improved newton type algorithm for adaptive implementation of pisarenko s harmonic retrieval method and its convergence analysis pisarenko s harmonic retrieval phr method is probably the first eigenstructure based algorithm for estimating the frequencies of sinusoids corrupted by additive white noise to develop an adaptive implementation of the phr method one group of authors has proposed a least squares type recursive algorithm in their algorithm they made approximations for both gradient and hessian the authors derive an improved algorithm where they use exact gradient and a different approximation for the hessian and analyze its convergence rigorously specifically they provide a proof for the local convergence and detailed arguments supporting the local instability of undesired stationary points computer simulations are used to verify the convergence performance of the new algorithm its performance is substantially better than that exhibited by its counterpart especially at low snr s

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

a fast recursive total least squares algorithm for adaptive fir filtering this paper proposes a new fast recursive total least squares n rtls algorithm to recursively compute the tls solution for adaptive finite impulse response fir filtering the n rtls algorithm is based on the minimization of the constrained rayleigh quotient c rq in which the last entry of the parameter vector is constrained to the negative one as analysis results on the convergence of the proposed algorithm we study the properties of the stationary points of the c rq the high computational efficiency of the new algorithm depends on the efficient computation of the fast gain vector fgv and the adaptation of the c rq since the last entry of the parameter vector in the c rq has been fixed as the negative one a minimum point of the c rq is searched only along the input data vector and a more efficient n rtls algorithm is obtained by using the fgv as compared with davila s rtls algorithms the n rtls algorithm saves the 6m number of multiplies divides and square roots mads the global convergence of the new algorithm is studied by lasalle s invariance principle the performances of the relevant algorithms are compared via simulations and the long term numerical stability of the n rtls algorithm is verified

neural network learning algorithms for tracking minor subspace in high dimensional data stream a novel random gradient based algorithm is developed for online tracking the minor component mc associated with the smallest eigenvalue of the autocorrelation matrix of the input vector sequence the five available learning algorithms for tracking one mc are extended to those for tracking multiple mcs or the minor subspace ms in order to overcome the dynamical divergence properties of some available random gradient based algorithms we propose a modification of the oja type algorithms called ojam which can work satisfactorily the averaging differential equation and the energy function associated with the ojam are given it is shown that the averaging differential equation will globally asymptotically converge to an invariance set the corresponding energy or lyapunov functions exhibit a unique global minimum attained if and only if its state matrices span the ms of the autocorrelation matrix of a vector data stream the other stationary points are saddle unstable points the globally convergence of ojam is also studied the ojam provides an efficient online learning for tracking the ms it can track an orthonormal basis of the ms while the other five available algorithms cannot track any orthonormal basis of the ms the performances of the relative algorithms are shown via computer simulations

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

least squares type algorithm for adaptive implementation of pisarenko s harmonic retrieval method pisarenko s harmonic retrieval method involves determining the minimum eigenvalue and the corresponding eigenvector of the covariance matrix of the observed random process recently thompson 9 suggested a constrained gradient search procedure for obtaining an adaptive version of pisarenko s method and his simulations have verified that the frequency estimates provided by his procedure were unbiased however the main cost of this technique was that the initial convergence rate could be very slow for certain poor initial conditions restating the constrained minimization as an unconstrained nonlinear problem we derived an alternative gauss newton type recursive algorithm which also used the second derivative matrix or hessian this algorithm may also be viewed as an approximate least squares algorithm simulations have been performed to compare this algorithm to a slight variant of thompson s original algorithm the most important conclusions are that the least squares type algorithm has faster convergence in the beginning while its convergence rate close to the true parameters depends on the signal to noise ratio of the input signal the approximate least squares algorithm resolves the sinusoids much faster than the gradient version

adaptation convergence of spectral estimation based on pisarenko harmonic retrieval proposed by thompson in 1 was a technique for adaptive spectral estimation particularly suited for tracking narrow band signals in broad band noise this technique represented a data adaptive variation of the block oriented pisarenko analysis technique the algorithm departs from similar adaptive approaches based on linear prediction in that its estimate of line frequency is unbiased with the level of additive white noise the original work offered a qualitative description of behavior and did not attempt the analysis of convergence characteristics necessary for practical application this paper presents a preliminary study of transient and steady state behavior indicating conditions for successful application while the characteristics parallel those of adaptive linear prediction in a general sense certain significant variations are evident in both global and local convergence behavior

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

on the variation of the spectra of matrices abstract several distances between the spectra of two matrices are discussed and compared optimal bounds are given which enable us to reduce certain bounds on the eigenvalue variation of matrices by a factor of about two the results of bhatia and mukherjea and of bhatia and friedland on the eigenvalue variation are derived in an elementary way using results of henrici on the spectral variation

a spectral study of the second order exceptional x1 jacobi differential expression and a related non classical jacobi differential expression abstract the exceptional x 1 jacobi differential expression is a second order ordinary differential expression with rational coefficients it was discovered by gomez ullate kamran and milson in 2009 in their work they showed that there is a sequence of polynomial eigenfunctions p n n 1 called the exceptional x 1 jacobi polynomials there is no exceptional x 1 jacobi polynomial of degree zero these polynomials form a complete orthogonal set in the weighted hilbert space l 2 1 1 w where w is a positive rational weight function related to the classical jacobi weight among other conditions placed on the parameters and it is required that 0 in this paper we develop the spectral theory of this expression in l 2 1 1 w we also consider the spectral analysis of the extreme non exceptional case namely when 0 in this case the polynomial solutions are the non classical jacobi polynomials p n 2 n 2 we study the corresponding jacobi differential expression in several hilbert spaces including their natural l 2 setting and a certain sobolev space s where the full sequence p n 2 n 0 is studied and a careful spectral analysis of the jacobi expression is carried out

performance analysis of adaptive eigenanalysis algorithms we present a rigorous analysis of several popular forms of short memory adaptive eigenanalysis algorithms using a stochastic averaging method a first order analysis shows that the algorithms do not have any equilibrium points despite published claims to the contrary through averaging analysis we show that they hover around an appropriate eigenvector a second order analysis is also given without the gaussian noise assumption and our results greatly outperform an earlier approximation in the literature the second order analysis has been of much interest in the offline study but in the dynamic adaptive case is uncommon

conditionally exactly solvable potentials and exceptional orthogonal polynomials it is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials some properties of such polynomials e g recurrence relation ladder operators differential equations etc have been obtained

on gradient adaptation with unit norm constraints in this correspondence we describe gradient based adaptive algorithms within parameter spaces that are specified by w 1 where spl middot is any vector norm we provide several algorithm forms and relate them to true gradient procedures via their geometric structures we also give algorithms that mitigate an inherent numerical instability for l sub 2 norm constrained optimization tasks simulations showing the performance of the techniques for independent component analysis are provided

adaptive multiple minor directions extraction in parallel using a pca neural network a principal component analysis pca neural network is developed for online extraction of the multiple minor directions of an input signal the neural network can extract the multiple minor directions in parallel by computing the principal directions of the transformed input signal so that the stability speed problem of directly computing the minor directions can be avoided to a certain extent on the other hand the learning algorithms for updating the net weights use constant learning rates this overcomes the shortcoming of the learning rates approaching zero in addition the proposed algorithms are globally convergent so that it is very simple to choose the initial values of the learning parameters this paper presents the convergence analysis of the proposed algorithms by studying the corresponding deterministic discrete time ddt equations rigorous mathematical proof is given to prove the global convergence the theoretical results are further confirmed via simulations

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

designing unimodular sequence sets with good correlations including an application to mimo radar a multiple input multiple output mimo radar system that transmits orthogonal waveforms via its antennas can achieve a greatly increased virtual aperture compared with its phased array counterpart this increased virtual aperture enables many of the mimo radar advantages including enhanced parameter identifiability and improved resolution practical radar requirements such as unit peak to average power ratio and range compression dictate that we use mimo radar waveforms that have constant modulus and good auto and cross correlation properties we present in this paper new computationally efficient cyclic algorithms for mimo radar waveform synthesis these algorithms can be used for the design of unimodular mimo sequences that have very low auto and cross correlation sidelobes in a specified lag interval and of very long sequences that could hardly be handled by other algorithms previously suggested in the literature a number of examples are provided to demonstrate the performances of the new waveform synthesis algorithms

moment representations of exceptional x1 orthogonal polynomials abstract we generalize the representations of x 1 exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries we start out directly from the darboux transformation allowing for a universal perspective rather than one dependent upon the particular system jacobi or type of laguerre polynomials we include a recursion formula for the adjusted moments and provide the initial adjusted moments for each system throughout we relate to the various examples of x 1 exceptional orthogonal polynomials we especially focus on and provide complete proofs for the jacobi and the type iii laguerre case as they are less prevalent in literature lastly we include a preliminary discussion explaining that the higher codimension setting becomes more involved the number of possibilities and choices is exemplified but only starts with the lack of a canonical flag

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed