generalized bayesian information criterion for source enumeration in array processing we investigate the problem of enumerating source signals impinging on an array of sensors in an information theoretic framework the conventional bayesian information criterion bic does not yield satisfactory performance for this problem because it only considers the density of the observations in order to remedy the limitations of the bic we propose a generalized bayesian information criterion gbic rule by incorporating the density of the sample eigenvalues or corresponding statistics such a density contains extra information and complements the density of the observations in constructing the gbic as a result two different expressions for the gbic are suggested simulation results validate the superiority of the proposed gbic over the conventional bic in terms of correctly determining the number of sources while their computational costs are comparable

an mdl algorithm for detecting more sources than sensors using outer products of array output 1991 2012 ieee in this paper we propose an algorithm for detecting the number m of gaussian sources received by an array of a number n n m of sensors this algorithm is based on the minimum description length mdl principle using the outer products of the array output we show that as long as the covariance matrix of the array output has the full rank n the covariance matrix of a vectorized outer product of the array output has the full rank n squared which meets a validity condition of the mdl algorithm we show by simulation that the mdl algorithm can perform substantially better than some relevant algorithms a necessary identifiability condition is also obtained for uncorrelated sources

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

estimation of the number of signals from features of the covariance matrix a supervised approach the purpose of this paper is to provide a fast and simplified detection test for use in the presence of a small number of sources from 0 2 which is able to accommodate correlated paths and nonwhite noise conventional eigenvalue based criteria are unable to do so for a uniform linear array using common sense arguments a small set of significant features of the covariance matrix are used as inputs to a neural net the nonlinear transfer function of the neural net is adjusted by supervised training to provide the discriminant functions for order selection in its outputs results from the net are then compared with conventional criteria and demonstrate superior performance in particular for correlated sources and small sample sizes training may be introduced for known nonwhite noise which serves to maintain high performance for reasonable correlation lengths

mmse based mdl method for robust estimation of number of sources without eigendecomposition it is well known that the conventional eigenvalue based minimum description length mdl approach for source number estimation suffers from high computational load and performs optimally only in the presence of spatially and temporally white noise to improve the robustness of the mdl methodology we propose to utilize the minimum mean square error mmse of the multistage wiener filter to calculate the required description length for encoding the observed data instead of relying on the eigenvalues of the data covariance matrix as there is no need to calculate the covariance matrix and its eigenvalue decomposition our derived mmse based mdl mmdl method is also more computationally efficient than the traditional counterparts numerical examples are included to demonstrate the robustness of the mmdl detector in nonuniform noise

a parametric method for determining the number of signals in narrow band direction finding a novel and more accurate method to determine the number of signals in the multisource direction finding problem is developed the information theoretic criteria of y yin and p krishnaiah 1988 are applied to a set of quantities which are evaluated from the log likelihood function based on proven asymptotic properties of the maximum likelihood estimation these quantities have the properties required by the criteria since the information theoretic criteria use these quantities instead of the eigenvalues of the estimated correlation matrix this approach possesses the advantage of not requiring a subjective threshold and also provides higher performance than when eigenvalues are used simulation results are presented and compared to those obtained from the nonparametric method given by h wax and t kailath 1985

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

time frequency projection filters and time frequency signal expansions we consider the problems of designing a linear time varying filter with a specified time frequency tf pass region and of constructing an orthonormal basis for the parsimonious expansion of signals located in a given tf support region these problems of tf filtering and tf signal expansion are reduced to the problem of designing a tf subspace i e a linear signal space comprising all signals located in a given tf legion specifically the tf filter is taken to be the orthogonal projection operator on the tf subspace we present an optimum design of tf subspaces that is based on the wigner distribution of a linear signal space and is an extension of the well known signal synthesis problem the optimum tf subspace is shown to be an eigenspace of the tf region and some properties of eigenspaces are discussed the performance of tf projection filters and tf signal expansions is studied both analytically and via computer simulation

the inclusion relation between sobolev and modulation spaces the inclusion relations between the lp sobolev spaces and the modulation spaces is determined explicitly as an application mapping properties of unimodular fourier multiplier ei d between lp sobolev spaces and modulation spaces are discussed

detection and localization of multiple sources in noise with unknown covariance the authors present a novel technique for the detection and localization of multiple sources in the presence of noise with unknown and arbitrary covariance the technique is applicable to coherent and noncoherent signals and to arbitrary array geometry and is based on rissanen s minimum description length mdl principle 1989 for model selection its computational load is comparable to that of analogous techniques for white noise simulation results demonstrating the performance of this technique are included

detection and localization of multiple sources via the stochastic signals model a method for the detection of coherent and noncoherent signals based on the application of j rissanen s 1978 minimum description length principle for model selection to the stochastic signals model is presented the detection and localization are done simultaneously with the location estimator coinciding with the maximum likelihood estimator derived by j f bohme 1986 the proposed method outperforms the method of m wax and i ziskind 1989 especially in the threshold region simulation results demonstrating the improved performance are included

singular values of compact pseudodifferential operators abstract this paper investigates the asymptotic decay of the singular values of compact operators arising from the weyl correspondence the motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay the problem is approached by using a gabor frame expansion of the symbol to construct an approximating finite rank operator this establishes a variety of sufficient conditions for the associated operator to be in a particular schatten class in particular an improvement of a sufficient condition of daubechies for an operator to be trace class is obtained in addition a new development and improvement of the calderon vaillancourt theorem in the context of the weyl correspondence is given additional results of this type are then obtained by interpolation

estimation of the number of sources in unbalanced arrays via information theoretic criteria estimating the number of sources impinging on an array of sensors is a well known and well investigated problem a common approach for solving this problem is to use an information theoretic criterion such as minimum description length mdl or the akaike information criterion aic the mdl estimator is known to be a consistent estimator robust against deviations from the gaussian assumption and nonrobust against deviations from the point source and or temporally or spatially white additive noise assumptions over the years several alternative estimation algorithms have been proposed and tested usually these algorithms are shown using computer simulations to have improved performance over the mdl estimator and to be robust against deviations from the assumed spatial model nevertheless these robust algorithms have high computational complexity requiring several multidimensional searches in this paper which is motivated by real life problems a systematic approach toward the problem of robust estimation of the number of sources using information theoretic criteria is taken an mdl type estimator that is robust against deviation from assumption of equal noise level across the array is studied the consistency of this estimator even when deviations from the equal noise level assumption occur is proven a novel low complexity implementation method avoiding the need for multidimensional searches is presented as well making this estimator a favorable choice for practical applications

esprit estimation of signal parameters via rotational invariance techniques an approach to the general problem of signal parameter estimation is described the algorithm differs from its predecessor in that a total least squares rather than a standard least squares criterion is used although discussed in the context of direction of arrival estimation esprit can be applied to a wide variety of problems including accurate detection and estimation of sinusoids in noise it exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure the technique when applicable manifests significant performance and computational advantages over previous algorithms such as mem capon s mlm and music

on information theoretic criteria for determining the number of signals in high resolution array processing an important problem in high resolution array processing is the determination of the number of signals arriving at the array information theoretic criteria provide a means to achieve this two commonly used criteria are the akaike information criterion aic and minimum descriptive length mdl criterion while the aic tends to overestimate even at a high signal to noise ratio snr the mdl criterion tends to underestimate at low or moderate snr by excluding irrelevant parameters a new log likelihood function has been chosen utilizing this new log likelihood function gives a set of more accurate estimates of the eigenvalues and in the establishment of modified information theoretic criteria which moderate the performance of the aic and the mdl criterion computer simulations confirm that the modified criteria have superior performance

positivity properties of phase plane distribution functions the aim of this paper is to compare the members of cohen s class of phase plane distributions with respect to positivity properties it is known that certain averages which are in a sense compatible with heisenberg s uncertainty principle of the wigner distribution over the phase plane yield non negative values for all states it is shown in this paper that the wigner distribution is unique in this respect among the members of cohen s class that have correct marginals or that satisfy moyal s formula for all states the subset of members of cohen s class not necessarily satisfying one of these two conditions with positivity properties comparable with those for the wigner distribution is shown to be rather small

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

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

boundedness of pseudodifferential operators on modulation spaces abstract we study classes of pseudodifferential operators which are bounded on large collections of modulation spaces the conditions on the operators are stated in terms of the l p q estimates for the continuous gabor transforms of their symbols in particular we show how these classes are related to the class of operators of grochenig and heil which is bounded on all modulation spaces

schatten classes for toeplitz operators with hilbert space windows on modulation spaces we prove that if 1 2 v 1 v 2 are appropriate j 1 2 and a in l p then the toeplitz operator tp h1 h2 a from m 2 omega 1 to m 2 omega 2 belongs to the schatten von neu

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

properly integral polynomials over the ring of integer valued polynomials on a matrix ring let d be a domain with fraction field k and let mn d be the ring of n n matrices with entries in d the ring of integer valued polynomials on the matrix ring mn d denoted intk mn d consists of those polynomials in k x that map matrices in mn d back to mn d under evaluation it has been known for some time that intq mn z is not integrally closed however it was only recently that an example of a polynomial in the integral closure of intq mn z but not in the ring itself appeared in the literature and the published example is specific to the case n 2 in this paper we give a construction that produces polynomials that are integral over intk mn d but are not in the ring itself where d is a dedekind domain with finite residue fields and n 2 is arbitrary we also show how our general example is related to p sequences for intk mn d and its integral closure in the case where d is a discrete valuation ring

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

mixed modulation spaces and their application to pseudodifferential operators abstract this paper uses frame techniques to characterize the schatten class properties of integral operators the main result shows that if the coefficients k m n of certain frame expansions of the kernel k of an integral operator are in l 2 p then the operator is schatten p class as a corollary we conclude that if the kernel or kohn nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space then the operator is schatten p class our corollary improves existing schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not schatten class

on the schatten von neumann properties of some pseudo differential operators abstract we obtain a number of explicit estimates for quasi norms of pseudo differential operators in the schatten von neumann classes s q with 0 q 1 the estimates are applied to derive semi classical bounds for operators with smooth or non smooth symbols

on accumulated spectrograms we study the eigenvalues and eigenfunctions of the time frequency localization operator h on a domain of the time frequency plane the eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain r indeed in analogy to the classical theory of landau slepianpollak the number of eigenvalues of h in 1 1 is equal to the measure of up to an error term depending on the perimeter of the boundary of our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues which we call the accumulated spectrogram form an approximate partition of unity of the given domain we derive asymptotic non asymptotic and weak l error estimates for the accumulated spectrogram as a consequence the domain can be approximated solely from the spectrograms of eigenfunctions without information about their phase

source enumeration for high resolution array processing using improved gerschgorin radii without eigendecomposition accurate detection of sources with low complexity is of considerable interest in practical applications of high resolution array processing this paper addresses a new computationally efficient method for source enumeration by using enhanced gerschgorin radii without eigendecompositofion the proposed method can calculate the gerschgorin radii in a more efficient manner in which the additive background noise can be efficiently suppressed and the computational complexity can be considerably reduced therefore the method is more accurate and computationally attractive furthermore the method does not rely on the eigenvalues of a covariance matrix or the signal noise power making it robust against deviations from the assumption of spatially white noise model numerical results are presented to demonstrate the performance of the method

the ring of integer valued polynomials on 2 2 matrices and its integral closure abstract let m n z denote the ring of n n matrices with integer entries and int q m n z q x the algebra of polynomials that preserve m n z i e polynomials for which f m m n z if m m n z the aim of this paper is to shed some light on this algebra by establishing two results the first general result is to show that when this algebra is localized at a prime p its integral closure can be identified with int q r n p the ring of polynomials in q x preserving the maximal order in a division algebra of dimension n 2 over q p the p adic numbers this is of interest computationally because one of the authors showed in an earlier paper that bhargava s method of p ordering can be extended to construct a regular basis for algebras of this type thus we have a method of computing an upper bound for the size of int q m n z our second result is a computational method for constructing a p local basis for int q m n z itself in the case n 2 in low degrees these results together give an estimate of the difference in size between int q m 2 z and its closure and in particular give constructions of polynomials in the latter but not in the former

maximum a posteriori maximum entropy order determination an instance crucial to most problems in signal processing is the selection of the order of a presupposed model examples are the determination of the putative number of signals present in white gaussian noise or the number of noise contaminated sources impinging on a passive sensor array it is shown that maximum a posteriori bayesian arguments coupled with maximum entropy considerations offer an operational and consistent model order selection scheme competitive with the minimum description length criterion

integer valued polynomials on algebras abstract let d be a domain with quotient field k and a a d algebra a polynomial with coefficients in k that maps every element of a to an element of a is called integer valued on a for commutative a we also consider integer valued polynomials in several variables for an arbitrary domain d and i an arbitrary ideal of d we show i adic continuity of integer valued polynomials on a for noetherian one dimensional d we determine spectrum and krull dimension of the ring int d a of integer valued polynomials on a we do the same for the ring of polynomials with coefficients in m n k the k algebra of n n matrices that map every matrix in m n d to a matrix in m n d

measurement of time variant linear channels the goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals in this paper it is shown that the spreading support of the operator that is the support of the symplectic fourier transform of the kohn nirenberg symbol of the operator has area less than one then the operator is identifiable if the spreading support is larger than one then the operator is not identifiable the shape of the support region is essentially arbitrary thereby proving a conjecture of bello the input signal considered is a weighted delta train where the weights are the window function of a finite gabor system whose elements satisfy a certain robust completeness property

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

robust multilinear tensor rank estimation using higher order singular value decomposition and information criteria model selection in tensor decomposition is important for real applications if the rank of the original data tensor is unknown and the observed tensor is noisy in the tucker model the minimum description length mdl or bayesian information criteria have been applied to tensors via matrix unfolding but these methods are sensitive to noise when the tensors have a multilinear low rank structure given by the tucker model in this study we propose new methods for improving the mdl so it is more robust to noise the proposed methods are justified theoretically by analyzing the multilinear low rank structure of tensors extensive experiments including numerical simulations and a real application to image denoising are provided to illustrate the advantages of the proposed methods

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

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

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

general asymptotic analysis of the generalized likelihood ratio test for a gaussian point source under statistical or spatial mismodeling this paper investigates the robustness of the generalized likelihood ratio test glrt for a far field gaussian point source given measurements from an array of sensors the performance of the glrt under two types of common modeling errors is investigated the first type is spatial mismodeling which relates to errors due to multipath effects or errors in the assumed number of sources i e deviation from the single point source assumption the second type is statistical mismodeling which relates to errors due to non gaussianity in either the noise or the signal i e deviation from the gaussian assumption it is shown that for some types of modeling errors the detector s performance improves and general conditions for such an improvement are found moreover for both types of errors the change in performance is analyzed and quantified this analysis shows that for a distributed source with small spatial spreading the degradation in performance is significant whereas for a constant modulus point source the performance improves simulations of various cases are shown to verify the analytical results

beurling type density theorems in the unit disk we consider two equivalent density concepts for the unit disk that provide a complete description of sampling and interpolation ina n the banach space of functionsf analytic in the unit disk with 1 z 2 n f z bounded this study reveals a nyquist density a sequence of points is roughly speaking a set of sampling if and only if its density in every part of the disk is strictly larger thann and it is a set of interpolation if and only if its density in every part of the disk is strictly smaller thann similar density theorems are also obtained for weighted bergman spaces

complex powers of hypoelliptic pseudodifferential operators abstract complex powers of a class of hypoelliptic pseudodifferential operators in r n as well as their heat kernels are studied an application to the schatten von neumann property of pseudodifferential operators is given

detection of signals by information theoretic criteria general asymptotic performance analysis detecting the number of sources is a well known and a well investigated problem in this problem the number of sources impinging on an array of sensors is to be estimated the common approach for solving this problem is to use an information theoretic criterion like the minimum description length mdl or the akaike information criterion aic although it has been gaining much popularity and has been used in a variety of problems the performance of information theoretic criteria based estimators for the unknown number of sources has not been sufficiently studied yet in the context of array processing the performance of such estimators were analyzed only for the special case of gaussian sources where no prior knowledge of the array structure if given is used based on the theory of misspecified models this paper presents a general asymptotic analysis of the performance of any information theoretic criterion based estimator and especially of the mdl estimator in particular the performance of the mdl estimator which assumes gaussian sources and structured array when applied to gaussian sources is analyzed in addition it is shown that the performance of a certain mdl estimator is not very sensitive to the actual distribution of the source signals however appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the mdl estimator simulation results show a good fit between the empirical and the theoretical results

detection of number of sources via exploitation of centro symmetry property forward backward fb averaging is a technique which can be employed in sensor array signal processing for exploiting a so called centro symmetry property of the signal subspace to improve parameter estimation accuracy although fb averaging effectively doubles the number of data samples it also introduces correlation among these data vectors this correlation complicates not only the performance analysis but also the detection of the number of sources when fb averaging is employed existing detection procedures such as sequential hypothesis sh likelihood ratio tests minimum description length mdl and an information criterion aic must be modified to correctly account for this preprocessing of the data herein proper exploitation of the centro symmetry property of the fb averaged covariance is described and a rigorous analysis is conducted it is shown that the general scheme of existing methods is still applicable to the fb averaging case but that the parameters e g degrees of freedom need to be adjusted

source number estimators using transformed gerschgorin radii we introduce effective uses of gerschgorin radii of the unitary transformed covariance matrix for source number estimation there are two approaches likelihood and heuristic used for developing the detection criteria the likelihood approach combines the gerschgorin radii to the well known source number detectors and improves their detection performances for gaussian and white noise processes it is verified that the gerschgorin likelihood estimators gle are consistent the gerschgorin aic yields a consistent estimate and the gerschgorin mdl criterion does not tend to underestimate for small or moderate data samples the heuristic approach applying the gerschgorin disk estimator gde developed from the projection concept overcomes the problem in cases of small data samples an unknown noise model and data dependency furthermore the detection performances of both approaches through the suggested rotations and averaging can be further improved finally the proposed and existing criteria are evaluated in various conditions by using simulated and measured experimental data

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics