instantaneous frequency estimation using the wigner distribution with varying and data driven window length the estimation of the instantaneous frequency if of a harmonic complex valued signal with an additive noise using the wigner distribution is considered if the if is a nonlinear function of time the bias of the estimate depends on the window length the optimal choice of the window length based on the asymptotic formulae for the variance and bias can be used in order to resolve the bias variance tradeoff however the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the if the goal of this paper is to develop an adaptive if estimator with a time varying and data driven window length which is able to provide quality close to what could be achieved if the smoothness of the if were known in advance the algorithm uses the asymptotic formula for the variance of the estimator only its value may be easily obtained in the case of white noise and relatively high sampling rate simulation shows good accuracy for the proposed adaptive algorithm

discrete time local polynomial approximation of the instantaneous frequency the local polynomial approximation lpa of the time varying phase is used to develop a new form of the fourier transform and the local polynomial periodogram lpp as an estimator of the instantaneous frequency if spl omega t of a harmonic complex valued signal the lpp is interpreted as a time frequency energy distribution over the t spl omega t spl omega sup 1 t spl omega sup m 1 t space where m is a degree of the lpa the variance and bias of the estimate are studied for the short and long time asymptotic behavior of the if estimates in particular it is shown that the optimal asymptotic mean squared errors of the estimates of spl omega sup k 1 t have orders o n sup 2k 1 and o n sup 2 m k 1 2m 3 k 1 2 m respectively for a polynomial spl omega t of the degree m 1 and arbitrary smooth spl omega t with a bounded mth derivative

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

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985

cut finite element methods for elliptic problems on multipatch parametric surfaces we develop a finite element method for the laplace beltrami operator on a surface described by a set of patchwise parametrizations the patches provide a partition of the surface and each patch is

the statistical performance of some instantaneous frequency estimators the authors examine the class of smoothed central finite difference scfd instantaneous frequency if estimators which are based on finite differencing of the phase of the analytic signal these estimators are closely related to if estimation via the periodic first moment with respect to frequency of discrete time frequency representations tfrs in l cohen s 1966 class the authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero crossing estimator and one based on linear regression on the signal phase it is found that the regression if estimator is biased and exhibits a large threshold for much of the frequency range by replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic scfd pscfd estimator which is unbiased and has a frequency independent variance yet retains the optimal performance and simplicity of the original estimator

fast orthogonalization algorithm and parallel architecture for ar spectral estimation based on forward backward linear prediction truncated qr methods have been shown to be comparable to svd based methods for the sinusoidal frequency estimation based on the forward backward linear prediction fblp model however without exploiting the special structure of the fblp matrix the qr decomposition qrd of the fblp matrix has a computational complexity on the order of 2 6m n n sup 2 3 o n sup 2 for a 2m n fblp matrix a fast algorithm to perform the qrd of the fblp matrix by exploiting its special toeplitz hankel form is proposed the computational complexity is reduced to 10n sup 2 4mn o n the fast algorithm can also be easily implemented by a linear systolic array reducing the number of time steps required to 2m 5n 4

a posteriori error estimates for mixed finite element approximations of parabolic problems we derive residual based a posteriori error estimates for parabolic problems on mixed form solved using raviart thomas nedelec finite elements in space and backward euler in time the error norm considered is the flux part of the energy i e weighted l 2 norm integrated over time in order to get an optimal order bound an elementwise computable post processed approximation of the scalar variable needs to be used this is a common technique used for elliptic problems the final bound consists of terms capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm

elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems we derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations the space discretization uses finite element spaces that are allowed to change in time our main tool is an appropriate adaptation of the elliptic reconstruction technique introduced by makridakis and nochetto we derive novel a posteriori estimates for the norms of l 0 t l2 and the higher order spaces l 0 t h1 and h1 0 t l2 with optimal orders of convergence

conditional mean and maximum likelihood approaches to multiharmonic frequency estimation the performance of an extended kalman filter ekf applied to the problem of estimating the assumed constant parameters fundamental frequency harmonic phases and amplitudes of a complex multiharmonic signal measured in noise is shown to be asymptotically i e as the number of measurements tends to infinity efficient the cramer rao cr bounds associated with the estimation problem are derived for the case where the measurements commence at an arbitrary time distinct from zero

the relationship between instantaneous frequency and time frequency representations the relationship between instantaneous frequency estimation based on the derivative of the phase of the analytic signal and the first moment of the general time frequency representation from cohen s 1989 class is given in the continuous and discrete time domains application of the standard linear definition of first moment to discrete time frequency representations leads to biased instantaneous frequency estimators with high variance to correct this problem it is shown that periodic circular definitions of moments must be used to account for the periodization of the frequency variable due to sampling

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

limitations on snr estimator accuracy we consider the samples of a pure tone in additive white gaussian noise awgn for which we wish to determine the signal to noise ratio snr defined here to be spl alpha a sup 2 2 spl sigma sup 2 where a is the tone amplitude and spl sigma sup 2 is the noise variance a and spl sigma sup 2 are assumed to be deterministic but unknown a priori if the variance of an unbiased estimator of spl alpha is spl sigma sub spl alpha spl circ sup 2 we show that at high snr the normalized standard a deviation satisfies the cramer rao lower bound crlb according to spl sigma sub spl alpha spl circ spl alpha spl ges spl radic 2 n where n is the number of independent observables used to obtain the snr estimate spl sigma spl circ

a new least squares stabilized nitsche method for cut isogeometric analysis we derive a new stabilized symmetric nitsche method for enforcement of dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis cutiga we consider c1 spli

an a posteriori error analysis of mixed finite element galerkin approximations to second order linear parabolic problems in this article a posteriori error estimates are derived for mixed finite element galerkin approximations to second order linear parabolic initial and boundary value problems using mixed elliptic reconstructions a posteriori error estimates in l infty l 2 and l 2 l 2 norms for the solution as well as its flux are proved for the semidiscrete scheme finally based on a backward euler method a completely discrete scheme is analyzed and a posteriori error bounds are derived which improves upon earlier results on a posteriori estimates of mixed finite element approximations to parabolic problems results of numerical experiments verifying the efficiency of the estimators have also been provided

semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals we propose a semiparametric approach to fundamental frequency estimation of an unknown periodic signal in additive white noise based on model selection our estimator maximizes a penalized version of the cumulated periodogram and is proved to be consistent and asymptotically efficient under very general conditions when the number of observations is fixed an implementation of this estimation method is proposed and illustrated on specific synthetic signals which arise in laser vibrometry we extend this method for estimating the fundamental frequencies of two periodic functions having different fundamental frequencies when the data consist of their sum and additive white noise we also compare the performances of our procedure with the so called microdoppler technique which is commonly used for laser vibrometry signals analysis we show on simulated data that the penalized cumulated periodogram yields an accurate estimation of the frequencies at very low signal to noise ratios

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

an a posteriori error estimate and adaptive timestep control for a backward euler discretization of a parabolic problem a simple algorithm for adaptive timestep control is presented for a backward euler discretization of a linear parabolic problem the algorithm is based on an a posteriori error estimate involving the computed approximate solution it is proved that with only very rough a priori information on the exact solution the algorithm will choose a sequence of timesteps for which the error will be controlled up to a constant uniformly in time on a given tolerance level

the phase of a vector perturbed by gaussian noise and differentially coherent receivers some results are presented regarding the asymptotic distribution of the phase of a vector perturbed by gaussian noise it is shown that for large signal to noise ratio the asymptotic distribution of the phase is of the tikhonov type this framework is then used for the synthesis of differentially coherent receiver structures one for m ary phase shift keying mpsk and the other for minimum shift keying msk the first structure bridges the performance gap between coherent and differentially coherent demodulation of mpsk the msk receiver uses matched filtering with differential demodulation

analysis of the variance threshold of kay s weighted linear predictor frequency estimator a theoretical approximation for the variance of kay s weighted linear predictor frequency estimator is derived from this expression an inequality describing the variance threshold of the estimator is found the window weights are then optimized to improve the variance numerical simulations demonstrate that the variance approximations are valid for medium to high signal to noise ratios or for large numbers of samples

cyclic barankin type bounds for non bayesian periodic parameter estimation in many practical periodic parameter estimation problems the appropriate performance criteria are periodic in the parameter space the existing mean square error mse lower bounds such as cramer rao bound crb and barankin type bounds do not provide valid lower bounds in such problems in this paper cyclic versions of the crb and the barankin type bounds hammersley chapman robbins and mcaulay seidman are derived for non bayesian parameter estimation the proposed bounds are lower bounds on the mean cyclic error mce of any cyclic unbiased estimator where the cyclic unbiasedness is defined by using lehmann unbiasedness these mce lower bounds can be readily obtained from existing mse lower bounds and thus can be easily calculated the cyclic barankin type bounds and the performance of the maximum likelihood ml estimator are compared in terms of mce in von mises distributed measurements problem and for frequency and amplitude estimation with gaussian noise in these problems the ml estimator is found to be cyclic unbiased

a unified approach to the stft tfds and instantaneous frequency cohen s class of time frequency representations tfrs is reformulated into a discrete time discrete frequency computer implementable form it is shown how in this form many of the properties of the continuous time continuous frequency formulation are either lost or altered intuitions applicable in the continuous time case do not necessarily carry over to the discrete time case examined the properties of the discrete variable formulation examined are the presence and form of cross terms instantaneous frequency estimation and relationships between cohen s class of tfrs a parameterized class of distributions which is a blending between the short time fourier transform stft and the wigner ville distribution the two main conclusions are that all tfrs of cohen s class implementable in the given form which includes all commonly used tfrs possess cross terms and that instantaneous frequency estimation using periodic moments of these tfrs is purposeless since simpler methods obtain the same result

a posteriori error estimate for the mixed finite element method a computable error bound for mixed finite element methods is established in the model case of the poisson problem to control the error in the h div x l 2 norm the reliable and efficient a posteriori error estimate applies e g to raviart thomas brezzi douglas marini and brezzi douglas fortin marini elements

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

estimation of multicomponent polynomial phase signals impinging on a multisensor array using state space modeling this contribution addresses the problem of estimating the parameters of multicomponent polynomial phase signals when impinging on a multisensor array an original approach is proposed based on a state space modelization of the signal and the application of an extended kalman filter for the state estimation the use of a multisensor array allows the exploitation of a spatial information and leads to the consideration of multiple filters with different observation equations computer simulations are used to demonstrate the effectiveness of the proposed algorithm

non bayesian periodic cramer rao bound the cramer rao bound crb is one of the most important tools for performance analysis in parameter estimation in many practical periodic parameter estimation problems the appropriate criterion is periodic in the parameter space however the crb does not provide a valid lower bound in such problems in this paper the periodic crb for non bayesian periodic parameter estimation is derived the proposed periodic crb is a lower bound on the mean square periodic error mspe of any periodic unbiased estimator where the periodic unbiasedness is defined by using lehmann unbiasedness it is shown that if there exists a periodic unbiased estimator which achieves the bound then the maximum likelihood produces it the periodic crb and the performance of some periodic unbiased estimators are compared in terms of mspe in a linear gaussian problem with modulo measurements and for phase estimation problems

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

graded parametric cutfem and cutiga for elliptic boundary value problems in domains with corners we develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neig

numerical methods for a model for compressible miscible displacement in porous media a nonlinear parabolic system is derived to describe compressible miscible displace ment in a porous medium the system is consistent with the usual model for incompressible miscible displacement two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture the concentration is treated by a galerkin method in both procedures while the pressure is treated by either a lalerkin method or by a parabolic mixed finite element method optimal order estimates in l2 and essentially optimal order estimates in lx are derived for the errors in the approximate solutions for both methods introduction we shall consider the single phase miscible displacement of one compressible fluid by another in a porous medium under the assumptions that no volume change results from the mixing of the components and that a pressure den sity relation exists for each component in a form that is independent of the mixing these equations of state will imply that the fluids are in the liquid state our model will represent a direct generalization of the model 3 4 7 that has been treated extensively for incompressible miscible displacement the reservoir s will be taken to be of unit thickness and will be identified with a bounded domain in r2 we shall omit gravitational terms for simplicity of exposi tion no significant mathematical questions arise when the lower order terms are included

kernel estimation of the instantaneous frequency considers kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise the expected estimation error consists of two terms the systematic bias error grows as the kernel halfwidth increases while the random error decreases for a nonmodulated signal g t the kernel halfwidth that minimizes the expected error is proportional to h spl sim spl sigma sup 2 n spl part sub t sup 2 g 2 sup 1 5 where spl sigma sup 2 is the noise variance and n is the number of measurements per unit time the author shows that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal a t exp i spl phi t for instantaneous frequency estimation the halfwidth which minimizes the expected error is larger h sub 1 3 spl sim spl sigma sup 2 a sup 2 n spl part sub t sup 3 e sup i spl phi t sup 2 sup 1 u7 since the optimal halfwidths depend on derivatives of the unknown function the authors initially estimate these derivatives prior to estimating the actual signal

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

phase based time domain estimation of the frequency and phase of a single sinusoid in awgn the role and applications of the additive observation phase noise model this paper presents the theoretical foundation for time domain phase based estimation of the frequency and phase of a single sinusoid in additive white gaussian noise awgn analogous to the theoretical foundation provided by rife and boorstyn for frequency domain fourier transform based estimation it is shown from the maximum a posteriori probability map and the maximum likelihood ml estimation principles that with the additive observation phase noise aopn due to the awgn being described by its a posteriori distribution conditioned on the received signal magnitude the received signal phase is a sufficient statistic for estimating the single sinusoid angle parameters using a geometric approach the exact statistical model for the aopn is derived where the a posteriori probability density function pdf and the corresponding a priori pdf are given by explicit closed form expressions that are valid for arbitrary signal to noise ratios snrs the a posteriori pdf is tikhonov and is of particular interest as it establishes the aopn model for phase based frequency phase map ml estimation in the time domain it is further illustrated that the results derived can yield various aopn models as special cases and the underlying physical insights and interconnections that exist among these models are revealed it is shown that the model derived by tretter is an ultimate specialization in the high snr limit of the aopn models developed here for high snr the a posteriori tikhonov pdf can be accurately approximated by a gaussian distribution which leads to the best linearized aopn model the applications of these aopn models to the design of linear estimators including the linear minimum mean square error lmmse estimator the linear minimum variance estimator and the lmmse implementation of the weighted phase averager are presented and their estimation performances are compared through computer simulations with the cramer rao lower bound crlb and the bayesian crlb as the benchmark to facilitate estimator design the a priori statistical models of the frequency and phase are proposed from the information theoretic perspective and an improved phase unwrapping algorithm over that given by fu and kam is presented it is shown that by incorporating all the information available in the aopn the estimation accuracy can be much improved

nonlinear processing and analysis of angular signals physical quantities referring to angles like vector direction color hue etc exhibit an inherently periodic nature due to this periodicity digital filters and edge operators proposed for data on the line cannot be applied on such data we introduce filters for angular signals circular mean circular median circular a trimmed mean circular modified trimmed mean particular emphasis is given to the circular median filter for which some interesting properties are derived we also use estimators of circular dispersion to introduce edge detectors for angular signals three variations for the extension of quasirange to circular data are proposed and expressions for their output pdf are derived these circular quasiranges have good and user controlled properties as edge detectors in noisy angular signals the performance of the proposed edge operators is evaluated on angular edges using certain quantitative criteria finally a series of experiments featuring one dimensional 1 d angular signals and hue images is used to illustrate the operation of the new filters and edge detectors

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and or corners summary the mapping method was introduced in jeong et al 2013 for highly accurate isogeometric analysis iga of elliptic boundary value problems containing singularities the mapping method is concerned with constructions of novel geometrical mappings by which push forwards of b splines from the parameter space into the physical space generate singular functions that resemble the singularities in other words the pullback of the singularity into the parameter space by the novel geometrical mapping a non uniform rational basis spline nurbs surface mapping becomes highly smooth one of the merits of iga is that it uses nurbs functions employed in designs for the finite element analysis however push forwards of rational nurbs may not be able to generate singular functions moreover the mapping method is effective for neither the k refinement nor the h refinement in this paper highly accurate stress analysis of elastic domains with cracks and or corners are achieved by enriched iga in which push forwards of nurbs via the design mapping are combined with push forwards of b splines via the novel geometrical mapping the mapping technique in a similar spirit of x fem or gfem we propose three enrichment approaches enriched iga for corners enriched iga for cracks and partition of unity iga for cracks copyright 2013 john wiley sons ltd

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

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

performance of quadratic time frequency distributions as instantaneous frequency estimators general performance analysis of the shift covariant class of quadratic time frequency distributions tfds as instantaneous frequency if estimators for an arbitrary frequency modulated fm signal is presented expressions for the estimation bias and variance are derived this class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear if however when the if is not a linear function of time then the estimate is biased cases of white stationary and white nonstationary additive noises are considered the well known results for the wigner distribution wd and linear fm signal and the spectrogram of signals whose if may be considered as a constant within the lag window are presented as special cases in addition we have derived the variance expression for the spectrogram of a linear fm signal that is quite simple but highly signal dependent this signal is considered in the cases of other commonly used distributions such as the born jordan and the choi williams distributions it has been shown that the reduced interference distributions outperform the wd but only in the case when the if is constant or its variations are small analysis is extended to the if estimation of signal components in the case of multicomponent signals all theoretical results are statistically confirmed

frequency estimation of a single complex sinusoid using a generalized kay estimator this letter introduces a generalized version of kay s estimator for the frequency of a single complex sinusoid in complex additive white gaussian noise the kay estimator is a maximum likelihood ml estimator at high signal to noise ratio snr based on differential phase measurements with a delay of one symbol interval in this letter the corresponding ml estimator with an arbitrary delay in the differential phase measurements is derived the proposed estimator reduces the variance at low snr compared with kay s original estimator for certain delay values explicit expressions for the window function and the corresponding high snr variance of the proposed generalized kay gk estimator are presented furthermore for some delay values the window function is nearly uniform and the implementation complexity is reduced compared with the original kay estimator for a delay value of two we show that the variance at asymptotically high snr approaches the cramer rao bound as the sequence length tends to infinity we also explore the effect of exchanging the order of summation and phase extraction for reduced complexity reasons the resulting generalized weighted linear predictor estimator and the gk estimator are compared with both autocorrelation based and periodogram based estimators in terms of computational complexity estimation range and performance at both low and high snrs

frequency estimation by phase unwrapping single frequency estimation is a long studied problem with application domains including radar sonar telecommunications astronomy and medicine one method of estimation called phase unwrapping attempts to estimate the frequency by performing linear regression on the phase of the received signal this procedure is complicated by the fact that the received phase is wrapped modulo and therefore must be unwrapped before the regression can be performed in this paper we propose an estimator that performs phase unwrapping in the least squares sense the estimator is shown to be strongly consistent and its asymptotic distribution is derived we then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem we derive a polynomial time algorithm that computes the least squares estimator the results of various simulations are described for different values of sample size and snr

elliptic reconstruction and a posteriori error estimates for parabolic problems it is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm l infty 0 t l 2 omega in this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order and regularity for piecewise polynomials of degree higher than one this technique may be regarded as the dual a posteriori counterpart of wheeler s elliptic projection method in the a priori error analysis

spectrographic measurement of instantaneous frequency and the time dependent weighted average instantaneous frequency instantaneous frequency if is an important signal characteristic arising in many fields it is a concept intimately linked to time frequency analysis where it can be obtained from a time frequency distribution tfd as the first conditional moment in frequency suggesting that the if is the average frequency at each time however this interpretation is questionable since it is well known that the if often ranges beyond the spectral support of the signal in addition to obtain the if from a spectrogram which is one possible tfd a very wideband and thus severely spectrally distorted spectrogram must be used more reasonable bandwidths are investigated and in particular give the conditions by which the first conditional moment of a spectrogram can indeed be interpreted as the average frequency at each time under these conditions namely when the spectrogram bandwidth is less than the frequency separation of the signal components in the time frequency plane the spectrogram yields not the usual if but a time dependent weighted average instantaneous frequency waif of the signal also while the if and waif are generally different sometimes they are the same in particular when there is symmetry in the time frequency spectrum of the signal in that case the first conditional spectral moments of both wideband and narrow band spectrograms are the same and interpretable as the average frequency at each time

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

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

a posteriori error estimates of finite element method for the time dependent navier stokes equations abstract in this paper we consider the posteriori error estimates of galerkin finite element method for the unsteady navier stokes equations by constructing the approximate navier stokes reconstructions the errors of velocity and pressure are split into two parts for the estimates of time part the energy method and other skills are used for the estimates of spatial part the well developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted more important the error estimates of time part can be controlled by the estimates of spatial part as a consequence the posteriori error estimates in l 0 t l 2 l 0 t h 1 and l 2 0 t l 2 norms for velocity and pressure are derived in both spatial discrete and time space fully discrete schemes

map ml estimation of the frequency and phase of a single sinusoid in noise the problem of estimating the frequency and carrier phase of a single sinusoid observed in additive white gaussian noise is addressed much of the work in the literature considers maximum likelihood ml estimation however the ml estimator given by the location of the peak of a periodogram in the frequency domain shown in d c rife and r r boorstyn single tone parameter estimation from discrete time observations ieee transactions on information theory vol it 20 pp 591 598 sep 1974 has a very high computational complexity this paper derives the explicit structure of the ml estimator for data processing in the time domain assuming only reasonably high signal to noise ratio snr the result of this approximate ml estimator shows that both the phase and the magnitude of the noisy signal samples are utilized in the estimator and the phase data alone as assumed in s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and s kay a fast and accurate single frequency estimator ieee transactions on acoustics speech and signal processing vol 39 pp 1203 1205 may 1991 is not a sufficient statistic the sample by sample iterative processing nature of the estimator enables us to propose a novel recursive phase unwrapping algorithm that allows the estimator to be implemented efficiently to facilitate the performance analysis a new linearized observation model for the instantaneous signal phase that is more accurate than that of s a tretter estimating the frequency of a noisy sinusoid by linear regression ieee transactions on information theory vol it 31 pp 832 835 nov 1985 and of s kay a fast and accurate single frequency estimator vol 39 pp 1203 1205 may 1991 is proposed this new model explains physically why the phase data are weighted by the magnitude information in the ml estimator moreover by incorporating a priori knowledge via the a priori probability density function of the unknown frequency and the carrier phase the explicit structure of the approximate maximum a posteriori probability map estimator is derived and the bayesian cramer rao lower bound bcrlb on the mean square error mse is obtained our analysis shows that the mse performance of the map estimator can approach the bcrlb very closely

space time adaptive algorithm for the mixed parabolic problem in this paper we present an a posteriori error estimator for the mixed formulation of a linear parabolic problem used for designing an efficient adaptive algorithm our space time discretization consists of lowest order raviart thomas finite element over graded meshes and discontinuous galerkin method with variable time step finally several examples show that the proposed method is efficient and reliable

time frequency analysis of seismic data using local attributes time frequency analysis is an important technology in seismic data processing and interpretation to localize frequency content in time we have developed a novel method for computing a time frequency map for nonstationary signals using an iterative inversion framework we calculated time varying fourier coefficients by solving a least squares problem that uses regularized nonstationary regression we defined the time frequency map as the norm of time varying coefficients time varying average frequency of the seismic data can also be estimated from the time frequency map calculated by our method we tested the method on benchmark synthetic signals and compared it with the well known s transform two field data examples showed applications of the proposed method for delineation of sand channels and for detection of low frequency anomalies

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

aspects of an adaptive hp finite element method adaptive strategy conforming approximation and efficient solvers the main components needed for an adaptive hp version finite element algorithm are discussed an adaptive hp refinement strategy effective methods for constructing conforming hp approximations and efficient solvers for the large ill conditioned systems of linear equations together these provide the methodology for an effective adaptive hp version algorithm the presentation emphasizes the links between the differing components showing how the algorithms may be implemented efficiently in practice the main principles are illustrated by use of concrete examples so that a non expert may develop their own adaptive hp code the performance of the whole algorithm is illustrated for some representative problems taken from linear elasticity

a posteriori error estimates in the maximum norm for parabolic problems we derive a posteriori error estimates in the l infty 0 t l infty omega norm for approximations of solutions to linear parabolic equations using the elliptic reconstruction technique introduced by makridakis and nochetto and heat kernel estimates for linear parabolic problems we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations we then establish a posteriori bounds for a fully discrete backward euler finite element approximation the elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators

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

mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities abstract the method of auxiliary mapping mam introduced by babuska and oh 2 is an effective method for dealing with singularities in elasticity 16 mam was extended to boundary element method bem in the framework of mesh free particle methods 17 reproducing polynomial particle methods 20 and also to infinite domain problems 18 similarly we consider nurbs geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems however the mapping techniques proposed in this paper are different from mam in order to generate singular shape functions mam uses conformal mappings that locally change the physical domain whereas the nurbs mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis moreover unlike mam the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned we prove error estimates in sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework

a posteriori error estimation and adaptivity for degenerate parabolic problems two explicit error representation formulas are derived for degenerate parabolic pdes which are based on evaluating a parabolic residual in negative norms the resulting upper bounds are valid for any numerical method and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient they are applied to a practical space time discretization consisting of c 0 piecewise linear finite elements over highly graded unstructured meshes and backward finite differences with varying time steps two rigorous a posteriori error estimates are derived for this scheme and used in designing an efficient adaptive algorithm which equidistributes space and time discretization errors via refinement coarsening a simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach and hints at the potentials and reliability of the proposed method