robust shrinkage affine projection sign adaptive filtering algorithms for impulsive noise environments two robust affine projection sign raps algorithms both of which minimize the mixed norm of l 1 and l 2 of the error signal are proposed the direction vector of the raps algorithms is obtained from the gradient of an l 1 norm based objective function while two related l 2 norm based minimization problems are solved to obtain the line search of the two raps algorithms the l 1 norm based direction vector reduces the impact of impulsive noise whereas the l 2 norm based line search produces an unbiased solution in the proposed algorithms in addition one of the two raps algorithms shares the data selective adaptation used in the set membership sm affine projection smap algorithm the proposed algorithms are shown to offer a significant improvement in the convergence speed as well as a significant reduction in the steady state misalignment relative to the pseudo affine projection sign paps algorithm in addition the proposed algorithms offer robust performance with respect to impulsive noise and improved tracking of the unknown system in comparison to that provided by the paps and affine projection sign aps algorithms these features of the proposed algorithms are demonstrated using simulation results in system identification and echo cancellation applications

a family of shrinkage adaptive filtering algorithms a family of adaptive filtering algorithms that uses a variable step size is proposed a variable step size is obtained by minimizing the energy of the noise free a posteriori error signal which is obtained by using a known l 1 hbox l 2 minimization formulation based on this methodology a shrinkage affine projection shap algorithm a shrinkage least mean squares shlms algorithm and a shrinkage normalized least mean squares shnlms algorithm are proposed the shap algorithm yields a significantly reduced steady state misalignment as compared to the conventional affine projection ap variable step size ap and set membership ap algorithms for the same convergence speed although the improvement is achieved at the cost of an increase in the average computational effort per iteration in the amount of 11 to 14 the shlms algorithm yields a significantly reduced steady state misalignment and faster convergence as compared to the conventional lms and variable step size lms algorithms similarly the shnlms algorithm yields a significantly reduced steady state misalignment and faster convergence as compared to the conventional normalized least mean squares nlms and set membership nlms algorithms

a feedback approach to the steady state performance of fractionally spaced blind adaptive equalizers this paper proposes a new approach to the analysis of the steady state performance of constant modulus algorithms cma which are among the most popular adaptive schemes for blind equalization a major feature of the proposed feedback approach is that it bypasses the need for working directly with the weight error covariance matrix in so doing approximate expressions for the steady state mean square error of several cm algorithms are derived including cma2 2 cma1 2 normalized cma and a new normalized cma variant with less bias a comparison among the various algorithms is also performed along with several simulation results the conclusions confirm the superior performance of cma2 2

steady state performance of spline adaptive filters recently a novel class of nonlinear adaptive filters called spline adaptive filters safs has been introduced and demonstrated to be very effective in many practical applications the learning rules of these architectures are based on the least mean square lms algorithm in order to provide theoretical foundation to the saf in this paper we provide a steady state performance evaluation in particular after the stochastic analysis of the mean behavior of the saf approach under the gaussian assumption the analytical derivation of the theoretical excess mean square error emse and the normalized misadjustment are derived and discussed the proposed analysis of emse and misadjustment is based on the energy conservation approach that has been extended to saf architecture the derived theoretical analysis allows to accurately predict the steady state performance therefore some properties for the correct choice of filter parameters are also provided experimental results demonstrate the effectiveness of the analysis results

ell _ p music robust direction of arrival estimator for impulsive noise environments a family of algorithms named lp music for direction of arrival doa estimation in impulsive noise is proposed the lp music estimator adopts the lp norm 1 p 2 of the residual fitting error matrix as the objective function for subspace decomposition rather than the frobenius norm that is used in the conventional music method although the matrix lp norm minimization based subspace decomposition will lead to a nonconvex optimization problem two iterative algorithms are designed for achieving efficient solutions the first algorithm is the iteratively reweighted singular value decomposition ir svd where the svd of a reweighted data matrix is performed in each iteration the second algorithm solves the nonconvex matrix lp norm minimization by alternating convex optimization two complex valued newton s methods with optimal step size in each iteration are devised to solve the resulting convex problem the convergence of the iterative procedure is also proved numerical results verify that the lp music methodology outperforms the standard music scheme and several existing outlier resistant doa estimation approaches in terms of resolution capability and estimation accuracy

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

analysis of error gradient adaptive linear estimators for a class of stationary dependent processes in many applications the training data to be processed by an adaptive linear estimator can be assumed to have a finite correlation length an exact analysis for this class of problems that yields the coefficient bias coefficient correlation matrix and mean square estimation error is obtained via a stochastic imbedding procedure a power series expansion in the gain parameter is used to obtain simplified expressions of order one for the above statistical moments these new expressions are shown to contain the terms that would result from an analysis based upon the assumption of independent training data plus additional terms arising from data correlation algorithm convergence properties are studied by identifying the appropriate matrix eigenvalues from the first order theory

variable explicit regularization in affine projection algorithm robustness issues and optimal choice a variable regularized affine projection algorithm vr apa is introduced without requiring the classical step size its use is supported from different points of view first it has the property of being hinfin optimal and it satisfies certain error energy bounds second the time varying regularization parameter is obtained by maximizing the speed of convergence of the algorithm although we first derive the vr apa for a linear time invariant lti system we show that the same expression holds if we consider a time varying system following a first order markov model we also find expressions for the power of the steady state error vector for the vr apa and the standard apa with no regularization parameter particularly we obtain quite different results with and without using the independence assumption between the a priori error vector and the measurement noise vector simulation results are presented to test the performance of the proposed algorithm and to compare it with other schemes under different situations an important conclusion is that the former independence assumption can lead to very inaccurate steady state results especially when high values of the projection order are used

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

tracking analysis of the sign algorithm in nonstationary environments a tracking analysis of the adaptive filters equipped with the sign algorithm and operating in nonstationary environments is presented under the assumption that the nonstationary can be modeled using a random disturbance it is shown that the long term time average of the mean absolute error is bounded and that there exists an optimal choice of the convergence constant mu which minimizes this quality applying the commonly used independence assumption and under the assumption that the nonstationarity is solely due to the time varying behavior of the optimal coefficients it is shown that the distributions of the successive coefficient misalignment vectors converge to a limiting distribution when the adaptive filter is used in the system identification mode finally under the additional assumption that the signals involved are zero mean and gaussian a set of nonlinear difference equations is derived that characterizes the mean and mean squared behavior of the filter coefficients and the mean squared estimation error during adaptation and tracking results of several experiments that show very good correlation with the theoretical analyses are presented

a unified approach to dynamic length algorithms for adaptive linear equalizers the number of taps is an important factor which affects nearly all the performance and complexity measures of adaptive equalizers recently dynamic length algorithms for the adaptive finite impulse response fir filters have attracted considerable attention and several methods have been proposed nevertheless the relationship among these techniques is still unclear which makes the performance comparison remaining difficult in this paper a unified explanation of these techniques is presented which summarizes all proposed algorithms and provides a platform for their performance comparison moreover a new type of dynamic length algorithms the varying threshold dynamic length algorithm vt dla is proposed simulation results show that the performance of the new algorithm outperforms that of existing algorithms

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

transient and steady state analysis of the affine combination of two adaptive filters in this paper we propose an approach to the transient and steady state analysis of the affine combination of one fast and one slow adaptive filters the theoretical models are based on expressions for the excess mean square error emse and cross emse of the component filters which allows their application to different combinations of algorithms such as least mean squares lms normalized lms nlms and constant modulus algorithm cma considering white or colored inputs and stationary or nonstationary environments since the desired universal behavior of the combination depends on the correct estimation of the mixing parameter at every instant its adaptation is also taken into account in the transient analysis furthermore we propose normalized algorithms for the adaptation of the mixing parameter that exhibit good performance good agreement between analysis and simulation results is always observed

spectral and pseudospectral approximations for the time fractional diffusion equation on an unbounded domain in this paper we consider the numerical approximation of the time fractional diffusion equation with variable coefficients on a semi infinite spatial domain a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed stability and convergence of the proposed scheme are rigorously established the scheme is unconditionally stable and convergent with order o 2 n 1 m 2 where n and m are the time step size polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analysed some numerical examples are presented to demonstrate the theoretical results

finite difference methods for the time fractional diffusion equation on non uniform meshes since fractional derivatives are integrals with weakly singular kernel the discretization on the uniform mesh may lead to poor accuracy the finite difference approximation of caputo derivative on non uniform meshes is investigated in this paper the method is applied to solve the fractional diffusion equation and a semi discrete scheme is obtained the unconditional stability and h 1 norm convergence are proved a fully discrete difference scheme is constructed with space discretization by compact difference method the error estimates are established for two kinds of nonuniform meshes numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods moreover a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution

convergence analysis of the binormalized data reusing lms algorithm normalized least mean squares algorithms for fir adaptive filtering with or without the reuse of past information are known to converge often faster than the conventional least mean squares lms algorithm this correspondence analyzes an lms like algorithm the binormalized data reusing least mean squares bndr lms algorithm this algorithm which corresponds to the affine projection algorithm for the case of two projections compares favorably with other normalized lms like algorithms when the input signal is correlated convergence analyses in the mean and in the mean squared are presented and a closed form formula for the mean squared error is provided for white input signals as well as its extension to the case of a colored input signal a simple model for the input signal vector that imparts simplicity and tractability to the analysis of second order statistics is fully described the methodology is readily applicable to other adaptation algorithms of difficult analysis simulation results validate the analysis and ensuing assumptions

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

mean square performance of a family of affine projection algorithms affine projection algorithms are useful adaptive filters whose main purpose is to speed the convergence of lms type filters most analytical results on affine projection algorithms assume special regression models or gaussian regression data the available analysis also treat different affine projection filters separately this paper provides a unified treatment of the mean square error tracking and transient performances of a family of affine projection algorithms the treatment relies on energy conservation arguments and does not restrict the regressors to specific models or to a gaussian distribution simulation results illustrate the analysis and the derived performance expressions

steady state mean square performance of the multichannel filtered x affine projection algorithm this paper provides a unified analysis of the steady state behavior of different multichannel filtered x affine projection algorithms when used in active noise control anc systems the analysis deals with two different filtering schemes the modified filtered x affine projection mfxap algorithm with the modified filtered x structure embedded that has been studied in previous works and the filtered x affine projection cfxap algorithm with the conventional filtered x structure embedded that becomes the main contribution of this work this study is based on energy conservation principles and does not require a specific signal distribution the derived theoretical models allow to accurately predict the steady state performance of the considered algorithms for moderate ap orders and low step sizes simulation results obtained in practical anc systems validate both the analysis and the achieved expressions showing a relative good match between theory and practice

analysis of the normalized lms algorithm with gaussian inputs the lms adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter for stability and convergence since the input power level is usually one of the statistical unknowns it is normally estimated from the data prior to beginning the adaptation process it is then assumed that the estimate is perfect in any subsequent analysis of the lms algorithm behavior in this paper the effects of the power level estimate are incorporated in a data dependent that appears explicitly within the algorithm the transient mean and second moment behavior of the modified lms nlms algorithm are evaluated taking into account the explicit statistical dependence of upon the input data the mean behavior of the algorithm is shown to converge to the wiener weight a constant coefficient matrix difference equation is derived for the weight fluctuations about the wiener weight the equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single frequency input in steady state for small expressions for the misadjustment error are also presented it is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the lms algorithm in the ale application the steady state weight fluctuations are shown to be mode dependent being largest at the frequency of the input

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

the least mean fourth lmf adaptive algorithm and its family new steepest descent algorithms for adaptive filtering and have been devised which allow error minimization in the mean fourth and mean sixth etc sense during adaptation the weights undergo exponential relaxation toward their optimal solutions time constants have been derived and surprisingly they turn out to be proportional to the time constants that would have been obtained if the steepest descent least mean square lms algorithm of widrow and hoff had been used the new gradient algorithms are insignificantly more complicated to program and to compute than the lms algorithm their general form is w j 1 w j 2 mu k epsilon j 2k 1 x j where w j is the present weight vector w j 1 is the next weight vector epsilon j is the present error x j is the present input vector mu is a constant controlling stability and rate of convergence and 2k is the exponent of the error being minimized conditions have been derived for weight vector convergence of the mean and of the variance for the new gradient algorithms the behavior of the least mean fourth lmf algorithm is of special interest in comparing this algorithm to the lms algorithm when both are set to have exactly the same time constants for the weight relaxation process the lmf algorithm under some circumstances will have a substantially lower weight noise than the lms algorithm it is possible therefore that a minimum mean fourth error algorithm can do a better job of least squares estimation than a mean square error algorithm this intriguing concept has implications for all forms of adaptive algorithms whether they are based on steepest descent or otherwise

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

discrete maximal regularity of time stepping schemes for fractional evolution equations in this work we establish the maximal ell p regularity for several time stepping schemes for a fractional evolution model which involves a fractional derivative of order alpha in 0 2 alpha ne 1 in time these schemes include convolution quadratures generated by backward euler method and second order backward difference formula the l1 scheme explicit euler method and a fractional variant of the crank nicolson method the main tools for the analysis include operator valued fourier multiplier theorem due to weis math ann 319 735 758 2001 doi 10 1007 pl00004457 and its discrete analogue due to blunck stud math 146 157 176 2001 doi 10 4064 sm146 2 3 these results generalize the corresponding results for parabolic problems

error energy bounds for adaptive gradient algorithms the paper establishes robustness optimality and convergence properties of the widely used class of instantaneous gradient adaptive algorithms the analysis is carried out in a purely deterministic framework and assumes no a priori statistical information it employs the cauchy schwarz inequality for vectors in an euclidean space and derives local and global error energy bounds that are shown to highlight as well as explain relevant aspects of the robust performance of adaptive gradient filters along the lines of h sup spl infin theory

a theory on the convergence behavior of the affine projection algorithm in this paper we present a theoretical convergence analysis of the affine projection algorithm apa based on the arguments of energy conservation although the apa and its convergence analysis have been widely studied the dependency of weight error vector on past noise is usually neglected for simplicity to obtain accurate theoretical results for the apa we here consider the dependency between the weight error vector and past noise in the mean square analysis presented by shin and sayed in mean square performance of a family of affine projection algorithms ieee transactions on signal processing vol 52 no 1 pp 90 102 january 2004 through this work we can also theoretically analyze the behavior of the periodic apa which updates its weights periodically simulation results show that our theoretical results coincide closely with simulations

finite element approximation of viscoelastic fluid flow with slip boundary condition we present a theoretical study of a creeping steady state isothermal flow of a viscoelastic fluid obeying an oldroyd type constitutive law with slip boundary condition the slip boundary condition is appropriate for problems that involve free boundaries and other examples where the usual no slip condition u 0 is not valid such as fiber spinning and microfluidics first we study the newtonian problem with slip boundary condition where the viscoelastic stress is added into the list of unknowns in addition the normal viscoelastic stress component associated with the slip boundary condition is introduced in order to balance its effects a second inf sup condition is proven to treat the discrete case we assume that the continuous solution of the non newtonian problem exists and is small and smooth approximating the extra stress velocity pressure and normal viscoelastic stress component via p 1 discontinuous p 2 continuous p 1 continuous and p 0 discontinuous elements respectively yields a stable finite element scheme finally via a fixed point argument we establish the existence of an approximate solution and derive error estimates

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

adaptive fractionally spaced blind cma equalization excess mse the performance of the constant modulus algorithm cma a reference algorithm for adaptive blind equalization is studied in terms of the excess mean square error emse due to the nonvanishing step size of the gradient descent algorithm an analytical approximation of emse is provided emphasizing the effect of the constellation size and resulting in design guidelines

optimal design of constrained fir filters without phase response specifications finite impulse response fir filters without phase response specifications have found many applications in signal processing and communications it is shown in this paper that the constrained lp magnitude error design of such an fir filter is equivalent to a constrained lp approximation of a given response with a phase equal to that of the optimal filter of the original problem an iterative method is then proposed to compute that phase by converting the nonconvex constrained lp magnitude error problem into a series of convex constrained lp elliptic frequency response error subproblems it is also shown that the solutions of these convex subproblems converge to a karush kuhn tucker point of the original problem the convergence and its model parameter and initial condition dependence are shown through the designs of fir filters without time domain constraints the iterative method is then applied to the minimax design of evidence filters nyquist filters and step response constrained filters and to the lp design of pulse shaping filters for ultra wideband systems design examples demonstrate that the proposed method obtains better filters in terms of magnitude error group delay and spectrum utilization efficiency than existing methods

iterative reweighted least squares design of fir filters develops a new iterative reweighted least squares algorithm for the design of optimal l sub p approximation fir filters the algorithm combines a variable p technique with a newton s method to give excellent robust initial convergence and quadratic final convergence details of the convergence properties when applied to the l sub p optimization problem are given the primary purpose of l sub p approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained l sub 2 approximation filters the new method can also be applied to the complex chebyshev approximation problem and to the design of 2d fir filters

steady state mse performance analysis of mixture approaches to adaptive filtering in this paper we consider mixture approaches that adaptively combine outputs of several parallel running adaptive algorithms these parallel units can be considered as diversity branches that can be exploited to improve the overall performance we study various mixture structures where the final output is constructed as the weighted linear combination of the outputs of several constituent filters although the mixture structure is linear the combination weights can be updated in a highly nonlinear manner to minimize the final estimation error such as in singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 lopes satorius and sayed 2006 bershad bermudez and tourneret 2008 and silva and nascimento 2008 we distinguish mixture approaches that are convex combinations where the linear mixture weights are constrained to be nonnegative and sum up to one singer and feder 1999 arenas garcia figueiras vidal and sayed 2006 affine combinations where the linear mixture weights are constrained to sum up to one bershad bermudez and tourneret 2008 and finally unconstrained linear combinations of constituent filters kozat and singer 2000 we investigate mixture structures with respect to their final mean square error mse and tracking performance in the steady state for stationary and certain nonstationary data respectively we demonstrate that these mixture approaches can greatly improve over the performance of the constituent filters our analysis is also generic such that it can be applied to inhomogeneous mixtures of constituent adaptive branches with possibly different structures adaptation methods or having different filter lengths

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

improving the tracking capability of adaptive filters via convex combination as is well known hessian based adaptive filters such as the recursive least squares algorithm rls for supervised adaptive filtering or the shalvi weinstein algorithm swa for blind equalization converge much faster than gradient based algorithms such as the least mean squares algorithm lms or the constant modulus algorithm cma however when the problem is tracking a time variant filter the issue is not so clear cut there are environments for which each family presents better performance given this we propose the use of a convex combination of algorithms of different families to obtain an algorithm with superior tracking capability we show the potential of this combination and provide a unified theoretical model for the steady state excess mean square error for convex combinations of gradient and hessian based algorithms assuming a random walk model for the parameter variations the proposed model is valid for algorithms of the same or different families and for supervised lms and rls or blind cma and swa algorithms

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds continuous approximation of the stress the author studies a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectively p 1 continuous p 2 continuous and p 1 continuous the streamline upwinding petrov galerkin method for the convection of the extra stress tensor is used suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution it is shown by a fixed point method that the approximate problem has a solution and an error bound is given

set membership binormalized data reusing lms algorithms this paper presents and analyzes novel data selective normalized adaptive filtering algorithms with two data reuses the algorithms the set membership binormalized lms sm bn drlms algorithms are derived using the concept of set membership filtering smf these algorithms can be regarded as generalizations of the previously proposed set membership nlms sm nlms algorithm they include two constraint sets in order to construct a space of feasible solutions for the coefficient updates the algorithms include data dependent step sizes that provide fast convergence and low excess mean squared error mse convergence analyzes in the mean squared sense are presented and closed form expressions are given for both white and colored input signals simulation results show good performance of the algorithms in terms of convergence speed final misadjustment and reduced computational complexity

adaptive volterra filters with evolutionary quadratic kernels using a combination scheme for memory control this paper proposes a new paradigm for adaptive volterra filtering using a time variant size of the quadratic kernel memory in order to optimally identify any unknown transversal second order nonlinear system to this end competing volterra structures of different sizes are employed in a hierarchical combination scheme so as to find the best configuration of the second order kernel memory using the already known diagonal coordinate representation the length and number of required quadratic kernel diagonals can be concurrently estimated by monitoring the combination performance subsequently the memory size of the involved models is dynamically increased or decreased following a set of intuitive rules since this automatic memory adaptation is performed along with the coefficient updates an efficient volterra filter is realized offering great flexibility and minimizing the risk of under or overmodeling any given quadratic nonlinearity besides the straightforward scheme a simplified version is presented greatly reducing the algorithmic demands this efficient version is based on a virtualization of the competing volterra filters by jointly using common coefficients and hence exhibits a computational complexity suitable for practical implementations the robust estimation performance of the approach is demonstrated by various examples for a nonlinear acoustic echo cancellation scenario involving stationary noise real speech signals and realistic volterra kernels

performance analysis of the deficient length lms adaptive algorithm in almost all analyzes of the least mean square lms finite impulse response fir adaptive algorithm it is assumed that the length of the adaptive filter is equal to that of the unknown system impulse response however in many practical situations a deficient length adaptive filter whose length is less than that of the unknown system is employed and analysis results for the sufficient length lms algorithm are not necessarily applicable to the deficient length case therefore there is an essential need to accurately quantify the behavior of the lms algorithm for realistic situations where the length of the adaptive filter is deficient in this paper we present a performance analysis for the deficient length lms adaptive algorithm for correlated gaussian input data and using the common independence assumption exact expressions that completely characterize the transient and steady state mean square performances of the algorithm are developed which lead to new insights into the statistical behavior of the deficient length lms algorithm simulation experiments illustrate the accuracy of the theoretical results in predicting the convergence behavior of the algorithm

a fast robust recursive least squares algorithm we present a fast robust recursive least squares frrls algorithm based on a recently introduced new framework for designing robust adaptive filters the algorithm is the result of minimizing a cost function subject to a time dependent constraint on the norm of the filter update although the characteristics of the exact solution to this problem are known there is no closed form solution in general however the approximate solution we propose is very close to the optimal one we also present some theoretical results regarding the asymptotic behavior of the algorithm the frrls is then tested in different environments for system identification and acoustic echo cancellation applications

fast and stable least squares approach for the design of linear phase fir filters we present a new numerical algorithm for solving the normal equations associated with the least squares design of linear phase fir filters the usual solution methods have a computational complexity of o n sup 3 moreover solving the normal equations with gaussian elimination commonly yields numerical errors especially when the filter is long here we convert a least squares method into the problem of constructing a system of orthonormal functions the proposed design algorithm needs only o n sup 2 computations and numerical errors can be reduced some examples are given to show the performance of the algorithm

a unified approach to the steady state and tracking analyses of adaptive filters most adaptive filters are inherently nonlinear and time variant systems the nonlinearities in the update equations tend to lead to difficulties in the study of their steady state performance as a limiting case of their transient performance this paper develops a unified approach to the steady state and tracking analyses of adaptive algorithms that bypasses many of these difficulties the approach is based on the study of the energy flow through each iteration of an adaptive filter and it relies on a fundamental error variance relation

distributed estimation over an adaptive incremental network based on the affine projection algorithm we study the problem of distributed estimation based on the affine projection algorithm apa which is developed from newton s method for minimizing a cost function the proposed solution is formulated to ameliorate the limited convergence properties of least mean square lms type distributed adaptive filters with colored inputs the analysis of transient and steady state performances at each individual node within the network is developed by using a weighted spatial temporal energy conservation relation and confirmed by computer simulations the simulation results also verify that the proposed algorithm provides not only a faster convergence rate but also an improved steady state performance as compared to an lms based scheme in addition the new approach attains an acceptable misadjustment performance with lower computational and memory cost provided the number of regressor vectors and filter length parameters are appropriately chosen as compared to a distributed recursive least squares rls based method

least squared error fir filter design with transition bands the authors propose the use of transition bands and transition functions in the ideal amplitude frequency response to allow the analytical design of optimal least squared error fir digital filters with an explicit control of the transition band edges design formulas are derived for approximations to ideal frequency responses which use pth order spline transition functions a mixed analytical and numerical method for zero error weight in the transition bands and passband and stopband error weighting functions with an integral squared error approximation are derived a variable order spline transition function is developed and a method for choosing the optimal order to minimize the integral squared approximation error is given

a robust variable step size lms type algorithm analysis and simulations a number of time varying step size algorithms have been proposed to enhance the performance of the conventional lms algorithm experimentation with these algorithms indicates that their performance is highly sensitive to the noise disturbance this paper presents a robust variable step size lms type algorithm providing fast convergence at early stages of adaptation while ensuring small final misadjustment the performance of the algorithm is not affected by existing uncorrelated noise disturbances an approximate analysis of convergence and steady state performance for zero mean stationary gaussian inputs and for nonstationary optimal weight vector is provided simulation results comparing the proposed algorithm to current variable step size algorithms clearly indicate its superior performance for cases of stationary environments for nonstationary environments our algorithm performs as well as other variable step size algorithms in providing performance equivalent to that of the regular lms algorithm

a new robust variable step size nlms algorithm a new framework for designing robust adaptive filters is introduced it is based on the optimization of a certain cost function subject to a time dependent constraint on the norm of the filter update particularly we present a robust variable step size nlms algorithm which optimizes the square of the a posteriori error we also show the link between the proposed algorithm and another one derived using a robust statistics approach in addition a theoretical model for predicting the transient and steady state behavior and a proof of almost sure filter convergence are provided the algorithm is then tested in different environments for system identification and acoustic echo cancelation applications

combination of recursive least p norm algorithms for robust adaptive filtering in alpha stable noise a method for adaptively minimizing the lp norm relying on the convex combination of two recursive least p norm rlpn filters is presented the approach is of interest when the noise is not gaussian for instance in the presence of impulsive or alpha stable s distributed noise in these cases the rlpn algorithm aiming at recursively minimizing the lp norm offers a more stable and robust solution than adaptive filtering schemes based on the minimization of the squared error however since the rlpn solution cannot be obtained in closed form for p 2 it is necessary to introduce some approximations that critically affect the filter behavior the main observed drawback is a poor convergence rate in nonstationary scenarios especially in the presence of abrupt changes in the model in this correspondence we show how this problem can be overcome by relying on convex combinations of two rlpn filters with long and short memories the proposed methods are empirically shown to outperform state of the art methods for this problem requiring just slightly higher computation than its close competitors

a recursive least m estimate algorithm for robust adaptive filtering in impulsive noise fast algorithm and convergence performance analysis this paper studies the problem of robust adaptive filtering in impulsive noise environment using a recursive least m estimate algorithm rlm the rlm algorithm minimizes a robust m estimator based cost function instead of the conventional mean square error function mse previous work has showed that the rlm algorithm offers improved robustness to impulses over conventional recursive least squares rls algorithm in this paper the mean and mean square convergence behaviors of the rlm algorithm under the contaminated gaussian impulsive noise model is analyzed a lattice structure based fast rlm algorithm called the huber prior error feedback least squares lattice h pef lsl algorithm is derived part of the h pef lsl algorithm was presented in icassp 2001 it has an order o n arithmetic complexity where n is the length of the adaptive filter and can be viewed as a fast implementation of the rlm algorithm based on the modified huber m estimate function and the conventional pef lsl adaptive filtering algorithm simulation results show that the transversal rlm and the h pef lsl algorithms have better performance than the conventional rls and other rls like robust adaptive algorithms tested when the desired and input signals are corrupted by impulsive noise furthermore the theoretical and simulation results on the convergence behaviors agree very well with each other

time stepping error bounds for fractional diffusion problems with non smooth initial data we apply the piecewise constant discontinuous galerkin method to discretize a fractional diffusion equation with respect to time using laplace transform techniques we show that the method is first order accurate at the nth time level t n but the error bound includes a factor t n 1 if we assume no smoothness of the initial data we also show that for smoother initial data the growth in the error bound as t n decreases is milder and in some cases absent altogether our error bounds generalize known results for the classical heat equation and are illustrated for a model problem

robust huber adaptive filter classical filtering methods are not optimal when the statistics of the signals violate the underlying assumptions behind the theoretical development most of the classical filtering theory like least squares filtering assumes gaussianity as its underlying distribution we present a new adaptive filter that is optimal in the presence of gaussian noise and robust to outliers this novel robust adaptive filter minimizes the huber objective function an estimator based on the huber objective function behaves as an l sub 1 norm estimator for large residual errors and as an l sub 2 norm estimator for small residual errors simulation results show the improved performance of the huber adaptive filter configured as a line enhancer over various nonlinear filters in the presence of impulsive noise and gaussian noise

comparison of rls lms and sign algorithms for tracking randomly time varying channels the performance of adaptive fir filters governed by the recursive least squares rls algorithm the least mean square lms algorithm and the sign algorithm sa are compared when the optimal filtering vector is randomly time varying the comparison is done in terms of the steady state excess mean square estimation error spl xi and the steady state mean square weight deviation spl eta it is shown that spl xi does not depend on the spread of eigenvalues of the input covariance matrix r in the cases of the lms algorithm and the sa while it does in the case of the rls algorithm in the three algorithms spl eta is found to be increasing with the eigenvalue spread the value of the adaptation parameter that minimizes spl xi is different from the one that minimizes spl eta it is shown that the minimum values of spl xi and spl eta attained by the rls algorithm are equal to the ones attained by the lms algorithm in any one of the three following cases 1 if r has equal eigenvalues 2 if the fluctuations of the individual elements of the optimal vector are mutually uncorrelated and have the same mean square value or 3 if r is diagonal and the fluctuations of the individual elements of the optimal vector have the same mean square value conditions that make the values of spl xi and spl eta of the lms algorithm smaller or greater than the ones of the rls algorithm are derived for gaussian input data the minimum values of spl xi and spl eta attained by the sa are found to exceed the ones attained by the lms algorithm by 1 db independently of r and the mutual correlation between the elements of the optimal vector

mean square performance of a convex combination of two adaptive filters combination approaches provide an interesting way to improve adaptive filter performance in this paper we study the mean square performance of a convex combination of two transversal filters the individual filters are independently adapted using their own error signals while the combination is adapted by means of a stochastic gradient algorithm in order to minimize the error of the overall structure general expressions are derived that show that the method is universal with respect to the component filters i e in steady state it performs at least as well as the best component filter furthermore when the correlation between the a priori errors of the components is low enough their combination is able to outperform both of them using energy conservation relations we specialize the results to a combination of least mean square filters operating both in stationary and in nonstationary scenarios we also show how the universality of the scheme can be exploited to design filters with improved tracking performance

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

transient analysis of adaptive filters with error nonlinearities the paper develops a unified approach to the transient analysis of adaptive filters with error nonlinearities in addition to deriving earlier results in a unified manner the approach also leads to new performance results without restricting the regression data to being gaussian or white the framework is based on energy conservation arguments and avoids the need for explicit recursions for the covariance matrix of the weight error vector

finite difference spectral approximations for the distributed order time fractional reaction diffusion equation on an unbounded domain the numerical approximation of the distributed order time fractional reaction diffusion equation on a semi infinite spatial domain is discussed in this paper a fully discrete scheme based on finite difference method in time and spectral approximation using laguerre functions in space is proposed the scheme is unconditionally stable and convergent with order o 2 2 n 1 m 2 where n and m are the time step size step size in distributed order variable polynomial degree and regularity in the space variable of the exact solution respectively a pseudospectral scheme is also proposed and analyzed some numerical examples are presented to demonstrate the efficiency of the proposed scheme

on the convergence behavior of the lms and the normalized lms algorithms it is shown that the normalized least mean square nlms algorithm is a potentially faster converging algorithm compared to the lms algorithm where the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics a very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the lms and nlms algorithms is proposed using this model answers can be obtained to questions for which no answers are currently available using other perhaps more realistic models examples are given to illustrate that even quantitatively the answers obtained can be good approximations it is emphasized that the convergence of the nlms algorithm can be speeded up significantly by employing a time varying step size the optimal step size sequence can be specified a priori for the case of a white input signal with arbitrary distribution

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

an affine projection algorithm with update interval selection this paper presents a mean square deviation msd analysis of the periodic affine projection algorithm p apa and two update interval selection methods to achieve improved performance in terms of the convergence and the steady state error the msd analysis of the p apa considers the correlation between the weight error vector and the measurement noise vector using this analysis it is verified that the update interval governs the trade off between the convergence rate and the steady state errors in the p apa to overcome this drawback the proposed apas increase the update interval when the adaptive filter reaches the steady state consequently these algorithms can reduce the overall computational complexity the simulation results show that the proposed apas perform better than the previous algorithms

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

low complexity constrained affine projection algorithms this paper proposes low complexity constrained affine projection cap algorithms the algorithms are suitable for linearly constrained filtering problems often encountered in communications systems the cap algorithms derived in this paper trade convergence speed and computational complexity in the same way as the conventional affine projection ap algorithm in addition data selective versions of the cap algorithm are derived based on the concept of set membership filtering the set membership constrained affine projection sm cap algorithms include several constraint sets in order to construct a space of feasible solutions for the coefficient updates the sm cap algorithms include a data dependent step size that provides fast convergence and low mean squared error the paper also discusses important aspects of convergence and stability of constrained normalized adaptation algorithms and shows that normalization may introduce bias in the final solution

a time domain feedback analysis of filtered error adaptive gradient algorithms this paper provides a time domain feedback analysis of gradient based adaptive schemes a key emphasis is on the robustness performance of the adaptive filters in the presence of disturbances and modeling uncertainties along the lines of h sup spl infin theory and robust filtering the analysis is carried out in a purely deterministic framework and assumes no prior statistical information or independence conditions it is shown that an intrinsic feedback structure can be associated with the varied adaptive schemes the feedback structure is motivated via energy arguments and is shown to consist of two major blocks a time variant lossless i e energy preserving feedforward path and a time variant feedback path the configuration is further shown to lend itself to analysis via a so called small gain theorem thus leading to stability and robustness conditions that require the contractivity of certain operators choices for the step size parameter in order to guarantee faster rates of convergence are also derived and simulation results are included to demonstrate the theoretical findings in addition the time domain analysis provided in this paper is shown to extend the so called transfer function approach to a general time variant scenario without any approximations

reduction of symmetric complex filters due to their linear phase property symmetric filters are an interesting class of finite impulse response fir filters moreover symmetric fir filters allow an efficient implementation in this paper we extend the classical definition of hermitian symmetry to a more general symmetry that is also applicable to complex filters this symmetry is called generalized hermitian symmetry we show the usefulness of this definition as it allows for a unified treatment of even and odd length filters central in this paper is a theorem on the reduction of generalized hermitian symmetric filters to hermitian symmetric filters both with finite precision coefficients a constructive proof of this theorem is presented and an associated procedure for reducing generalized hermitian symmetric filters is derived two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs finally all three examples show that a special instance of the generalized hermitian symmetric filters with finite precision coefficients may have lower arithmetic costs than the hermitian symmetric filter from which it is derived

quantized kernel least mean square algorithm in this paper we propose a quantization approach as an alternative of sparsification to curb the growth of the radial basis function structure in kernel adaptive filtering the basic idea behind this method is to quantize and hence compress the input or feature space different from sparsification the new approach uses the redundant data to update the coefficient of the closest center in particular a quantized kernel least mean square qklms algorithm is developed which is based on a simple online vector quantization method the analytical study of the mean square convergence has been carried out the energy conservation relation for qklms is established and on this basis we arrive at a sufficient condition for mean square convergence and a lower and upper bound on the theoretical value of the steady state excess mean square error static function estimation and short term chaotic time series prediction examples are presented to demonstrate the excellent performance

adaptive stepsize selection for tracking in a regime switching environment we consider the problem of using a stochastic approximation algorithm to perform online tracking in a non stationary environment characterised by abrupt regime changes the primary contribution of this paper is a new approach for adaptive stepsize selection that is suitable for this type of non stationarity our approach is pre emptive rather than reactive and is based on a strategy of maximising the rate of adaptation subject to a constraint on the probability that the iterates fall outside a pre determined range of acceptable error the basis for our approach is provided by the theory of weak convergence for stochastic approximation algorithms

on the convergence of blind adaptive equalizers for constant modulus signals this paper studies the behavior of the error sequence of stop anti go variants of two adaptive blind equalizers namely cma2 2 and sato s 1975 algorithm it is shown that for transmitted signals with constant modulus spl gamma the equalizer output can be made to lie within the circle of radius spl gamma spl radic c infinitely often for some value of c that is only slightly larger than one

an l_1 method for the design of linear phase fir digital filters this paper considers the design of linear phase finite impulse response digital filters using an l1 optimality criterion the motivation for using such filters as well as a mathematical framework for their design is introduced it is shown that l1 filters possess flat passbands and stopbands while keeping the transition band comparable to that of least squares filters the uniqueness of l1 based filters is explored and an alternation type theorem for the optimal frequency response is derived an efficient algorithm for calculating the optimal filter coefficients is proposed which may be viewed as the analogue of the celebrated remez exchange method a comparison with other design techniques is made demonstrating that the l1 approach may be a good alternative in several applications

from stochastic processes to numerical methods we have introduced a new explicit numerical method based on a discrete stochastic process for solving a class of fractional partial differential equations that model reaction subdiffusion the scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks we show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest this limiting procedure guarantees the consistency of the numerical scheme the positivity of the solution and stability results are simply obtained provided that the underlying process is well posed we also show that the method can be applied to standard reaction diffusion equations this work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations including fractional partial differential equations

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

performance analysis of the linearly constrained constant modulus algorithm based multiuser detector this paper quantifies the adaptive performance of a blind adaptive multiuser detector mud based on the linearly constrained constant modulus algorithm lccma in both a stationary and nonstationary channel a framework is developed to apply the feedback analysis method to analyzing adaptive mud schemes a closed form expression for the excess mean square error emse of lccma blind adaptive mud in a cdma communications system is derived for both of the steady state and tracking cases the effects of additive noise and multiple access interference are considered a transient analysis is performed that predicts the learning curve of the adaptive filter computer simulation is used to verify the accuracy of the analysis

transient analysis of data normalized adaptive filters this paper develops an approach to the transient analysis of adaptive filters with data normalization among other results the derivation characterizes the transient behavior of such filters in terms of a linear time invariant state space model the stability of the model then translates into the mean square stability of the adaptive filters likewise the steady state operation of the model provides information about the mean square deviation and mean square error performance of the filters in addition to deriving earlier results in a unified manner the approach leads to stability and performance results without restricting the regression data to being gaussian or white the framework is based on energy conservation arguments and does not require an explicit recursion for the covariance matrix of the weight error vector

shrinkage linear and widely linear complex valued least mean squares algorithms for adaptive beamforming in this paper shrinkage linear complex valued least mean squares sl clms and shrinkage widely linear complex valued least mean squares swl clms algorithms are devised for adaptive beamforming by exploiting the relationship between the noise free a posteriori and a priori error signals the sl clms method is able to provide a variable step size to update the weight vector for the adaptive beamformer significantly enhancing the convergence speed and decreasing the steady state misadjustment on the other hand besides adopting a variable step size determined by minimizing the square of the augmented noise free a posteriori errors the swl clms approach exploits the noncircular properties of the signal of interest which considerably improves the steady state performance simulation results are presented to illustrate their superiority over the clms complex valued normalized lms variable step size recursive least squares rls algorithms and their corresponding widely linear based schemes additionally our proposed algorithms are more computationally efficient than the rls solutions though they may have a slightly slower convergence rate

adaptive fuzzy filtering in a deterministic setting many real world applications involve the filtering and estimation of process variables this study considers the use of interpretable sugeno type fuzzy models for adaptive filtering our aim in this study is to provide different adaptive fuzzy filtering algorithms in a deterministic setting the algorithms are derived and studied in a unified way without making any assumptions on the nature of signals i e process variables the study extends in a common framework the adaptive filtering algorithms usually studied in signal processing literature and p norm algorithms usually studied in machine learning literature to semilinear fuzzy models a mathematical framework is provided that allows the development and an analysis of the adaptive fuzzy filtering algorithms we study a class of nonlinear lms like algorithms for the online estimation of fuzzy model parameters a generalization of the algorithms to the p norm is provided using bregman divergences a standard tool for online machine learning algorithms

numerical analysis of nonlinear subdiffusion equations we present a general framework for the rigorous numerical analysis of time fractional nonlinear parabolic partial differential equations with a fractional derivative of order alpha in 0 1 in time it relies on three technical tools a fractional version of the discrete gronwall type inequality discrete maximal regularity and regularity theory of nonlinear equations we establish a general criterion for showing the fractional discrete gronwall inequality and verify it for the l1 scheme and convolution quadrature generated by backward difference formulas further we provide a complete solution theory e g existence uniqueness and regularity for a time fractional diffusion equation with a lipschitz nonlinear source term together with the known results of discrete maximal regularity we derive pointwise l 2 omega norm error estimates for semidiscrete galerkin finite element solutions and fully discrete solutions which are of order o h 2 up to a logarithmic factor and o tau alpha

optimum step size of sign algorithm for nonstationary adaptive filtering the adaptive filtering sign algorithm is analyzed in the case of nonstationary and correlated data the excess time average absolute estimation error eaae is used as a performance index of the algorithm it is proved that the eaae is the sum of two terms one proportional to the algorithm step size mu and the other proportional to 1 mu the optimum step size that minimizes the eaae is found equal to square root q p with q being the power of fluctuations of the optimal filter weights and p being the power of filter input this formula is derived under the assumption that increments of optimal filter weights are white computer simulations are used to examine the validity of the formula for other models of variation of the optimal filter simulation results show a good fit with the obtained optimum step size formula

singularity analysis and asymptotics of bernoulli sums the asymptotic analysis of a class of binomial sums that arise in information theory can be performed in a simple way by means of singularity analysis of generating functions the method developed extends the range of applicability of singularity analysis techniques to combinatorial sums involving transcendental elements like logarithms or fractional powers

rational spline nonstandard finite difference scheme for the solution of time fractional swift hohenberg equation abstract in this paper we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time fractional swift hohenberg equation in the sense of riemann liouville derivative via fourier method the method is convergent and unconditionally stable also we investigated the existence and uniqueness of the proposed method numerical results are demonstrated to validate the applicability and the theoretical results

robust rls in the presence of correlated noise using outlier sparsity relative to batch alternatives the recursive least squares rls algorithm has well appreciated merits of reduced complexity and storage requirements for online processing of stationary signals and also for tracking slowly varying nonstationary signals however rls is challenged when in addition to noise outliers are also present in the data due to e g impulsive disturbances existing robust rls approaches are resilient to outliers but require the nominal noise to be white an assumption that may not hold in e g sensor networks where neighboring sensors are affected by correlated ambient noise prewhitening with the known noise covariance is not a viable option because it spreads the outliers to noncontaminated measurements which leads to inefficient utilization of the available data and unsatisfactory performance in this correspondence a robust rls algorithm is developed capable of handling outliers and correlated noise simultaneously in the proposed method outliers are treated as nuisance variables and estimated jointly with the wanted parameters identifiability is ensured by exploiting the sparsity of outliers which is effected via regularizing the least squares ls criterion with the l1 norm of the outlier vectors this leads to an optimization problem whose solution yields the robust rls estimates for low complexity real time operation a suboptimal online algorithm is proposed which entails closed form updates per time step in the spirit of rls simulations demonstrate the effectiveness and improved performance of the proposed method in comparison with the nonrobust rls and its state of the art robust renditions