bandwidth truncation for chebyshev polynomial and ultraspherical chebyshev galerkin discretizations of differential equations the petrov galerkin ultraspherical polynomial chebyshev polynomial discretization of the highest derivative of a differential equation is a diagonal matrix the same is true for fourier galerkin discretizations nevertheless the spectral discretizations of simple problems like u x x q x u f x are usually dense matrices the villain is the multiplication matrix the galerkin representation of a term like q x u x unfortunately this part of the galerkin matrix is dense however if the ode coefficient q x has a chebyshev or fourier series that converges much more rapidly than u x then it is possible to realize great cost savings at no loss of accuracy by truncating the full n n galerkin matrix to a banded matrix where the bandwidth m n one of our themes is that when the spectral series for q x and u x have similar rates of convergence as is almost universal when a nonlinear equation is linearized for a newton krylov iteration such accuracy lossless truncation is impossible nonlinearity is but one of many causes of this sort of solution coefficient equiconvergence when bandwidth truncation is possible though our second theme is to show that a modest amount of floating point operations and memory can be saved by an unsymmetric truncation in which the number of elements retained to the left of the main diagonal is roughly double the number kept to the right our second improvement is to replace the m term spectral series for q x by its m 2 m 2 chebyshev pade rational approximation this sometimes allow one to halve the matrix bandwidth reducing the linear algebra costs by a factor of four

spatial spectrum estimation in a location sector multiple narrowband source localization using arbitrarily configured arrays and spatial spectrum estimation is considered a new eigenspace based approach which uses projections onto a particular vector or vector set in the estimated noise only subspace is described several closest vector estimators are developed by using different measures of closeness first closest is a novel full dimensional element space approach to spatial spectrum estimation which has important performance advantages relative to pertinent established spatial spectrum estimators it incorporates a priori knowledge of the array manifold over a location sector of interest to provide snr spectral resolution thresholds which are lower than those of min norm for some arrays substantially lower second relationships between the closest approach and several established approaches to spatial spectrum estimation are established for a linear equispaced array min norm is shown to be a special case of the closest approach one which is based on projection onto a noise only subspace vector which is close to the array manifold over the entire field of view

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

performance analysis of music employing conjugate symmetric beamformers if one incorporates a beamformer composed of conjugate centro symmetric weight vectors as a preprocessor to an eigenstructure direction finding algorithm a real valued decomposition can be employed to estimate the noise and signal subspaces from the sample covariance matrix the effect of employing the real processing methodology on the angle estimation performance of beamspace music is explored specifically the distribution of the real valued signal subspace eigenvectors is derived and used in an asymptotic analysis of the bias and variance of the music estimator the theoretical analysis shows that processing the real part of the beamspace sample covariance matrix offers significant performance gains in addition to the obvious computational benefit relative to the conventional complex valued procedure particularly in the case of correlated sources monte carlo simulations are included to verify the theoretical expressions a trade off study of the estimation accuracy versus the desire to provide adequate rejection of unwanted signals in a sector based interrogation scheme for various beamforming architectures is also presented

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

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

direction of arrival estimation for temporally correlated narrowband signals signal direction of arrival doa estimation using an array of sensors has been the subject of intensive research and development during the last two decades efforts have been directed to both better solutions for the general data model and to develop more realistic models so far many authors have assumed the data to be independent and identically distributed i i d samples of a multivariate statistical model although this assumption reduces the complexity of the model it may not be true in certain situations where signals show temporal correlation some results are available on the temporally correlated signal model in the literature the temporally correlated stochastic cramer rao bound crb has been calculated and an instrumental variable based method called iv ssf is introduced also it has been shown that temporally correlated crb is lower bounded by the deterministic crb in this paper we show that temporally correlated crb is also upper bounded by the stochastic i i d crb we investigate the effect of temporal correlation of the signals on the best achievable performance we also show that the iv ssf method is not efficient and based on an analysis of the crb propose a variation in the method which boosts its performance simulation results show the improved performance of the proposed method in terms of lower bias and error variance

efficient application of music algorithm under the coexistence of far field and near field sources this correspondence is concerned with source localization and classification for scenarios where both the far field and near field narrowband sources may coexist we propose an efficient music based solution that requires neither a multidimensional search nor high order statistics hos we also derive the stochastic cramer rao bound crb for the problem under consideration the performance of the proposed method is compared with an existing method and with the crb

performance analysis of music type high resolution estimators for direction finding in correlated and coherent scenes an asymptotic analysis is presented of a class of high resolution estimators for resolving correlated and coherent plane waves in noise these estimators are in turn constructed from certain eigenvectors associated with spatially smoothed or unsmoothed covariance matrices generated from a uniform array the analysis is first carried out for the smoothed case and from this the conventional i e unsmoothed multiple signal classification music scheme follows as a special case independent of the total number of sources present in the scene the variance of the conventional music estimator along the true arrival angles is shown to be zero within a first order approximation the bias expressions in the smoothed case are used to obtain a resolution threshold for two coherent equipowered plane waves in white noise and the result is compared to the one derived by kaveh et al 1986 for two uncorrelated equipowered plane waves

the effect of source number underestimation on music location estimates an analytical expression for the variance of music source location estimates is derived under the assumption that the detection step has already taken place resulting in an underestimation of the source number the analytic predictions are compared with monte carlo simulation results for beam space music the derived variance expression is more general than those reported to date and is also valid for other situations where the asymptotic correlation matrix yields biased source estimates

design of two music like estimators based on bias minimization two classes of music like estimators are considered one class called weighted norm music possesses an optimizing functional or null spectrum which is the product of the music null spectrum and an angle dependent weight the second class which is denoted the d sub r estimator has an optimizing functional that is dependent on a parameter r and is a generalized distance between two particular vectors in the signal subspace it is shown that the asymptotic mean square errors of these estimators are the same as music by determining an appropriate weight based on a derived large sample expression for the estimator bias a weighted norm music estimator is found that gives zero bias of order n sup 1 where n is the sample size using an approximate relation between the two types of estimators under consideration a data dependent parameter r spl theta is derived for the d sub r estimator which results in small bias over a wide range of signal to noise ratios snrs for two closely spaced sources

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

bias analysis of the music location estimator the authors present a rigorous bias analysis of the music location estimator and they derive an accurate and concise bias expression the analysis is based on the second order taylor series expansion of the derivative of the null spectrum properties of the null spectrum and statistics of the estimated signal eigenvectors it is proven that in the derivation the remainder term in the second order taylor series can be dropped but the second order terms cannot be simulations verify that the bias expression is valid over a wide range of signal to noise ratios snrs extending down into the resolution threshold region of music although asymptotic this expression can be accurately applied to a limited number of snapshot cases the utility of the expression is shown by using it in a study of music location estimator characteristics estimate bias and standard deviation are compared for variations in snr numbers of sensors and snapshots and source correlation music resolvability and estimator performance bounds are addressed accounting for bias

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

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

esprit a subspace rotation approach to estimation of parameters of cisoids in noise the application of a subspace invariance approach esprit to the estimation of parameters frequencies and powers of cisoids in noise is described esprit exploits an underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets the new approach has several advantages including improved resolution over pisarenko s technique for harmonic retrieval

recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary bessel polynomials abstract a formula expressing explicitly the derivatives of ordinary bessel polynomials of any degree and for any order in terms of the ordinary bessel polynomials themselves is proved another explicit formula which expresses the ordinary bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original ordinary bessel coefficients is also given a formula for the ordinary bessel coefficients of the moments of one single ordinary bessel polynomial of certain degree is proved formula for the ordinary bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its ordinary bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between generalized bessel and ordinary bessel polynomials is described explicit formula for these coefficients between jacobi and ordinary bessel polynomials is given of which the ultraspherical polynomials and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre ordinary bessel and hermite ordinary bessel are also developed

an analysis of weighted eigenspace methods in the presence of sensor errors among all direction finding methods eigenspace based spectral methods are of interest because they are known to yield asymptotically unbiased and efficient estimates these algorithms require knowledge of the array geometry and its elements characteristics in real systems only approximate knowledge of the required array characterization is available there is valid concern that deviation of nominal sensor gain and phase values can result in degradation of doa estimates we analyze the effects of sensor gain and phase perturbation on weighted eigenspace methods wems we quantify the sensitivity of wems by first and second order statistical analyses simulation results are included to validate our analytical results

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

performance analysis of the mvdr spatial spectrum estimator the performance of the minimum variance distortionless response spectrum estimator is analyzed finite data effects and the sensitivity of the method to random perturbations in the signal model and in the noise covariance matrix are studied the snapshots are assumed to be complex independent identically distributed gaussian vectors an expression for the exact model asymptotic bias is also derived analytical expressions for the variance and the bias of the estimator are derived and compared with simulation results these expressions are then employed to study the characteristics of the estimator

measure concentration for a class of random processes let x x i i be a stationary random process with a countable alphabet and distribution q let q x k 0 denote the conditional distribution of x x 1 x 2 x n given the k length past write d 1 x 1 0 if 1 x 1 and d 1 x 1 1 otherwise we say that the process x admits a joining with finite distance u if for any two past sequences k 0 k 1 0 and x k 0 x k 1 x 0 there is a joining of q k 0 and q x k 0 say dist 0 x 0 k 0 x k 0 such that the main result of this paper is the following inequality for processes that admit a joining with finite distance

bias and variance of direction of arrival estimates from music min norm and fine we derive variance and bias expressions of direction of arrival doa estimates for min norm and fine these expressions for arbitrarily configured arrays are valid and accurate over wide ranges of snr and number of snapshots using these expressions we illustrate that both min norm and fine have smaller estimate bias and further fine has comparable estimate variance compared to music

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

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

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

performance analysis of root music the authors analyze the performance of root music a variation of the music algorithm for estimating the direction of arrival doa of plane waves in white noise in the case of a linear equispaced sensor array the performance of the method is analyzed by examining the perturbation in the roots of the polynomial formed in the intermediate step of root music in particular asymptotic results for the mean squared error in the estimates of the direction of arrival are derived simplified expressions are presented for the one and two source case and compared to those obtained for least squares esprit computer simulations are also presented and they are in close agreement with the theory an important outcome of this analysis is the fact that the error in the signal zeros has a largely radial component this provides an explanation as to why the root music is superior to the spectral music algorithm

applications of cumulants to array processing ii non gaussian noise suppression the main motivation of using higher order statistics in signal processing applications has been their insensitivity to additive colored gaussian noise the main objection to those methods is their possible vulnerability to non gaussian noise the authors investigate the effects of non gaussian ambient noise on cumulant based direction finding systems using the interpretation for the information provided by cumulants for array processing applications described in dogan and mendek they first demonstrate the suppression of uncorrelated non gaussian noise that has spatially varying statistics then they indicate methods to suppress spatially colored non gaussian noise using cumulants and an additional sensor whose measurement noise component is independent of the noise components of the original array measurements they also indicate the noise suppression properties of the virtual esprit algorithm proposed in dogan and mendel in addition they propose a method that combines second and fourth order statistics together in order to suppress spatially colored non gaussian noise finally they also illustrate how to suppress spatially colored non gaussian noise when the additional sensor measurement is not available simulations are presented to verify the results

a statistical resolution theory of the ar method of spectral analysis the autoregressive ar method of spectral analysis is widely used in diverse areas for its solid theoretical foundation interesting physical interpretation computational efficiency and more importantly high resolution capability various aspects of its statistical performance have been investigated however the resolution probability that provides the most rigorous description of the spectrum resolution capability is still not available in the literature in this paper by formulating the resolution event in the framework of statistical decision theory and directly determining its probability from its characteristic function we obtain an exact asymptotic formula for the probability of resolution on this basis we determine the limiting resolving behavior of the sample ar spectrum and develop the corresponding geometrical insight in the parametric space simulation and numerical results are also presented to confirm and illustrate the theory

performance analysis of the enhanced minimum variance spatial spectrum estimator the performance of the enhanced minimum variance emv spatial spectrum estimator is studied expressions are presented fur the variance and bias of direction of arrival estimators based on general functions of the covariance matrix and these expressions are then used to study the emv estimator finite data effects and random perturbations on the ideal modeling assumptions are studied separately

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

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

probability of resolution of the music algorithm the music algorithm is well known for its high resolution capability and various aspects of its statistical performance have been investigated however rigorous asymptotic analysis of one of its most important performance measures the probability of resolution is not available yet we analyze the probability of the music algorithm resolving two spatially separated signal sources in the context of array processing by formulating the resolution problem in the framework of statistical decision theory and directly determining the probability density function pdf of the indefinite and singular quadratic form that defines the resolution event we arrive at an exact asymptotic formula for the probability of resolution this is accomplished by a multistep procedure computer simulations have been performed to confirm the validity of the theory

an adaptive pseudo spectral method for reaction diffusion problems the spectral interpolation error was considered for both the chebyshev pseudo spectral and galerkin approximations a family of functionals i sub r u with the property that the maximum norm of the error is bounded by i sub r u j sub r where r is an integer and j is the degree of the polynomial approximation was developed these functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize i sub r u the number of collocation points is then chosen to maintain a prescribed error bound the method is illustrated by various examples from combustion problems in one and two dimensions

transformations of matrices into banded form abstract a class of full or triangular matrices r is described for which there exist banded matrices b such that the product br is also banded the banded matrices yield recursion relations for solving systems of linear equations examples of such matrices arising from second derivative operators acting on orthogonal function expansions are used to illustrate the main theorem and its application practical considerations in efficient implementation are discussed