a new approach to frequency analysis with amplified harmonics summary for stationary time series having a mixed spectrum we introduce a new method for estimating the discrete frequencies based on amplification of the sine waves in the time series and on minimization of least squares regressions on amplified waves under the assumption that the continuous part of the spectrum arises from a stationary autoregressive moving average process we show that the estimators for the discrete frequencies obtained with this method are strongly consistent more precisely as the serial length n tends to infinity their bias almost surely converges to zero with the rate n 3 2 log n 5 6 2 we also establish a central limit theorem which shows that these estimators have exactly the same asymptotic variance as that given by whittle s method which is known to yield the most accurate frequency estimates however the new technique has two advantages over whittle s method first it yields an algorithm that is both robust against improper starting estimates and computationally efficient secondly it gives rise to a simple tool based on its amplified harmonics for identifying the number of sine waves in a time series a simulation study is also reported that gives statistical results on the properties of the new method on simulated series as well as comparative results with whittle s method

linear prediction approach for efficient frequency estimation of multiple real sinusoids algorithms and analyses based on the linear prediction property of sinusoidal signals two constrained weighted least squares frequency estimators for multiple real sinusoids embedded in white noise are proposed in order to achieve accurate frequency estimation the first algorithm uses a generalized unit norm constraint while the second method employs a monic constraint the weighting matrices in both methods are a function of the frequency parameters and are obtained in an iterative manner for the case of a single real tone with sufficiently large data samples both estimators provide nearly identical frequency estimates and their performance approaches crame spl acute r rao lower bound crlb for white gaussian noise before the threshold effect occurs algorithms for closed form single tone frequency estimation are also devised computer simulations are included to corroborate the theoretical development and to contrast the estimator performance with the crlb for different frequencies observation lengths and signal to noise ratio snr conditions

recursive identification of switched arx systems we consider the problem of recursively identifying the parameters of a deterministic discrete time switched auto regressive exogenous sarx model under the assumption that the number of models the model orders and the mode sequence are unknown the key to our approach is to view the identification of multiple arx models as the identification of a single though more complex lifted dynamical model built by applying a polynomial embedding to the input output data we show that the dynamics of this lifted model do not depend on the value of the discrete state or the switching mechanism and are linear on the so called hybrid model parameters therefore one can identify the parameters of the lifted model using a standard recursive identifier applied to the embedded input output data the estimated hybrid model parameters are then used to build a polynomial whose derivatives at a regressor give an estimate of the parameters of the arx model generating that regressor the estimated arx model parameters are shown to converge exponentially to their true values under a suitable persistence of excitation condition on a projection of the embedded input output data such a condition is a natural generalization of the well known result for arx models although our algorithm is designed for perfect input output data our experiments also evaluate its performance as a function of the level of noise for different choices of the number of models and model orders we also present an application to temporal video segmentation

statistical analysis of music and subspace rotation estimates of sinusoidal frequencies consideration is given to the analysis of the large sample second order properties of multiple signal classification music and subspace rotation sur methods such as esprit for sinusoidal frequency estimation explicit expressions for the covariance elements of the estimation errors associated with either method are derived these expressions of covariances are then used to analyze and compare the statistical performances of the music and sur estimation sure methods both music and sure are based on the eigendecomposition of a sample data covariance matrix the expressions for the estimation error variances derived are used to study the dependence of music and sure performances on the dimension of the data covariance matrix used

on the construction of infinitely many mutually noncongruent examples of minimal imbeddings of s 2n 1 into cp n n geqslant 2

a generalized least squares method for frequency estimation the application of the generalized least squares gls method to the estimation of the frequencies of sinusoids in additive colored noise is discussed an iterative algorithm is proposed starting from the assumption that the parameter vector in the model of the sinusoids is symmetric the algorithm utilizes an adaptive strategy for contracting the poles of the estimated signal model expressions for the probability limit and the asymptotic variance of the estimates are derived for the single sinusoid case possible convergence points and the asymptotic behavior of the algorithm in the case of low snrs signal to noise ratios are analyzed extensive simulation results show that the algorithm represents a simple and reliable tool for practical applications

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

new results on employing cumulants for retrieving sinusoids in colored non gaussian noise a novel analysis is presented which shows that signals consisting of sinusoids plus non gaussian noise have a new signal to noise ratio nsnr associated with their fourth order cumulants this nsnr is a multiple of the original snr when the noise is colored and the original snr is equal to or greater than one therefore employing cumulants for harmonic retrieval problems is now an approach capable of handling not only gaussian noise but also colored non gaussian noise the presented analysis is verified via computer simulations

instrumental variable methods for arma spectral estimation the modified yule walker technique of arma spectral estimation is shown to be a special case of the instrumental variable method of system identification several recursive instrumental variable algorithms are proposed for adaptive spectral estimation including a new algorithm for the overdetermined case an efficient lattice algorithm is presented for solving the modified yule walker equations given the sample correlation coefficients the performance of some of the algorithms is illustrated by simulation results

tracking switched dynamic network topologies from information cascades contagions such as the spread of popular news stories or infectious diseases propagate in cascades over dynamic networks with unobservable topologies however social signals such as product purchase time or blog entry timestamps are measurable and implicitly depend on the underlying topology making it possible to track it over time interestingly network topologies often jump between discrete states that may account for sudden changes in the observed signals the present paper advocates a switched dynamic structural equation model to capture the topology dependent cascade evolution as well as the discrete states driving the underlying topologies conditions under which the proposed switched model is identifiable are established leveraging the edge sparsity inherent to social networks a recursive ell 1 norm regularized least squares estimator is put forth to jointly track the states and network topologies an efficient first order proximal gradient algorithm is developed to solve the resulting optimization problem numerical experiments on both synthetic data and real cascades measured over the span of one year are conducted and test results corroborate the efficacy of the advocated approach

enhanced puma for direction of arrival estimation and its performance analysis direction of arrival doa estimation is a problem of significance in many applications in practice due to the occurrence of coherent signals and or when the number of available snapshots is small it is a challenge to find doas accurately this problem is revisited here through a new enhanced principal singular vector utilization for modal analysis epuma doa estimation approach which improves the threshold performance by first generating p k doa candidates for k sources where p geq k and then judiciously selecting k of them the asymptotic variance of epuma is theoretically derived and numerical results are provided to validate the asymptotic analysis and illustrate the practical merits of epuma

a general structural equation model with dichotomous ordered categorical and continuous latent variable indicators a structural equation model is proposed with a generalized measurement part allowing for dichotomous and ordered categorical variables indicators in addition to continuous ones a computationally feasible three stage estimator is proposed for any combination of observed variable types this approach provides large sample chi square tests of fit and standard errors of estimates for situations not previously covered two multiple indicator modeling examples are given one is a simultaneous analysis of two groups with a structural equation model underlying skewed likert variables the second is a longitudinal model with a structural model for multivariate probit regressions

estimation of all pole model parameters from noise corrupted sequences least squares estimates of the parameters of all pole signal models are biased when applied to noise corrupted all pole sequences this bias has been shown to be proportional to the inverse of the signal to noise ratio an autocorrelation like procedure is shown to increase under some circumstances the signal to noise ratio while retaining the pole locations when the noise is white the signal to noise ratio improvement varies for each mode in the all pole sequence with modes corresponding to pole locations close to the unit circle showing the most improvement a signal to noise ratio cutoff exists for each mode below which no improvement can be obtained this cutoff depends on the proximity of the pole location to the unit circle approaching zero as the pole nears the origin this cutoff corresponds to the point at which a mode s spectral value just equals the level of the noise floor this procedure can be applied recursively until an acceptable signal to noise ratio is obtained the all pole parameters can then be estimated from the processed sequence using least squares estimation

a fast efficient technique for the estimation of frequency summary a technique is presented for the estimation of the frequency of a sinusoid in the presence of noise the technique is based on fitting arma 2 2 models iteratively in a special way the estimator is shown to be strongly consistent and as efficient as the least squares estimator of the frequency or the periodogram maximizer a simple accelerated version of the technique is shown to converge in a small number of iterations the number depending on the accuracy of the initiator of the procedure the results of a number of simulations are reported

a robust approach for high resolution frequency estimation a robust estimation method for frequencies of received signals is considered the influence function of a robust estimate is derived for the sinusoidal signals the variance of the robust estimate is also derived using the influence function it is found that the robust estimate attains the cramer rao lower bound crlb for the contaminated gaussian distribution it is of order o n sup 3 and is close to the crlb for the perfect gaussian distribution the authors introduce some basic definitions for the high resolution frequency estimation method and prove that the robust estimate has the high resolution property this property is also confirmed by numerical simulations

on the construction of codimension two minimal immersions of exotic spheres into euclidean spheres it is nowadays a well known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non diffeomorphic to spheres such manifolds were christened exotic spheres by their founder milnor 223 the structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in 20 for example it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres in differential geometry and geometric variational theory the study of closed minimal submanifolds of the euclidean n sphere s 1 is directly related to that of the local structure of singularities of minimal subvarieties in the general riemannian setting therefore closed minimal submanifolds of s 1 are not only interesting nice geometric objects by themselves but are also of general theoretical importance in the study of geometric measure theory 16 18 24 among various problems on the closed minimal submanifolds of euclidean spheres the following two problems naturally distinguish themselves as especially interesting namely

a new order determination technique for arma processes it is generally recognized that when performing spectral estimation via the modeling approach autoregressive moving average arma models can provide estimates of higher resolution than auto gressive models alone 1 2 however there is lacking a systematic method for determining the orders of an arma process from its output of the very few order determination techniques in existence most are trial and error based the akaike information criterion for arma processes is one such example this paper presents a new method that determines the arma orders in one pass it is a modification of the method of chow 6 and uses the gram schmidt orthonormalization procedure to determine linear dependency of the columns in an order determination the theoretical development is verified by simulation which also includes results on order determination from output sequences alone

estimating frequency by interpolation using fourier coefficients the periodogram of a time series that contains a sinusoidal component provides a crude estimate of its frequency parameter the maximizer over the fourier frequencies being within o t sup 1 of the frequency as the sample size t increases in the paper a technique for obtaining an estimator that has root mean square error of order t sup 3 2 is presented which involves only the fourier components of the time series at three frequencies the asymptotic variance of the estimator varies between roughly the asymptotic variance of the maximizer of the periodogram over all frequencies the cramer rao lower bound and three times this variance the advantage of the new estimator is its computational simplicity

estimating the steady state frequency of a sine wave burst with extremely short record length we present a technique based on modified linear prediction for accurate estimation of the steady state frequency of a sine wave burst with extremely short record length the estimation procedure presented here makes use of automatic windowing of the burst location within the observation interval such a windowing technique eliminates the transition effects which accompany the estimators derived using the traditional lpc technique to implement this windowing a set of artificial state variables given by the product of output samples and their delayed versions is used the advantages of this method are demonstrated using a computer simulation

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

employing symmetry constraints for improved frequency estimation by eigenanalysis methods the problem of extracting sinusoid signals from noisy observations made at equally spaced times is considered eigenanalysis methods such as pisarenko s method and the extended prony method find the eigenvector with minimum eigenvalue of a suitably chosen matrix and then obtain the complex sinusoids as the roots of the polynomial that has the components of the eigenvector as coefficients for the sinusoids to be undamped it is necessary that the roots lie on the unit circle and hence that the eigenvector be conjugate symmetric it is shown how this symmetry constraint can be incorporated into eigenanalysis estimation methods in a routine way the practical importance of the constraint is investigated by cramer rao variance bound calculations and by simulation the following conclusions are made 1 the symmetry constraint is straightforward to implement and reduces the amount of computation required 2 the relative reduction in variance of the constrained over the unconstrained frequency estimators

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

on the construction of infinitely many congruence classes of imbedded closed minimal hypersurfaces in s n 1 for all n geq 3

optimal high order yule walker estimation of sinusoidal frequencies the asymptotic properties of sinusoidal frequency estimators based on the high order yule walker hoyw equations were analyzed recently the results of that analysis are used to propose two classes of frequency estimators one class uses singular value decomposition and the other uses a sparse matrix solution both classes entail two estimation steps the first step generates initial estimates which are used to obtain an optimal weighting matrix and the second step generates an optimally weighted estimate each two step method produces asymptotically minimum variance estimates over all estimators of their class the implementation of the proposed estimators is described in detail and numerical examples are presented to evaluate their performance

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

asymptotic properties of the high order yule walker estimates of sinusoidal frequencies the asymptotic properties of the high order yule walker hoyw estimator of sinusoidal frequencies are analyzed an explicit formula for the covariance matrix of the hoyw frequency estimation error is derived the effects on estimation accuracy of the number of yw equations the model order and a certain weighting matrix are investigated analytically the optimally weighted hoyw estimator is described the analysis presented together with the already existing empirical evidence provides enough motivation for the use of an overdetermined and high order yule walker estimator

a parameter estimation approach to estimation of frequencies of sinusoids it is shown that with uniform sampling a sample of the sum of m sinusoids at time nt can be uniquely expressed as a linear combination of the 2m samples at times n 1 t n 2m t the 2m coefficients of linear combination are also the coefficients of a 2m order polynomial whose roots are e pmj omega i omega i being the frequencies of the sinusoids given the samples of sinusoids plus noise a consistent estimate of the 2m coefficients are obtained by the instrumental variable method of parameter estimation it is also possible to track time varying frequencies by a recursive algorithm that exponentially weighs out the past data simulation examples of one and two sinusoids are given

iterative inverse filtering approach to the estimation of frequencies of noisy sinusoids a signal modeling approach is proposed to reduce the problem of estimating frequencies of noisy sinusoids to the parameter estimation problem both white and colored noise are considered the maximum likelihood approach leads to an algorithm computationally inefficient for low values of signal to noise ratio based on the equation error formulation an iterative inverse filtering algorithm is derived in the case of colored noise and a generalized least squares algorithm in the case of white noise it is shown on the basis of numerous experimental results that the iterative inverse filtering algorithm provides highly accurate estimates of unknown frequencies for low values of signal to noise ratio even in the case of a small number of sampling points

on instrumental variable estimation of sinusoid frequencies and the parsimony principle multiple sinusoids in noise can be modeled as an arma process with the ar parameters satisfying certain symmetry relations according to the parsimony principle the constraints on the ar parameters should he taken into account to get improved estimation accuracy it is shown in this note that when estimating the ar parameters by a general instrumental variable method such a parsimony does not necessarily apply however the parsimony principle does hold when an optimal instrumental variable method is used

existence and uniqueness for a class of cauchy problems with characteristic initial manifolds on donne un theoreme d existence unicite pour une grande classe d equations d ordre 2 quasilineaires a donnees de cauchy sur une variete qui est caracteristique au sens ou l equation est singuliere d ordre 1 le long de la variete initiale