an approach to time series analysis and arma spectral estimation this paper presents an approach to time series analysis and arma spectral estimation from only the output data corrupted by noise it is shown that the generalized not well known modified yule walker myw equations hold when the residual is some correlated noise to solve such equations a new version of the generalized least squares gls method is proposed yielding ar parameter estimates with higher accuracy this gls method can also be used to enhance the estimation accuracy of ar parameters for short and noisy data in case of the myw equation holding theoretically furthermore a simple procedure for improving ma parameter estimates is studied our approach as cadzow s singular value decomposition svd method has provided significantly higher performance spectral estimates for a low order arma model than those obtained via usual techniques which are based upon direct solution of the myw equations and which use a high order model without considering an additive noise

estimation of close sinusoids in colored noise and model discrimination this paper considers the estimation of the two close dominant frequencies in the signal when it is known a priori that the observation is a sum of two close sinusoids and an additive colored noise whose spectral density is unknown earlier attempts have assumed the additive noise to be independent next we develop decision rules for checking whether the observed signal has only one sinusoid or two close sinusoids all of the earlier studies assumed that there are two close sinusoids in the signal another model discrimination problem considered is the determination of the causal structure of the observed periodicity a rule is given to test whether the observation comes from a sinusoid plus additive noise possible colored or the observation comes from a stationary autoregressive model we also present a numerical study showing the efficacy of the robust estimation procedure for estimating the two frequencies and the decision rules for checking the number of dominant frequencies and the causal mechanism finally we compare the proposed method to well known methods used for two close sinusoids in white noise

lieb thirring constant on the sphere and on the torus abstract we prove on the sphere s 2 and on the torus t 2 the lieb thirring inequalities with improved constants for orthonormal families of scalar and vector functions

estimation of decay rates of transient signals the problem of estimating time constants of multiple exponentials in additive white gaussian noise is investigated it is shown that existing linear production based techniques which perform very well for decay rate estimation of signals with different frequencies fail when applied to the case of decaying exponentials at a single frequency an alternate minimum weighted norm method is proposed it is shown that this new technique achieves acceptable estimates of time constants at a signal to noise ratio of 20 db a significant improvement over existing techniques

recursive updating the eigenvalue decomposition of a covariance matrix the author addresses the problem of computing the eigensystem of the modified hermitian matrix given the prior knowledge of the eigensystem of the original hermitian matrix specifically an additive rank k modification corresponding to adding and deleting blocks of data to and from the covariance matrix is considered an efficient and parallel algorithm which makes use of a generalized spectrum slicing theorem is derived for computing the eigenvalues the eigenvector can be computed explicitly in terms of the solution of a much reduced k k homogeneous hermitian system the overall computational complexity is shown to be improved by an order of magnitude from o n sup 3 to o n sup 2 k where n n is the size of the covariance matrix it is pointed out that these ideas can be applied to adaptive signal processing applications such as eigen based techniques for frequency or angle of arrival estimation and tracking specifically adaptive versions of the principal eigenvector method and the total least squares method are derived

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

variance of least squares estimators for a damped sinusoidal process the variance of least squares estimators for the parameter estimation of a damped sinusoidal process is analyzed based on first order perturbation analytical expressions for the variances of the frequency damping factor amplitude and phase estimators are derived explicit expressions are given for both damped and undamped single mode cases the effect of mode separation on the accuracy is investigated through the two mode case the dependence of the variances on number of data points model order signal to noise ratio and mode separation is investigated both analytically and numerically for practical applications extensive monte carlo simulation results are given to verify enhance and illustrate the analytical expressions

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

on the zeros of the linear prediction error filter for deterministic signals the zeros of a linear prediction error filter polynomial for a class of deterministic signals that are a sum of samples of m exponentially damped undamped sinusoids is studied it is assumed that n samples are available for processing and that they are uncorrupted by noise it is shown that the exponent parameters of the m signals can be determined from m zeros called signal zeros of an lth degree prediction error filter polynomial l m if l lies in between m and n m n m 2 in a special case the rest of the l m zeros of the filter polynomial switch are called extraneous zeros are shown to be approximately uniformly distributed with in the unit circle regardless of the type of exponentials in the signal if the prediction filter coefficients are chosen to have minimum euclidean length the results obtained provide insight into the estimation problem with noisy data

redundant and on line cordic for unitary transformations two sided unitary transformations of arbitrary 2 spl times 2 matrices are needed in parallel algorithms based on jacobi like methods for eigenvalue and singular value decompositions of complex matrices this paper presents a two sided unitary transformation structured to facilitate the integrated evaluation of parameters and application of the typically required transformations using only the primitives afforded by cordic thus enabling significant speedup in the computation of these transformations on special purpose processor array architectures implementing jacobi like algorithms we discuss implementation in nonredundant cordic to motivate and lead up to implementation in the redundant and on line enhancements to cordic both variable and constant scale factor redundant cfr cordic approaches are detailed and it is shown that the transformations may be computed in 10n spl delta time where n is the data precision in bits and spl delta is a constant accounting for accumulated on line delays a more area intensive approach using a novel on line cordic encoded angle summation difference scheme reduces computation time to 6n spl delta the area time complexities involved in the various approaches are detailed

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

efficient numerically stabilized rank one eigenstructure updating signal processing rank one eigenvalue decomposition evd updating is well suited to the problem of tracking time varying subspaces previously published rank one evd updating algorithms suffer from a linear buildup of roundoff error which makes them impractical for a large number of recursive updates in the present work the authors develop a numerical stabilization technique which eliminates the error buildup problem in a computationally efficient manner and makes the rank one evd update a practical numerical tool for online computation a simplified eigenvalue iteration is also given it reduces the complexity of the algorithm somewhat as well as the computation time simulations are presented to illustrate numerical performance

statistical analysis of effective singular values in matrix rank determination a major problem in using svd singular value decomposition as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end conference regions are derived for the perturbed singular values of matrices with noisy observation data the analysis is based on the theories of perturbations of singular values and statistical significance test threshold bounds for perturbation due to finite precision and i i d random models are evaluated in random models the threshold bounds depend on the dimension of the matrix the noisy variance and predefined statistical level of significance results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given

geometric methods for determining system poles from transient response the problem treated is that of identifying the poles of a finite order system by observing its transient decay after cessation of input for a limited time using possibly multiple observation points and experimental repetition various approaches are studied having the common characteristic that a homogeneous matrix equation must be solved several techniques that have been given scant attention in the literature are consolidated by a geometric treatment together with new results including an analytical treatment of the consequences of assuming an excessively high system order derivation of a statistically unbiased estimate for an intermediate parameter in the solution new theorems on error effects a recipe for effective use of the singular value decomposition and a new method for suppression of extraneous poles

singular value decomposition and improved frequency estimation using linear prediction linear prediction based lp methods for fitting multiple sinusoid signal models to observed data such as the forward backward fblp method of nuttall 5 and ulrych and clayton 6 are very ill conditioned the locations of estimated spectral peaks can be greatly affected by a small amount of noise because of the appearance of outliers lp estimation of frequencies can be greatly improved at low snr by singular value decomposition svd of the lp data matrix the improved performance at low snr is also better than that obtained by using the eigenvector corresponding to the minimum eigenvalue of the correlation matrix as is done in pisarenko s method and its variants

the reconstruction of a band limited function and its fourier transform from a finite number of samples at arbitrary locations by singular value decomposition a method for the stable interpolation of a bandlimited function known at sample instants with arbitrary locations in the presence of noise is given singular value decomposition is used to provide a series expansion that in contrast to the method of sampling functions permits simple identification of vectors in the minimum norm space poorly represented in the sample values three methods miller regularization least squares estimation and maximum a posteriori estimation are given for obtaining regularized reconstructions when noise is present the singular value decomposition svd method is used to interrelate these methods examples illustrating the technique are given

threshold bounds in svd and a new iterative algorithm for order selection in ar models the problem of order determination of ar autoregressive models using singular value decomposition svd is reexamined from a statistical point of view thresholds for distinguishing between significant and nonsignificant singular values are derived and a novel iterative algorithm for order selection in ar models is presented simulation results show the technique to be very effective when a small number of samples is available

matrix fraction approach for finite state aerodynamic modeling a least square procedure for the finite state approximation of the aerodynamic matrix is introduced which is more efficient and or simpler to use than those currently available this is accomplished by starting from the approximation of the aerodynamic matrix as nd 1 or d 1 n where n and d are matrices with polynomial dependence on complex reduced frequency p three different finite state realizations based on the same matrix polynomial approximation are presented the advantages of the approach and problems encountered in using the method are discussed numerical results are included

total least squares approach for frequency estimation using linear prediction the resolution of the estimated closely spaced frequencies of the multiple sinusoids degrades as the signal to noise ratio snr of the received signal becomes low this resolution can be improved by using the total least squares tls method in solving the linear prediction lp equation this approach makes use of the singular value decomposition svd of the augmented matrix for low rank approximation to reduce the noise effect from both the observation vector and the lp data matrix simultaneously comparison is made to the principle eigenvector pe method of tufts and kumaresan both on theoretical and experimental grounds the tls algorithm exhibits superior performance over the pe method where low rank approximation is applied to the data matrix only

on characteristic exponents in turbulence ruelle has found upper bounds to the magnitude and to the number of non negative characteristic exponents for the navier stokes flow of an incompressible fluid in a domain the latter is particularly important because it yields an upper bound to the hausdorff dimension of attracting sets however ruelle s bound on the number has three deficiences i it relies on some unproved conjectures about certain constants ii it is valid only in dimensions 3 and not 2 iii it is valid only in the limit in this paper these deficiences are remedied and in addition the final constants in the inequality are improved

a nonlinear filter for estimating a sinusoidal signal and its parameters in white noise on the case of a single sinusoid a nonlinear filter is proposed for estimating a complex sinusoidal signal and its parameters frequency amplitude and phase from measurements corrupted by white noise this filter is derived by applying an extended complex kalman filter eckf to a nonlinear stochastic system whose state variables are a function of its frequency and a sample of an original signal and then proof of the stability is given in the case of a single complex sinusoid simulations demonstrate that the proposed nonlinear filter is effective as a method for estimating a single complex sinusoid and its frequency under a low signal to noise ratio snr in addition the effect of the initial condition in the filter on frequency estimation is also discussed

correction to rotation and gray scale transform invariant texture classification using spiral resampling subband decomposition and hidden markov model this paper proposes a new texture classification algorithm that is invariant to rotation and gray scale transformation first we convert two dimensional 2 d texture images to one dimensional 1 d signals by spiral resampling then we use a quadrature mirror filter qmf bank to decompose sampled signals into subbands in each band we take high order autocorrelation functions as features features in different bands which form a vector sequence are then modeled as a hidden markov model bmm during classification the unknown texture is matched against all the models and the best match is taken as the classification result simulations showed that the highest correct classification rate for 16 kinds of texture was 95 14

linear prediction approach to direction estimation of cyclostationary signals in multipath environment we investigate the estimation of the directions of arrival doa of closely spaced narrowband cyclostationary signals in the presence of multipath propagation by exploiting the spatial and temporal properties of most communication signals we propose a new cyclic forward backward linear prediction fblp approach for coherent signals impinging on a uniform linear array ula in the proposed algorithm the evaluation of the cyclic array covariance matrix is avoided and the difficulty of choosing the optimal time lag parameter is alleviated as a result the proposed approach has two advantages 1 the computational load is relatively reduced and 2 the robustness of estimation is significantly improved the performance of the proposed approach is confirmed through numerical examples and it is shown that this approach is superior in resolving the closely spaced coherent signals with a small length of array data and at relatively low signal to noise ratio snr