactly at true frequencies when noise exists we provide an explicit estimate on the perturbation of the noise space correlation function in terms of noise level dimension d the minimum separation among frequencies the maximum and minimum amplitudes while frequencies are separated by 2 rayleigh length rl at each direction as a by product the maximum and minimum non zero singular values of the multidimensional vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes under the 2 rl separation condition if noise is i i d gaussian we show that perturbation of the noise space correlation function decays like sqrt log bf n bf n as the sample size bf n increases when the separation among frequencies drops below 2 rl our numerical experiments show that the noise tolerance of music obeys a power law with the minimum separation of frequencies

performance of high resolution frequencies estimation methods compared to the cramer rao bounds an explicit expression for the cramer rao bounds crbs and a calculation of the statistical perturbation of the covariance matrix due to additive noise are presented the results are applied to a statistical efficiency analysis of the main frequency estimation methods based on eigenvalue decomposition for the covariance matrix in order to characterize the perturbation of the signal subspace only the component of the perturbation of the eigenvectors orthogonal to the subspace is considered this gives a simpler and more significant form of the error covariance the treatment includes the cases of forward backward and moving averages the crb and estimation variances are calculated in the presence of additive random noise but for a given set of amplitudes characterized by their sample covariance matrix this approach is more realistic for the evaluation of efficiency in the small sample case

consistency and mse performance of music based doa of a single source in white noise with randomly missing data music is a popular algorithm for estimating the direction of arrival doa in array signal processing applications in this paper we analyze the performance of the music algorithm for a single source system in the presence of noisy and missing data when only a random subset of the entries in the data matrix are observed we prove consistency of the doa estimate when signal from a single source is impinging on low coherence arrays and derive an analytic expression for the mean squared error mse performance of music for the case of uniform linear arrays in the large array and relatively large sample setting our analysis is mathematically justified in both the sample rich and deficient regimes the expression for the mse is a simple function of array geometry signal to noise ratio snr the fraction of entries observed and the ratio of the number of sensors to number of snapshots we derive a phase transition threshold which separates a regime where music is consistent from a regime where music is inconsistent this threshold depends upon the snr the probability of observing entries in the data matrix and number of sensors and snapshots in a simple manner which we make explicit

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

extended derivations of music in the presence of steering vector errors we present two extensions of music to the case in which the steering vector is imprecisely known and show that the extended methods lead to the same direction of arrival doa estimates as music this somewhat surprising result provides a more general motivation of music than those currently available and shows that music possesses a certain degree of inherent robustness to steering vector errors

orthogonality of residual polynomials used in minimax polynomial preconditioning a polynomial from bbb p n bbb p n the set of polynomials of degree less or equal n is called minimax residual polynomial on a compact set e subset bbb r if it has least max norm on e among all polynomials from bbb p n with fixed lowest coefficient or with two fixed lowest coefficients it is pointed out that recently published results on orthogonality of minimax residual polynomials on two intervals by h jiang 5 are direct consequences of results of the author on orthogonality properties of classical minimal polynomials with respect to the max norm in fact as is demonstrated even more general and stronger results hold

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

detection of the number of coherent signals by the mdl principle an approach is presented to the problem of detecting the number of sources impinging on a passive sensor array that is based on j rissanen s 1983 minimum description length mdl principle the approach is applicable to any type of sources including the case of sources which are fully correlated referred to as the coherent signals case two slightly different detection criteria are derived both requiring the estimation of the locations of the sources the first is tailored to the detection problem per se whereas the second is tailored to the combined detection estimation problem consistency of the two criteria is proved and their performance is demonstrated by computer simulations

application of cumulants to array signal processing v sensitivity issues for pt iv see ibid vol 45 p 2265 76 1997 virtual esprit vespa imposes a slight constraint on the array i e only a pair of identical sensors are required we follow the classical procedure to define the sensitivity of vespa with respect to each model parameter and derive closed forms for the sensitivity formulas we justify our derivations by comparing the theoretical formulas with the results from monte carlo simulations additionally we demonstrate by simulations that vespa is more robust to model mismatches than esprit in most situations

performance prediction of maximum likelihood direction of arrival estimation in the presence of modeling errors this paper provides new analytic expressions for the root mean square rms error and bias of the maximum likelihood ml direction of arrival doa estimator in the presence of steering vector modeling errors previous work has provided a first order approximation of these performance measures which is valid for small modeling errors in order to take into account larger errors and provide tools for designing an experimental setup a more accurate but still easy to use performance analysis is necessary for such an investigation the doa estimation errors are expressed as a hermitian form with a stochastic vector composed of the modeling errors closed form expressions relating the bias and rms errors to the statistical moments of the modeling error are then deduced from the statistics of this hermitian form simulations are provided to illustrate the usefulness of the theoretical results

decaying turbulence in the generalised burgers equation we consider the generalised burgers equation frac partial u partial t f u frac partial u partial x nu frac partial 2 u partial x 2 0 t geqq 0 x in s 1 where f is strongly convex and is small and positive we obtain sharp estimates for sobolev norms of u upper and lower bounds differ only by a multiplicative constant then we obtain sharp estimates for the dissipation length scale and the small scale quantities which characterise the decaying burgers turbulence i e the structure functions and the energy spectrum the proof uses a quantitative version of an argument by aurell et al j fluid mech 238 467 486 1992 note that we are dealing with decaying as opposed to stationary turbulence thus our estimates are not uniform in time however they hold on a time interval t1 t2 where t1 and t2 depend only on f and the initial condition and do not depend on the viscosity these results allow us to obtain a rigorous theory of the one dimensional burgers turbulence in the spirit of kolmogorov s 1941 theory in particular we obtain two results which hold in the inertial range on one hand we explain the bifractal behaviour of the moments of increments or structure functions on the other hand we obtain an energy spectrum of the form k 2 these results remain valid in the inviscid limit

parameter estimation of superimposed signals using the em algorithm a computationally efficient algorithm for parameter estimation of superimposed signals based on the two step iterative em estimate and maximize with an e step and an m step algorithm is developed the idea is to decompose the observed data into their signal components and then to estimate the parameters of each signal component separately the algorithm iterates back and forth using the current parameter estimates to decompose the observed data better and thus increase the likelihood of the next parameter estimates the application of the algorithm to the multipath time delay and multiple source location estimation problems is considered

asymptotically perfect wideband focusing of multiring circular arrays the concept of coherent focusing of sensor arrays introduced by wang and kaveh led to the development of high performance and computationally efficient algorithms for wideband direction finding and beamforming nonetheless the quality of focusing depends on the understanding and the proper exploitation of specific array manifold properties circular arrays exhibit uniform isotropic performance over the entire azimuthal range and allow the use of fast algorithms originally developed for uniform linear arrays by decomposing the manifold into circular harmonics coherent wideband focusing of circular arrays suffers from ambiguities noise warping and numerical ill conditioning in this work it is shown that a fast orthonormal beamspace transformation combining the responses of two concentric rings can perfectly focus wideband sources at all azimuth angles in the circular harmonic domain in the limit of infinite number of sensors the information loss after focusing is minimized through an analytical approach the proposed focusing scheme is computationally very efficient and can be directly extended to multiring circular arrays that are arranged into a set of nested two ring subarrays to cover very large bandwidths

weighted subspace fitting for general array error models model error sensitivity is an issue common to all high resolution direction of arrival estimators much attention has been directed to the design of algorithms for minimum variance estimation taking only finite sample errors into account approaches to reduce the sensitivity due to array calibration errors have also appeared in the literature herein one such approach is adopted that assumes that the errors due to finite samples and model errors are of comparable size a weighted subspace fitting method for very general array perturbation models is derived this method provides minimum variance estimates under the assumption that the prior distribution of the perturbation model is known interestingly the method reduces to the wsf mode estimator if no model errors are present vice versa assuming that model errors dominate the method specializes to the corresponding model errors only subspace fitting method unlike previous techniques for model errors the estimator can be implemented using a two step procedure if the nominal array is uniform and linear and it is also consistent even if the signals are fully correlated the paper also contains a large sample analysis of one of the alternative methods namely mapprox it is shown that mapprox also provides minimum variance estimates under reasonable assumptions

intermittency of height fluctuations in stationary state of the kardar parisi zhang equation with infinitesimal surface tension in 1 1 dimensions the kardar parisi zhang kpz equation with infinitesimal surface tension dynamically develops sharply connected valley structures within which the height derivative is not continuous we discuss the intermittency issue in the problem of stationary state forced kpz equation in 1 1 dimensions it is proved that the moments of height increments c a behave as x 1 x 2 xi a with xi a a for length scales x 1 x 2 sigma the length scale sigma is the characteristic length of the forcing term we have checked the analytical results by direct numerical simulation

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements the problem of estimating the frequencies phases and amplitudes of sinusoidal signals is considered a simplified maximum likelihood gauss newton algorithm which provides asymptotically efficient estimates of these parameters is proposed initial estimates for this algorithm are obtained by a variation of the overdetermined yule walker method and periodogram based procedure use of the maximum likelihood gauss newton algorithm is not however limited to this particular initialization method some other possibilities to get suitable initial estimates are briefly discussed an analytical and numerical study of the shape of the likelihood function associated with the sinusoids in noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure some numerical examples are presented to illustrate the performance of the proposed estimation procedure comparison to the performance corresponding to the cramer rao lower bound is also presented using a simple expression for the asymptotic cramer rao bound covariance matrix derived in the paper

maximum likelihood localization of multiple sources by alternating projection an algorithm referred to as apm for computing the maximum likelihood estimator of the locations of simple sources in passive sensor arrays is presented the algorithm is equally applicable to the case of coherent signals and to the case of a single snapshot the algorithm is iterative the maximum of the likelihood function is computed by successive approximations the convergence of the algorithm to the global maximum is demonstrated for a variety of scenarios the key to this global convergence is the initialization scheme

optimal utilization of signal free samples for array processing in unknown colored noise fields many algorithms for direction of arrival doa estimation require the noise covariance matrix to be known or to possess a known structure in many cases the noise covariance is in fact estimated from separate measurements this paper addresses the combined effects of finite sample sizes both in the estimated noise covariance matrix and in the data with signals present it is assumed that a batch of signal free samples is available in addition to the signal containing samples no assumption is made on the structure of the noise covariance in this paper the asymptotic covariance of the weighted subspace fitting wsf algorithm is derived for the case in which the data are whitened using an estimated noise covariance the expression obtained suggests an optimal weighting that improves performance compared to the standard choice in addition a new method based on covariance matching is proposed both methods are asymptotically statistically efficient the cramer rao lower bound crb on the covariance of the estimate for the data model is also derived monte carlo simulations show promising small sample performance for the two new methods and confirm the asymptotic results

array processing in correlated noise fields based on instrumental variables and subspace fitting accurate signal parameter estimation from sensor array data is a problem which has received much attention in the last decade a number of parametric estimation techniques have been proposed in the literature in general these methods require knowledge of the sensor to sensor correlation of the noise which constitutes a significant drawback this difficulty can be overcome only by introducing alternative assumptions that enable separating the signals from the noise in some applications the raw sensor outputs can be preprocessed so that the emitter signals are temporally correlated with correlation length longer than that of the noise an instrumental variable iv approach can then be used for estimating the signal parameters without knowledge of the spatial color of the noise a computationally simple iv approach has recently been proposed by the authors herein a refined technique that can give significantly better performance is derived a statistical analysis of the parameter estimates is performed enabling optimal selection of certain user specified quantities a lower bound on the attainable error variance is also presented the proposed optimal iv method is shown to attain the bound if the signals have a quasideterministic character

sensor array processing based on subspace fitting algorithms for estimating unknown signal parameters from the measured output of a sensor array are considered in connection with the subspace fitting problem the methods considered are the deterministic maximum likelihood method ml esprit and a recently proposed multidimensional signal subspace method these methods are formulated in a subspace fitting based framework which provides insight into their algebraic and asymptotic relations it is shown that by introducing a specific weighting matrix the multidimensional signal subspace method can achieve the same asymptotic properties as the ml method the asymptotic distribution of the estimation error is derived for a general subspace weighting and the weighting that provides minimum variance estimates is identified the resulting optimal technique is termed the weighted subspace fitting wsf method numerical examples indicate that the asymptotic variance of the wsf estimates coincides with the cramer rao bound the performance improvement compared to the other techniques is found to be most prominent for highly correlated signals

a performance analysis of subspace based methods in the presence of model error ii multidimensional algorithms for pt i see ibid vol 40 no 7 p 1758 74 1992 in pt i the performance of the music algorithms for narrowband direction of arrival doa estimation when the array manifold and noise covariance are not correctly modeled was investigated this analysis is extended to multidimensional subspace based algorithms including deterministic or conditional maximum likelihood md music weighted subspace fitting wsf mode and esprit a general expression for the variance of the doa estimates that can be applied to any of the above algorithms and to any of a wide variety of scenarios is presented optimally weighted subspace fitting algorithms are presented for special cases involving random unstructured errors of the array manifold and noise covariance it is shown that one dimensional music outperforms all of the above multidimensional algorithms for random angle independent array perturbations

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

the effects of array calibration errors on df based signal copy performance this paper studies the effect of array calibration errors on the performance of various direction finding df based signal copy algorithms unlike blind copy methods this class of algorithms requires an estimate of the directions of arrival doas of the signals in order to compute the copy weight vectors under the assumption that the observation time is sufficiently long the following algorithms are studied classical beamforming least squares total least squares linearly constrained minimum variance beamforming and structured stochastic estimation expressions for the mean square error of the signal estimates are derived as a function of the calibration errors for both the case where the doas are known precisely and for the case where the doas must be estimated

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

detection and estimation in sensor arrays using weighted subspace fitting the problem of signal parameter estimation of narrowband emitter signals impinging on an array of sensors is addressed a multidimensional estimation procedure that applies to arbitrary array structures and signal correlation is proposed the method is based on the recently introduced weighted subspace fitting wsf criterion and includes schemes for both detecting the number of sources and estimating the signal parameters a gauss newton type method is presented for solving the multidimensional wsf and maximum likelihood optimization problems the global and local properties of the search procedure are investigated through computer simulations most methods require knowledge of the number of coherent noncoherent signals present a scheme for consistently estimating this is proposed based on an asymptotic analysis of the wsf cost function the performance of the detection scheme is also investigated through simulations

gauss tchebycheff quadrature formulas it is well known that the tchebycheff weight function 1 x 2 1 2 is the only weight function up to a linear transformation for which then point gauss quadrature formula has equal weights for alln in this paper we describe explicitly all weight functions which have the property that then k point gauss quadrature formula has equal weights for allk where n k n 1 n 2 is an arbitrary subsequence of furthermore results on the possibility of tchebycheff quadrature on several intervals are given

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

probability distribution functions for the random forced burgers equation statistical properties of solutions of the random forced burgers equation have been a subject of intensive studies recently see refs 1 6 of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces 1 it is also hoped that the field theoretic techniques developed for the burgers equation will eventually be useful for understanding more complex phenomena such as turbulence in this paper we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced burgers equation we will consider the problem

a dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos capturing all scales of random modes on independent grids in stochastic computations or uncertainty quantification methods the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion the size of this system increases drastically when the number of independent random variables and or order of polynomial chaos expansions increases this is invariably the case for large scale simulations and or problems involving steep gradients and other multiscale features such features are variously reflected on each solution component or random uncertainty mode requiring the development of adaptive methods for their accurate resolution in this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space refinement on physical space that allows all scales of each solution component to be refined independently of the rest we exemplify this using the convection diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method thus we establish a new elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

probability densities in strong turbulence abstract in this work we using mellin s transform combined with the gaussian large scale boundary condition calculate probability densities pdfs of velocity increments p r u r velocity derivatives p u r and the pdf of the fluctuating dissipation scales q re where re is the large scale reynolds number the resulting expressions strongly deviate from the log normal pdf p l r u r often quoted in the literature it is shown that the probability density of the small scale velocity fluctuations includes information about the large integral scale dynamics which is responsible for the deviation of p r u r from p l r u r an expression for the function d h of the multifractal theory free from spurious logarithms recently discussed in u frisch m martins afonso a mazzino v yakhot j fluid mech 542 2005 97 is also obtained

multidimensional potential burgers turbulence we consider the multidimensional generalised stochastic burgers equation in the space periodic setting frac partial mathbf u partial t nabla f mathbf u cdot nabla mathbf u nu delta mathbf u nabla eta quad t geq 0 mathbf x in mathbb t d mathbb r mathbb z d under the assumption that u is a gradient here f is strongly convex and satisfies a growth condition is small and positive while is a random forcing term smooth in space and white in time for solutions u of this equation we study sobolev norms of u averaged in time and in ensemble each of these norms behaves as a given negative power of these results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence namely the averages of the increments and of the energy spectrum these quantities behave as a power of the norm of the relevant parameter which is respectively the separation l in the physical space and the wavenumber k in the fourier space our bounds do not depend on the initial condition and hold uniformly in nu we generalise the results obtained for the one dimensional case in 10 confirming the physical predictions in 4 30 note that the form of the estimates does not depend on the dimension the powers of nu mathbf k ell are the same in the one and the multi dimensional setting

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed