estimation of amplitude and phase parameters of multicomponent signals this paper considers the problem of estimating signals consisting of one or more components of the form a t e sup j spl phi t where the amplitude and phase functions are represented by a linear parametric model the cramer rao bound crb on the accuracy of estimating the phase and amplitude parameters is derived by analyzing the crb for the single component case if is shown that the estimation of the amplitude and the phase are decoupled numerical evaluation of the crb provides further insight into the dependence of estimation accuracy on signal to noise ratio snr and the frequency separation of the signal components a maximum likelihood algorithm for estimating the phase and amplitude parameters is also presented its performance is illustrated by monte carlo simulations and its statistical efficiency is verified

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

estimation and statistical analysis of exponential polynomial signals we derive and analyze a suboptimal algorithm for estimating the unknown parameters of an exponential polynomial signal eps model the eps model represents a signal as an exponent raised to a complex polynomial of time that is the phase and the logarithm of the amplitude are approximated as finite order polynomials of time

analysis of polynomial phase signals by the integrated generalized ambiguity function the aim of this work is the performance analysis of a method for the detection and parameter estimation of mono or multicomponent polynomial phase signals pps embedded in white gaussian noise and based on a generalized ambiguity function the proposed method is shown to be asymptotically efficient for second order pps and nearly asymptotically efficient for third order ppss the method presents some advantages with respect to similar techniques like the polynomial phase transform for example in terms of i a closer approach to the cramer rao lower bounds ii a lower snr threshold iii a better capability of discriminating multicomponent signals

performance bounds for polynomial phase parameter estimation with nonuniform and random sampling schemes estimating the parameters of a cisoid with an unknown amplitude and polynomial phase using uniformly spaced samples can result in ambiguous estimates due to nyquist sampling limitations it has been shown previously that nonuniform sampling has the advantage of unambiguous estimates beyond the nyquist frequency however the effect of sampling on the cramer rao bounds is not well known this paper first derives the maximum likelihood estimators and cramer rao bounds for the parameters with known arbitrary sampling times it then outlines two methods for incorporating random sampling times into the lower variance bounds describing one in detail it is then shown that for a signal with additive white gaussian noise the bounds for the estimation with nonuniform sampling tend toward those of uniform sampling thus nonuniform sampling overcomes the ambiguity problems of uniform sampling without incurring the penalty of an increased variance in parameter estimation

fundamental initial frequency and frequency rate estimation of random amplitude harmonic chirps we consider the problem of estimating the fundamental initial frequency and frequency rate of a linear chirp with random amplitudes harmonic components we develop an iterative nonlinear least squares estimator which involves a large number of computations as it requires high resolution search in the initial frequency and frequency rate parameter space as an alternative we suggest two suboptimal low complexity estimators the first is based on the high order ambiguity function which reduces the problem to a one dimensional search the second method applies our recently published harmonic separate estimate method which was used for constant amplitude harmonic chirps we present modifications of both methods for harmonic chirps with random amplitudes we also provide a framework for estimating the number of harmonic components numerical simulations show that the iterative nonlinear least squares estimator achieves its asymptotic accuracy in medium to high signal to noise ratio while the two sub optimal low complexity estimators perform well in high signal to noise ratio real data examples demonstrate the performance of the harmonic separate estimate method on random amplitude real life signals

a manifold independent approach to understanding transport in stochastic dynamical systems abstract we develop a new collection of tools aimed at studying stochastically perturbed dynamical systems specifically in the setting of bi stability that is a two attractor system it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero noise case barriers in the phase space pseudo barriers thus creating a pre heteroclinic tangency chaos like behavior the stochastic dynamical system has a corresponding frobenius perron operator with a stochastic kernel which describes how densities of initial conditions move under the noisy map thus in studying the action of the frobenius perron operator we learn about the transport of the map we have employed a galerkin ulam like method to project the frobenius perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space graph theoretic methods allow us to re order the resulting finite dimensional markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo barriers of the stochastic dynamics our toolbox allows us to find 1 regions of high activity of transport 2 flux across pseudo barriers and also 3 expected time of escape from pseudo basins some of these quantities are also possible via the manifold dependent stochastic melnikov method but melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available our methods are unique in that they can essentially be considered as a black box of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures we use here a model of childhood diseases to showcase our methods our tools will allow us to make specific observations of 1 loss of reducibility between basins with increasing noise 2 identification in the phase space of active regions of stochastic transport 3 stochastic flux which essentially completes the heteroclinic tangle

modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps abstract we model a two dimensional open fluid flow that has temporally irregular time dependence by a random map n 1 m n n where on each iterate n the map m n is chosen from an ensemble we show that a tracer distribution advected through a chaotic region can be entrained on a set that becomes fractal as time increases theoretical and numerical results on the multifractal dimension spectrum are presented

new smoothed time frequency rate representations for suppressing cross terms in this paper to suppress the cross terms in a time frequency rate representation tfrr we propose four kinds of smoothed tfrrs stfrrs based on high resolution tfrr hr tfrr and cubic phase function we first define two types of ambiguity functions of tfrr af tfrr and give a relationship between af tfrrs and tfrrs then a mask is devised to extract the auto terms and suppress the cross terms in af tfrr domain finally four stfrrs are obtained based on the masked af tfrrs stfrrs are suitable for a signal composed of multiple linear frequency modulated components or cubic phase components with small third order parameters to analyze a signal composed of multiple cubic phase components with large third order parameters we propose smoothed time chirp rate representations stcrrs in the same way as stfrrs numerical examples demonstrate that stfrrs and stcrrs can suppress the cross terms effectively

estimating parameters of multiple wideband polynomial phase sources in sensor arrays we consider the problem of estimating the parameters of multiple wideband polynomial phase signal pps sources in sensor arrays a new maximum likelihood ml direction of arrival doa estimator is introduced and the exact cramer rao bound crb is derived for the general case of multiple constant amplitude polynomial phase sources since the proposed exact ml estimator is computationally intensive an approximate solution is proposed originating from the analysis of the log likelihood ll function in the single chirp signal case as a result a new form of spatio temporal matched filter referred to as the chirp beamformer is derived which is applicable to well separated sources that have distinct time frequency or and spatial signatures this beamforming approach requires solving a three dimensional 3 d optimization problem and therefore enjoys essentially simpler implementation than that entailed by the exact ml simulation results are presented illustrating the performance of the estimators and validating our theoretical crb analysis

stochastic climate dynamics random attractors and time dependent invariant measures a b s t r a c t this article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s the nonlinear deterministic and the linear stochastic one this unification via the theory of random dynamical systems rds allows one to consider the detailed geometric structure of the random attractors associated with nonlinear stochastically perturbed systems we report on high resolution numerical studies of two idealized models of fundamental interest for climate dynamics the first of the two is a stochastically forced version of the classical lorenz model the second one is a low dimensional nonlinear stochastic model of the el nino southern oscillation enso these studies provide a good approximation of the two models global random attractors as well as of the time dependent invariant measures supported by these attractors the latter are shown to have an intuitive physical interpretation as random versions of sinai ruelle bowen srb measures

characteristic spaces and rigidity for analytic hilbert modules abstract it is well known that a polynomial in one variable is completely determined by its zeros counting multiplicities we generalize this result to an ideal of polynomials in several variables by introducing the characteristic spaces of the ideal one finds that the ideal is completely determined by its characteristic spaces on a characteristic set in particular a primary ideal is completely determined by its characteristic space at any zero point some straightforward applications of the above results yield the algebraic reduction theorem for analytic hilbert modules in several variables also we obtain some general rigidity results for analytic hilbert modules by using the techniques of af envelopes of analytic hilbert modules

dynamic range of the detectable parameters for polynomial phase signals using multiple lag diversities in high order ambiguity functions two lag diversities in the high order ambiguity functions for single component polynomial phase signals pps was explored by zhou and wang see ieee signal processing lett vol 4 p 240 42 1997 and signal processing vol 65 no 2 p 1452 55 1998 the lag diversity enlarges the dynamic range of the detectable parameters for pps in this paper we first find a connection between the above multiple lag diversity problem and the multiple undersampling problem in the frequency detection using discrete fourier transform dft using the connection and some results on the multiple undersampling problem we recently obtained we prove that the dynamic range obtained by zhou and wang is already the maximal one for the detectable parameters for single component pps furthermore the dynamic range for the detectable parameters for multicomponent pps is given when multiple lag diversities are used we show that the maximal dynamic range is reached when the number of the lags in the high order ambiguity function haf is at least twice of the number of the single components in a multicomponent pps more lags than twice the number of single components do not increase the dynamic range

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

bi instability and the global role of unstable resonant orbits in a driven laser driven class b lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior we describe the onset of global heteroclinic connections which give rise to chaotic saddles these form the precursor topology which creates both localized homoclinic chaos as well as global mixed mode heteroclinic chaos to locate the relevant periodic orbits creating the precursor topology approximate maps are derived using matched asymptotic expansions and subharmonic melnikov theory locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser 2000 published by elsevier science b v

nonparametric estimation of instantaneous frequency the local polynomial approximation of time varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex valued harmonic signal given by discrete time observations with a noise the considered estimators are high order nonparametric generalizations of the short time fourier transform and the wigner ville distribution the asymptotic variance and bias of the estimates are obtained

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

bootstrap modeling of a class of nonstationary signals the problem of modeling polynomial phase signals is considered techniques based on the bootstrap and multiple hypotheses tests for optimal model selection of constant amplitude polynomial phase signals embedded in stationary noise are developed phase parameter estimators based on both the least squares method and the polynomial phase transform are used the proposed techniques are compared with existing ones including akaike s 1978 1974 information criterion and rissanen s 1978 minimum description length criterion and are shown to outperform these procedures for the considered small sample sizes

parameter estimation for locally linear fm signals using a time frequency hough transform an estimator for the phase parameters of mono and multicomponent fm signals with both good numerical properties and statistical performance is proposed the proposed approach is based on the hough transform of the pseudo wigner ville time frequency distribution pwvd it is shown that the numerical properties of the estimator can be improved by varying the pwvd window length the effect of the window time extent on the statistical performance of the estimator is delineated experimental data is used for validation of the statistical properties

estimation of chirp radar signals in compound gaussian clutter a cyclostationary approach signal detection of known within a complex scaling rank one waveforms in non gaussian distributed clutter has received considerable attention we expand on published solutions to consider the case of rank one waveforms that have some unknown parameters i e signal amplitude initial phase doppler shift and doppler rate of change the contribution of this paper is the derivation and performance analysis of two joint estimators of doppler shift and doppler rate the chirp embedded in correlated compound gaussian clutter one solution is based on the maximum likelihood ml principle and the other one on target signal second order cyclostationarity the hybrid cramer rao lower bounds hcrlbs and a large sample closed form expression for the mean square estimation error only for the doppler shift are also derived numerical examples are provided to show the behavior of the proposed estimator under different non gaussian clutter scenarios

analysis and synthesis of multicomponent signals using positive time frequency distributions a new approach to the analysis and reconstruction of multicomponent nonstationary signals from their time frequency distribution tfd is presented specifically we consider a tfd based on the recently introduced minimum cross entropy principle mce this positive tfd is cross terms free and hence has an advantage over the family of bilinear distributions based on the mce tfd a new algorithm for reconstructing the phase and amplitude parameters of each component of the signal is developed to evaluate the accuracy of the algorithm monte carlo simulations are presented and compared with the corresponding cramer rao bound it is shown that the new algorithm is superior to presently available methods in both efficiency and performance it is concluded that together with the mce tfd representation the proposed approach provides a powerful tool for analysis of nonstationary multicomponent signals embedded in additive gaussian noise

on the dirichlet space for finitely connected regions this paper is devoted to the study of the dirichlet space dir g for finitely connected regions g we are particularly interested in the algebra of bounded multiplication operators on this space results in different directions are obtained one direction deals with the structure of closed subspaces invariant under all bounded multiplication operators in particular we show that each such subspace contains a bounded function for regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by z must be invariant under all bounded multiplication operators and furthermore it is of the form p dir g where p is a polynomial with all its roots lying in g another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators typical results are if f e dir g and f is bounded away from zero then f is cyclic on the other hand if the zero set of the radial limit function of f on the boundary has positive logarithmic capacity then f is not cyclic also some other sufficient conditions for a function to be cyclic are given lastly we study transitive operator algebras containing all bounded multiplication operators we prove that they are dense in the algebra of all bounded operators in the strong operator topology

characterization of nonstationary chaotic systems nonstationary dynamical systems arise in applications but little has been done in terms of the characterization of such systems as most standard notions in nonlinear dynamics such as the lyapunov exponents and fractal dimensions are developed for stationary dynamical systems we propose a framework to characterize nonstationary dynamical systems a natural way is to generate and examine ensemble snapshots using a large number of trajectories which are capable of revealing the underlying fractal properties of the system by defining the lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots we show that the kaplan yorke formula which is fundamental in nonlinear dynamics remains valid most of the time even for nonstationary dynamical systems

analytic hilbert spaces over the complex plane in this paper we study the structure of quasi invariant subspaces of analytic hilbert spaces over the complex plane we especially investigate when two quasi invariant subspaces are similar or unitarily equivalent for an analytic hilbert space over the complex plane

efficient estimation of a narrow band polynomial phase signal impinging on a sensor array the parameters of interest of a polynomial phase signal observed by a sensor array include the direction of arrival and the polynomial coefficients the direct maximum likelihood estimation of these parameters requires a nonlinear multidimensional search in this paper we present a two step estimation approach the estimation requires only a one dimensional search in the direction of arrival space and involves a simple least squares solution for the polynomial coefficients the efficiency of the estimates is corroborated by monte carlo simulations

estimation and classification of polynomial phase signals the measurement of the parameters of complex signals with constant amplitude and polynomial phase measured in additive noise is considered a novel new integral transform that is adapted for signals of this type is introduced this transform is used to derive estimation and classification algorithms that are simple to implement and that exhibit good performance the algorithms are extended to constant amplitude and continuous nonpolynomial phase signals

escape from attracting sets in randomly perturbed systems the dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science previous studies of escape in chaotic systems have rather focused on the case of unbounded noise usually assumed to have gaussian distribution in this paper we address the problem of escape induced by bounded noise we show that the dynamics of escape from an attractor s basin is equivalent to that of a closed system with an appropriately chosen hole using this equivalence we show that there is a minimum noise amplitude above which escape takes place and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition we verify our analytical predictions through numerical simulations of two well known two dimensional maps with noise

driving a conceptual model climate by different processes snapshot attractors and extreme events in a low order chaotic model of global atmospheric circulation the effects of driving i e time dependent periodic chaotic and noisy forcing are investigated with particular interest in extremal behavior an approach based on snapshot attractors formed by a trajectory ensemble is applied to represent the time dependent likelihood of extreme events in terms of a physical observable a single trajectory based framework on the other hand is used to determine the maximal value and the kurtosis of the distribution of the same observable we find the most significant effect of the driving on the magnitude relative frequency and variability of extreme events when its characteristic time scale becomes comparable to that of the model climate extreme value statistics is pursued by the method of block maxima and found to follow weibull distributions deterministic drivings result in shape parameters larger in modulus than stochastic drivings but otherwise strongly dependent on the particular type of driving the maximal effects of deterministic drivings are found to be more pronounced both in magnitude and variability of the extremes than white noise and the latter has a stronger effect than red noise

a high resolution time frequency rate representation and the cross term suppression in this paper we propose a high resolution time frequency rate representation hr tfrr and derive its analytical formula this formula implies that the hr tfrr has a narrower frequency rate support than the cubic phase function cpf therefore the hr tfrr has a higher frequency rate fr resolution than the cpf and can distinguish closer components in the time frequency rate domain due to the bilinear transform the hr tfrr suffers the cross term when the instantaneous frequency functions of the components are cross or very close in order to suppress the cross term we further propose a smoothed hr tfrr shr tfrr by introducing an fr window to the hr tfrr which is expressed in the convolution form finally numerical examples demonstrate that the hr tfrr and the shr tfrr outperform their counterparts in frequency rate resolution and cross term suppression respectively in addition satisfactory performance of the proposed methods in phase parameter estimation and real world signal analysis is given

polynomial phase signal analysis based on the polynomial derivatives decompositions an exact decomposition of the derivatives of any order of a polynomial spl phi t is proposed in terms of spl phi t t sub 0 spl phi t t sub n this result allows us to introduce generalized time frequency distributions for studying signals having a polynomial phase and a constant amplitude in order to determine the degree and the coefficients of the corresponding phase the relationships between these distributions and the already known polynomial distributions i e the polynomial phase transform and the polynomial wigner ville distribution are discussed illustrations by example are proposed

invariant subspaces quasi invariant subspaces and hankel operators abstract in this paper using the theory of hilbert modules we study invariant subspaces of the bergman spaces on bounded symmetric domains and quasi invariant subspaces of the segal bargmann spaces we completely characterize small hankel operators with finite rank on these spaces

parameter estimation of phase modulated signals using bayesian unwrapping parametric estimation of phase modulated signals pms in additive white gaussian noise is considered the prohibitive computational expense of maximum likelihood estimation for this problem has led to the development of many suboptimal estimators which are relatively inaccurate and cannot operate at low signal to noise ratios snrs in this paper a novel technique based on a probabilistic unwrapping of the phase of the observations is developed the method is capable of more accurate estimation and operates effectively at much lower snrs than existing algorithms this is demonstrated in monte carlo simulations

fractal snapshot components in chaos induced by strong noise in systems exhibiting transient chaos in coexistence with periodic attractors the inclusion of weak noise might give rise to noise induced chaotic attractors when the noise amplitude exceeds a critical value an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set a further increase of noise leads to a fuzzy nonfractal pattern by means of the concept of snapshot attractors and random maps we point out that the fuzzy pattern can be decomposed into well defined fractal components the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories the pattern of the snapshot attractor and its characteristic numbers such as the finite time lyapunov exponents and numerically evaluated fractal dimensions change continuously in time we find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size the validity of the kaplan yorke formula is also investigated a superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise induced attractor at the same noise strength

identifiability and aliasing in polynomial phase signals polynomial phase signals have numerous applications including radar sonar geophysics and radio communication many techniques for estimating the parameters of polynomial phase signals have been described in the literature despite the significant interest aliasing of polynomial phase parameters has not been fully clarified we address the problem of identifiability and aliasing in polynomial phase signals we fully describe the region in which aliasing does not occur for polynomial phase signals of any order we call this the identifiable region we find that this region is the voronoi region of a lattice generated by the coefficients of a set of polynomials known as the integer valued polynomials we show how aliasing can be resolved by solving the nearest lattice point problem we discuss some of the consequences of these results on a popular estimator for polynomial phase signals that is based on the discrete polynomial phase transform dppt it is shown that the range of parameters suitable for the dppt estimator is very small compared to the identifiable region

discrete chirp fourier transform and its application to chirp rate estimation the discrete fourier transform dft has found tremendous applications in almost all fields mainly because it can be used to match the multiple frequencies of a stationary signal with multiple harmonics in many applications wideband and nonstationary signals however often occur one of the typical examples of such signals is chirp type signals that are usually encountered in radar signal processing such as synthetic aperture radar sar and inverse sar imaging due to the motion of a target the radar return signals are usually chirps and their chirp rates include the information about the target such as the location and the velocity in this paper we study discrete chirp fourier transform dcft which is analogous to the dft besides the multiple frequency matching similar to the dft the dcft can be used to match the multiple chirp rates in a chirp type signal with multiple chirp components we show that when the signal length n is prime the magnitudes of all the sidelobes of the dcft of a quadratic chirp signal are 1 whereas the magnitude of the mainlobe of the dcft is spl radic n with this result an upper bound for the number of the detectable chirp components using the dcft is provided in terms of signal length and signal and noise powers we also show that the n point dcft performs optimally when n is a prime

instantaneous frequency estimation of quadratic and cubic fm signals using the cross polynomial wigner ville distribution the cross polynomial wigner ville distribution x pwvd is applied to instantaneous frequency if estimation of polynomial fm signals the x pwvd is used in a simple iterative algorithm similar to the one proposed by boashash and o shea see ibid vol 41 no 3 p 1439 1993 for the cross wigner ville distribution x wvd the advantage of the x pwvd over the polynomial wigner ville distribution is that the former is linear with respect to the observed signal and as a result it attains the cramer rao bound at lower signal to noise ratios snrs

classification of nonstationary narrowband signals using segmented chirp features and hidden gauss markov models a method is provided for classifying finite duration signals with narrow instantaneous bandwidth and dynamic instantaneous frequency if in this method events are partitioned into nonoverlapping segments and each segment is modeled as a linear chirp forming a piecewise linear if model the start frequency chirp rate signal energy and noise energy are estimated in each segment the resulting sequences of frequency and rate features for each event are classified by evaluating their likelihood under the probability density function pdf corresponding to each narrowband class hypothesis the class conditional pdfs are approximated using continuous state hidden gauss markov models hgmms whose parameters are estimated from labeled training data previous hgmm algorithms are extended by dynamically weighting the output covariance matrix by the ratio of the estimated signal and noise energies from each segment this covariance weighting discounts spurious features from segments with low signal to noise ratio snr making the algorithm more robust in the presence of dynamic noise levels and fading signals the classification algorithm is applied in a simulated three class cross validation experiment for which the algorithm exhibits percent correct classification greater than 97 as low as 7 db snr

adaptive time varying cancellation of wideband interferences in spread spectrum communications based on time frequency distributions the paper proposes an adaptive method for suppressing wideband interferences in spread spectrum ss communications the proposed method is based on the time frequency representation of the received signal from which the parameters of an adaptive time varying interference excision filter are estimated the approach is based on the generalized wigner hough transform as an effective way to estimate the instantaneous frequency of parametric signals embedded in noise the performance of the proposed approach is evaluated in the presence of linear and sinusoidal fm interferences plus white gaussian noise in terms of the snr improvement factor and bit error rate ber

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

kernel design for time frequency signal analysis using the radon transform the authors present a kernel design technique based on using the radon transform of the modulus of the ambiguity function of the signal for determination of angles and distances of radially distributed contents of the autoterms in the ambiguity domain the proposed kernel effectively reduces the cross terms and noise for linear fm signals the result is a tool for high resolution time frequency representation of nonstationary primarily linear fm signals

joint model order selection and parameter estimation of chirps with harmonic components we consider the problem of jointly determining the number of harmonic components of a fundamental linear chirp and estimating its parameters i e its initial frequency and frequency rate given time samples of the observed signal common model order criteria select the number of harmonics based on the maximum likelihood estimator we develop exact and approximated maximum likelihood estimators of these parameters to avoid an exhaustive search in the initial frequency frequency rate space involved by those estimators we propose an alternative low complexity two step estimation method the first step separates the signal to its harmonic components then in the second step the parameters of interest are estimated using least squares method given the phases of the harmonic components the method is compared to the exact and approximated maximum likelihood estimators and to the well known high order ambiguity function based method numerical simulations and real data examples demonstrate that the proposed low complexity method can successfully replace the maximum likelihood estimator in the model order criteria at moderate to high signal to noise ratio since the estimates obtained by the proposed method achieve the cramer rao lower bound at these signal to noise ratios

finite codimensional invariant subspaces of bergman spaces abstract for a large class of bounded domains in c we describe thosefinite codimensional subspaces of the bergman space that are invariant undermultiplication by z using different techniques for certain domains in cn wedescribe those finite codimensional subspaces of the bergman space that areinvariant under multiplication by all the coordinate functions fix a positive integer n and let v denote lebesgue volume measure on cn sothat if n 1 then v is just area measure let q c c be a domain which asusual means that 0 is a nonempty open connected subset of c for an analyticfunction from 0 to c and 1 p oo the norm n p is defined by wfh r jn f pdv the bergman space l q is defined to be the set of analytic functions from fi toc such that n p oo our goal in this paper is to describe the closed finite codimensional subspaces oflpa 0 that are invariant under multiplication by the coordinate functions zi

a low complexity blind estimator of narrowband polynomial phase signals consider the problem of estimating the parameters of multiple polynomial phase signals observed by a sensor array in practice it is difficult to maintain a precisely calibrated array the array manifold is then assumed to be unknown and the estimation is referred to as blind estimation to date only an approximated maximum likelihood estimator amle was suggested for blindly estimating the polynomial coefficients of each signal however this estimator requires a multidimensional search over the entire coefficient space instead we propose an estimation approach which is based on two steps first the signals are separated using a blind source separation technique which exploits the constant modulus property of the signals then the coefficients of each polynomial are estimated using a least squares method applied to the unwrapped phase of the estimated signal this estimator does not involve any search in the coefficient spaces the computational complexity of the proposed estimator increases linearly with respect to the polynomial order whereas that of the amle increases exponentially simulation results show that the proposed estimator achieves the cramer rao lower bound at moderate or high signal to noise ratio

invariant subspaces with finite codimension in bergman spaces let omega be a domain in the complex plane denote by l p roman a omega 1 le p infty the bergman space over omega the author presents a description of finite codimensional space e subset l p roman a omega such that ze subset e under some conditions on omega an analogous result is due to n s axler en and n p bourdon en same journal bf306 1988 no 2 805 817 mr0933319 89f 46051 for an arbitrary bounded domain in c there are described those finite codimensional subspaces of the bergman space that are invariant under multiplication by z

characterization of polynomial functions and application to time frequency analysis we propose necessary and sufficient conditions for a function spl phi t to be a polynomial and establish a formula that allows us to compute the value of the derivative spl phi t of a polynomial at any point in terms of the values of this polynomial for this purpose we establish that any polynomial spl phi t of degree spl les q can be represented as a linear combination of spl phi t t sub 1 q t t sub q where t sub 1 t sub q denote arbitrary parameters a similar property is established for the derivative spl phi t some signal processing applications of these results are discussed

the discrete polynomial phase transform the discrete polynomial phase transform dpt is a new tool for analyzing constant amplitude polynomial phase signals the main properties of the dpt are its ability to identify the degree of the phase polynomial and to estimate its coefficients the transform is robust to deviations from the ideal signal model such as slowly varying amplitude additive noise and nonpolynomial phase the authors define the dpt derive its basic properties and use it to develop computationally efficient estimation and detection algorithms a statistical accuracy analysis of the estimated parameters is also presented

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

estimating the doa and the polarization of a polynomial phase signal using a single polarized vector sensor this paper introduces a novel algorithm to estimate the direction of arrival doa and the polarization of a completely polarized polynomial phase signal of an arbitrary degree the algorithm utilizes a polarized vector sensor comprising a spatially collocated six component electromagnetic vector sensor a dipole triad or a loop triad this esprit based algorithm is based on a time invariant matrix pencil pair derived from the time delayed data sets collected by a single polarized vector sensor the high order difference function of the signal s phase constructs the invariant factor used in the esprit algorithm the steering vector is estimated from the signal subspace eigenvector of the data correlation matrix following which the closed form doa and polarization can be obtained given the degree of the polynomial phase signal this approach resolves the two dimensional azimuth elevation angle and the polarization of the source and requires neither a priori knowledge of the polynomial phase signal s coefficients nor a priori knowledge of the polynomial phase signal s frequency spectrum the efficacy of the proposed algorithm is verified by monte carlo simulations estimation accuracies of the doa and the polarization parameters are evaluated by the closed form cramer rao bounds which are independent of the polynomial coefficients the degree of the polynomial phase signal and the azimuth angle of the source

the cramer rao lower bound for signals with constant amplitude and polynomial phase the authors derive the cramer rao lower bound crlb for complex signals with constant amplitude and polynomial phase measured in additive gaussian white noise the exact bound requires numerical inversion of an ill conditioned matrix while its o n sup 1 approximation is free of matrix inversion the approximation is tested for several typical parameter values and is found to be excellent in most cases the formulas derived are of practical value in several radar applications such as electronic intelligence systems elint for special pulse compression radars and motion estimation from doppler measurements consequently it is of interest to analyze the best possible performance of potential estimators of the phase coefficients as a function of signal parameters the signal to noise ratio the sampling rate and the number of measurements this analysis is carried out

estimating random amplitude polynomial phase signals a cyclostationary approach modeling of a class of nonstationary signals with randomly time varying amplitude and parametric polynomial phase is addressed a novel approach is proposed for the estimation of the time varying phase by exploiting the higher order cyclostationarity of these signals the method does not require nonlinear search is easy to implement and yields consistent estimates for the parameters the resulting algorithms are theoretically tolerant to a large class of noises including additive stationary non gaussian noise and any gaussian noise simulation examples supporting the theory are provided

noise induced unstable dimension variability and transition to chaos in random dynamical systems results are reported concerning the transition to chaos in random dynamical systems in particular situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle which can be expected in any periodic window of a nonlinear dynamical system under noise the asymptotic attractor of the system can become chaotic as characterized by the appearance of a positive lyapunov exponent generic features of the transition include the following 1 the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions unstable dimension variability and this nonhyperbolicity develops as soon as the attractor becomes chaotic 2 for systems described by differential equations the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero lyapunov exponent after the noisy attractor becomes chaotic and 3 the largest lyapunov exponent becomes positive from zero in a continuous manner and its scaling with the variation of the noise amplitude is algebraic formulas for the scaling exponent are derived in all dimensions numerical support using both low and high dimensional systems is provided

analysis of multicomponent lfm signals by a combined wigner hough transform the aim of the paper is the performance evaluation of a method for the analysis of mono or multicomponent linear frequency modulation lfm signals based on the hough transform of the wigner ville distribution of the signals a closed form expression is found for the signal to noise ratio and the parameter estimation accuracy the overall method as any nonlinear method exhibits a threshold effect nevertheless it is shown to be asymptotically efficient and offers a good rejection capability of the cross terms

dynamics of leaking hamiltonian systems in order to understand the dynamics in more detail in particular for visualizing the space filling unstable foliation of closed chaotic hamiltonian systems we propose to leak them up the cutting out of a finite region of their phase space the leak through which escape is possible leads to transient chaotic behavior of nearly all the trajectories the never escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically it is an approximant of the full hamiltonian foliation the better the smaller the leak is the escape rate depends sensitively on the orientation of the leak even if its area is fixed the applications for chaotic advection for chemical reactions superimposed on hydrodynamical flows and in other branches of physics are discussed

analysis of multicomponent polynomial phase signals while the theory of estimation of monocomponent polynomial phase signals is well established the theoretical and methodical treatment of multicomponent polynomial phase signals mc ppss is limited in this paper we investigate several aspects of parameter estimation for mc ppss and derive the crameacuter rao bound we show the limits of existing techniques and then propose a nonlinear least squares nls approach we also motivate the use the nelder mead simplex algorithm for minimizing the nonlinear cost function the slight increase in computational complexity is a tradeoff for improved mean square error performance which is evidenced by simulation results

on polynomial phase signals with time varying amplitudes we address the parameter estimation problem for a class of nonstationary signals modeled as polynomial phase signals with time varying amplitudes exponentially damped polynomial phase signals are treated as a special case and are analyzed in detail high order instantaneous moments provide the basic analytical tool but links are shown to exist with either the usually employed fft based technique or the high resolution kumaresan tufts 1982 music and matrix pencil methods asymptotic properties of the relevant estimators are established cramer rao lower bounds on the amplitude and phase parameter estimates are derived and computer simulations are carried out to evaluate the performance of various schemes we focus on parametric modeling of am fm signals mainly because parametric techniques offer parsimony and allow for theoretically unlimited resolution

parameter estimation for random amplitude chirp signals we consider the problem of estimating the parameters of a chirp signal observed in multiplicative noise i e whose amplitude is randomly time varying two methods for solving this problem are presented first an unstructured nonlinear least squares approach nls is proposed it is shown that by minimizing the nls criterion with respect to all samples of the time varying amplitude the problem reduces to a two dimensional 2 d maximization problem a theoretical analysis of the nls estimator is presented and an expression for its asymptotic variance is derived it is shown that the nls estimator has a variance that is very close to the cramer rao bound the second approach combines the principles behind the high order ambiguity function hbf and the nls approach it provides a computationally simpler but suboptimum estimator a statistical analysis of the haf based estimator is also carried out and closed form expressions are derived for the asymptotic variance of the haf estimators based on the data and on the squared data numerical examples attest to the validity of the theoretical analyzes and establish a comparison between the two proposed methods

multicomponent polynomial phase signal analysis using a tracking algorithm we describe an efficient technique analyzing signals that comprise a number of polynomial phase components the technique is based on a previously proposed multiple frequency tracker which is an algorithm for recursive estimation of parameters of multiple sine waves in noise it has a relatively low snr threshold and moderate computational complexity

split radix algorithms for arbitrary order of polynomial time frequency transforms the polynomial time frequency transform is one of important tools for estimating the coefficients of the polynomial phase signals ppss with the maximum likelihood method the transform converts a one dimensional 1 d data sequence into a multidimensional output array from which the phase coefficients of the data sequence are estimated a prohibitive computational load is generally needed for high order polynomial phase signals although the 1 d fast fourier transform fft algorithm can be used based on the split radix concept this paper derives a fast algorithm for arbitrary order of polynomial time frequency transforms to significantly reduce the computational complexity comparisons on the computational complexity needed by various algorithms are also made to show the merits of the proposed algorithm

advection in chaotically time dependent open flows the passive advection of tracer panicles is considered in open two dimensional incompressible flows with chaotic time dependence as illustrative examples we investigate flows produced by chaotically moving ideal point vortices the advection problem can be seen as a chaotic scattering process in a chaotically driven hamiltonian system studying the motion of tracer ensembles we present numerical evidence for the existence of a bounded chaotic set containing infinitely many aperiodic trajectories never leaving the mixing region of the flow these ensembles converge to filamental patterns which however do not follow self similar scaling nevertheless they possess a fractal dimension after averaging over several finite time realizations of the flow we propose random maps as simple models of the phenomenon

improving polynomial phase parameter estimation by using nonuniformly spaced signal sample methods this paper investigates the computationally efficient parameter estimation of polynomial phase signals embedded in noise many authors have previously proposed multilinear analysis methods which operate on uniformly spaced samples of the signal such methods include the higher order ambiguity functions hafs the polynomial wigner ville distributions pwvds and the higher order phase hp functions this paper investigates the use of multilinear methods which operate on nonuniformly spaced signal samples it is seen that the relaxation of the requirement to use uniformly spaced samples in the analysis can lead to significant performance improvements a theoretical analysis and simulations are presented in support of these claims

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

difference equation representation of chirp signals and instantaneous frequency amplitude estimation we present a time varying coefficient difference equation representation for sinusoidal signals with time varying amplitudes and frequencies we first obtain a recursive equation for a single chirp signal then using this result we obtain time varying coefficient difference equation representations for signals composed of multiple chirp signals we analyze these equations using the skew shift operators we show that the phases of the poles of the difference equations produce instantaneous frequencies if and the magnitudes are proportional to the ratio of successive values of the instantaneous amplitudes ia then algorithms are presented for the estimation of instantaneous frequencies and instantaneous amplitudes for multicomponent signals composed of chirps using the difference equation representation the first algorithm we propose is based on the skew shift operators next we derive the conditions under which we can use the so called frozen time approach we propose an algorithm for if and ia estimation based on the frozen time approach then we propose an automatic signal separation method finally we apply the proposed algorithms to single and multicomponent signals and compare the results with some existing methods

estimation of multicomponent polynomial phase signals impinging on a multisensor array using state space modeling this contribution addresses the problem of estimating the parameters of multicomponent polynomial phase signals when impinging on a multisensor array an original approach is proposed based on a state space modelization of the signal and the application of an extended kalman filter for the state estimation the use of a multisensor array allows the exploitation of a spatial information and leads to the consideration of multiple filters with different observation equations computer simulations are used to demonstrate the effectiveness of the proposed algorithm

stochastic perturbations in open chaotic systems random versus noisy maps we investigate the effects of random perturbations on fully chaotic open systems perturbations can be applied to each trajectory independently white noise or simultaneously to all trajectories random map we compare these two scenarios by generalizing the theory of open chaotic systems and introducing a time dependent conditionally map invariant measure for the same perturbation strength we show that the escape rate of the random map is always larger than that of the noisy map in random maps we show that the escape rate ensuremath kappa and dimensions d of the relevant fractal sets often depend nonmonotonically on the intensity of the random perturbation we discuss the accuracy bias and precision variance of finite size estimators of ensuremath kappa and d and show that the improvement of the precision of the estimations with the number of trajectories n is extremely slow ensuremath propto 1 mathrm ln n we also argue that the finite size d estimators are typically biased general theoretical results are combined with analytical calculations and numerical simulations in area preserving baker maps