orthogonal polynomials and the finite toda lattice the choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed it is then shown that the same induced time dependence for finitely supported distributions which gives the ith kp flow under the dual isomorphism induces the ith flow of the toda hierarchy on the matrix the corresponding solution is an n particle finite nonperiodic toda solution where n is the cardinality of the support of c plus the sum of the orders of the highest derivative taken at each point

isospectral flows related to frobenius stickelberger thiele polynomials the isospectral deformations of the frobenius stickelberger thiele fst polynomials introduced in spiridonov et al commun math phys 272 139 165 2007 are studied for a specific choice of the deformation of the spectral measure one is led to an integrable lattice fst lattice which is indeed an isospectral flow connected with a generalized eigenvalue problem in the second part of the paper the spectral problem used previously in the study of the modified camassa holm mch peakon lattice is interpreted in terms of the fst polynomials together with the associated fst polynomials resulting in a map from the mch peakon lattice to a negative flow of the finite fst lattice furthermore it is pointed out that the degenerate case of the finite fst lattice unexpectedly maps to the interlacing peakon ode system associated with the two component mch equation studied in chang et al adv math 299 1 35 2016

on an integrable discretization of the rayleigh quotient gradient system and the power method with a shift an integrable discretization of the rayleigh quotient gradient system is established the solution of the discrete gradient system is described explicitly and converges exponentially to the same equilibrium point as that of the continuous gradient system for arbitrary large difference step size it is shown that the discrete gradient system is essentially equivalent to the power method with a shift of origin for calculating the largest eigenvalue the power method is then proved to be a discrete gradient method

generalized factorial schur functions biorthogonal rational functions and the rii chain generalized factorial schur functions are introduced through a newton type interpolation they are applied to construct a class of biorthogonal rational functions and a special solution to its spectral transformation chain the rii chain a two parameter deformation of the schur type functions is also discussed

toda type differential equations for the recurrence coefficients of orthogonal polynomials and freud transformation transformations of the measure of orthogonality for orthogonal polynomials namely freud transformations are considered jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations dynamics of the coefficients are described by generalized toda equations the classical toda lattice equations are the simplest special case of dynamics of the coefficients under the freud transformation of the measure of orthogonality also dynamics of hankel determinants its minors and other notions corresponding to the orthogonal polynomials are studied

on schur flows for the finite schur dmkdv flows a non local poisson structure is introduced and shown to be linked via backlund darboux transformations to linear and quadratic poisson structures for the toda lattice two different lax representations for the schur flows are used one to construct backlund darboux transformations and the other to solve the cauchy problem via the trigonometric moment problem

strong consistency of the over and underdetermined lse of 2 d exponentials in white noise we consider the problem of least squares estimation of the parameters of two dimensional 2 d exponential signals observed in the presence of an additive noise field when the assumed number of exponentials is incorrect we consider both the case where the number of exponential signals is underestimated and the case where the number of exponential signals is overestimated in the case where the number of exponential signals is underestimated we prove the almost sure convergence of the least squares estimates lse to the parameters of the dominant exponentials in the case where the number of exponential signals is overestimated the estimated parameter vector obtained by the least squares estimator contains a subvector that converges almost surely to the correct parameters of the exponentials

two algorithms for adaptive retrieval of slowly time varying multiple cisoids in noise two algorithms for tracking parameters of slowly varying multiple complex sine waves cisoids in noise the multiple frequency tracker and the adaptive notch filter are described for high signal to noise ratio snr the properties of the algorithms i e stability noise rejection and tracking speed are studied analytically using a linear filter approximation technique the tradeoff between noise rejection and tracking error for both algorithms is shown to be similar different choices of the design variables are discussed namely i minimal mean square estimation error for random walk modeled frequency variations and ii minimal stationary estimation variance subject to a given tracking delay

calculating laplace transforms in terms of the toda molecule a method for calculating a continued fraction expansion of laplace transforms is presented in terms of the tau function of the continuous time toda molecule equation and its integrable discretization the z transform and a discrete analogue of the laplace transform of a set of finite samples are also formulated

cmv the unitary analogue of jacobi matrices we discuss a number of properties of cmv matrices by which we mean the class of unitary matrices studied recently by cantero moral and velazquez we argue that they play an equivalent role among unitary matrices to that of jacobi matrices among all hermitian matrices in particular we describe the analogues of well known properties of jacobi matrices foliation by co adjoint orbits a natural symplectic structure algorithmic reduction to this shape lax representation for an integrable lattice system ablowitz ladik and the relation to orthogonal polynomials as offshoots of our analysis we will construct action angle variables for the finite ablowitz ladik hierarchy and describe the long time behavior of this system 2006 wiley periodicals inc

maximum likelihood estimation of 2 d superimposed exponential signals the problem of maximum likelihood estimation of the parameters i e frequencies amplitudes and noise variance of 2 d superimposed exponential signals is considered in this paper a polynomial rooting approach is proposed to obtain the estimates of the 2 d frequencies strong consistency and limiting distribution are established for the estimates of the parameters furthermore the covariance matrix of the limiting distribution attains the cramer rao lower bound

fine estimators of two dimensional parameters and application to spatial shift estimation this paper presents a fast technique for fine estimation of two dimensional 2 d parameters based on a parabolic interpolation of the same ambiguity function samples and aimed at block oriented estimation of the spatial shift between pairs of images in video sequences expressions for the bias and variance of the position error and the prediction error are derived the method is tested using a synthetically generated autocorrelation function varying the directionality and the eccentricity factor in order to compare the performance of the proposed 2 d estimator to the case of two separate one dimensional 1 d estimators the method has also been applied in vision systems evidencing encouraging results for estimating the parameters of sophisticated global motion models from real images

approximate maximum likelihood algorithms for two dimensional frequency estimation of a complex sinusoid starting with the maximum likelihood ml formulation three iterative algorithms for approximate ml frequency estimation of a two dimensional 2 d complex sinusoid in white gaussian noise are developed mean and variance analyses of the proposed methods are provided which show that they are approximately unbiased and their performance achieves cramer rao lower bound crlb at sufficiently high signal to noise ratio snr conditions computer simulation results are included to corroborate the theoretical development as well as to contrast the performance of the proposed algorithms with kay s estimators and the crlb

schur flow for orthogonal polynomials on the unit circle and its integrable discretization a one parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered the corresponding dynamics of the schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi discrete modified kdv equation namely the schur flow a discrete analogue of the miura transformation is found an integrable discretization of the schur flow enables us to compute a pade approximation of the caratheodory functions or equivalently to compute a perron caratheodory continued fraction in a polynomial time

bootstrap modeling of a class of nonstationary signals the problem of modeling polynomial phase signals is considered techniques based on the bootstrap and multiple hypotheses tests for optimal model selection of constant amplitude polynomial phase signals embedded in stationary noise are developed phase parameter estimators based on both the least squares method and the polynomial phase transform are used the proposed techniques are compared with existing ones including akaike s 1978 1974 information criterion and rissanen s 1978 minimum description length criterion and are shown to outperform these procedures for the considered small sample sizes

speckle reducing anisotropic diffusion this paper provides the derivation of speckle reducing anisotropic diffusion srad a diffusion method tailored to ultrasonic and radar imaging applications srad is the edge sensitive diffusion for speckled images in the same way that conventional anisotropic diffusion is the edge sensitive diffusion for images corrupted with additive noise we first show that the lee and frost filters can be cast as partial differential equations and then we derive srad by allowing edge sensitive anisotropic diffusion within this context just as the lee 1980 1981 1986 and frost 1982 filters utilize the coefficient of variation in adaptive filtering srad exploits the instantaneous coefficient of variation which is shown to be a function of the local gradient magnitude and laplacian operators we validate the new algorithm using both synthetic and real linear scan ultrasonic imagery of the carotid artery we also demonstrate the algorithm performance with real sar data the performance measures obtained by means of computer simulation of carotid artery images are compared with three existing speckle reduction schemes in the presence of speckle noise speckle reducing anisotropic diffusion excels over the traditional speckle removal filters and over the conventional anisotropic diffusion method in terms of mean preservation variance reduction and edge localization

least squares estimation of 2 d sinusoids in colored noise asymptotic analysis this paper considers the problem of estimating the parameters of real valued two dimensional 2 d sinusoidal signals observed in colored noise this problem is a special case of the general problem of estimating the parameters of a real valued homogeneous random field with mixed spectral distribution from a single observed realization of it the large sample properties of the least squares ls estimator of the parameters of the sinusoidal components are derived making no assumptions on the type of the probability distribution of the observed field it is shown that if the disturbance field satisfies a combination of conditions comprised of a strong mixing condition and a condition on the order of its uniformly bounded moments the normalized estimation error of the ls estimator is consistent asymptotically normal with zero mean and a normalized asymptotic covariance matrix for which a simple expression is derived it is further shown that the ls estimator is asymptotically unbiased the normalized asymptotic covariance matrix is block diagonal where each block corresponds to the parameters of a different sinusoidal component assuming further that the colored noise field is gaussian the ls estimator of the sinusoidal components is shown to be asymptotically efficient

near maximal 2 d sinusoid identification from limited data we describe a novel method to identify a large number of two dimensional 2 d sinusoids from very limited data starting with the noiseless exponential superposition model we derive the 2 d linear prediction equations for the data and present a general 2 d exponential identification algorithm based on one dimensional 1 d rooting techniques then we enhance this general algorithm for the special case of 2 d sinusoid identification by using conjugate reverse data to increase the number of identifiable sinusoids and by using fft to avoid polynomial rooting noise is handled by least squares solutions of the linear prediction equations and the polynomial equations numerical simulations demonstrate that our algorithm provides a competitive alternative to existing algorithms

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

the asymptotic distribution of sums of extreme values from a regularly varying distribution on donne une description complete de la distribution asymptotique de la somme des k n valeurs extremes superieures x n 1 kd n n x n 2 kd n n x n n

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

dsf chain for biorthogonal polynomials and its application to matrix eigenvalue problems abstract a discrete time integrable system called the dsf chain is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete schur flow the dsf algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed a discrete analogue of the miura transformation between the discrete schur flow and the qd algorithm is constructed two special solutions to the dsf chain related to q special functions are presented

determinantal solutions of solvable chaotic systems it is shown that two solvable chaotic systems the arithmetic harmonic mean ahm algorithm and the ulam von neumann uvn map have determinantal solutions an additional formula for certain determinants and riccati difference equations play a key role in both cases two infinite hierarchies of solvable chaotic systems are presented which have determinantal solutions

an efficient approach for two dimensional parameter estimation of a single tone in this paper parameter estimation of a two dimensional 2 d single damped real complex tone in the presence of additive white gaussian noise is addressed by utilizing the rank one property of the 2 d noise free data matrix the damping factor and frequency for each dimension are estimated in a separable manner from the principal left and right singular vectors according to an iterative weighted least squares procedure the remaining parameters are then obtained straightforwardly using standard least squares the biases as well as variances of the damping factor and frequency estimates are also derived which show that they are approximately unbiased and their performance achieves crame r rao lower bound crlb at sufficiently large signal to noise ratio snr and or data size conditions we refer the proposed approach to as principal singular vector utilization for modal analysis puma which performs estimation in a fast and accurate manner the development and analysis can easily be adapted for a tone which is undamped in at least one dimension furthermore comparative simulation results with several conventional 2 d estimators and crlb are included to corroborate the theoretical development of the puma approach as well as to demonstrate its superiority

relative stability of trimmed sums this paper gives extensions of mori s strong law for r s n s n x n 1 x n r where s n x1 x2 x n x i are iidrv s and x i n is x i arranged in decreasing order of absolute magnitude the methods differ from mori s continuity of the distribution of the x i is assumed throughout necessary and sufficient conditions for relative stability r s n b n 1 a s for some b n including a generalised condition of spitzer s and a dominated ergodic theorem are proved a one sided version of the relative stability results is also given a theorem of kesten s is generalised to show that if r s n a n b n is bounded almost surely for constants a n b n then r s n alpha n b n xrightarrow p 0 for some n a corollary to this is that if r s n b n is bounded away from 0 and a s then r s n is relatively stable this generalises a result of chow and robbins apart from the continuity assumption

a generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system abstract a generalized eigenvalue algorithm for a certain class of tridiagonal matrix pencils is presented the algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil the convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system

factorial type schur functions orthogonal rational functions and discrete dressing chains a relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on schur type symmetric functions a system of orthogonal rational functions is constructed using a multiparameter deformation of the schur functions spectral equations for the orthogonal rational functions are derived by using properties of the schur type symmetric functions the compatibility condition of the spectral equations induces a discrete dressing chain which is a toda type discrete integrable system describing dressing transformations for orthogonal rational functions

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

improved approximate time domain ml estimators of chirp signal parameters and their performance analysis we derive an improved approximate maximum likelihood ml estimator for the parameters of a single component chirp signal in the time domain which has much lower computational complexity than that of the traditional estimators in the frequency domain we term it the zero order amplitude weighted phase based estimator awpe it is a weighted linear combination of the phases of the received signal samples where the weights are determined by the amplitudes of the received signals it achieves better performance than the only existing time domain estimator in the literature that only exploits the phase information a phase unwrapping pu algorithm is needed to recover the phase information correctly from the received signal samples several pu algorithms are compared and analyzed a new robust pu algorithm based on the first order phase differences of the consecutive received signal samples is proposed this new unwrapping algorithm has smaller pu failure probability than that based directly on the phases of the received signal samples the need for pu can even be removed by using the second order phase differences of the consecutive received signal samples two additional structures for the ml based awpe using the phase differences are derived starting from using the zero order awpe obtained the first is based on the first order phase differences and the second on the second order phase differences when the pu algorithm works perfectly i e no phase unwrapping failure occurs the estimators based on phase differences do not perform as well as the original one based on absolute phase information due to the increased noise variance this is also applicable when the chirp parameters take on very small values however when no knowledge of the ranges of the chirp parameters is available and the possible pu failures are taken into account the estimators based on the phase differences could perform better performance analysis of the estimators is given based on an improved phase noise model it is easy to verify that the estimates obtained are unbiased the cramer rao lower bound crlb on the mean square error mse is derived the mse performance of the estimators approaches the crlb at high signal to noise ratios the variance performance of the estimators based on phase differences is analyzed whose validity is demonstrated by simulation results

covariance shaping least squares estimation a new linear estimator is proposed which we refer to as the covariance shaping least squares csls estimator for estimating a set of unknown deterministic parameters x observed through a known linear transformation h and corrupted by additive noise the csls estimator is a biased estimator directed at improving the performance of the traditional least squares ls estimator by choosing the estimate of x to minimize the weighted total error variance in the observations subject to a constraint on the covariance of the estimation error so that we control the dynamic range and spectral shape of the covariance of the estimation error the presented csls estimator is shown to achieve the cramer rao lower bound for biased estimators furthermore analysis of the mean squared error mse of both the csls estimator and the ls estimator demonstrates that the covariance of the estimation error can be chosen such that there is a threshold snr below which the csls estimator yields a lower mse than the ls estimator for all values of x as we show some of the well known modifications of the ls estimator can be formulated as csls estimators this allows us to interpret these estimators as the estimators that minimize the total error variance in the observations among all linear estimators with the same covariance

statistical analysis of pisarenko s method for sinusoidal frequency estimation statistical analysis is performed for pisarenko s method of sinusoidal frequency estimation by using the periodogram technique under the assumption that the data consist of several sinusoids plus white noise we give an asymptotic expression of the error variance of the sinusoidal frequency estimator also a modified method is proposed this is computationally efficient but the statistical property is not improved we also present simulation results and compare them to the theoretical ones lastly we point out that the pisarenko s method can be implemented as an on line bias compensated least squares algorithm thus avoiding the eigenvalue and eigenvector calculations

a fast and accurate single frequency estimator a novel frequency estimator for a single complex sinusoid in complex white gaussian noise is proposed the estimator is more computationally efficient that the optimal maximum likelihood estimator yet attains equally good performance at moderately high signal to noise ratios the estimator is shown to be related to the linear prediction estimator this relationship is used to reveal why the linear prediction estimator does not attain the cramer rao bound even at high signal to noise ratios

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

cmv matrices in random matrix theory and integrable systems a survey we present a survey of recent results concerning a remarkable class of unitary matrices the cmv matrices we are particularly interested in the role they play in the theory of random matrices and integrable systems throughout the paper we also emphasize the analogies and connections to jacobi matrices

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

hypercomplex signals a novel extension of the analytic signal to the multidimensional case the construction of gabor s 1946 complex signal which is also known as the analytic signal provides direct access to a real one dimensional 1 d signal s local amplitude and phase the complex signal is built from a real signal by adding its hilbert transform which is a phase shifted version of the signal as an imaginary part to the signal since its introduction the complex signal has become an important tool in signal processing with applications for example in narrowband communication different approaches to an n d analytic or complex signal have been proposed in the past we review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n d this extension leads to a new definition of local phase which reveals information on the intrinsic dimensionality of the signal the different approaches are unified by expressing all of them as combinations of the signal and its partial and total hilbert transforms examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown an example is provided for the two dimensional 2 d hypercomplex signal which shows how the novel phase concept can be used in texture segmentation

an exact direct method of sinusoidal parameter estimation derived from finite fourier integral of differential equation in this paper we propose a novel method for estimating the parameters frequency amplitude and phase of real sinusoids to derive the estimator we start from the characteristic differential equation of a sinusoid to remove differentials and obtain an algebraic relation for frequency we introduce finite period weighted integrals of the differential equation which become equivalent to the differential equation when a sufficient number of weight functions are applied as weight functions we show that fourier kernels have excellent properties terms related to integral boundaries are readily eliminated observations are provided by fourier coefficients and the relation becomes independently accurate for multiple sinusoids if they are sufficiently spaced we solve the obtained equations in two ways one is for approaching to the cramer rao lower bound crlb and the other is for enhancing the interference rejection capability also methods are proposed to calculate the weighted integrals from sampled signals with an improved accuracy proposed algorithms are examined under noise and sinusoidal interference error variances are compared with the crlb and other fast fourier transform fft based methods

estimating the frequency of a noisy sinusoid by linear regression corresp the estimation of the parameters of a sinusoid from observations of signal samples corrupted by additive noise is investigated at high signal to noise ratios the additive noise is viewed as an equivalent phase noise suggesting frequency and phase estimation by linear regression on the signal phase the variances of the regression estimates are shown to achieve the cramer rao bounds a formula for the variance of the regression frequency estimator is derived in terms of the noise power spectrum a simple formula for the variance with 1 f 2 phase noise is presented

strong limit theorems for weighted quantile processes on donne une description complete du comportement presque sur des versions ponderees du processus quantile uniforme

algebro geometric solutions of the baxter szegő difference equation we derive theta function representations of algebro geometric solutions of a discrete system governed by a transfer matrix associated with an extension of the trigonometric moment problem studied by szeg and baxter we also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro geometric solutions

estimating two dimensional frequencies by matrix enhancement and matrix pencil a new method called the matrix enhancement and matrix pencil memp method is presented for estimating two dimensional 2 d frequencies in the memp method an enhanced matrix is constructed from the data samples and then the matrix pencil approach is used to extract out the 2 d sinusoids from the principal eigenvectors of the enhanced matrix the memp method yields the estimates of the 2 d frequencies efficiently without solving the roots of a 2 d polynomial or searching in a 2 d space it is shown that the memp method can be faster than a 2 d fft method if the number of the 2 d sinusoids is much smaller than the data set simulation results are provided to show that the accuracy of the memp method can be very close to the cramer rao lower bound

analytic estimation of subsample spatial shift using the phases of multidimensional analytic signals in this correspondence a method of analytic subsample spatial shift estimation based on an a priori n d signal model is proposed the estimation uses the linear phases of n analytic signals defined with the multidimensional hilbert transform this estimation proposes i an analytic solution to the n d shift estimation and ii an estimation without processing complex cross correlation function or cross spectra between signals contrary to most phase shift estimators the method provides better performance in estimating subsample shifts than two classical estimators one using the maximum of cross correlation function and the other seeking the zero of the complex correlation function phase two delay estimators using the in phase and quadrature phase components of signals are also compared to our estimator like most estimators using the complex signal phases the estimator proposed herein presents the advantage of unaltered accuracy when low sampled signals are used moreover we show that this method can be applied to motion tracking with ultrasound images thus included in a block based motion estimation method and tested with ultrasound data this estimator provides an analytical solution to the translation estimation problem

on the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics we shall disclose a relationship between the almost sure stability of weighted empirical distribution functions and sums of order statistics first we obtain an extension of a theorem due to csaki on the almost sure stability of the standardized uniform empirical distribution function this result is then shown to be an essential tool to derive a characterization of the almost sure stability of the sum of k nupper order statistics from a sample of n independent observations from a distribution with positive support in the domain of attraction of a non normal stable law where 1 k n n and k n as n

continued fractions and integrable systems stieltjes solution of the classical moment problem is the forerunner of inverse spectral theory in the modern theory of completely integrable systems the method of inverse scattering is used to obtain explicit solutions of a class of hamiltonian systems which arise as isospectral deformations of a linear operator in this lecture we explain how stieltjes s formulas are used to obtain explicit solutions to a number of completely integrable systems including discrete reductions of some nonlinear partial differential equations and the isospectral deformations of jacobi matrices a k a toda flows

high snr perturbation in single 2 d frequency estimate using matrix pencil an exact and most revealing expression of the high signal to noise snr first order perturbation variance of the single 2d frequency estimate using the matrix enhancement and matrix pencil memp method is provided this expression is a general form consistent with several previously reported expressions for the 1d frequency case based on this expression the statistical efficiency of the memp method is discussed the optimum efficiency of the memp method is shown to be achieved by its 1d specialized version

an efficient two dimensional frequency estimator the problem of two dimensional frequency estimation of a sinusoid embedded in complex white gaussian noise is addressed a frequency estimator and its statistical performance are derived the estimator is demonstrated to be an efficient estimator it achieves the cramer rao bound for high signal to noise ratios the results of a computer simulator which compares the variance with that of the maximum likelihood estimator and the theoretical two dimensional cramer rao bound are presented

estimating two dimensional frequencies by a cumulant based fblp method this correspondence presents a higher order statistics hos based modified version of the two dimensional forward backward linear prediction 2 d fblp method the modified method is suitable for estimating 2 d frequencies of a 2 d data set observed in white or colored gaussian noise w cgn of unknown covariance simulation results are provided to verify the effectiveness of the proposed method

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations