array interpolation and doa mse reduction interpolation or mapping of data from a given real array to data from a virtual array of more suitable geometry is well known in array signal processing this operation allows arrays of any geometry to be used with fast direction of arrival doa estimators designed for linear arrays in an earlier companion paper a first order condition for zero doa bias under such mapping was derived and was also used to construct a design algorithm for the mapping matrix that minimized the doa estimate bias this bias minimizing theory is now extended to minimize not only bias but also to consider finite sample effects due to noise and reduce the doa mean square error mse an analytical first order expression for mapped doa mse is derived and a design algorithm for the transformation matrix that minimizes this mse is proposed generally doa mse is not reduced by minimizing the size of the mapping errors but instead by rotating these errors and the associated noise subspace into optimal directions relative to a certain gradient of the doa estimator criterion function the analytical mse expression and the design algorithm are supported by simulations that show not only conspicuous mse improvements in relevant scenarios but also a more robust preprocessing for low signal to noise ratios snrs as compared with the pure bias minimizing design developed in the previous paper

estimating functions of probability distributions from a finite set of samples this paper addresses the problem of estimating a function of a probability distribution from a finite set of samples of that distribution a bayesian analysis of this problem is presented the optimal properties of the bayes estimators are discussed and as an example of the formalism closed form expressions for the bayes estimators for the moments of the shannon entropy function are derived then numerical results are presented that compare the bayes estimator to the frequency counts estimator for the shannon entropy we also present the closed form estimators all derived elsewhere for the mutual information mathrm ensuremath chi 2 covariance and some other statistics c 1995 the american physical society

direction of arrival estimation for nonuniform sensor arrays from manifold separation to fourier domain music methods in this paper the problem of spectral search free direction of arrival doa estimation in arbitrary nonuniform sensor arrays is addressed in the first part of the paper we present a finite sample performance analysis of the well known manifold separation ms based root music technique then we propose a new class of search free doa estimation methods applicable to arrays of arbitrary geometry and establish their relationship to the ms approach our first technique is referred to as fourier domain fd root music and is based on the fact that the spectral music function is periodic in angle it uses the fourier series to expand this function and reformulate the underlying doa estimation problem as an equivalent polynomial rooting problem our second approach applies the zero padded inverse fourier transform to the fd root music polynomial to avoid the polynomial rooting step and replace it with a simple line search our third technique refines the fd root music approach by using weighted least squares approximation to compute the polynomial coefficients the proposed techniques are shown to offer substantially improved performance to complexity tradeoffs as compared to the ms technique

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

direction of arrival estimation using mode with interpolated arrays presents a generalization of the mode direction finding algorithm using the interpolated array approach the proposed algorithm is statistically efficient for uncorrelated correlated and even coherent sources and requires only a modest amount of computations the use of array interpolation makes it possible to easily incorporate calibration data into the estimation procedure the performance of the algorithm was evaluated by computer simulation verifying its statistical efficiency

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

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

transmit array interpolation for doa estimation via tensor decomposition in 2 d mimo radar in this paper we propose a two dimensional joint transmit array interpolation and beamspace design for planar array mono static multiple input multiple output radar for direction of arrival doa estimation via tensor modeling our underlying idea is to map the transmit array to a desired array and suppress the transmit power outside the spatial sector of interest in doing so the signal to noise ratio is improved at the receive array then we fold the received data along each dimension into a tensorial structure and apply tensor based methods to obtain doa estimates in addition we derive a closed form expression for doa estimation bias caused by interpolation errors and argue for using a specially designed look up table to compensate for the bias the corresponding cramer rao bound crb is also derived simulation results are provided to show the performance of the proposed method and compare its performance to crb

preprocessing for direction finding with minimal variance degradation numerous authors have advocated the use of preprocessing in high resolution direction of arrival doa algorithms the benefits cited include reduced computation improved performance in spatially colored noise and enhanced resolution the authors identify the preprocessing matrices that provide minimum variance estimates of doa for a number of models and algorithms they examine the cramer rao bound crb for gaussian signals the crb for deterministic signals and the asymptotic variance of the music estimator for preprocessed data they also study the effect of array manifold errors on the direction estimates as expected the optimal preprocessor requires knowledge of the source directions however they show that performance that is close to optimal can be obtained with only approximate knowledge of the source directions with an error not exceeding the array beamwidth if the design rules outlined in this paper are used

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

which is the better entropy expression for speech processing s log s or log s in maximum entropy spectral analysis mesa one maximizes the integral of logs f where s f is a power spectrum the resulting spectral estimate which is equivalent to that obtained by linear prediction and other methods is popular in speech processing applications an alternative expression s f logs f is used in optical processing and elsewhere this paper considers whether the alternative expression leads to spectral estimates useful in speech processing we investigate the question both theoretically and empirically the theoretical investigation is based on generalizations of file two estimates the generalizations take into account prior estimates of the unknown power spectrum it is shown that both estimates result from applying a generalized version of the principle of maximum entropy but they differ concerning the quantities that are treated as random variables the empirical investigation is based on speech synthesized using the different spectral estimates although both estimates lead to intelligible speech speech based on the mesa estimate is qualitatively superior

real valued music for efficient direction estimation with arbitrary array geometries most of the existing methods for direction of arrival doa estimation are based on numerical characteristics behind the entire array output covariance matrix aocm since the aocm is generally a complex matrix those approaches require tremendous complex computations accordingly this paper addresses the problem of doa estimation with real valued computations by considering the real part of aocm r aocm and the imaginary part of aocm i aocm separately it is shown that the null space of r aocm and that of i aocm are the same subspace which coincides with the intersection of the original noise subspace and its conjugate subspace using such a mathematical fact a novel real valued music rv music estimator with a real valued subspace decomposition on only r acom or i aocm instead of the entire acom is derived compared with most state of the art unitary algorithms suitable for only centro symmetric arrays csas the proposed technique can be used with arbitrary array geometries unlike conventional music with exhaustive spectral search rv music involves a limited search over only half of the total angular field of view with a real valued noise subspace and hence reduces the complexity by 75 theoretical performance analysis on the mean square error mse and numerical simulations demonstrate that rv music shows a very close accuracy to the standard music

performance analysis of spatial smoothing with interpolated arrays the interpolated spatial smoothing algorithm is a computationally efficient method for estimating the directions of arrival doas of signals some of which may be perfectly correlated it extends the spatial smoothing method to arbitrary array geometries a statistical performance analysis of the algorithm is presented closed form expressions for the covariance matrix of the doa estimation errors are derived using a perturbation analysis evaluating these expressions for specific cases and comparing them to the cramer rao lower bound for the doa estimates provides insight into the statistical efficiency of this algorithm the formulas for the error covariance are quite general and can be specialized to provide results for other doa estimation algorithms as well

direction finding for wide band signals using an interpolated array the authors derive a new direction finding algorithm for multiple wideband signals received by an arbitrary array and analyze its performance using an interpolation technique they generate a set of virtual arrays each for a different frequency band having the same array manifold the convergence matrices of these arrays are added to produce a composite covariance matrix direction of arrival doa estimates are obtained by eigendecomposition of this composite covariance matrix using the narrowband music algorithm or its variants closed form expressions for the asymptotic covariance matrix of the doa estimation errors are derived using a perturbation analysis evaluated for specific cases and compared with the cramer rao lower bound special attention is given to correlated and coherent signals the formulas for the error covariance are quite general and can be modified to provide results for other wideband doa estimation algorithms

line array beamforming using linear prediction for aperture interpolation and extrapolation it is demonstrated that linear prediction can be successfully utilized in conjunction with otherwise conventional narrow band line array beamforming for aperture extrapolation apex where there are significant snr signal to noise ratio and resolution advantages compared to the conventional approach aperture interpolation where faulty sensor data are replaced and a combined technique alpinex the performance of all three methods is investigated in connection with both simulated and real data the apex technique is shown to be amenable to a simple theoretical treatment further the approach is demonstrated to offer a practical performance comparable to the well known high resolution technique music when the observation time is short

array interpolation and bias reduction interpolation mapping of data from a given antenna array onto the output of a virtual array of more suitable configuration is well known in array signal processing this operation allows arrays of any geometry to be used with fast direction of arrival doa estimators designed for linear arrays conditions for preserving doa error variance under such mappings have been derived by several authors however in many cases such as omnidirectional signal surveillance over multiple octaves systematic mapping errors will dominate over noise effects and cause significant bias in the doa estimates to prevent mapping errors from unduly affecting the doa estimates this paper uses a geometrical interpretation of a taylor series expansion of the doa estimator criterion function to derive an alternative design of the mapping matrix verifying simulations show significant bias reduction in the doa estimates compared with previous designs the key feature of the proposed design is that it takes into account the orthogonality between the manifold mapping errors and certain gradients of the estimator criterion function with the new design mapping of narrowband signals between dissimilar array geometries over wide sectors and large frequency ranges becomes feasible

doa estimation via manifold separation for arbitrary array structures in this paper we consider the manifold separation technique mst which stems from the wavefield modeling formalism developed for array processing mst is a method for modeling the steering vector of antenna arrays of practical interest with arbitrary 2 d or 3 d geometry it is the product of a sampling matrix dependent on the antenna array only and a vandermonde structured coefficients vector depending on the wavefield only this allows fast direction of arrival doa algorithms designed for linear arrays to be used on arrays with arbitrary configuration in real world applications the calibration measurements used to determine the sampling matrix are corrupted by noise this impairs the performance of mst based algorithms in particular we study the effect of noisy calibration measurements on subspace based doa algorithms using mst expressions describing the error in the doa estimates due to calibration noise and truncation are derived this allows predicting the performance of mst based algorithms in real world applications the analysis is verified by simulations we established a link between the optimal number of selected modes and the statistics of calibration noise we analyze the modeling error when mst is used for 1 d azimuth doa estimation

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

estimating probability densities from short samples a parametric maximum likelihood approach a parametric method similar to autoregressive spectral estimators is proposed to determine the probability density function pdf of a random set the method proceeds by maximizing the likelihood of the pdf yielding estimates that perform equally well in the tails as in the bulk of the distribution it is therefore well suited for the analysis short sets drawn from smooth pdfs and stands out by the simplicity of its computational scheme its advantages and limitations are discussed

on the resolution probability of music in presence of modeling errors the problem of resolving closely spaced signal sources using an antenna array remains a difficult one although several estimation methods are available in the literature when the array correlation matrix is known the resolution capability of subspace algorithms is infinitely high however in the presence of modeling errors the resolution deteriorates even for a known correlation matrix in this paper we analyze the music method by way of three different definitions of the resolution assuming gaussian circular random modeling errors we determine the corresponding expressions of the probability of source resolution versus the model mismatch a first series of simulations validates the mathematical expression of the three resolution probabilities a second series of simulations is used to select among them the tightest one to the empirical one the results are useful e g for determining the necessary antenna calibration accuracy to achieve a target performance