performance analysis of an adaptive notch filter with constrained poles and zeros a statistical analysis of the sine wave frequency estimate obtained by the adaptive notch filter anf recently proposed by a nehorai ibid vol assp 33 p 983 96 aug 1985 is presented asymptotically valid expressions for the bias and mean square error of the frequency estimate are provided the analysis verifies the performance exhibited by the anf and clarifies the role of some user variables such as the notch bandwidth and the number of data samples processed

computation of the exponential of large sparse skew symmetric matrices in this paper we consider methods for evaluating both exp a and exp tau a q 1 where rm exp cdot is the exponential function a is a sparse skew symmetric matrix of large dimension q1 is a given vector and tau is a scaling factor the proposed method is based on two main steps a is factorized into its tridiagonal form h by the well known lanczos iterative process and then exp a is derived making use of an effective schur decomposition of h the procedure takes full advantage of the sparsity of a and of the decay behavior of exp h several applications and numerical tests are also reported

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

tracking properties of a gradient based second order adaptive iir notch filter with constrained poles and zeros gradient type adaptive iir notch filters have many attractive merits for various real life applications since they require a small number of computations and yet demonstrate practical performance however it is generally quite difficult to assess their performance analytically their tracking properties in particular have not yet been investigated in this paper the tracking performance of a plain gradient pg algorithm is analyzed in detail for a second order adaptive iir notch filter with constrained poles and zeros which takes a linear chirp signal as its input first two sets of difference equations for the frequency tracking error and mean square error mse are established in the sense of convergence in the mean and convergence in the mean square respectively closed form expressions for the asymptotic tracking error and mse are then derived from these difference equations an optimum step size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking mse it is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size which is an interesting property for a real valued adaptive filtering algorithm extensive simulations are performed to support the analytical findings

spectrum estimation notch filters and music a novel extension of the multiple signal classification music algorithm for the frequency estimation problem is proposed in this work it is shown that the music algorithm is a data dependent optimum notch filter design technique when the input data are passed through this optimum notch filter the signal to noise ratio snr of the output is minimized the zeros of the optimum notch filter are then used to estimate the frequencies in the classical music algorithm this optimization is carried out over the set of all finite impulse response fir filters having a fixed order the framework of this paper allows the use of rational notch filters the fact that the rational filters have sharper spectral characteristics is used to obtain improved estimates particularly at low snr some interesting properties of the proposed algorithm are derived and further perspectives on the implementation aspects are provided the analytical predictions are substantiated using numerical simulation results

a minimal parameter adaptive notch filter with constrained poles and zeros a new algorithm is presented for adaptive notch filtering and parametric spectral estimation of multiple narrow band or sine wave signals in an additive broad band process the algorithm is of recursive prediction error rpe form and uses a special constrained model of infinite impulse response iir with a minimal number of parameters the convergent filter is characterized by highly narrow bandwidth and uniform notches of desired shape for sufficiently large data sets the variances of the sine wave frequency estimates are of the same order of magnitude as the cramer rao bound results from simulations illustrate the performance of the algorithm under a wide range of conditions

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

performance analysis of the sign algorithm for a constrained adaptive iir notch filter adaptive infinite impulse response iir notch filters are very attractive in terms of their reasonable performances and low computational requirements generally it is very difficult to assess their performances analytically due to their iir nature this paper analyzes in detail the steady state performance of the sign algorithm sa for a well known adaptive iir notch filter with constrained poles and zeros slow adaptation and gaussianity of the notch filter output are assumed for the sake of analysis two difference equations are first established for the convergences in the mean and mean square in the vicinity of the steady state of the algorithm steady state estimation error or bias and mean square error mse of the sa are then derived in closed forms a coarse stability bound is also derived for the algorithm theory based comparison between the algorithm and the plain gradient pg algorithm is done in some detail extensive simulations are conducted to demonstrate the validity of the analytical results for both slow and relatively fast adaptations

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

optimal instrumental variable estimation and approximate implementations the accuracy properties of instrumental variables iv methods are investigated extensions such as prefiltering of data and use of additional instruments are included in the analysis the parameter estimates are shown to be asymptotically gaussian distributed an explicit expression is given for the covariance matrix of their distribution the covariance matrix is then taken as a multivariable measure of accuracy it is shown how it can be optimized by an appropriate selection of instruments and prefilter the so obtained optimal instrumental variable estimates cannot be used directly since the true system and the statistical properties of the disturbance must be known in order to compute the optimal instruments and prefilters a multistep procedure consisting of three or four simple steps is then proposed as a way to overcome this difficulty this procedure includes modeling of the disturbance as an arma process using a statistically efficient method such as a prediction error method the statistical properties of the estimates obtained with the multistep procedure are also analyzed those estimates are shown to be asymptotically gaussian distributed as well the covariance matrix of the estimation errors is compared to that corresponding to a prediction error method for some model structures these two approaches give the same asymptotic accuracy the conclusion is that the multistep procedure which is quite easy to implement and also has nice uniqueness properties can be viewed as an interesting alternative to prediction error methods

analysis of a general recursive prediction error identification algorithm a general class of algorithms for recursive identification of stochastic dynamical systems is studied in this class the discrepancy between the measured output and the output predicted from previous data according to a candidate model the prediction error is minimized over the model set using a stochastic approximation approach it is proved that this class of methods has the same convergence properties as its off line counterparts under mild and general assumptions these assumptions do not for example include stationary conditions or conditions that the true system can be exactly represented within the model set the considered class of methods contains as special cases several well known algorithms other common recursive identification methods like the extended least squares method and the extended kalman filter can be interpreted as approximate prediction error methods with simplified gradient calculations therefore the approach taken here may also serve as a basis for unified description of many recursive identification methods

technical communique convergence analysis of recursive identification algorithms with forgetting factor a convergence analysis of a modified version of the least squares recursive identification algorithm with forgetting factor is given it is shown that the parametric distance converges to a zero mean random variable it is also shown that under persistent excitation condition on both system input and output the condition number of the adaptation gain matrix is bounded the variance of the parametric distance is bounded by the product of the noise variance times the upper bound of the condition number of the gain matrix this is done by normalizing the measurement vector entering in the identification algorithm and by using a forgetting factor verifying l t 0

normalisers of subgroups of the modular group abstract let g be a subgroup of finite index of the modular group and let n g be the normaliser of g in psl2 r in this article we give an algorithm that determines n g

new multiplicative perturbation bounds for the generalized polar decomposition abstract some new frobenius norm bounds of the unique solution to certain structured sylvester equation are derived based on the derived norm upper bounds new multiplicative perturbation bounds are provided both for subunitary polar factors and positive semi definite polar factors some previous results are then improved

arma spectral estimation for noisy signals observed through a linear system spectral estimation for autoregressive moving average arma processes observed through a known linear system is considered an algorithm based on a system identification parameter estimation technique known as the recursive prediction error method is used for spectral estimation the algorithm is formulated in the state space to incorporate the constraints imposed by the linear system the linear system may represent the distortion introduced by the sensor or the channel several simulation examples are presented to illustrate the performance of the estimator

a recursive maximum likelihood algorithm for arma line enhancement optimal filtering of narrow band signals corrupted by noise requires an infinite impulse response iir filter a recursive maximum likelihood algorithm is proposed for adaptively adjusting the coefficients of an iir line enhancer the performance of this adaptive iir filter is analyzed and compared to that of a finite impulse response filter preliminary simulation results are presented to illustrate the behavior of the proposed algorithm under various signal to noise conditions

stochastic approximation method with gradient averaging for unconstrained problems a stochastic approximation algorithm is studied in which random gradient estimates are averaged by an auxiliary filter convergence of the algorithm is proved under a rather general noise condition a practical version of the method is described and numerical results are given

factorized all pass based iir adaptive notch filters this paper introduces a new family of infinite impulse response adaptive notch filters that forms multiple notches using a second order factorization of an all pass transfer function the new orthogonal realization is amenable for adaptive filtering to obtain the unknown frequencies of interest two new adaptive filtering algorithms are presented that can achieve fast convergence at low computational cost local convergence analysis for the new algorithms is performed and a detailed discussion of their properties is provided the new all pass based notch realization introduces a different compromise between bias and signal to noise ratio snr when compared with realizations previously reported in the literature specifically it achieves lower bias than other approaches at low snr this property is particularly attractive for the estimation and tracking of multiple sinusoids furthermore the bias can be made arbitrarily small or can be accurately estimated and compensated for extensive computer simulations are provided to illustrate the performance of the proposed adaptive notch filters in terms of bias speed of convergence and tracking capability

grassmannian spectral shooting we present a new numerical method for computing the pure point spectrum associated with the linear stability of coherent structures in the context of the evans function shooting and matching approach all the relevant information is carried by the flow projected onto the underlying grassmann manifold we show how to numerically construct this projected flow in a stable and robust manner in particular the method avoids representation singularities by in practice choosing the best coordinate patch representation for the flow as it evolves the method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed for large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization we demonstrate this by comparing the two methods in three applications boussinesq solitary waves autocatalytic travelling waves and the ekman boundary layer

tracking characteristics of the constrained iir adaptive notch filter the tracking behavior of a constrained iir notch filter the coefficients of which are estimated using a constant step size gauss newton algorithm is studied using weak convergence theory and the concept of prescaling it is shown that the mean behavior of the adaptive filter coefficients can be studied by using an ordinary differential equation ode approximate and simple closed form results are derived for the tracking behavior of a second order notch filter computer simulations are presented to substantiate the analysis

analysis and performance evaluation of an adaptive notch filter an adaptive notch filter is derived by using a general prediction error framework the proposed infinite impulse response filler has a special structure that guarantees the desired transfer characteristics the filter coefficients are updated by a version of the recursive maximum likelihood algorithm the convergence properties of the algorithm and its asymptotic behavior are discussed and its performance is evaluated by simulation results

gradient based adaptive iir notch filtering for frequency estimation the use of gradient based algorithms with infinite impulse response iir notch filtering for estimating sinusoids imbedded in noise is investigated two notch filter model structures are presented the first is for applications where two signal sources with correlated noise components can be assessed the second can be used in situations where only one composite signal source is available error surface analysis indicates that second order modules are unimodal and result in guaranteed convergence higher order modules are multimodal and require judicious choice of initial parameter estimates simulation results are included to demonstrate the performance characteristics of both filter structures

a globally convergent frequency estimator online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications in this paper the authors provide a solution to the problem of ensuring a globally convergent estimation more specifically they propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features 1 all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values 2 the authors obtain a simple tuning procedure for the estimator design parameters which trades off the adaptation tracking capabilities with noise sensitivity ensuring exponential stability of the desired orbit and 3 transient performance is considerably enhanced even for small or large frequencies as witnessed by extensive simulations to reveal some of the stability instability mechanisms of the existing algorithms and motivate our modifications the authors make appeal to a novel nonlinear state dependent time scaling the main advantage of working in the new time scale is that they remove the coupling between the parameter update law and the filter itself decomposing the system into a feedback form where the required modifications to ensure stability become apparent even though they limit their attention here to the simplest case of a single constant frequency without noise the algorithm is able to track time varying frequencies preserve local stability in the presence of multiple sinusoids and is robust with respect to noise

a filon type asymptotic approach to solving highly oscillatory second order initial value problems abstract in this paper we analyze and design filon type asymptotic methods for solving highly oscillatory second order initial value problems q t mq t f t q t q t where m is a non singular and diagonalizable matrix having large eigenvalues and m 1 we first derive filon type asymptotic methods for highly oscillatory linear systems and then extend the methods to highly oscillatory nonlinear systems the integrators are a blend of the existing filon type methods and asymptotic methods based on the variation of constants formula it is shown that the new integrators are convergent and the accuracy of the new integrators is improved as m grows large this point is significant and powerful for effectively dealing with highly oscillatory systems numerical experiments are carried out and the numerical results show the advantage and efficiency of our new integrators as compared with existing numerical methods for highly oscillatory problems from the scientific literature

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

adaptive stochastic filters with no strict positive real condition a new adaptive stochastic filter structure is introduced which avoids the strict passivity test used as a sufficient condition for convergence required by existing adaptive schemes the proposed algorithm consists of three stages in the first stage an autoregressive model is fitted and the residue obtained is used as an estimate of the noise in the second stage an autoregressive recursive moving average model is fitted using the residual of the first stage a modified residual is then filtered using a parameter and the model obtained from the second stage to generate an improved estimate of the noise in the third stage this improved estimate of the noise is used to obtain a better autoregressive moving average model it is shown that the proposed algorithm will also reduce the bias in the estimated parameters the simulation results given show that the proposed filter compares favorably to the algorithm introduced by mayne and clark and also landau this filter is then applied to the adaptive line enhancement sinusoidal detection and adaptive spectral estimation problems to illustrate its usefulness

an adaptive notch filter with improved tracking properties an analysis of the properties of an adaptive notch filter anf applied to time varying frequency tracking is presented starting from the derivation of an expression for anf output power asymptotically optimal values for the pole contraction and forgetting factors are derived for recursive prediction error rpe type anf algorithms based on the obtained results a new anf algorithm that includes adaptation of both pole contraction and forgetting factors is proposed the given experimental results confirm the theoretical conclusions and show that the proposed algorithm is highly efficient in practice

recurrence relations and convergence theory of the generalized polar decomposition on lie groups the subject matter of this paper is the analysis of some issues related to generalized polar decompositions on lie groups this decomposition depending on an involutive automorphism a is equivalent to a factorization of z g g being a lie group as z xy with a x x 1 and y y and was recently discussed by munthe kaas quispel and zanna together with its many applications to numerical analysis it turns out that contrary to x t log x an analysis of y t log y is a very complicated task in this paper we derive the series expansion for y t log y obtaining an explicit recurrence relation that completely defines the function y t in terms of projections on a lie triple system p and a subalgebra t of the lie algebra g and obtain bounds on its region of analyticity the results presented in this paper have direct application among others to linear algebra integration of differential equations and approximation of the exponential

a recursive maximum likelihood algorithm for arma spectral estimation a spectral estimation technique is presented for autoregressive moving average arma processes the technique is based on a parameter estimation technique known as the rec ursive maximum likelihood rml method the recursive spectral estimation algorithm is presented and its asymptotic properties are discussed simulation results are presented to illustrate the performance of the estimator for various types of data

efficient computation of the matrix exponential by generalized polar decompositions in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the method of generalized polar decompositions which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation we develop techniques valid for a range of lie groups the orthogonal group the symplectic group lorentz isotropy and scaling groups however the gpd approach is equally promising in a more general context even when lie group structure is not at issue our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential

the parameterized sr algorithm for symplectic butterfly matrices the sr algorithm is a structure preserving algorithm for computing the spectrum of symplectic matrices any symplectic matrix can be reduced to symplectic butterfly form a symplectic matrix b in butterfly form is uniquely determined by 4n 1 parameters using these 4n 1 parameters we show how one step of the symplectic sr algorithm for b can be carried out in o n arithmetic operations compared to o n 3 arithmetic operations when working on the actual symplectic matrix moreover the symplectic structure which will be destroyed in the numerical process due to roundoff errors when working with a symplectic butterfly matrix will be forced by working just with the parameters

iterative filtering for multiple frequency estimation it is well known that prony s least squares estimator gives inconsistent estimates for multiple frequency estimation in a recent attempt to diminish this problem dragosevic and stankovic 1989 couple the least squares method of autoregressive ar estimation with an iterative filtering scheme discussed by kay 1988 using an all pole filter but the inconsistency still persists this paper attacks the chronic inconsistency with a general approach of parametric filtering that unifies and extends the previous work it is shown that the inconsistency can be eliminated with an appropriately parametrized filter the clue for the correct parametrization comes from a formula for the bias of the least squares ar estimator the fact of the matter is that as long as a filter satisfies the parametrization requirement consistent estimates can be obtained from the least squares ar estimator on the basis of the filtered data in particular the all pole filter considered by dragosevic and stankovic can be easily reparametrized so that it too satisfies the parametrization requirement and thus leads to a consistent estimator experimental results show that the modified method has a higher resolution than the discrete fourier transform and that its overall performance is quite remarkable

a stable and efficient adaptive notch filter for direct frequency estimation we investigate the problem of direct frequency estimation a new adaptive algorithm is developed for constrained pole zero notch filters the basic characteristic of this algorithm is that the notch filter is cascaded and hence purely parametrized by the notching frequencies this makes the algorithm much simpler and hence more efficient and more robust such that the poles and zeros are automatically located on the respective constrained circle no matter if the algorithm is implemented with infinite or finite precision the algorithm is analyzed and compared with some existing algorithms several numerical examples are given as well as the corresponding simulations

a structural view of stability in adaptive iir filters this correspondence deals with the stability issue in adaptive iir filters from a structural view a new condition under which the bounded input bounded output bibo stability of the filters can be ensured is derived based on this result a normal state space realization for second order iir filters is proposed it is shown that the adaptive iir filter realized with such a structure is bibo stable no matter it is implemented with infinite or finite precision this property is very important for real time applications though both the proposed and the normalized lattice based structures are bibo stable the former is simpler and has minimal pole sensitivities two adaptive iir filter structures are developed based on a cascade form of second order filters the first structure is fit for the filters whose poles are all complex and the second one is for the constrained notch filters simulations are given to demonstrate the performance of the proposed structures

maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements the problem of estimating the frequencies phases and amplitudes of sinusoidal signals is considered a simplified maximum likelihood gauss newton algorithm which provides asymptotically efficient estimates of these parameters is proposed initial estimates for this algorithm are obtained by a variation of the overdetermined yule walker method and periodogram based procedure use of the maximum likelihood gauss newton algorithm is not however limited to this particular initialization method some other possibilities to get suitable initial estimates are briefly discussed an analytical and numerical study of the shape of the likelihood function associated with the sinusoids in noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure some numerical examples are presented to illustrate the performance of the proposed estimation procedure comparison to the performance corresponding to the cramer rao lower bound is also presented using a simple expression for the asymptotic cramer rao bound covariance matrix derived in the paper

stationary points of the recursive generalized least squares algorithm for adaptive notch filtering possible convergence points of the generalized least squares adaptive notch filtering algorithm are analytically derived for the multiple sinusoid case showing explicitly the dependence of the asymptotic bias on the pole contraction factor signal to noise ratio and the true model parameters the results for symmetric and nonsymmetric parameterization are compared

unbiased estimation of a sinusoid in colored noise via adapted notch filters standard notch models for the estimation of a sinusoid in noise are characterized by two poles on the unit circle and two zeros aligned with the poles these models provide biased estimates of the unknown frequency the bias increasing with the zero pole distance on the other hand keeping the zeros far from the poles has the beneficial effect of making the identification algorithm less sensitive to the adopted initialization in this paper we remove the pole zero alignment constraint the extra degree of freedom thus obtained is used to alleviate the traditional trade off between unbiasedness of the estimates and robustness against poor initializations the proposed technique applies to the case of both white and colored additive noise

sr and sz algorithms for the symplectic butterfly eigenproblem abstract sr and sz algorithms for the symplectic generalized eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed a 2 n 2 n symplectic butterfly matrix has 8 n 4 generically nonzero entries which are determined by 4 n 1 parameters while the sr algorithm operates directly on the matrix entries the sz algorithm works with the 4 n 1 parameters the algorithms are made more compact and efficient by using laurent polynomials instead of standard polynomials to drive the iterations

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

volume preserving algorithms for source free dynamical systems in this paper we first expound why the volume preserving algorithms are proper for numerically solving source free systems and then prove all the conventional methods are not volume preserving secondly we give a general method of constructing volume preserving difference schemes for source free systems on the basis of decomposing a source free vector field as a finite sum of essentially 2 dimensional hamiltonian fields and of composing the corresponding essentially symplectic schemes into a volume preserving one lastly we make some special discussions for so called separable source free systems for which arbitrarily high order explicit revertible volume preserving schemes can be constructed

funnel filters a new class of filters for frequency estimation of harmonic signals second order notch filters are commonly used for frequency estimation purposes they are characterized by a single design parameter the value of which is selected so as to achieve a suitable compromise between tracking performance and bias variance errors in this paper a new class of filters of fourth order is presented the funnel shaped filters or funnel filters for short these filters enables one to estimate harmonic signals with enhanced tracking capability without any deterioration of the steady state performance

exponential convergence of adaptive identification and control algorithms output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting or spanning condition on the system inputs together with several other standard conditions an adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions

adaptive notch filtering for the retrieval of sinusoids in noise in this paper an adaptive notch filter is developed employing a frequency domain and time domain analysis for the enhancement and tracking of sinusoids in additive noise colored or white the notch filter is implemented as a constrained infinite impulse response filter with the constraint enforced by a single parameter termed the debiasing parameter the resulting notch filter requires few parameters facilitates the formation of the desired band rejection filter response and also leads to various useful implementations cascade parallel for the adaptation of the filter coefficients the stochastic gauss newton algorithm is used the convergence of this updating procedure is established by studying the associated differential equation also it is shown that the structure present in the problem enables truncation of the gradient thereby reducing the complexity of adapting the filter coefficients simulation results are presented to substantiate the analysis and to demonstrate the potential of the notch filtering technique

groups commensurable with the modular group abstract let a and b be maximal subgroups of psl 2 r that commensurable with psl 2 z s and t are commensurable with each other if s t is of finite index in both s and t in this article we determine the index a a b and the level of a b appendix a

least squares yule walker and overdetermined yule walker estimation of ar parameters a monte carlo analysis of finite sample properties a monte carlo analysis of the accuracy properties of least squares ls yule walker yw and overdetermined yule walker oyw methods for estimating the parameters of autoregressive ar processes is presented comparisons of the estimated finite sample accuracy to the theoretical asymptotic accuracy are included it is shown that considerable differences may occur in some cases choice of the number of equations in the yw system of equations is discussed some remarks concerning the feasiblity and usefulness of an analytical study of the finite sample accuracy properties are also included

the extended kalman filter as a pulmonary blood flow estimator pulmonary blood flow is represented as a time varying parameter in a simple model of the bodily uptake of gases that are soluble in blood the pulmonary blood flow is estimated on line using ljung s modification of the extended kalman filter the experimental technique for estimation of pulmonary blood flow is based on zwart s soluble gas method modified to permit time varying ventilation and pulmonary perfusion and also to permit binary dynamic forcing of the inhaled soluble gas computer simulations were performed to determine the convergence time and to choose values of algorithmic parameters experiments done in humans gave agreement to within the accuracy of that measured invasively by a thermal dilution technique the method shows promise for use in many biomedical applications including patient monitoring physiological experimentation and clinical heart function tests

closed forms for the exponential mapping on matrix lie groups based on putzer s method we present closed forms for the exponential of some infinitesimal generators of lie groups which play an important role in physics and engineering applications these explicit forms are based on the putzer s method we also compare this methodology and results with related work by other authors

recursive identification algorithms for right matrix fraction description models multivariable identification algorithms are usually designed only for left matrix fraction description lmfd models in this paper we consider recursive identification algorithms for right matrix fraction description rmfd models with diagonal denominator matrices the algorithms are of prediction error pe and model reference mr type results from simulations illustrate the performance of the algorithms

adaptive algorithms for joint time delay estimation and iir filtering a new class of gradient based algorithms for joint adaptive time delay estimation and iir filtering for the recursive identification of systems with long pure delays is introduced the algorithms update an adaptive delay estimate and the coefficients of an iir filter using familiar gradient descent methods in addition interpolation of the input sequence is used to obtain delays that are not constrained to integer numbers of samples three algorithms using a direct form filter parametrization and one using a normalized lattice filter are derived and computer simulations are used to demonstrate their convergence and tracking properties

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

stability bounds and steady state coefficient variance for a second order adaptive iir notch filter in this correspondence we present a stochastic convergence analysis of a second order adaptive iir notch filter the approach is based on a linearization of the gradient in the vicinity of the optimal solution expressions for stability bounds on the algorithm step size and the steady state coefficient variance are derived these closed form analytical results are verified by computer simulations for different signal and noise conditions

multiple fully adaptive notch filter design based on allpass sections we develop a canonical adaptive cascade structure iir notch filter to detect and track multiple time varying frequencies in additive white gaussian noise the algorithm uses allpass frequency transformation filters and a truncated gradient simulations indicate that our algorithm is computationally simple converges rapidly and has good frequency resolution

an improved lattice based adaptive iir notch filter a novel lattice based adaptive infinite impulse response iir notch filter is developed which features independent tuning of the notch frequency and attenuation bandwidth the internal structure is based on planar rotators ensuring reliable numerical behaviour and high processing rates in cordic environments a simple update law allows a simpler implementation than previously proposed designs rather than minimizing an output error cost function the algorithm is designed to achieve a stable associated differential equation resulting in a globally convergent unbiased frequency estimator in the single sinusoid case independent of the notch filter bandwidth using a second order structure in the multiple sinusoid case unbiased estimation of one of the input frequencies is achieved by thinning the notch bandwidth the tracking behavior is superior to conventional output error designs and the estimation of extremal frequencies is less prone to overflow instability than previously reported structures

a method for approximation of the exponential map in semidirect product of matrix lie groups and some applications in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the decomposition of lie algebra as the semidirect product of two lie subspaces and an application of the baker campbell hausdorff formula our results extend the results in iserles and zanna 2005 2 zanna and munthe kaas 2001 02 4 to a range of lie groups the lie group of all solid motions in euclidean space the lorentz lie group of all solid motions in minkowski space and the group of all invertible upper triangular matrices in our method the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods in addition by this method the approximated matrix exponential belongs to the corresponding lie group

recursive parameter estimation for noisy autoregressive signals corresp the problem of recursively estimating the unknown parameters of a scalar autoregressive ar signal observed in additive white noise including signal power and noise variance is considered a state space model in a canonical but noninnovations form is used to represent the noisy ar signal an algorithm based on a system identification parameter estimation technique known as the recursive prediction error method is presented for recursive parameter estimation two simulation examples illustrate the effectiveness of the proposed algorithm

on rademacher s conjecture congruence subgroups of genus zero of the modular group we list all genus zero congruence subgroups of psl2 z there are altogether 132 of them up to conjugation in psl2 z geometrical invariants genus v2 v3 number of cusps index in psl2 z fundamental polygons farey symbol and independent generators of all such groups are determined generators of the function fields associated to such groups are also determined http www math nus edu sg matlml

tracking analysis of an adaptive notch filter with constrained poles and zeros this paper analyzes the asymptotic tracking properties of an adaptive notch filter anf with pole zero constraints for the cancellation or retrieval of multiple time varying sine waves in additive noise the asymptotic mean square error mse is analyzed using the methods of ljung and gunnarsson 1990 when the variations in the underlying frequencies are assumed to be sufficiently small closed form expressions for the mse are derived as functions of the tuning variables of the algorithm the results give insight into the operational properties of the algorithm and are used in order to minimize the mse with respect to the tuning variables computer simulations confirm the validity of the derived results