on smoothed rank profile tests in eigenstructure methods for directions of arrival estimation we propose a statistical procedure known as the smoothed rank profile srp test that can be applied to an array covariance matrix to determine the source coherency structure and solvability of the directions of arrival estimation problem in the presence of coherent sources the srp is the rank profile of a telescoping series of matrices extracted from the observed array covariance matrix

a data based enumeration technique for fully correlated signals presents a novel method for estimating the number of signals impinging on a uniform linear array using observed sensor data unlike other algorithms that apply rissanen s minimum description length mdl principle to the observed data for source enumeration this method applies it to the prediction errors of a linear model that has been fitted to an appropriate data matrix it is a 1d method that achieves improved performance even for fully correlated signals over contemporary approaches particularly with short data records and closely spaced signals asymptotic consistency is shown and substantiating simulation examples are included

chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

on the methods for solving yule walker equations the three well known fast algorithms for the solution of yule walker equations the levinson euclidean and berlekamp massey algorithms are reviewed and the relationship between each of them and the pade approximation problem is shown the algorithms are classified with reference to three criteria namely 1 the path they follow in the pade table 2 the organization of the computations one pass or two pass and 3 the auxiliary variables used this classification shows that the set of known classical algorithms is not complete and the missing variants are proposed with these variants of the berlekamp massey and euclid algorithms both forward and backward predictors can be obtained without additional cost a unified representation of the two pass algorithms is given in such a way that the application of the divide and conquer strategy becomes straightforward a general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided

on a generalization of the szego levinson recurrence and its application in lossless inverse scattering predictor polynomials corresponding to nested toeplitz matrices are known to be connected by the szego levinson recurrence relation a generalization of that result where the relevant reduction process for toeplitz matrices of decreasing order is defined by an elementary one parameter linear transformation is addressed the descending and ascending versions of the corresponding generalized szego levinson recurrence relations are discussed in detail in particular these relations are shown to be essentially the same as the extraction formulas for canonical schur and brune sections in the dewilde dym 1984 recursive solution of the lossless inverse scattering problem some extensions of the levinson algorithm for linear prediction and of the schur cohn algorithm for polynomial stability test are presented

centrosymmetric and centro skewsymmetric toeplitz plus hankel matrices and bezoutians abstract in the paper algebraic properties of centrosymmetric and centro skewsymmetric toeplitz plus hankel t h matrices are investigated it is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related toeplitz systems with symmetric or skewsymmetric right hand sides in the singular case the related toeplitz matrices determine the kernel of the t h matrices finally centrosymmetric and centro skewsymmetric t h bezoutians are characterized

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a fast algorithm for solving a toeplitz system of equations a fast algorithm for the solution of a toeplitz system of equations is presented the algorithm requires order n log n 2 computations where n is the number of equations for banded toeplitz matrices the order of computations is reduced to only n log n m log m 2 where 2m is the maximum number of nonzero principal subdiagonals of the toeplitz matrix the algorithm is in essence a fast implementation of the trench algorithm in reverse thus the algorithm involves imbedding of the given matrix in a cyclic matrix and a fast hd half divisor algorithm to compute the first row of the inverse matrix the desired solution is then obtained directly from the first row by applying fast fourier transform techniques in order n log n computations finally the extension of the algorithm to block toeplitz matrices is also presented

an extension of the split levinson algorithm and its relatives to the joint process estimation problem it is shown that the split levinson algorithm the split schur algorithm and the split lattice algorithm to compute the reflection coefficients of the optimal linear prediction filter for a discrete time stationary stochastic process can be extended to the more general case of the joint process estimation problem the new algorithms are essentially based on well defined recurrence relations for symmetric prediction filters and symmetric estimation filters they are more economical than the standard methods in terms of storage space and number of arithmetic operations

the levinson recurrence and fast algorithms for solving toeplitz systems of linear equations this work brings together classical polynomial theory as it relates to the levinson recurrence for a hermitian toeplitz operator and matrix theory as it relates to the class of hermitian centro hermitian matrices a new computationally efficient alternative is presented to the levinson recurrence on either the hermitian or skew hermitian polynomial spaces this approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is hermitian toeplitz or real symmetric toeplitz analysis of the computational complexity of the algorithms presented is also performed and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best known recursive algorithms they also provide further insight into the mathematical properties of the structurally rich toeplitz matrices

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

computationally efficient stepup and stepdown procedures for real predictor coefficients algorithms are presented for the stepup and stepdown problem for the real case it is shown that the split levinson algorithm and similar algorithms can be used to reduce the multiplicative complexity of the problem considerably

on linear prediction models constrained to have unit modulus poles and their use for sinusoidal frequency estimation linear prediction models with their poles restricted to the unit circle can be efficiently determined by using the levinson or the split levinson algorithm a simple proof of this property is presented the prime application of linear prediction models constrained to have unit modulus poles is the estimation of sinusoidal frequencies the consistency properties of the corresponding frequency estimates are analyzed it is shown that in the presence of noise the estimates are inconsistent an explicit expression for the asymptotic bias is provided

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

symmetric centrosymmetric matrix vector multiplication abstract we present a method for the multiplication of an arbitrary vector by a symmetric centrosymmetric matrix requiring 5 4 n 2 o n floating point operations rather than the 2n 2 operations needed in the case of an arbitrary matrix combining this method with trench s algorithm for toeplitz matrix inversion yields a method for solving toeplitz systems with the same complexity as levinson s algorithm

some properties of classes of real self reciprocal polynomials abstract the purpose of this paper is twofold firstly we investigate the distribution simplicity and monotonicity of the zeros around the unit circle and real line of the real self reciprocal polynomials r n z 1 z z 2 z n 1 z n n 2 and r secondly as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self reciprocal polynomials s n z k 0 n s n k z k n 2 with s n 0 s n n 1 s n n k s n k 1 k k 1 2 n 2 when n is odd and s n n k s n k 1 k k 1 2 n 2 1 s n n 2 n 2 when n is even lie on the unit circle solving then an open problem given by kim and park in 2008

a fast algorithm for nevanlinna pick interpolation a new fast algorithm is derived for the scalar nevanlinna pick interpolation problem this fast algorithm requires approximately 28 fewer multiplications than the classical algorithm the algorithm is based on the three term recurrence similar to that of the so called split algorithms for linear prediction

bowtie factors of toeplitz matrices by means of split algorithms the bowtie factorization of a toeplitz correlation matrix r and its inverse is introduced the term bowtie describes the pattern of the factoring matrices several mathematical properties of bowtie matrices are described including their close relation with block triangular matrices the bowtie factors of the inverse r can be computed using a vector version of the split levinson algorithm restricted to orders with the same parity as n the dimension of r the bowtie factors of r can be computed using split vector versions of the schur algorithm the orthogonal factorization of the underlying data matrix y can also be computed by the split versions of the lattice algorithm producing a bowtie orthogonal factor of y

interactions on manifolds and the construction of material structure abstract a theory of surface interactions on manifolds is presented as a framework for force and stress theory pertaining to materials with micro structure the dependence of surface forces on the normal in the case of euclidean geometry is replaced by dependence on the tangent hyperplane and its orientation odd quantities are used to model interactions that satisfy newton s law of action and reaction stresses are defined as vector valued forms of order m 1 where m is the dimension of the body manifold it is shown how a stress field induces surface forces for the various bodies the special case of a scalar valued balance law is presented with some additional detail finally we consider the way material structures are induced by balance laws

performance of spatial smoothing algorithms for correlated sources the problem of identifying the angles of arrival of a set of plane waves impinging on a narrowband array of sensors and related spectral analysis problems have been addressed with a large number of algorithms one of the most popular techniques is the multiple signal classification music the major shortcoming of the music algorithm is that it performs poorly when the sources are highly correlated fortunately two algorithms exist to overcome this problem spatial smoothing ss and forward backward spatial smoothing fbss the performance of the ss technique depends on signal bearings and spatial separation for the same smoothing fbss can offer improved performance but this depends on the signal phases numerical results for the variance of the algorithms are given to illustrate the points made

detection of coherent signals using weighted subspace smoothing a new approach to detecting the number of coherent signals incident upon a uniformly spaced linear array is presented the approach combines a modified spatial smoothing scheme and a modified mdl criterion the modified spatial smoothing scheme referred to as weighted subspace smoothing is actually a generalization of the post smoothing approach proposed by krim and proakis in 1994 it is shown that the noise eigenvalues obtained with weighted subspace smoothing are more accurate than those obtained with the original spatial smoothing we thus attempt to improve the detection performance using these more accurate eigenvalues a novel modification of the minimum description length mdl criterion is proposed to accomplish this computer simulations are presented to study the performance improvement of this new approach

doa estimation for mixed signals in the presence of mutual coupling in this paper a new direction of arrival doa estimation method is proposed in the presence of unknown mutual coupling the impinging signals are a mixture of signals in different correlation degrees which are uncorrelated partially correlated or coherent with each other the method proceeds in three steps the noncoherent uncorrelated or partially correlated signals are firstly estimated with unknown mutual coupling then these estimates are utilized to get the mutual coupling coefficients finally by eliminating the contributions of the noncoherent signals and compensating the mutual coupling effect only coherent signals remain to be estimated this method is simple but effective and does not need iterative processing simulation results demonstrate the effectiveness and performance of the proposed method

optimum block adaptive filtering algorithms using the preconditioning technique we propose three block adaptive algorithms using the preconditioning technique the toeplitz preconditioned optimum block adaptive toba algorithm employs a preconditioner assumed to be toeplitz the symmetric successive overrelaxation ssor preconditioned optimum block adaptive soba algorithm uses a product of triangular matrices as a preconditioner and the circulant preconditioned oba coba algorithm is based on a circulant preconditioner it is also shown that their tracking properties and convergence rates are superior to those of the oba algorithm the self orthogonizing block adaptive filter sobaf and the normalized frequency domain oba nfoba algorithm

lebesgue perturbation of a quasi definite hermitian functional the positive definite case abstract in this work we study the problem of orthogonality with respect to a sum of measures or functionals first we consider the case where one of the functionals is arbitrary and quasi definite and the other one is the lebesgue normalized functional next we study the sum of two positive measures the first one is arbitrary and the second one is the lebesgue normalized measure and we obtain some relevant properties concerning the new measure finally we consider the sum of a bernstein szeg measure and the lebesgue measure in this case we obtain more simple explicit algebraic relations as well as the relation between the corresponding szeg s functions

on biorthogonality of hermitian and skew hermitian szego levinson polynomials a system of polynomials biorthogonal with respect to a given weight function which is not necessarily even is constructed from a corresponding system of szego or levinson polynomials two term coupled recurrences are presented for the biorthogonal polynomial system and it is shown that these recurrences reduce to a pair of three term uncoupled recurrences in the case that the unit circle weight function is even several observations relative to recent linear prediction algorithms utilizing hermitian and skew hermitian levinson polynomials are discussed

relations between univalent and orthogonal polynomials two sequences of polynomials are studied one satisfies a three term recurrence relation for specific parameters and another a para orthogonality property using the fact that these polynomials have their zeros lying on the unit circle and some other properties we establish a criterion in order that the polynomials be univalent in the open unit disk

linear interpolation lattice the analysis and synthesis filters of linear prediction theory are extended to include linear interpolation a noncausal lattice filter for linear interpolation is obtained from the linear interpolation recursion the results provide insight into the interrelation of prediction and interpolation and of one and two sided representations of one dimensional markov sequences

synthesis of extremal wavelet generating filters using gaussian quadrature orthogonal wavelets can be generated from finite impulse response quadrature mirror filters these filters are also used in perfect reconstruction filter banks this paper addresses the problem of efficiently synthesizing such filters a class of extremal filters is defined by the property that their magnitude spectrum maximizes an integral criterion it is found that these filters are characterized by their zeros on the unit circle which frequently can be obtained from a set of orthogonal polynomials a family of filters is constructed that minimize the subband aliasing energy and can generate wavelets with an arbitrary number of vanishing moments the algorithm for generating these filters makes use of the levinson recursions gaussian quadrature and a fast version of euclid s algorithm similar to other methods for constructing quadrature mirror filters the spectral factorization of a polynomial is the computationally expensive part of this algorithm

simple and efficient nonparametric method for estimating the number of signals without eigendecomposition inspired by the computational simplicity and numerical stability of qr decomposition a nonparametric method for estimating the number of signals without eigendecomposition mense is proposed for the coherent narrowband signals impinging on a uniform linear array ula by exploiting the array geometry and its shift invariance property to decorrelate the coherency of signals through subarray averaging the number of signals is revealed in the rank of the qr upper trapezoidal factor of the autoproduct of a combined hankel matrix formed from the cross correlations between some sensor data since the infection of additive noise is defused signal detection capability is improved a new detection criterion is then formulated in terms of the row elements of the qr upper triangular factor when finite array data are available and the number of signals is determined as a value of the running index for which this ratio criterion is maximized where the qr decomposition with column pivoting is also used to improve detection performance the statistical analysis clarifies that the mense detection criterion is asymptotically consistent furthermore the proposed mense algorithm is robust against the array uncertainties including sensor gain and phase errors and mutual coupling and against the deviations from the spatial homogeneity of noise model the effectiveness of the mense is verified through numerical examples and the simulation results show that the mense is superior in detecting closely spaced signals with a small number of snapshots and or at relatively low signal to noise ratio snr

finite precision arithmetic and the split schur algorithms the split schur algorithms of p delsarte and y genin 1987 represent methods of computing reflection coefficients that are computationally more efficient in terms of multiplications than the conventional schur algorithm by a constant factor the authors investigate the use of fixed point binary arithmetic with quantization due to rounding in the implementation of the symmetric and antisymmetric split schur algorithms it is shown through a combination of analysis and simulation that the errors in the reflection coefficient estimates due to quantization are large when the input signal is either a narrowband high pass signal or a narrowband low pass signal

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

a symmetric algorithm for toeplitz systems abstract we derive an algorithm for real symmetric toeplitz systems with an arbitrary right hand side it differs from the levinson algorithm in that the solution is built up from its middle component s outwards rather than from top to bottom we then exploit the symmetry of this method by solving separately for the even and odd parts of the right hand side of the system on a sequential machine the complexity of our algorithm for a system of order n is 7 2 n 2 o n flops compared to 4n 2 o n flops for levinson s algorithm the algorithm can be extended to nonsymmetric systems just like levinson s algorithm

source association doa and fading coefficients estimation for multipath signals this paper addresses the source association sa direction of arrival doa and fading coefficients fcs estimation problem in multipath environment first we establish a rank reduction property for a multipath signal model with the existence of multiple groups of coherent signals subsequently based on this property effective algorithms for sa doa and fcs estimation have been developed the proposed doa and fcs estimation methods exploit the multipath structure information to achieve improved accuracy the new doa estimation methods work well even in the case that the doas of the multipath signals associated with different sources are nearly overlapped meanwhile the new methods are applicable to arbitrary array geometry while without decreasing the effective array aperture then the stochastic cramer rao bound on doa and fcs estimation of multipath model mcrb exploiting the multipath structure information is derived in closed form numerical simulations have been provided to demonstrate the effectiveness of the proposed methods

detection of the number of coherent signals by the mdl principle an approach is presented to the problem of detecting the number of sources impinging on a passive sensor array that is based on j rissanen s 1983 minimum description length mdl principle the approach is applicable to any type of sources including the case of sources which are fully correlated referred to as the coherent signals case two slightly different detection criteria are derived both requiring the estimation of the locations of the sources the first is tailored to the detection problem per se whereas the second is tailored to the combined detection estimation problem consistency of the two criteria is proved and their performance is demonstrated by computer simulations

splitting of some s domain stability test algorithms some classic s domain algorithms are first expressed in a unified form from this all possible split forms are then derived and classified in particular attention is focused on two families of algorithms which include the standard routh algorithm and an s domain levinson like algorithm respectively the related split forms are analysed and compared their complexity can be reduced by suitably choosing some scaling factors it turns out that the usual form of the routh algorithm which is intrinsically split and the simplified version of the split levinson like algorithm are substantially equivalent and superior to any other possible split algorithm

split algorithms for skewsymmetric toeplitz matrices with arbitrary rank profile split levinson type and schur type algorithms for the solutions of linear systems with a nonsingular skewsymmetric toeplitz matrix are designed in contrast to previous ones the algorithms work for any nonsingular skewsymmetric toeplitz matrix moreover generalizations of zw and wz factorizations of skewsymmetric toeplitz matrices related to the new split algorithms are presented

two dimensional direction estimation for a mixture of noncoherent and coherent signals this paper deals with the two dimensional 2 d direction of arrival doa estimation of a mixture of noncoherent including uncorrelated and partially correlated and coherent i e fully correlated narrowband signals impinging on a planar sensor array composed of two parallel uniform linear arrays ulas an oblique projection based approach for 2 d direction estimation opade is proposed by using some cross correlations between the received array data in the proposed opade the oblique projection is utilized to isolate the coherent signals from the noncoherent ones and the effect of additive noise is alleviated while the computationally intensive eigendecomposition is avoided and the estimated elevation and azimuth angles are paired automatically further an iterative alternating scheme is presented to improve the estimation accuracy of the oblique projector and hence that of the doas of coherent signals the cramer rao lower bound crb for the mixture of noncoherent and coherent signals is also derived explicitly where the prior knowledge of the signal correlation is incorporated into the 2 d doa estimation of noncoherent signals finally the effectiveness of the opade and the theoretical analysis are substantiated through numerical examples

a general linear prediction approach to the complex split inversion algorithm a general linear prediction framework is developed for the derivation of the complex split levinson algorithm which leads naturally to a structural interpretation of the redundancy present in the classical levinson durbin algorithm a novel order recursive predictor structure is presented it is conceptually simpler than a lattice realization since it propagates only one generalized prediction error when the usual prediction error is desired it can be recovered through a simple terminal stage

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

recurrence relations for the even and odd characteristic polynomials of a symmetric toeplitz matrix in a recent paper computation of the smallest even and odd eigenvalues of a symmetric positive definite toeplitz matrix siam j matrix anal appl 25 2004 949 963 melman proved a recurrence relation of the even and odd characteristic polynomials of a real symmetric toeplitz matrix t on which a symmetry exploiting method for computing the smallest eigenvalue of t can be based in this note we present a proof of the recurrence relation which is less technical and more transparent

immittance domain levinson algorithms several computationally efficient versions of the levinson algorithm for solving linear equations with toeplitz and quasi toeplitz matrices are presented motivated by a new stability test the new versions require half the number of multiplications and the same number of additions as the conventional form of the levinson algorithm the saving is achieved by using three term rather than two term recursions and propagating them in an impedance admittance or immittance domain rather than the conventional scattering domain one of the recursions coincides with recent results of p delsarte and y genin ieee trans acoust speech signal proc vol assp 34 p 470 8 june 1986 on split levinson algorithms for symmetric toeplitz matrices where the efficiency is gained by using the symmetric and skew symmetric versions of the usual polynomials this special structure is lost in the quasi toeplitz case but one still can obtain similar computational reductions by suitably using three term recursions in the immittance domain

polynomials generated by a three term recurrence relation bounds for complex zeros abstract this paper deals with the zeros of polynomials generated by a certain three term recurrence relation the main objective is to find bounds in terms of the coefficients of the recurrence relation for the regions where the zeros are located in most part the zeros are explored through an eigenvalue representation associated with a corresponding hessenberg matrix applications to szego polynomials para orthogonal polynomials and polynomials with non zero complex coefficients are considered

fast algorithms for estimation and signal processing an inverse scattering framework the inverse scattering problem for a lossless transmission line has been used as a framework for deriving the levinson and schur fast algorithms for toeplitz matrices here the same framework is used to derive and attach novel simple scattering interpretations to the lattice algorithm and the split algorithms this demonstrates that the scattering and interpretation is more fundamental to fast signal processing algorithms than previously noted

levinson type extensions for non toeplitz systems it is shown levinson s basic principle for the solution of normal equations which are of toeplitz form may be extended to the case where these equations do not possess this specific symmetry the method is illustrated by application to various examples which are chosen so that the coefficient matrix possesses various symmetries specifically the solution of the normal equations when the associated matrix is the doubly symmetric non toeplitz covariance matrix is considered next the solution of extended yule walker equations where the coefficient matrix is toeplitz but nonsymmetric is obtained finally the approach is illustrated by considering the determination of the prediction error operator when the normal equations are of symmetric toeplitz form

a two step even odd split levinson algorithm for toeplitz systems abstract we use a two step durbin method rather than the single step version in the even odd split levinson algorithm for strongly nonsingular real symmetric toeplitz systems with arbitrary right hand side thereby slightly reducing the complexity of this algorithm to 19 8 n 2 o n flops on a sequential machine we also present extensive numerical results comparing several levinson type methods

monotonicity of zeros for a class of polynomials including hypergeometric polynomials we study the monotonicity of zeros in connection with perturbed recurrence coefficients of polynomials satisfying certain three term recurrence relations of frobenius type these recurrence relations are the key ingredient for the tridiagonal approach developed by delsarte and genin to solve the standard linear prediction problem as a particular case we consider the askey para orthogonal polynomials on the unit circle 2 f 1 n a b i 2 a 1 z a b r extending a recent result about the monotonicity of their zeros with respect to the parameter b finally the consequences of our results in the theory of orthogonal polynomials on the real line are discussed

fast algorithms for toeplitz and hankel matrices abstract the paper gives a self contained survey of fast algorithms for solving linear systems of equations with toeplitz or hankel coefficient matrices it is written in the style of a textbook algorithms of levinson type and schur type are discussed their connections with triangular factorizations pade recursions and lanczos methods are demonstrated in the case in which the matrices possess additional symmetry properties split algorithms are designed and their relations to butterfly factorizations are developed

new levinson schur and lattice type algorithms for linear phase filtering the ls estimate of the impulse response of fir filters with linear phase is known to be given by a structured linear set of equations three new algorithms are presented for the efficient solution of such systems one is the unwindowed extension of a previously derived levinson type structurally symmetric algorithm the second is a novel exact least squares lattice algorithm for time recursive processing the third is a schur type structurally symmetric algorithm with high degree of parallelism the new structures have considerably lower computational complexity and more parsimonious parametrization compared to previously derived algorithms this is because in contrast to the previously derived algorithms the new algorithms are developed so as to respect the symmetry which is intrinsic in the linear phase problem

positive interpolatory quadrature formulas and para orthogonal polynomials we establish a relation between quadrature formulas on the interval 1 1 that approximate integrals of the form j f 11 f x x dx and szego quadrature formulas on the unit circle that approximate integrals of the form i f f ei d the functions x and are assumed to be weight functions on 1 1 and respectively and are related by cos sin it is well known that the nodes of szego formulas are the zeros of the so called para orthogonal polynomials bn z n z n z 1 n z and n z being the orthogonal and reciprocal polynomials respectively with respect to the weight function furthermore for 1 we have recently obtained gauss type quadrature formulas on 1 1 see bultheel et al j comput appl math 132 1 2000 1 in this paper making use of the para orthogonal polynomials with 1 a one parameter family of interpolatory quadrature formulas with positive coefficients for j f is obtained along with error expressions for analytic integrands finally some illustrative numerical examples are also included

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

mse based regularization approach to direction estimation of coherent narrowband signals using linear prediction this paper addresses the problem of directions of arrival doas estimation of coherent narrowband signals impinging on a uniform linear array ula when the number of signals is unknown by using an overdetermined linear prediction lp model with a subarray scheme the doas of coherent signals can be estimated from the zeros of the corresponding prediction polynomial although the corrected least squares cls technique can be used to improve the accuracy of the lp parameters estimated from the noisy array data the inversion of the resulting matrix in the cls estimation is ill conditioned and then the cls estimation becomes unstable to combat this numerical instability we introduce multiple regularization parameters into the cls estimation and show that determining the number of coherent signals is closely related to the truncation of the eigenvalues an analytical expression of the mean square error mse of the estimated lp parameters is derived and it is clarified that the number of signals can be determined by comparing the optimal regularization parameters with the corresponding eigenvalues an iterative regularization algorithm is developed for estimating directions without any a priori knowledge where the number of coherent signals and the noise variance are estimated from the noise corrupted received data simultaneously

oblique projection based enumeration of mixed noncoherent and coherent narrowband signals in many application scenarios the signals impinging on the array of sensors comprise the noncoherent uncorrelated and or partially correlated signals and the coherent signals with several groups due to multipath propagation in this paper we consider the problem of estimating the number of noncoherent narrowband signals and that of coherent narrowband signals with multiple groups impinging on a planar sensor array composed of two parallel uniform linear arrays ulas and an oblique projection based enumerator for the mixed signals opems is proposed by utilizing the qr decomposition based ratio criterion qrrc for rank determination of a matrix in the opems the number of noncoherent signals and that of coherent signals in each group are estimated separately where only the elevation angles of noncoherent signals are estimated to isolate the coherent signals from the noncoherent ones and the computationally intensive and time consuming eigendecomposition procedure is avoided the consistency of the proposed opems is analyzed and its effectiveness is verified through numerical examples

matrix representations of split bezoutians abstract inverses of symmetric or skewsymmetric toeplitz matrices as well as of centrosymmetric or centro skewsymmetric toeplitz plus hankel matrices can be represented as sums of two split bezoutians which are highly structured matrices since all of their rows and columns are symmetric or skewsymmetric vectors thus it is desirable to find matrix representations for split bezoutians b this is the main aim of the present paper recursion formulas for the entries of b are presented bases of very simple split bezoutians or of sparse matrices are constructed and b is represented as a corresponding linear combination moreover matrix representations of gohberg semencul type are established

euclid algorithm orthogonal polynomials and generalized routh hurwitz algorithm this paper is concerned with the relations between the euclid algorithm the theory of orthogonal polynomials and the problem of locating the zeros of a complex polynomial with respect to the imaginary axis in particular a simple generalized routh hurwitz algorithm is proposed which allows one to determine in any situation the numbers of zeros of an arbitrary complex polynomial in the right half plane on the imaginary axis and hence in the left half plane moreover it turns out that this algorithm yields as a side result a well defined factorization of the considered polynomial furthermore as a straightforward consequence of the adopted approach two presumably original algorithms are put into light which involve linear arithmetic operations only a polynomial nonnegativity test on the real axis and a characterization of positive real functions

forces and the existence of stresses in invariant continuum mechanics in an invariant formulation of pth grade continuum mechanics forces are defined as elements of the cotangent bundle of the banach manifold of c p embeddings of the body in space it is shown that forces can be represented by measures which generalize the stresses of continuum mechanics the mathematical representation procedure makes the restriction of forces to subbodies possible the local properties of the stress measures are examined for the case where stresses are given in terms of smooth densities it is shown that the structure of forces agrees with the form of forces one assumes in the traditional formulation and the equilibrium differential equations are obtained

on the feasibility of impulse reflection response data for the two dimensional inverse scattering problem layer stripping algorithms for inverse scattering problems are very fast but have the reputation of being numerically unstable especially when applied to noisy data the goal of this paper is to provide an explicitly discrete framework for layer stripping algorithms for the two dimensional 2 d schrodinger equation inverse scattering problem we determine when 2 d layer stripping algorithms are numerically stable explain why they are stable and specify exactly the discrete problem they solve when they are stable we reformulate the 2 d schrodinger equation as a multichannel two component wave system by fourier transforming the schrodinger equation in the lateral spatial variable discretization results in new 2 d layer stripping algorithms which incorporate multichannel transmission effects this leads to an important new feasibility condition on impulse reflection response data for stability of these algorithms a 2 d discrete schrodinger equation is defined and analogous results are obtained numerical examples illustrate the new results especially how rendering noisy data feasible stabilizes layer stripping algorithms

fast triangular factorization of the sum of quasi toeplitz and quasi hankel matrices abstract the literature contains several recent fast algorithms for the triangular factorization of strongly regular toeplitz plus hankel matrices in this paper we study the rather more general sum of quasi toeplitz and quasi hankel matrices both hermitian and non hermitian quasi toeplitz and quasi hankel matrices are those that are congruent to toeplitz and hankel matrices in a special sense the derivation is based on the concept of displacement structure and its intimate relation to the schur reduction procedure for triangular factorization various special cases covering displacement ranks from two to eight are considered several other problems e g factorization of the inverse matrix solution of exact or overdetermined linear systems can be reduced to the direct factorization problem

reduced rank mdl method for source enumeration in high resolution array processing this paper proposes a reduced rank minimum description length mdl method to enumerate the incident waves impinging on a uniform linear array ula first a new observation data and a reference signal are formed from sensor data by means of the shift invariance property of the ula a cross correlation between them is calculated which is able to capture signal information and efficiently suppress additive noise second the normalized cross correlation is used as initial information for a recursion procedure to quickly partition the observation data into two orthogonal components in a signal subspace and a reduced rank noise subspace the components in the noise subspace are employed to calculate the total code length that is required to encode the observation data finally the model with the shortest code length namely the minimum description length is chosen as the best model unlike the traditional mdl methods this method partitions the observation data into the cleaner signal and noise subspace components by means of the recursion procedure avoiding the estimation of a covariance matrix and its eigendecomposition thus the method has the advantage of computational simplicity its performance is demonstrated via numerical results

comparison between some stability criteria of discrete time filters routh s algorithm is written in the form of polynomial recursion and transposed for testing stability in the wide sense of discrete time linear systems the algorithm obtained is a three term recursion of symmetric polynomials this recursion easily gives both the coefficients appearing in the continued fraction expansion associated to a given polynomial in the z domain and a computation scheme for the bilinear transformation of polynomials finally this recursion is compared to other algorithms already known for testing stability

toeplitz properties of the block matrices encountered in the processing of spatiotemporal signals various signal processing applications such as system identification fir finite impulse response wiener filtering linear prediction and spectral estimation are known to lead to the solution of toeplitz systems for the case of one dimensional signals and to toeplitz block toeplitz systems for the case of two dimensional signals the authors investigate such systems for the case of l dimensional signals with l 2 and emphasize their block and persymmetrical nature such important characteristics allow the use of fast algorithms for signals of any dimension the case of three dimensional signals is treated in detail not only because it is of special interest for video applications but also because the reasoning used for this case can be straightforwardly applied to higher dimension signals

immittance versus scattering domain fast algorithms for non hermitian toeplitz and quasi toeplitz matrices abstract the classical algorithms of schur and levinson are efficient procedures to solve sets of hermitian toeplitz linear equations or to invert the corresponding coefficient matrices they propagate pairs of variables that may describe incident and scattered waves in an associated cascade of layered media model and thus they can be viewed as scattering domain algorithms it was recently found that a certain transformation of these variables followed by a change from two term to three term recursions results in reduction in computational complexity in the abovementioned algorithms roughly by a factor of two the ratio of such pairs of transformed variables can be interpreted in the above layered media model as an im pedance or ad mittance hence the name immittance domain variables this paper provides extensions for previous immittance schur and levinson algorithms from hermitian to non hermitian matrices it considers both toeplitz and quasi toeplitz matrices matrices with certain hidden toeplitz structure and compares two and three term recursion algorithms in the two domains the comparison reveals that for non hermitian matrices the algorithms are equally efficient in both domains this observation adds new comprehension to the source and value of algorithms in the immittance domain the immittance algorithms like the scattering algorithms exploit the quasi toeplitz structure to produce fast algorithms however unlike the scattering algorithms they can respond also to symmetry of the underlying matrix when such extra structure is present and yield algorithms with improved efficiency

computation of the newton step for the even and odd characteristic polynomials of a symmetric positive definite toeplitz matrix we compute the newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite toeplitz matrix as the reciprocal of the trace of an appropriate matrix we show that after the yule walker equations are solved this trace can be computed in 0 n additional arithmetic operations which is in contrast to existing methods which rely on a recursion requiring o n 2 additional arithmetic operations

source enumeration in a correlated signal environment the authors present a simple method for determining the number of signals impinging on a uniform linear array that is applicable even in the extreme case of fully correlated signals this technique uses what they term modified rank sequences which is a modification of the construction implicit in the matrix decomposition method of di 1981 they prove that if a particular rank sequence stabilizes the last two terms of the sequence are equal to a value strictly less then the common row size of the defining block matrices then this value equals the number of signals provided that the number of signals has not exceeded a bresler macovski 1986 type bound using the above characterization of stability they formulate an algorithm that either determines the number of signals or indicates that the resolution capability of the algorithm has been exceeded they also provide theorems that show that under certain conditions a rank sequence can stabilize to a value strictly less than the number of signals this result allows them to find simple counterexamples to all of the existing rank sequence methods

performance analysis of a split path lms adaptive filter for ar modeling a split path adaptive filter is proposed for extracting the model parameters of an autoregressive process the structure is composed of two linear phase filters connected in parallel one antisymmetric and the other symmetric the two filters are adapted independently on a sample by sample basis using the least mean square lms algorithm the performance of the system in terms of convergence speed and excess mean square error is analyzed in detail and comparisons with the conventional transversal structure are made theoretical analysis and experimental results show that the model can provide a much faster rate of convergence at the expense of only a moderate increase in computation two methods for choosing control parameters for the split path adaptive filter are also suggested to improve further the convergence behavior

interactions in general continua with microstructure a satisfactory axiomatic treatment of the interaction among bodies modelled as standard continua was achieved by noll almost thirty years ago and was presented in the seminal paper 1 variously reprinted a full review of the matter is also available in the first chapters of the treatise 2

symmetric solutions and eigenvalue problems of toeplitz systems algorithms and properties of symmetric solutions of a toeplitz system are studied it is shown that the numbers of positive and negative eigenvalues associated with symmetric antisymmetric eigenvectors are the same as the numbers of positive and negative predictor errors of symmetric antisymmetric filters based on the odd symmetric solutions and the property that all roots of symmetric and antisymmetric filters are on the unite circle a method for pisarenko s decomposition is introduced compared with some other methods it reduces the number of iterations and the computational cost in each iteration considerably

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

levinson type algorithms for polynomial fitting and for cholesky and q factors of hankel and vandermonde matrices this paper presents levinson 1947 type algorithms for i polynomial fitting ii obtaining a q decomposition of vandermonde matrices and a cholesky factorization of hankel matrices iii obtaining the inverse of hankel matrices the algorithm for the least squares solution of hankel systems of equations requires 3n sup 2 9n 3 multiply and divide operation mdo the algorithm for obtaining an orthogonal representation of an m spl times n vandermonde matrix x and computing the cholesky factors f of hankel matrices requires 5mn n sup 2 2n 3m mdo and the algorithm for generating the inverse of hankel matrices requires 3 n sup 2 n 2 2 mdo our algorithms have been tested by means of fitting of polynomials of various orders and fortran versions of all subroutines are provided in the appendix

high resolution velocity spectra using eigenstructure methods stacking spectra provide maximum likelihood estimates for the stacking velocity or for the ray parameter of well separated reflections in additive white noise however the resolution of stacking spectra is limited by the aperture of the array and the frequency of the data despite these limitations parametric spectral estimation methods achieve better resolution than does stacking to improve resolution the parametric methods introduce a parsimonious model for the spectrum of the data in particular when the data are modeled as the superposition of wavefronts the properties of the eigenstructure of the data covariance matrix can be used to obtain high resolution spectra the traditional stacking spectra can also be expressed as a function of the data covariance matrix and directly compared to the eigenstructure spectra the superiority of the latter in separating closely interfering reflections is then apparent from a simple geometric interpretation eigenstructure methods were originally developed