statistical analysis of the music algorithm in the presence of modeling errors taking into account the resolution probability this paper considers the statistical performance of the music method under the condition that two closely spaced sources impinging on an array of sensors are effectively resolved i e the spectrum exhibits two peaks in the neighborhood of the true directions of arrival doa the music algorithm is known to have an infinite resolution power in theory however in the presence of modeling errors sources can not be resolved with certainty even if the array correlation matrix is perfectly known the focus of this paper is to predict the bias and variance of the doa estimates taking into account the possible resolution failure of music this performance prediction based on our recent mathematical investigation is new to the best of our knowledge a general mathematical framework to derive closed form expressions of the bias and variance versus the model mismatch conditioned on a general statistical resolution test is proposed in order to illustrate our mathematical approach statistical tests with one and two conditions respectively are investigated the accuracy of the performance prediction is illustrated in a simulation study it is found that the proposed approach outperforms the classical technique which ignores the possible resolution failure of the music algorithm therefore our results provide better tools for determining the necessary antenna calibration accuracy to achieve some targeted specifications on the estimator performance

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

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 polyconvex integrand euler lagrange equations and uniqueness of equilibrium in this manuscript we are interested in stored energy functionals w defined on the set of d d matrices which not only fail to be convex but satisfy rm lim det xi rightarrow 0 w xi infty we initiate a study which we hope will lead to a theory for the existence and uniqueness of minimizers of functionals of the form e mathbf u int omega w nabla mathbf u mathbf f cdot mathbf u rm d x as well as their euler lagrange equations the techniques developed here can be applied to a class of functionals larger than those considered in this manuscript although we keep our focus on polyconvex stored energy functionals of the form w xi f xi h rm det xi such that rm lim t rightarrow 0 h t infty which appear in the study of ogden material we present a collection of perturbed and relaxed problems for which we prove uniqueness results then we characterize these minimizers by their euler lagrange equations

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

on the asymptotic performance analysis of subspace doa estimation in the presence of modeling errors case of music this paper provides a new analytic expression of the bias and rms error root mean square error of the estimated direction of arrival doa in the presence of modeling errors in first order approximations of the rms error are derived which are accurate for small enough perturbations however the previously available expressions are not able to capture the behavior of the estimation algorithm into the threshold region in order to fill this gap we provide a second order performance analysis which is valid in a larger interval of modeling errors to this end it is shown that the doa estimation error for each signal source can be expressed as a ratio of hermitian forms with a stochastic vector containing the modeling error then an analytic expression for the moments of such a hermitian forms ratio is provided finally a closed form expression for the performance bias and rms error is derived simulation results indicate that the new result is accurate into the region where the algorithm breaks down

a sensitivity analysis of the music algorithm a first order analysis of the sensitivity of the music algorithm to system errors which cause differences between the array manifold used by music and the true array manifold is presented the effect of such errors on the direction of arrival estimates is investigated the ability of the music algorithm to resolve two closely spaced sources in the presence of phase and gain errors in the array elements or the receivers or errors in the element locations is analyzed a formula for computing the failure threshold of music as a function of source separation and other system parameters is derived and tested by simulation the analysis assumes that the exact covariance matrix of array element outputs is known the infinite data case

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

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

performance analysis of time reversal music in this paper we study the performance of multiple signal classification music in computational time reversal tr applications the analysis builds upon classical results on first order perturbation of singular value decomposition the closed form of mean squared error mse matrix of tr music is derived for the single frequency case in both multistatic co located and non co located scenarios the proposed analysis is compared with cramer rao lower bound crlb and it is exploited for comparison of tr music when linear and nonlinear multiple scattering is present finally a numerical analysis is provided to confirm the theoretical findings

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

fast and efficient time reversal imaging using space frequency propagator method it is known that the eigenvalue decomposition and singular value decomposition svd of multi static data matrices mdms are the basis of time reversal tr imaging however the computational loads of the svd can be significantly high when the mdms are large the computational cost of the propagator method pm without any svd of mdms is much lower than that of svd based methods in this paper to reduce the computational complexity in tr imaging first a novel space frequency propagator method sf pm for tr imaging in large mdms is proposed the computational complexity of the proposed sf pm is then analyzed moreover the theoretical analysis is presented to show that the sf pm has a significantly lower computational complexity than the conventional svd based approach finally the reasonable imaging results and superior computational efficiency of the sf pm are verified by 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

an aronsson type approach to extremal quasiconformal mappings abstract we study c 2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a localized form of extremality in the spirit of the work of g aronsson on absolutely minimizing lipschitz extensions we also prove short time existence for smooth solutions of a gradient flow of qc diffeomorphisms associated to the extremal problem

music maximum likelihood and cramer rao bound further results and comparisons the problem of determining the direction of arrival of narrowband plane waves using sensor arrays and the related problem of estimating the parameters of superimposed signals from noisy measurements are studied a number of results have been recently presented by the authors on the statistical performance of the multiple signal characterization music and the maximum likelihood ml estimators for the above problems this work extends those results in several directions first it establishes that in the class of weighted music estimators the unweighted music achieves the best performance i e the minimum variance of estimation errors in large samples next it derives the covariance matrix of the ml estimator and presents detailed analytic studies of the statistical efficiency of music and ml estimators these studies include performance comparisons of music and mle with each other as well as with the ultimate performance corresponding to the cramer rao bound finally some numerical examples are given which provide a more quantitative study of performance for the problem of finding two directions with uniform linear sensor arrays

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

perturbation analysis for subspace decomposition with applications in subspace based algorithms subspace decomposition has been exploited in different applications due to perturbations from various sources such as finite data samples and measurement noise perturbations arise in subspaces therefore some loss is introduced to performance of subspace based algorithms although first order perturbation results have been proposed in the literature and applied to various problems up to second order perturbation analysis can provide more accurate analytical results and is studied in this paper based on the orthogonality principle perturbations of subspaces and singular values or eigenvalues are derived explicitly as functions of a perturbation in the objective matrix up to the second order respectively all in closed forms it is shown that by keeping only the first order terms the derived results reduce to those from existing approaches examples to apply the proposed results to both matrix computation and subspace based channel estimation are provided to verify our analysis

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

on the sensitivity of time reversal imaging techniques to model perturbations time reversal tr techniques involve physical or synthetic retransmission of signals acquired by a set of transceivers in a time reversed fashion and can be used for a host of applications including ultrawideband imaging of obscured targets this work investigates the performance of some tr based techniques under non ideal conditions and is divided into three main parts in the first part we investigate the robustness of tr based imaging techniques via the decomposition of tr operator dort algorithm and the multiple signal classification music algorithm under clutter and additive noise in the second part we examine the effect of tr invariance breaking due to losses in the intervening media on the performance of both dort and tr based music in the third part effects of translational perturbations of the tr array are examined as a class of model perturbations these ensuing effects are then exploited in a controlled manner as a means to extract second order spatial statistics of the probed media

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

time reversal imaging with multiple signal classification considering multiple scattering between the targets the time reversal imaging with multiple signal classification method for the location of point targets developed within the framework of the born approximation in lehman and devaney transmission mode time reversal super resolution imaging j acoust soc am 113 2742 2753 2003 is generalized to incorporate multiple scattering between the targets it is shown how the same method can be used in the location of point targets even if there is multiple scattering between them on the other hand both the conventional images and the calculated values of the target scattering amplitudes are scattering model dependent

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

decomposition of the time reversal operator detection and selective focusing on two scatterers the decomposition of the time reversal operator dort method is a selective detection and focusing technique using an array of transmit receive transducers it relies on the theory of iterative time reversal mirrors which was presented by prada et al c prada j l thomas and m fink j acoust soc am 97 62 71 1995 the time reversal operator was defined as k k where is the frequency means complex conjugate and k is the transfer matrix of the array of l transducers insonifying a time invariant scattering medium it was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers the dort method consists in determining these eigenvectors and using them for the selective focusing this paper presents a complete analysis of this method in the case of two scatterers the

first order perturbation analysis of singular vectors in singular value decomposition singular value decomposition svd is an important technique in signal processing in the literature the perturbation analysis of svd has been well documented in the context of subspace decomposition the contribution of the signal subspace to the perturbation of the singular vectors that span the signal subspace is often ignored as it is treated as second and higher order terms and thus the first order perturbation is typically given as the column span of the noise subspace in this correspondence we show that not only the noise subspace but also the signal subspace contribute to the first order perturbation of the singular vectors we further show that the contribution of the signal subspace does not impact on the performance analysis of algorithms that rely on the signal subspace for parameter estimation but it affects the analysis of algorithms that depends on the individual basis vectors for the latter we also give a condition under which the contribution of the signal subspace to the perturbation of singular vectors may be ignored in the statistical sense numerical examples are provided to validate our theoretic claims

a new family of spline based biorthogonal wavelet transforms and their application to image compression in this paper we design a new family of biorthogonal wavelet transforms and describe their applications to still image compression the wavelet transforms are constructed from various types of interpolatory and quasiinterpolatory splines the transforms use finite impulse response and infinite impulse response filters that are implemented in a fast lifting mode

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

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