the envelope of spectral power for stochastic processes it is well known that the maximum likelihood ml spectral estimate represents an envelope on spectral power which is consistent with estimated covariance lags however in computing the envelope the ml method allows for spectra corresponding to complex valued stochastic processes we study the respective envelope for processes constrained to be real valued at the outset the resulting envelope is less conservative than the one provided by the ml method a closed form expression exists for the new envelope

optimization based computation of analytic interpolants of bounded complexity this paper provides a unifying algorithm for computing any analytic interpolant of bounded complexity such computation can be performed by solving an optimization problem due to a theorem by georgiou and lindquist this optimization problem is numerically solvable by a continuation method the proposed numerical algorithm is useful among other cases for designing a low degree controller for a benchmark problem in robust control the algorithm unifies previously developed algorithms for the caratheodory extension and the nevanlinna pick interpolation to one for more general interpolation problems

large character sums pretentious characters and the polya vinogradov theorem in 1918 polya and vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate one of the main results of this paper provides a substantial improvement of the polya vinogradov bound for characters of odd bounded order in 1977 montgomery and vaughan showed how the polya vinogradov inequality may be sharpened assuming the generalized riemann hypothesis we give a simple proof of their estimate and provide an improvement for characters of odd bounded order the paper also gives characterizations of the characters for which the maximal character sum is large and it finds a hidden structure among these characters

skew prime polynomial matrices the polynomial model approach abstract we examine the concept of skew primeness of polynomial matrices in terms of the associated polynomial model it is shown that skew primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism this basic result is then applied to the special case of nonsingular polynomial matrices we investigate the nonuniqueness of skew complements of a skew prime pair it is shown that the space of equivalence classes of skew complements is in bijective correspondence with a finite dimensional linear space finally the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly

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

a nearly optimal method to compute the truncated theta function its derivatives and integrals a poly log time method to compute the truncated theta function its derivatives and integrals is presented the method is elementary rigorous explicit and suited for computer implementation we repeatedly apply the poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition the method relies on the periodicity of the complex exponential which enables the suitable normalization of the arguments and on the self similarity of the gaussian which ensures that we still obtain a truncated theta function after each application of the poisson summation in other words our method relies on modular properties of the theta function applications to the numerical computation of the riemann zeta function and to nding the number of solutions of waring type diophantine equations are discussed

least squares approximation of structured covariances state covariances of linear systems satisfy certain constraints imposed by the underlying dynamics these constraints dictate a particular structure of state covariances however sample covariances almost always fail to have the required structure the renewed interest in using state covariances for estimating the power spectra of inputs gives rise to the approximation problem in this note the structured covariance least squares problem is formulated and the lyapunov type matricial linear constraint is converted into an equivalent set of trace constraints efficient unconstrained maximization methods capable of solving the corresponding dual problem are developed

on the estimation of structured covariance matrices this paper discusses a method for estimating the covariance matrix of a multivariate stationary process w generated as the output of a given linear filter fed by a stationary process y the estimated covariance matrix must satisfy two constraints it must be positive semi definite and it must be consistent with the fact that w is the output of the given linear filter it turns out that these constraints force the estimated covariance to lie in the intersection of a cone with a linear space while imposing only the first of the two constraints is rather straightforward guaranteeing that both are satisfied is a non trivial issue to which quite a bit of attention has already been devoted in the literature our approach extends the method for estimating the toeplitz covariance matrix of order m of a process y based on the biased spectral estimator stoica moses 1997 this extension is based on the characterization of the output covariance matrix in terms of the filter parameters and the sequence of covariance lags of the input process after introducing our estimation method we propose a comparison performance between this one and other methods proposed in the literature simulation results show that our approach constitutes a valid estimation procedure

uncertainty bounds for spectral estimation the purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second order statistics the family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the true power spectrum our aim is to quantify the size of this uncertainty set using suitable notions of distance and in particular to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the true power spectrum since the uncertainty set may contain power spectra with lines and discontinuities it is natural to quantify distances in the weak topology the topology defined by continuity of moments we provide examples of such weakly continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty we then consider certain high resolution techniques which utilize filter banks for preprocessing and compute worst case a priori uncertainty bounds solely on the basis of the filter dynamics this allows the a priori tuning of the filter banks for improved resolution over selected frequency bands

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

fast methods to compute the riemann zeta function the riemann zeta function on the critical line can be computed using a straightforward application of the riemann siegel formula schonhage s method or heath brown s method the complexities of these methods have exponents 1 2 3 8 and 1 3 respectively in this article three new fast and potentially practical methods to compute zeta are presented one method is very simple its complexity has exponent 2 5 a second method relies on this author s algorithm to compute quadratic exponential sums its complexity has exponent 1 3 the third method which is our main result employs an algorithm developed here to compute cubic exponential sums with a small cubic coecient its complexity has exponent 4 13 approximately 0 307

zeros of partial sums of l functions abstract we consider a certain class of multiplicative functions f n c let f s n 1 f n n s be the associated dirichlet series and f n s n n f n n s be the truncated dirichlet series in this setting we obtain new halasz type results for the logarithmic mean value of f more precisely we prove estimates for the sum n 1 x f n n in terms of the size of f 1 1 log x and show that these estimates are sharp as a consequence of our mean value estimates we establish non trivial zero free regions for the partial sums f n s in particular we study the zero distribution of partial sums of the dedekind zeta function of a number field k more precisely we give some improved results for the number of zeros up to height t as well as new zero density results for the number of zeros up to height t lying to the right of re s where 1 2

on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities we consider the georgiou lindquist constrained approximation of spectra in the kullback leibler sense we propose an alternative iterative algorithm to solve the corresponding convex optimization problem the lagrange multiplier is computed as a fixed point of a nonlinear matricial map simulation indicates that the algorithm is extremely effective

optimal parameters for 2 cyclic aor assume email protected c n x n is a 2 cyclic consistently ordered matrix and j is denoted as its associated block jacobi iteration matrix we consider the 2 cyclic aor method for solving the consistent linear systems ax b in the case that s j 2 is either nonnegative or nonpositive we give detailed discussion and derive definite expressions on optimal parameters and spectral radius by efficient method moreover we give some numerical examples

the caratheodory fejer pisarenko decomposition and its multivariable counterpart when a covariance matrix with a toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity the singular summand corresponds to the covariance of a purely deterministic component of a time series whereas the identity corresponds to white noise this is the caratheacuteodory fejeacuter pisarenko cfp decomposition in the present paper we study multivariable analogs for block toeplitz matrices as well as for matrices with the structure of state covariances of finite dimensional linear systems which include block toeplitz ones to this end we develop theory which addresses questions of existence uniqueness and realization of multivariable power spectra possibly having deterministic components we characterize state covariances which admit only a deterministic input power spectrum and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part we then show that multivariable decomposition of a state covariance in accordance with a deterministic component white noise hypothesis for the input does not exist in general we finally reinterpret the cfp dictum and consider replacing the scalar multiple of the identity by a covariance of maximal trace which is admissible as a summand the summand can be either block diagonal corresponding to white noise or have a short range correlation structure corresponding to a moving average component the trace represents the maximal variance energy that can be accounted for by a process at the input e g noise with the aforementioned structure and this maximal solution can be computed via convex optimization the decomposition of covariances and spectra according to the range of their time domain correlations is an alternative to the cfp dictum with potentially great practical significance

zero free regions for dirichlet series in this paper we are interested in explicit zero free discs for some dirichlet series and we also study a general beurling nyman criterion for l functions our results generalize and improve previous results obtained by n nikolski and by a de roton as a concrete application we get for example a beurling nyman type criterion for the siegel zero problem

on the geometry of maximum entropy problems we show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinite dimensional maximum entropy problems concerning probability distributions spectral densities and covariance matrices these include burg s spectral estimation method and dempster s covariance completion as well as various recent generalizations of the above we then apply this orthogonality principle to the new problem of completing a block circulant covariance matrix when an a priori estimate is available

fourier uniformity of bounded multiplicative functions in short intervals on average let lambda denote the liouville function we show that as x rightarrow infty begin aligned int x 2x sup alpha left sum x 0 fixed but arbitrarily small previously this was only known for theta 5 8 for smaller values of theta this is the first non trivial case of local fourier uniformity on average at this scale we also obtain the analogous statement for non pretentious 1 bounded multiplicative functions we illustrate the strength of the result by obtaining cancellations in the sum of lambda n lambda n h lambda n 2h over the ranges h x theta and n x and where lambda is the von mangoldt function

corrigenda optimum accelerated overrelaxation method in a special case math comp 36 1981 183 187 mr 82c 65016 by g avdelas and the author

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

an extension of the accelerated overrelaxation aor theory for linear systems with heritian matrix abstract in this paper the convergence theory of the accelerated overrelaxation aor method for the solution of the linear system ax b where a is hermitian ah a is studied under the general splitting a d e eh where d is hermitian and e 1 2 d a s with sh s the purpose of the paper is to cover the case where d is hermitian and negative definite the case where d is hermitian and positive definite was covered in 8

tunable line spectral estimators based on state covariance subspace analysis subspace methods for spectral analysis can be adapted to the case where state covariance of a linear filter replaces the traditional toeplitz matrix formed out of a partial autocorrelation sequence of a time series this observation forms the basis of a new framework for spectral analysis the goal of this paper is to quantify potential advantages in working with state covariance data instead of the autocorrelation sequence to this end we identify tradeoffs between resolution and robustness in spectral estimates and how these are affected by the filter dynamics the approach leads to a novel tunable high resolution frequency estimator

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

modeling continuous time processes via input to state filters a direct algorithm to estimate continuous time arma carma models is proposed in this paper in this approach we first pass the observed data through an input to state filter and compute the state covariance matrix the properties of the state covariance matrix are then exploited to estimate the half spectrum of the observed data at a set of user defined points on the right half plane finally the continuous time parameters are obtained from the half spectrum estimates by solving an analytic interpolation problem with a positive real constraint as shown by simulations the proposed algorithm delivers much more reliable estimates than indirect modeling approaches which rely on estimating an intermediate discrete time model

a globally convergent matricial algorithm for multivariate spectral estimation in this paper we first describe a matricial newton type algorithm designed to solve the multivariable spectrum approximation problem we then prove its global convergence finally we apply this approximation procedure to multivariate spectral estimation and test its effectiveness through simulation simulation shows that in the case of short observation records this method may provide a valid alternative to standard multivariable identification techniques such as matlab s pem and matlab s n4sid

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

information matrix and d optimal design with gaussian inputs for wiener model identification we present a closed form expression for the fischer s information matrix associated with the identification of wiener models in the derivation we assume that the input signal is gaussian the analysis allows the linear sub system in the wiener model to have a generic rational transfer function of arbitrary order it also allows the static nonlinearity of the wiener model to be a polynomial of arbitrary degree in addition we show how this analysis can be used to design tractable algorithms for d optimal input design the idea is further extended to design optimal inputs consisting of a sequence of gaussian signals with different mean values and variances by combining gaussian inputs with different means we can tune the amplitude distribution of the input to achieve the best identification accuracy in d optimal sense the analytical results are also illustrated with some numerical simulations

generalizing caratheodory s uniqueness of harmonic parameterization to n dimensions consider a sum of f exponentials in n dimensions and let i sub n be the number of equispaced samples taken along the nth dimension it is shown that if the frequencies or decays along every dimension are distinct and spl sigma sub n 1 sup n i sub n spl ges 2f n 1 then the parameterization in terms of frequencies decays amplitudes and phases is unique the result can be viewed as generalizing a classic result of caratheodory to n dimensions the proof relies on a recent result regarding the uniqueness of low rank decomposition of n way arrays

on sinusoid estimation in nonstationary noise this note presents a sinusoid estimation algorithm which will converge to the spectrum of the sinusoid process in nonstationary additive noise the algorithm is in the framework of the tangential nevanlinna pick setting the procedure is a generalization of capon s maximum likelihood estimate and some classical limit theorem for orthogonal polynomials the method is used in an experimental example to find the natural frequencies of a lightly damped structure

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

signal estimation via selective harmonic amplification music redux the technique known as multiple signal classification music is a semi empirical way to obtain pseudo spectra that highlight the spectral energy distribution of a time series it is based on a certain canonical decomposition of a toeplitz matrix formed out of an estimated autocorrelation sequence the purpose of this paper is to present an analogous canonical decomposition of the state covariance matrix of a stable linear filter driven by a given time series accordingly the paper concludes with a modification of music the new method starts with filtering the time series and then estimating the covariance of the state of the filter this step in essence improves the signal to noise ratio snr by amplifying the contribution to the actual value of the state covariance of a selected harmonic interval where spectral lines are expected to reside then the method capitalizes on the canonical decomposition of the filter state covariance to retrieve information on the location of possible spectral lines the framework requires uniformly spaced samples of the process

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

on the convergence of an efficient algorithm for kullback leibler approximation of spectral densities this paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a la byrnes georgiou lindquist the approximation is pursued with respect to the kullback leibler pseudo distance which gives rise to a convex optimization problem after developing the variational analysis we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem based on the iteration of a nonlinear map in a bounded subset of the dual space our main result is the proof of local convergence of the latter established as a consequence of the central manifold theorem supported by numerical evidence we conjecture that in the mentioned bounded set the convergence is actually global

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

multiple ergodic theorems for arithmetic sets we establish results with an arithmetic flavor that generalize the polyno mial multidimensional szemeredi theorem and related multiple recurrence and conver gence results in ergodic theory for instance we show that in all these statements we can restrict the implicit parameter n to those integers that have an even number of dis tinct prime factors or satisfy any other congruence condition in order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions these averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors

a maximum entropy enhancement for a family of high resolution spectral estimators structured covariances occurring in spectral analysis filtering and identification need to be estimated from a finite observation record the corresponding sample covariance usually fails to possess the required structure this is the case for instance in the byrnes georgiou lindquist three like tunable high resolution spectral estimators there the output covariance of a linear filter is needed to initialize the spectral estimation technique the sample covariance estimate however is usually not compatible with the filter in this paper we present a new systematic way to overcome this difficulty the new estimate is obtained by solving an ancillary problem with an entropic type criterion extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances

solution of the general moment problem via a one parameter imbedding this paper presents a computational theory for the general scalar moment problem the formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics and in nonuniform multidimensional sampling given a finite set of moments the theory provides a test for the existence of a positive measure which is consistent with such data at the same time the theory also provides a characterization of all such consistent positive measures it should be noted that classical results e g in the theory of the trigonometric moment problem hamburger stieljes nevanlinna pick interpolation etc are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels the centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function spl psi which plays the role of a free parameter the differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure for each spl psi the fixed point determines a corresponding measure suitable choice of spl psi allows recovering any measure which is consistent with the data the fixed point of the differential equation corresponds to an extremum of an entropy like functional and the differential equation is constructed via an appropriate homotopy that follows changes in the lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments

variance error interpolation and experiment design we investigate how the variance error associated with the prediction error identification is related to the power spectral densities of the input and the additive noise at the output let f e i w be the ratio of the input power spectral density psd to the output noise psd we characterize the set of all functions f for which the variance error remains constant this analysis results a minimal finite dimensional affine parameterization of the variance error this parameterization connects our analysis with the theory of nevanlinna pick interpolation it is shown that the set of all f for which the variance error remains constant can be characterized by the solutions of a nevanlinna pick interpolation problem this insight has interesting consequences in optimal input design where it is possible to use some recent tools in analytic interpolation theory to tune shape of the input psd to suit certain needs

further results on the convergence of the pavon ferrante algorithm for spectral estimation in this paper we provide a detailed analysis of the global convergence properties of an extensively studied and extremely effective fixed point algorithm for the kullback leibler approximation of spectral densities proposed by pavon and ferrante in their paper on the georgiou lindquist approach to constrained kullback leibler approximation of spectral densities our main result states that the algorithm globally converges to one of its fixed points

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records

convergence rate of newton s method for l 2 spectral estimation in the paper we prove the holder continuous property of the jacobian of the function generated from the dual of the power spectrum estimation problem it follows that the convergence of the newton method for the problem is at least of order where m is the order of the trigonometric bases this result theoretically confirms the numerical observation by potter 1990 and cole and goodrich 1993