resolution of a system of fuzzy polynomial equations using the grobner basis the occurrence of imprecision in the real world is inevitable due to some unexpected situations the imprecision is often involved in any engineering design process the imprecision and uncertainty are often interpreted as fuzziness fuzzy systems have an essential role in the uncertainty modelling which can formulate the uncertainty in the actual environment in this paper a new approach is proposed to solve a system of fuzzy polynomial equations based on the grobner basis in this approach first the h cut of a system of fuzzy polynomial equations is computed and a parametric form for the fuzzy system with respect to the parameter of h is obtained then a grobner basis is computed for the ideal generated by the h cuts of the system with respect to the lexicographical order using faugere s algorithm i e f 4 algorithm the grobner basis of the system has an upper triangular structure therefore the system can be solved using the forward substitution hence all the solutions of the system of fuzzy polynomial equations can easily be obtained finally the proposed approach is compared with the current numerical methods some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods

analytical blind channel identification a novel analytical blind single input single output siso identification algorithm is presented based on the noncircular second order statistics of the output it is shown that statistics of order higher than two are not mandatory to restore identifiability our approach is valid for instance when the channel is excited by phase shift keying psk inputs it is shown that the channel taps need to satisfy a polynomial system of degree 2 and that identification amounts to solving the system we describe the algorithm that is able to solve this particular system entirely analytically thus avoiding local minima computer results eventually show the robustness with respect to noise and to channel length overdetermination identifiability issues are also addressed

on the virtual array concept for higher order array processing for about two decades many fourth order fo array processing methods have been developed for both direction finding and blind identification of non gaussian signals one of the main interests in using fo cumulants only instead of second order so ones in array processing applications relies on the increase of both the effective aperture and the number of sensors of the considered array which eventually introduces the fo virtual array concept presented elsewhere and allows in particular a better resolution and the processing of more sources than sensors to still increase the resolution and the number of sources to be processed from a given array of sensors new families of blind identification source separation and direction finding methods at an order m 2q q spl ges 2 only have been developed recently in this context the purpose of this paper is to provide some important insights into the mechanisms and more particularly to both the resolution and the maximal processing capacity of numerous 2qth order array processing methods whose previous methods are part of by extending the virtual array concept to an arbitrary even order for several arrangements of the data statistics and for arrays with space angular and or polarization diversity

icar a tool for blind source separation using fourth order statistics only the problem of blind separation of overdetermined mixtures of sources that is with fewer sources than or as many sources as sensors is addressed in this paper a new method called independent component analysis using redundancies in the quadricovariance icar is proposed in order to process complex data this method without any whitening operation only exploits some redundancies of a particular quadricovariance matrix of the data computer simulations demonstrate that icar offers in general good results and even outperforms classical methods in several situations icar i succeeds in separating sources with low signal to noise ratios ii does not require sources with different second order or and first order spectral densities iii is asymptotically not affected by the presence of a gaussian noise with unknown spatial correlation iv is not sensitive to an over estimation of the number of sources

numerical solution of a system of fuzzy polynomials by fuzzy neural network in this paper a new approach for solving systems of fuzzy polynomials based on fuzzy neural network fnn is presented this method can also lead to improve numerical methods in this work an architecture of fuzzy neural networks is also proposed to find a real root of a system of fuzzy polynomials if exists by introducing a learning algorithm finally we illustrate our approach by numerical examples

inversion of polynomial systems and separation of nonlinear mixtures of finite alphabet sources in this contribution multiple input multiple output mimo mixing systems are considered which are instantaneous and nonlinear but polynomial we first address the problem of invertibility searching the inverse in the class of polynomial systems it is shown that grobner bases techniques offer an attractive solution for testing the existence of an exact inverse and computing it by noticing that any nonlinear mapping can be interpolated by a polynomial on a finite set we tackle the general nonlinear case relying on a finite alphabet assumption of the input source signals theoretical results on polynomials allow us to represent nonlinear systems as linear combinations of a finite set of monomials we then generalize the first results to give a condition for the existence of an exact nonlinear inverse the proposed method allows to compute this inverse in polynomial form in the light of the previous results we go further to the blind source separation problem it is shown that for sources in a finite alphabet the nonlinear problem is tightly connected with both problems of under determination and of dependent sources we concentrate on the case of two binary sources for which an easy solution can be found by simulation this solution is compared to techniques borrowed from classification methods

a generalization of joint diagonalization criteria for source separation in the field of blind source separation joint diagonalization based approaches constitute an important framework leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices jade and simultaneous third order tensor diagonalization stotd algorithms however they are often restricted to the case of cumulants of order four in this paper we extend the results leading to jade and stotd to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results this also leads us to generalize some links between contrast functions and joint diagonalization criteria on which these algorithms are based in turn one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast moreover for the two generalized jade and stotd contrasts the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression whatever the cumulant order finally some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint diagonalization of cumulant matrices

on nonlinear equations for fuzzy mappings in probabilistic normed spaces in this paper we introduce the concept of probabilistic contractor couple which simplifies and weakens that of probabilistic g contractor given by cho et al fuzzy sets and systems 110 2000 115 122 by using this concept we study the existence and iteration convergence of solutions for set valued nonlinear operator equations and nonlinear equations of fuzzy mappings in menger pn spaces the results presented in this paper improve and extend the corresponding results of cho et al in a sense

malykhin s problem abstract we construct a model of zfc where every separable frechet topological group is metrizable this solves a 1978 problem of v i malykhin

sequential blind identification of underdetermined mixtures using a novel deflation scheme in this brief we consider the problem of blind identification in underdetermined instantaneous mixture cases where there are more sources than sensors a new blind identification algorithm which estimates the mixing matrix in a sequential fashion is proposed by using the rank 1 detecting device blind identification is reformulated as a constrained optimization problem the identification of one column of the mixing matrix hence reduces to an optimization task for which an efficient iterative algorithm is proposed the identification of the other columns of the mixing matrix is then carried out by a generalized eigenvalue decomposition based deflation method the key merit of the proposed deflation method is that it does not suffer from error accumulation the proposed sequential blind identification algorithm provides more flexibility and better robustness than its simultaneous counterpart comparative simulation results demonstrate the superior performance of the proposed algorithm over the simultaneous blind identification algorithm

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

newton s method for solving a system of fuzzy nonlinear equations abstract in this paper we propose the numerical solution for a system of fuzzy nonlinear equations by newton s method the fuzzy quantities are presented in parametric form some numerical illustrations are given to show the efficiency of algorithm

parallel factor analysis in sensor array processing this paper links multiple invariance sensor array processing mi sap to parallel factor parafac analysis which is a tool rooted in psychometrics and chemometrics parafac is a common name for low rank decomposition of three and higher way arrays this link facilitates the derivation of powerful identifiability results for mi sap shows that the uniqueness of single and multiple invariance esprit stems from uniqueness of low rank decomposition of three way arrays and allows tapping on the available expertise for fitting the parafac model the results are applicable to both data domain and subspace mi sap formulations the paper also includes a constructive uniqueness proof for a special parafac model

ch with no ostaszewski spaces this is the published version also available here http dx doi org 10 1090 s0002 9947 99 02407 1 first published in trans amer math soc in 1999 published by the american mathematical society

blind identification and source separation in 2 spl times 3 under determined mixtures under determined mixtures are characterized by the fact that they have more inputs than outputs or with the antenna array processing terminology more sources than sensors the problem addressed is that of identifying and inverting the mixture which obviously does not admit a linear inverse identification is carried out with the help of tensor canonical decompositions on the other hand the discrete distribution of the sources is utilized for performing the source extraction the under determined mixture being either known or unknown the results presented in this paper are limited to two dimensional 2 d mixtures of three sources

on the behavior of current second and higher order blind source separation methods for cyclostationary sources most of the second order so and higher order ho blind source separation methods developed in the 1990s aim at blindly separating statistically independent sources that are assumed zero mean stationary and ergodic nevertheless in many situations of practical interest such as in radiocommunication contexts the sources are nonstationary and very often quasi cyclostationary digital modulations in these conditions it becomes important to wonder whether the performance of these current so and ho blind source separation methods which have been developed for stationary sources may be affected by the potential nonstationarity of the latter limiting the analysis to the so and fourth order fo cumulant based blind source separation methods the purpose of this paper is to bring some answers to this important question through the behavior analysis of the empirical so and fo cumulants estimator in the presence of zero mean quasi cyclostationary sources

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

fourth order blind identification of underdetermined mixtures of sources fobium for about two decades numerous methods have been developed to blindly identify overdetermined p spl les n mixtures of p statistically independent narrowband nb sources received by an array of n sensors these methods exploit the information contained in the second order so the fourth order fo or both the so and fo statistics of the data however in practical situations the probability of receiving more sources than sensors increases with the reception bandwidth and the use of blind identification bi methods able to process underdetermined mixtures of sources for which p n may be required although such methods have been developed over the past few years they all present serious limitations in practical situations related to the radiocommunications context for this reason the purpose of this paper is to propose a new attractive bi method exploiting the information contained in the fo data statistics only that is able to process underdetermined mixtures of sources without the main limitations of the existing methods provided that the sources have different trispectrum and nonzero kurtosis with the same sign a new performance criterion that is able to quantify the identification quality of a given source and allowing the quantitative comparison of two bi methods for each source is also proposed in the paper finally an application of the proposed method is presented through the introduction of a powerful direction finding method built from the blindly identified mixture matrix

a finite step global convergence algorithm for the parameter estimation of multichannel ma processes an iterative algorithm for the identification of multichannel moving average ma processes using higher order statistics is proposed it is shown that the algorithm has a finite step global convergence property three multichannel ma models including one nonminimum phase ma model are estimated by this algorithm with satisfactory performances

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

parafac based blind estimation of possibly underdetermined convolutive mimo systems in this paper we consider the problem of blind identification of a convolutive multiple input multiple output mimo system with no outputs and ni inputs while many methods have been proposed to blindly identify convolutive mimo systems with no ges ni overdetermined very scarce results exist for the case of underdetermined all of which refer to systems that either have some special structure or special and values in this paper we show that as long as independent of whether the system is overdetermined or underdetermined we can always find the appropriate order of statistics that guarantees identifiability of the system response within trivial ambiguities we also propose an algorithm to reach the solution that consists of parallel factorization parafac of a way tensor containing th order statistics of the system outputs followed by an iterative scheme for a certain order of statistics we provide the description of the class of identifiable mimo systems we also show that this class can be expanded by applying parafac decomposition to a pair of tensors instead of one tensor the proposed approach constitutes a novel scheme for estimation of underdetermined systems and improves over existing approaches for overdetermined systems

linear sigma additivity and some applications we show that countable increasing unions preserve a large family of well studied covering properties which are not necessarily additive using this together with infinite combinatorial methods and simple forcing theoretic methods we explain several phenomena settle problems of just miller scheepers and szeptycki 1996 gruenhage and szeptycki 2005 tsaban and zdomskyy 2008 and tsaban 2006 2007 and construct topological groups with very strong combinatorial properties

exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses amp minimum shift keying and offset qpsk are two well known modulations which can be interpreted as a set of time phase shifted am pulses we show in this paper that any constant amplitude binary phase modulation can also be expressed as a sum of a finite number of time limited amplitude modulated pulses amp decomposition new methods for computing autocorrelation and power frequency spectrum are derived which give very simple results for half integer index modulations we also show that the signal can be built with good accuracy using only one optimized pulse main pulse this synthesis is particularly satisfactory for modulations that have good spectral characteristics and or low index

blind parafac receivers for ds cdma systems this paper links the direct sequence code division multiple access ds cdma multiuser separation equalization detection problem to the parallel factor parafac model which is an analysis tool rooted in psychometrics and chemometrics exploiting this link it derives a deterministic blind parafac ds cdma receiver with performance close to non blind minimum mean squared error mmse the proposed parafac receiver capitalizes on code spatial and temporal diversity combining thereby supporting small sample sizes more users than sensors and or less spreading than users interestingly parafac does not require knowledge of spreading codes the specifics of multipath interchip interference doa calibration information finite alphabet constant modulus or statistical independence whiteness to recover the information bearing signals instead parafac relies on a fundamental result regarding the uniqueness of low rank three way array decomposition due to kruskal 1977 1988 and generalized herein to the complex valued case that guarantees identifiability of all relevant signals and propagation parameters these and other issues are also demonstrated in pertinent simulation experiments

fast algorithms for mutual information based independent component analysis this paper provides fast algorithms to perform independent component analysis based on the mutual information criterion the main ingredient is the binning technique and the use of cardinal splines which allows the fast computation of the density estimator over a regular grid using a discretized form of the entropy the criterion can be evaluated quickly together with its gradient which can be expressed in terms of the score functions both offline and online separation algorithms have been developed our density entropy and score estimators also have their own interest

concise formulae for the cumulant matrices of a random vector concise formulae are given for the cumulant matrices of a real valued zero mean random vector up to order 6 in addition to usual matrix operations they involve only the kronecker product the vec operator and the commutation matrix orders 5 and 6 are provided here for the first time the same method as provided in the paper can be applied to compute higher orders an immediate consequence of these formulae is to return 1 upper bounds on the rank of the cumulant matrices and 2 the expression of the sixth order moment matrix of a gaussian vector due to their conciseness the proposed formulae also have a computational advantage as compared to the repeated use of leonov and shiryaev formula

fourth order criteria for blind sources separation various criteria based on contrast functions as well as higher order statistics are used for solving the problem of blind separation of sources in the paper for the case of instantaneous mixtures of two sources it is proved that the minimization or the cancellation of a really simple criterion a fourth order cross cumulant leads to a set of solutions whose spurious ones can be simply cancelled by using a decorrelation

time frequency approach to underdetermined blind source separation this paper presents a new time frequency tf underdetermined blind source separation approach based on wigner ville distribution wvd and khatri rao product to separate n non stationary sources from m m n mixtures first an improved method is proposed for estimating the mixing matrix where the negative value of the auto wvd of the sources is fully considered then after extracting all the auto term tf points the auto wvd value of the sources at every auto term tf point can be found out exactly with the proposed approach no matter how many active sources there are as long as n 2m 1 further discussion about the extraction of auto term tf points is made and finally the numerical simulation results are presented to show the superiority of the proposed algorithm by comparing it with the existing ones

a study of the rank ambiguity issues in direction of arrival estimation we first extend a theorem on linear independence of steering vectors proposed by godara and cantoni to include more array sensor scenarios we then show that an array can have pairwise linearly independent steering vectors even when all its intersensor spacings are more than spl lambda 2 where spl lambda is the wavelength of the signals we next propose a theorem for characterizing rank 2 ambiguity which is applicable to direction of arrival estimation applications where the array sensor locations are fixed and known subsequently we identify a class of three sensor arrays and a class of uniform circular arrays that have pairwise linearly independent steering vectors and are free of rank 2 ambiguity we also show that collinearity of sensors uniformity in intersensor spacings the dimensions of intersensor spacings or a combination of some or all of these may give rise to linearly dependent steering vectors in particular we demonstrate that for a m sensor array m linearly dependent steering vectors exist if the aperture is greater than m 1 2 spl lambda 2 or when at least m 1 2 1 sensors are collinear

kruskal s permutation lemma and the identification of candecomp parafac and bilinear models with constant modulus constraints candecomp parafac cp analysis is an extension of low rank matrix decomposition to higher way arrays which are also referred to as tensors cp extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angle carrier estimation to blind multiuser detection the uniqueness of cp decomposition is not fully understood yet despite its theoretical and practical significance toward this end we first revisit kruskal s permutation lemma which is a cornerstone result in the area using an accessible basic linear algebra and induction approach the new proof highlights the nature and limits of the identification process we then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank these new conditions explain a curious example provided recently in a previous paper by sidiropoulos who showed that kruskal s condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the possibly pretransformed component matrices as another interesting application of the permutation lemma we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus cm constraints thus providing an interesting link to and unification with cp

a fast fixed point algorithm for independent component analysis we introduce a novel fast algorithm for independent component analysis which can be used for blind source separation and feature extraction we show how a neural network learning rule can be transformed into a fixedpoint iteration which provides an algorithm that is very simple does not depend on any user defined parameters and is fast to converge to the most accurate solution allowed by the data the algorithm finds one at a time all nongaussian independent components regardless of their probability distributions the computations can be performed in either batch mode or a semiadaptive manner the convergence of the algorithm is rigorously proved and the convergence speed is shown to be cubic some comparisons to gradient based algorithms are made showing that the new algorithm is usually 10 to 100 times faster sometimes giving the solution in just a few iterations

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

blind identification of fir systems excited by discrete alphabet inputs an algorithm for the identification of finite impulse response fir system parameters from output measurements for systems excited by discrete alphabet inputs is described the approach taken is algebraic it does not rely directly on the statistical properties of the measurements but rather it essentially solves the nonlinear equations appearing in the problem by converting them to equivalent linear equations using the discrete alphabet property of the input signal the proposed algorithm was tested by computer simulations and some of these simulations are illustrated

blind underdetermined mixture identification by joint canonical decomposition of ho cumulants a new family of cumulant based algorithms is proposed in order to blindly identify potentially underdetermined mixtures of statistically independent sources these algorithms perform a joint canonical decomposition cand of several higher order cumulants through a cand of a three way array with special symmetries these techniques are studied in terms of identifiability performance and numerical complexity from a signal processing viewpoint the proposed methods are shown i to have a better estimation resolution and ii to be able to process more sources than the other classical cumulant based techniques second from a numerical analysis viewpoint we deal with the convergence speed of several procedures for three way array decomposition such as the acdc scheme we also show how to accelerate the iterative cand algorithms by using differently the symmetries of the considered three way array next from a multilinear algebra viewpoint the paper aims at giving some insights on the uniqueness of a joint cand of several hermitian multiway arrays compared to the cand of only one array this allows us as a result to extend the concept of virtual array va to the case of combination of several vas

numerical solution of fuzzy polynomials by fuzzy neural network in this paper an architecture of fuzzy neural networks is proposed to find a real root of fuzzy polynomials if exists by introducing a learning algorithm we proposed a learning algorithm from the cost function for adjusting of crisp weights finally we illustrated our approach by numerical examples

applications of cumulants to array processing i aperture extension and array calibration an interpretation for the use of cumulants in narrowband array processing problems is proposed it is shown how fourth order cumulants of multichannel observations increase the directional information compared with second order statistics based on the interpretation it is shown how cumulants can be used to increase the effective aperture of an arbitrary antenna array the amount of partial information necessary to jointly calibrate an arbitrary array and estimate the directions of far field sources is also investigated it is proven that the presence of a doublet and use of fourth order cumulants is sufficient to accomplish this task the proposed approach is computationally efficient and more general than covariance based algorithms that have addressed the calibration problem under constraints a class of beamforming techniques is proposed to recover the source waveforms proposed estimation procedures are based on cumulants which bring insensitivity to the spatial correlation structure of additive gaussian measurement noise simulations are provided to illustrate the use of the proposed algorithms

blind identification of underdetermined mixtures by simultaneous matrix diagonalization in this paper we study simultaneous matrix diagonalization based techniques for the estimation of the mixing matrix in underdetermined independent component analysis ica this includes a generalization to underdetermined mixtures of the well known sobi algorithm the problem is reformulated in terms of the parallel factor decomposition parafac of a higher order tensor we present conditions under which the mixing matrix is unique and discuss several algorithms for its computation

blind identification of underdetermined mixtures based on the characteristic function the complex case blind identification of underdetermined mixtures can be addressed efficiently by using the second characteristic function caf of the observations our contribution is twofold first we propose the use of a levenberg marquardt algorithm herein called lemacaf as an alternative to an alternating least squares algorithm known as alescaf which has been used recently in the case of real mixtures of real sources second we extend the caf approach to the case of complex sources for which the previous algorithms are not suitable we show that the complex case involves an appropriate tensor stowage which is linked to a particular tensor decomposition an extension of the lemacaf algorithm called then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition in our simulation results we first provide performance comparisons between third and fourth order versions of alescaf and lemacaf in various situations involving bpsk sources then a performance study of is carried out considering 4 qam sources these results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level

neural network approach to solving fuzzy nonlinear equations using z numbers in this article the fuzzy property is described by means of the z number as the coefficients and variables of the fuzzy equations this alteration for the fuzzy equation is appropriate for system modeling with z number parameters in this article the fuzzy equation with z number coefficients and variables is tended to be used as the models for the uncertain systems the modeling issue related to the uncertain system is to obtain the z number coefficients and variables of the fuzzy equation nevertheless it is extremely hard to get the z number coefficients of the fuzzy equations in this article in order to model the uncertain nonlinear systems a novel structure of the multilayer neural network is utilized in such a manner that it is able to obtain the z number coefficients of the fuzzy equation the suggested technique is validated by some examples with applications

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

adaptive algorithms to track the parafac decomposition of a third order tensor the parafac decomposition of a higher order tensor is a powerful multilinear algebra tool that becomes more and more popular in a number of disciplines existing parafac algorithms are computationally demanding and operate in batch mode both serious drawbacks for on line applications when the data are serially acquired or the underlying model changes with time adaptive parafac algorithms that can track the sought decomposition at low complexity would be highly desirable this is a challenging task that has not been addressed in the literature and the topic of this paper given an estimate of the parafac decomposition of a tensor at instant t we propose two adaptive algorithms to update the decomposition at instant t 1 the new tensor being obtained from the old one after appending a new slice in the time dimension the proposed algorithms can yield estimation performance that is very close to that obtained via repeated application of state of art batch algorithms at orders of magnitude lower complexity the effectiveness of the proposed algorithms is illustrated using a mimo radar application tracking of directions of arrival and directions of departure as an example

sensor array signal processing via a procrustes rotations based eigenanalysis of the esprit data pencil the authors propose the procrustes rotation based esprit algorithm pro esprit for estimating characteristic parameters associated with a scene of radiating sources given data derived from two identical sensor arrays the x and y arrays which are displaced relative to one another pro esprit uses the inherent redundancy built into the esprit array system to compensate for deviations from the ideal array model this redundancy is manifested by two fundamental properties of the noiseless x and y data matrices comprising the esprit data pencil a they have the same row space the signal subspace and 2 they have the same column space the source subspace these properties are utilized by invoking two primary tools 1 the solution to the procrustes problem for optimally approximating an invariant subspace rotation and 2 the total least squares method for perturbing each of two estimates of a common subspace in a minimal fashion until the two perturbed spaces are the same

fourth order cumulant based blind identification of underdetermined mixtures in this paper we study two fourth order cumulant based techniques for the estimation of the mixing matrix in underdetermined independent component analysis the first method is based on a simultaneous matrix diagonalization the second is based on a simultaneous off diagonalization the number of sources that can be allowed is roughly quadratic in the number of observations for both methods explicit expressions for the maximum number of sources are given simulations illustrate the performance of the techniques

nonsymmetrical contrasts for sources separation in this paper the problem of the blind separation of independent sources is considered our approach relies on high order inverse criteria after generalizing the definition of classical contrast functions we exhibit a wide class of generally nonsymmetrical functions that will be called generalized contrasts and whose maximization is proved to be a sufficient condition for source separation we also establish a connection with cumulant matching showing an equivalence between the two approaches then in the case of two sources a statistical study of the estimated parameter based on one of these new contrasts is presented finally computer simulations illustrate the results and demonstrate all the interest we can find in considering a nonsymmetrical contrast

newton s method for solving fuzzy nonlinear equations in this paper we propose the numerical solution for a fuzzy nonlinear equation by newton s method the fuzzy quantities are presented in parametric form some numerical illustrations are given to show the efficiency of algorithms

analytical method for blind binary signal separation the blind separation of multiple co channel binary digital signals using an antenna array involves finding a factorization of a data matrix x into x as where all entries of s are 1 or 1 it is shown that this problem can be solved exactly and noniteratively via a certain generalized eigenvalue decomposition as indicated by simulations the algorithm is robust in the presence of noise an interesting implication is that certain cluster segmentation problems can be solved using eigenvalue techniques

cumulant based independence measures for linear mixtures this paper deals with independence measures for linear mixtures of mutually independent random variables such measures also known as contrasts constitute useful criteria in solving blind source separation problems by making use of the schur convexity properties we show that it is possible to define a wide ranging class of contrast functions based on the auto cumulants of the components of the random vector being considered among the most appealing characteristics of these new contrast functions is that they can be used to combine cumulants of different orders in a flexible way furthermore extensions of existing cross cumulant base contrasts are proposed finally some particularization of our approach to measures of decorrelation is considered a general characterization of these decorrelation measures using strictly schur convex functions is provided

positive definite combination of symmetric matrices this correspondence presents a new algorithm to find a positive definite combination of some given real symmetric matrices when this problem has solutions we show that it can be written as an optimization problem an iterative algorithm using a relaxation type method is proposed to solve this optimization problem we also prove that this algorithm has a finite step global convergence property a comparison by computer simulations shows that our algorithm is much faster than another one generally proposed in literature to find such a combination

maximum likelihood estimation of constellation vectors for blind separation of co channel bpsk signals and its performance analysis in this paper we present a method for blind separation of co channel bpsk signals arriving at an antenna array this method consists of two parts the maximum likelihood constellation estimation and assignment we show that at high snr the maximum likelihood constellation estimation is well approximated by the smallest distance clustering algorithm which we proposed earlier on heuristic grounds we observe that both these methods for estimating the constellation vectors perform very well at high snr and nearly attain cramer rao bounds using this fact and noting that the assignment algorithm causes negligible error at high snr we derive upper bounds on the probability of bit error for the above method at high snr these upper bounds fall very rapidly with increasing snr showing that our constellation estimation assignment approach is very efficient simulation results are given to demonstrate the usefulness of the bounds

partitioning pairs of countable ordinals on montre que les paires d ordinaux denombrables peuvent etre colorees avec une infinite non denombrable de couleurs de telle sorte que tout ensemble non denombrable contienne des paires de chaque couleur

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

simultaneous schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems this paper presents a new jacobi type method to calculate a simultaneous schur decomposition ssd of several real valued nonsymmetric matrices by minimizing an appropriate cost function thereby the ssd reveals the average eigenstructure of these nonsymmetric matrices this enables an r dimensional extension of unitary esprit to estimate several undamped r dimensional modes or frequencies along with their correct pairing in multidimensional harmonic retrieval problems unitary esprit is an esprit type high resolution frequency estimation technique that is formulated in terms of real valued computations throughout for each of the r dimensions the corresponding frequency estimates are obtained from the real eigenvalues of a real valued matrix the ssd jointly estimates the eigenvalues of all r matrices and thereby achieves automatic pairing of the estimated r dimensional modes via a closed form procedure that neither requires any search nor any other heuristic pairing strategy moreover we describe how r dimensional harmonic retrieval problems with r spl ges 3 occur in array signal processing and model based object recognition applications

smoothed analysis for tensor methods in unsupervised learning smoothed analysis is a powerful paradigm in overcoming worst case intractability in high dimensional data analysis and unsupervised learning while polynomial time smoothed analysis guarantees have been obtained for worst case intractable problems like tensor decomposition and learning mixtures of gaussians such guarantees have been hard to obtain for several other important problems in data analysis a core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries that are given by low degree polynomials of a few base underlying random variables in this work we address this challenge by obtaining high confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind we then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in high dimensional data analysis

source separation when the input sources are discrete or have constant modulus in this paper we present a new method for the source separation problem when some prior information on the input sources is available more specifically we study the situation where the distributions of the input signals are discrete or are concentrated on a circle the method is based on easy properties of hankel forms and on the divisibility of gaussian distributions in both situations we prove that the estimator converges in absence of noise or if we know the first moments of the noise up to its scale moreover in the absence of noise the estimate converges with a finite number of observations

line search and trust region strategies for canonical decomposition of semi nonnegative semi symmetric 3rd order tensors numerical solutions are proposed to fit the candecomp parafac cp model of real three way arrays when the latter are both nonnegative and symmetric in two modes in other words a semi nonnegative indscal analysis is performed the nonnegativity constraint is circumvented by means of changes of variable into squares leading to an unconstrained problem in addition two globalization strategies are studied namely line search and trust region regarding the former a global plane search scheme is considered it consists in computing for a given direction one or two optimal stepsizes depending on whether the same stepsize is used in various updating rules moreover we provide a compact matrix form for the derivatives of the objective func tion this allows for a direct implementation of several iterative algorithms such as conjugate gradient levenberg marquardt and newton like methods in matrix programming environments like matlab our numerical results show the advantage of our optimization strategies when combined with a priori information such as partial symmetry

multiple invariance esprit a subspace fitting formulation of the esprit problem is presented that provides a framework for extending the algorithm to exploit arrays with multiple invariances in particular a multiple invariance mi esprit algorithm is developed and the asymptotic distribution of the estimates is obtained simulations are conducted to verify the analysis and to compare the performance of mi esprit with that of several other approaches the excellent quality of the mi esprit estimates is explained by recent results which state that under certain conditions subspace fitting methods of this type are asymptotically efficient

confac decomposition approach to blind identification of underdetermined mixtures based on generating function derivatives this work proposes a new tensor based approach to solve the problem of blind identification of underdetermined mixtures of complex valued sources exploiting the cumulant generating function cgf of the observations we show that a collection of second order derivatives of the cgf of the observations can be stored in a third order tensor following a constrained factor confac decomposition with known constrained structure in order to increase the diversity we combine three derivative types into an extended third order confac decomposition a detailed uniqueness study of this decomposition is provided from which easy to check sufficient conditions ensuring the essential uniqueness of the mixing matrix are obtained from an algorithmic viewpoint we develop a confac based enhanced line search confac els method to be used with an alternating least squares estimation procedure for accelerated convergence and also analyze the numerical complexities of two confac based algorithms namely confac als and confac els in comparison with the levenberg marquardt lm based algorithm recently derived to solve the same problem simulation results compare the proposed approach with some higher order methods our results also corroborate the advantages of the confac based approach over the competing lm based approach in terms of performance and computational complexity

an approach to h control of fuzzy dynamic systems this paper presents an approach to h control of a class of continuous time nonlinear systems which can be represented by a fuzzy dynamic model a suitable piecewise differentiable quadratic lyapunov function is used to establish asymptotic stability and h performance of the closed loop fuzzy control system a constructive algorithm is developed to obtain the h feedback control law which involves solving a set of suitable algebraic riccati equations an example is given to illustrate the application of the method

differential fast fixed point algorithms for underdetermined instantaneous and convolutive partial blind source separation this paper concerns underdetermined linear instantaneous and convolutive blind source separation bss i e the case when the number of observed mixed signals is lower than the number of sources we propose partial bss methods which separate supposedly nonstationary sources of interest while keeping residual components for the other supposedly stationary ldquonoiserdquo sources these methods are based on the general differential bss concept that we introduced before in the instantaneous case the approach proposed in this paper consists of a differential extension of the fastica method which does not apply to underdetermined mixtures in the convolutive case we extend our recent time domain fast fixed point c fica algorithm to underdetermined mixtures both proposed approaches thus keep the attractive features of the fastica and c fica methods our approaches are based on differential sphering processes followed by the optimization of the differential nonnormalized kurtosis that we introduce in this paper experimental tests show that these differential algorithms are much more robust to noise sources than the standard fastica and c fica algorithms

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

performance analysis of the total least squares esprit algorithm the asymptotic distribution of the estimation error for the total least squares tls version of esprit is derived the application to a uniform linear array is treated in some detail and a generalization of esprit to include row weighting is discussed the cramer rao bound crb for the esprit problem formulation is derived and found to coincide with the asymptotic variance of the tls esprit estimates through numerical examples a comparison of this method to least squares esprit music and root music as well as to the crb for a calibrated array is also presented tls esprit is found to be competitive with the other methods and the performance is close to the calibrated crb for many cases of practical interest for highly correlated signals however the performance deviates significantly from the calibrated crb simulations are included to illustrate the applicability of the theoretical results to a finite number of data

detection and estimation in sensor arrays using weighted subspace fitting the problem of signal parameter estimation of narrowband emitter signals impinging on an array of sensors is addressed a multidimensional estimation procedure that applies to arbitrary array structures and signal correlation is proposed the method is based on the recently introduced weighted subspace fitting wsf criterion and includes schemes for both detecting the number of sources and estimating the signal parameters a gauss newton type method is presented for solving the multidimensional wsf and maximum likelihood optimization problems the global and local properties of the search procedure are investigated through computer simulations most methods require knowledge of the number of coherent noncoherent signals present a scheme for consistently estimating this is proposed based on an asymptotic analysis of the wsf cost function the performance of the detection scheme is also investigated through simulations

algebraic joint zero diagonalization and blind sources separation this paper adresses the problem of the joint zero diagonalization of a given set of matrices we establish the identiflability conditions of the zero diagonalizer and we propose a new algebraical algorithm based on the reformulation of the initial problem into a joint diagonalization problem the zero diagonalizer is not constrained to be unitary computer simulations illustrate the behavior of the algorithm moreover as an application we show that the blind separation of correlated sources can be performed applying this algorithm to a particular set of spatial quadratic time frequency distribution matrices in this case computer simulations are also provided in order to illustrate the performances of the proposed algorithm and to compare it with other existing ones

high order contrasts for independent component analysis this article considers high order measures of independence for the independent component analysis problem and discusses the class of jacobi algorithms for their optimization several implementations are discussed we compare the proposed approaches with gradient based techniques from the algorithmic point of view and also on a set of biomedical data

katetov s problem in 1948 miroslav katetov showed that if the cube x 3 of a compact space x satisfies the separation axiom t 5 then x must be metrizable he asked whether x 3 can be replaced by x 2 in this metrization result in this note we prove the consistency of this implication

general approach to blind source separation this paper identifies and studies two major issues in the blind source separation problem separability and separation principles we show that separability is an intrinsic property of the measured signals and can be described by the concept of m row decomposability introduced in this paper we also show that separation principles can be developed by using the structure characterization theory of random variables in particular we show that these principles can be derived concisely and intuitively by applying the darmois skitovich theorem which is well known in statistical inference theory and psychology some new insights are gained for designing blind source separation filters

complex random vectors and ica models identifiability uniqueness and separability in this paper the conditions for identifiability separability and uniqueness of linear complex valued independent component analysis ica models are established these results extend the well known conditions for solving real valued ica problems to complex valued models relevant properties of complex random vectors are described in order to extend the darmois skitovich theorem for complex valued models this theorem is used to construct a proof of a theorem for each of the above ica model concepts both circular and noncircular complex random vectors are covered examples clarifying the above concepts are presented

consistent noisy independent component analysis we study linear factor models under the assumptions that factors are mutually independent and independent of errors and errors can be correlated to some extent under the factor non gaussianity second to fourth order moments are shown to yield full identification of the matrix of factor loadings we develop a simple algorithm to estimate the matrix of factor loadings from these moments we run monte carlo simulations and apply our methodology to data on cognitive test scores and financial data on stock returns

blind separation of synchronous co channel digital signals using an antenna array i algorithms we propose a maximum likelihood ml approach for separating and estimating multiple synchronous digital signals arriving at an antenna array at a cell site the spatial response of the array is assumed to be known imprecisely or unknown we exploit the finite alphabet property of digital signals to simultaneously estimate the array response and the symbol sequence for each signal uniqueness of the estimates is established for bpsk signals we introduce a signal detection technique based on the finite alphabet property that is different from a standard linear combiner computationally efficient algorithms for both block and recursive estimation of the signals are presented this new approach is applicable to an unknown array geometry and propagation environment which is particularly useful in wireless communication systems simulation results demonstrate its promising performance

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

non orthogonal joint diagonalization in the least squares sense with application in blind source separation approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation bss algorithms a common measure of the attained diagonalization of the set is the weighted least squares wls criterion however most well known algorithms are restricted to finding an orthogonal diagonalizing matrix relying on a whitening phase for the nonorthogonal factor often such an approach implies unbalanced weighting which can result in degraded performance we propose an iterative alternating directions algorithm for minimizing the wls criterion with respect to a general not necessarily orthogonal diagonalizing matrix under some mild assumptions we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations we distinguish between hermitian and symmetrical problems using bss simulations results we demonstrate the improvement in estimating the mixing matrix resulting from the relaxation of the orthogonality restriction

blind identification of overcomplete mixtures of sources biome the problem of blind identification of linear mixtures of independent random processes is known to be related to the diagonalization of some tensors this problem is posed here in terms of a non conventional joint approximate diagonalization of several matrices in fact a congruent transform is applied to each of these matrices the left transform being rectangular full rank and the right one being unitary the application in antenna signal processing is described and suboptimal numerical algorithms are proposed