asymptotic theory for estimators of high order statistics of stationary processes high order cumulants and high order spectra play an important role in the theory of stationary processes for nonlinear processes it is quite challenging to develop an asymptotic theory for their estimators this paper presents a systematic asymptotic theory for estimators of high order moments for a general class of stationary processes using the framework of functional dependence measures in particular we prove the asymptotic normality of the estimators and establish a uniform convergence rate we also provide a sufficient condition for the summability of high order cumulants based on the latter we prove consistency and asymptotic normality of the third order spectra or bispectrum estimators under mild and easily verifiable conditions

a single blind controlled competition among tests for nonlinearity and chaos interest has been growing in testing for nonlinearity or chaos in economic data but much controversy has arisen about the available results this paper explores the reasons for these empirical difficulties we designed and ran a single blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series the data generating mechanisms include linear processes chaotic recursions and nonchaotic stochastic processes and both large and small samples were included in the experiment the data series were produced in a single blind manner by the competition manager and sent by e mail without identifying information to the experiment participants each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test the results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos

a test for aliasing using bispectral analysis abstract aliasing is a signal confounding problem that arises when a continuous time signal is sampled at a rate slower than twice the highest frequency component of a fourier series representation of the signal aliasing can be especially serious for social science time series applications since the sampling designs used to construct most social science data bases are fixed by considerations other than the nature of the underlying continuous time mechanisms after collecting sampled data it is of value to test the observations for the presence of aliasing it is shown that the nature of the support set of a sampled band limited stationary signal can be used to motivate an amended version of the hinich bispectrum test for gaussianity hinich 1982 as a test for aliasing

the complex cepstrum of higher order cumulants and nonminimum phase system identification a computationally efficient identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase system the method is based on the idea of computing the complex cepstrum of higher order cumulants of the system output in particular the differential cepstrum parameters of the nonminimum phase impulse response are estimated directly from higher order cumulants by least squares solution or two dimensional fft operations the method reconstructs the minimum phase and maximum phase impulse response components separately it is flexible enough to be applied on autoregressive ar moving average ma or arma system without a priori knowledge of the type of the system benchmark simulation examples demonstrate the effectiveness of the method even with short length data records

the long step method of analytic centers for fractional problems we develop a long step surface following version of the method of analytic centers for the fractional linear problem min t 0 t 0 b x a x eh b x ek x eg whereh is a closed convex domain k is a convex cone contained in the recessive cone ofh g is a convex domain andb a are affine mappings tracing a two dimensional surface of analytic centers rather than the usual path of centers allows to skip the initial centering phase of the path following scheme the proposed long step policy of tracing the surface fits the best known overall polynomial time complexity bounds for the method and at the same time seems to be more attractive computationally than the short step policy which was previously the only one giving good complexity bounds

nonlinear digital filter realization by distributed arithmetic it is assumed that a nonlinear shift invariant digital filter is represented by a truncated discrete volterra series a matrix notation based on the properties of the kronecker product of matrices is derived for the nonlinear operator of order k in the volterra expansion the matrix notation shows how the well known distributed arithmetic originally proposed for linear digital filters can be adopted for the realization of nonlinear operators the characteristics of combinatorial and memory oriented structures are compared and the problem of complexity reduction is discussed

arma parameter estimation using only output cumulants several algorithms are developed to estimate the parameters of a causal nonminimum phase autoregressive moving average arma p q system which is excited by an unobservable independently identically distributed non gaussian process the output is contaminated by additive colored gaussian noise of unknown power spectral density a fundamental result is presented pertaining to the identifiability of ar parameters based on the yule walker equations drawn from a specific set of p 1 1 d slices of the kth k 2 order output cumulant several ma parameter estimation algorithms are developed one method uses q 1 d slices of the output cumulant a second method uses only two 1 d cumulant slices these methods do not involve computation of the residual i e ar compensated time series or polynomial factorization multidimensional versions of the various algorithms are presented a simulation study demonstrating the effectiveness of the algorithms is included

on polynomiality of the method of analytic centers for fractional problems we establish polynomial time convergence of the method of analytic centers for the fractional programming problemt min x g tb x a x k whereg n is a closed and bounded convex domain k m is a closed convex cone anda x g n b x g k are regular enough say affine mappings

time series modeling by a second order hammerstein system in this paper we discuss the modeling of an observed time series as the output of a second order hammerstein model driven by an independent and identically distributed i i d gaussian noise the modeling is discussed from the point of view of the hypothesis testing ht of statistical indexes derived from higher order statistics hos three points of view can be adopted in ht the more general approach is to not make any assumption on the index statistics notably on the variance and thus to use the t 2 hotelling test the second point of view involves calculating a theoretical variance in order to define normalized variables whose statistics are known the last approach is a mixed version of the others two these three approaches are applied to statistical indexes derived from linear combination of slices of spectra from the third order up the fifth order some subcases need to use the derivatives of the polyspectra since general relationships are null and cannot be used for the ht the second part of this paper is devoted to linear and quadratic kernel identification using the relationship defined for ht definition as for the ht several subcases have to be discussed using either the polyspectra or the derivatives of polyspectra

phase reconstruction from bispectrum slices we propose a new method for the reconstruction of the fourier phase of a complex lti system based on the principal arguments of any pair of horizontal consecutive bispectrum slices of the system output since principal bispectrum arguments only are required there is no need for two dimensional 2 d phase unwrapping the reconstructed phase differs from the true one by a constant integer multiples of 2 spl pi and a linear phase component corresponding to an integer time delay the ability to choose the location of the two slices enables us to avoid low signal to noise ratio snr bispectral regions which usually occur in the case of bandlimited systems

sampling and distortion tradeoffs for bandlimited periodic signals in this paper the optimal sampling strategies uniform or nonuniform and distortion tradeoffs for gaussian bandlimited periodic signals with additive white gaussian noise are studied our emphasis is on characterizing the optimal sampling locations as well as the optimal presampling filter to minimize the reconstruction distortion we first show that to achieve the optimal distortion no presampling filter is necessary for any arbitrary sampling rate then we provide a complete characterization of optimal distortion for low and high sampling rates with respect to the signal bandwidth we also provide bounds on the reconstruction distortion for rates in the intermediate region it is shown that nonuniform sampling outperforms uniform sampling for low sampling rates in addition the optimal nonuniform sampling set is robust with respect to missing sampling values on the other hand for the sampling rates above the nyquist rate the uniform sampling strategy is optimal

blind deconvolution of discrete linear systems we study the blind deconvolution problem in the case where the input noise has a finite discrete support and the transfer linear system is not necessarily minimum phase we propose a new family of estimators built using algebraic considerations the estimates are consistent under very wide assumptions the input signal need not be independently distributed the cardinality of the finite support may be estimated simultaneously we consider in particular ar systems in this case we prove that the estimator of the parameters is perfect a s with a finite number of observations

second order polynomial estimators from uncertain observations using covariance information this paper presents recursive least mean squared error second order polynomial filtering and fixed point smoothing algorithms to estimate a signal from uncertain observations when only the information on the moments up to fourth order of the signal and observation noise is available the estimators require the autocovariance and crosscovariance functions of the signal and their second order powers in a semidegenerate kernel form and the probability that the signal exists in the observed values

a digital method of modeling quadratically nonlinear systems with a general random input without assuming particular statistics of the input a practical digital method of estimating linear and quadratic transfer functions of a nonlinear time invariant system that can be described by volterra series of up to second order is presented the method is tested and validated by analyzing input output data of a known quadratically nonlinear system it is used when there is little knowledge about the input statistics or the input is non gaussian it is also noted that the ordinary coherence functions cannot be used in explaining the input output power transfer relationship of a quadratic system excited by a non gaussian input signal with respect to the practical application of the method the relationship between the mean square errors involved in the transfer function estimates and the number of averages taken from the spectral estimation is qualitatively discussed

time domain approaches to quadratic phase coupling estimation bicoherence is a quantity used to estimate the contribution of second order nonlinearity to the power in bifrequencies the conventional approach to the problem uses second and third order periodograms for bicoherence estimation time domain approaches that use autocorrelation and third order cumulant samples or their estimates are developed these techniques are suitable for use in situations where high resolution approaches are required for power spectrum and bispectrum estimation

on the von neumann economic growth problem we study the complexity of the von neumann economic growth problem gamma ast max gamma mid exists y ne 0 y ge 0 b gamma ay ge 0 where a and b are given two nonnegative and rational m n matrices and a has no all zero column let the binary data length of a and b be l we develop an interior point algorithm to generate a i such that 2 1 i in om nl minm nt iterations where each iteration solves a system of m n linear equations

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

invertibility conditions of a class of moment matrices and applications we consider a multivariable polynomial with random variables as arguments all the monoids of this polynomial are arranged to define a random vector for which we study the correlation matrix we establish necessary and sufficient invertibility conditions of this matrix and investigate some applications

system reconstruction from higher order spectra slices we consider the problem of system reconstruction from higher order spectra hos slices we establish that the impulse response of a complex system can be reconstructed up to a scalar and a shift based on any pair of hos slices as long as the distance between the two slices satisfies a certain condition one slice is sufficient for the reconstruction in the case of a real system we propose a cepstrum based method for system reconstruction we also propose a new method for the reconstruction of the system fourier phase based on the phase of any odd indexed bispectrum slice being able to choose the slices to be used in the reconstruction allows us to avoid bispectrum regions dominated by noise

deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes nongaussian linear processes are considered it is shown that the phase of the transfer function can be estimated under broad conditions this is not true of gaussian linear processes and in this sense gaussian linear processes are atypical the asymptotic behavior of a phase estimate is determined the phase estimates make use of bispectral estimates these ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase a number of computational illustrations are given

blind identification of lti zmnl lti nonlinear channel models a simple method is proposed for blind identification of discrete time nonlinear models consisting of two linear time invariant lti subsystems separated by a polynomial type zero memory nonlinearity zmnl of order n the lti zmnl lti model the linear subsystems are allowed to be of nonminimum phase nmp though the first lti can be completely identified only if it is of minimum phase with a circularly symmetric gaussian input the linear subsystems can be identified using simple cepstral operations on a single 2 d slice of the n 1 th order polyspectrum of the output signal the linear subsystem of an lti zmnl model can be identified using only a 1 d moment or polyspectral slice if it is of minimum phase the zmnl coefficients are not identified and need not be known the order n of the nonlinearity can in principle be estimated from the output signal the methods are analytically simple computationally efficient and possess noise suppression characteristics computer simulations are presented to support the theory

on the design of nonlinear discrete time predictors corresp the problem of minimum mean squared error prediction of a discrete time random process using a nonlinear filter consisting of a zero memory nonlinearity followed by a linear filter is studied classes of random processes for which the best predictor is realizable using a nonlinear filter of the above form are discussed for those random processes for which the best predictor is not realizable using the above nonlinear filter an iterative procedure is presented for finding a suboptimal nonlinear filter special attention is directed to the case where the nonlinearity is a polynomial also a noniterative approach based on nonlinear regression is presented

on non linear serial dependencies in stock returns abstract this paper considers the h p tests of linearity and gaussianity of daily stock returns and their implications for non linear models of those returns

on measuring the system coherency of quadratically nonlinear systems results on modeling quadratic systems described by volterra series up to order two are presented specifically expressions for the linear and quadratic transfer functions and the system coherency are derived it is shown that all the three quantities can be independently computed in the discrete frequency domain in terms of various spectral moments

identification of non minimum phase linear stochastic systems abstract the identification of time invariant non minimum phase stochastic systems driven by non gaussian white noise is considered given only the noisy observations of the system output a two step procedure is proposed in the first step a spectrally equivalent system is estimated using a standard technique that exploits only the second order statistics of the measurements in the second step a partial set of 4th order cumulants of the measurements is exploited to resolve the location of the system zeros knowledge of the probability distribution of the driving noise is not required strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are also presented in support of the theory

mixed phase wavelet estimation using fourth order cumulants in recent years methods have been developed in the field of high order statistics that can reliably estimate a mixed phase wavelet from the noisy output of a convolutional process these methods use high order covariance functions of the data called cumulants which retain phase information and allow recovery of the wavelet the assumption is that the reflection coefficient series is a non gaussian stationary and statistically independent random process the method described in this paper uses the fourth order cumulant of the data and a moving average noncausal parametric model for the wavelet the fourth order moment function of this wavelet matches the fourth order cumulant of the data in a minimum mean squared error sense numerical simulations demonstrate that the method can accurately estimate mixed phase wavelets even when the reflectivity has a distribution close to gaussian three seismic data examples demonstrate possible uses of the method in the first two an average source wavelet is est

distortion rate function of sub nyquist sampled gaussian sources the amount of information lost in sub nyquist sampling of a continuous time gaussian stationary process is quantified we consider a combined source coding and sub nyquist reconstruction problem in which the input to the encoder is a noisy sub nyquist sampled version of the analog source we first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate limited version of its samples this expression is a function of the sampling frequency and the average number of bits describing each sample it is given as the sum of two terms minimum mean square error in estimating the source from its noisy but otherwise fully observed sub nyquist samples and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source we extend this result to multi branch uniform sampling where the samples are available through a set of parallel channels with a uniform sampler and a pre sampling filter in each branch further optimization to reduce distortion is then performed over the pre sampling filters and an optimal set of pre sampling filters associated with the statistics of the input signal and the sampling frequency is found this results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering these results thus unify the shannon whittaker kotelnikov sampling theorem and shannon rate distortion theory for gaussian sources

nonparametric identification of hammerstein systems a discrete time nonlinear hammerstein system is identified and the correlation and frequency domain methods for identification of its linear subsystem are presented the main results concern the estimation of the nonlinear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data it is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable pointwise as well as global consistencies are established for lipschitz characteristics the rate of the convergence in probability is o n sup 1 3

fourier series based nonminimum phase model for statistical signal processing in this paper a parametric fourier series based model fsbm for or as an approximation to an arbitrary nonminimum phase linear time invariant lti system is proposed for statistical signal processing applications where a model for lti systems is needed based on the fsbm a minimum phase linear prediction error lpe filter for amplitude estimation of the unknown lti system together with the cramer rao cr bounds is presented then an iterative algorithm for obtaining the optimum lpe filter with finite data is presented that is also an approximate maximum likelihood algorithm when data are gaussian then three iterative algorithms using higher order statistics hos with finite non gaussian data are presented to estimate parameters of the fsbm followed by some simulation results as well as some experimental results with real speech data to support the efficacy of the proposed algorithms using the fsbm finally we draw some conclusions

efficient implementations of quadratic digital filters efficient implementation structures of quadratic filters are introduced in this paper using systolic arrays distributed arithmetic and linear convolutions with multipliers the implementations are based on matrix decompositions and consist of a set of parallel 1 d fir filters in cascade with a set of sequential square in add out type of operations the structures are compared using various realistic figures of merit such as data throughput delay cost that is proportional to chip area and variance of roundoff errors in the filter output assuming fixed point arithmetic for comparison purposes the direct and biglieri quadratic structures are included in this study evaluating the comparative results the conclusion drawn is that the matrix decomposition realization of a quadratic filter implemented with systolic arrays seems to be offering the best compromise among the various conflicting figures of merit

nonlinear infinite extent interpolation of stationary processes the problem of minimum mean square interpolation of discrete time stationary stochastic processes using a system consisting of a finite number p of nonlinear instantaneous transformations followed by a p input one output infinite extent linear filter is studied the expressions of the optimal interpolator and of the approximation error are derived and generalize the corresponding relations known in linear interpolation theory some extensions including the estimation problem with several missing observations are also investigated

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

blind identifiability of a quadratic stochastic system quadratic systems appear in various fields of signal processing in particular in detection and estimation in this paper we establish some conditions to determine the order and identify the coefficients of any discrete and finite extent quadratic system driven by a sequence of independent and identically distributed random variables the input sequence is assumed unobservable and the conditions are based on some properties of the output cumulant functions up to the third order necessary and sufficient conditions ensuring the determination of the order are given a sufficient condition of uniqueness to recover the parameters is presented lastly some examples are given

tensor product basis approximations for volterra filters the paper studies approximations for a class of nonlinear filters known as volterra filters although the volterra filter provides a relatively simple and general representation for nonlinear filtering it is often highly overparameterized due to the large number of parameters the utility of the volterra filter is limited the overparameterization problem is addressed in the paper using a tensor product basis approximation tpba in many cases a volterra filter may be well approximated using the tpba with far fewer parameters hence the tpba offers considerable advantages over the original volterra filter in terms of both implementation and estimation complexity furthermore the tpba provides useful insight into the filter response the paper studies the crucial issue of choosing the approximation basis several methods for designing an appropriate approximation basis and error bounds on the resulting mean square output approximation error are derived certain methods are known to be nearly optimal

on the linear minimum mean squared error estimation of an undersampled wide sense stationary random process we consider the problem of linearly estimating in the sense of minimum mean squared error a wide sense stationary process in noise given uniformly spaced samples where the sampling interval is such that significant aliasing occurs we derive the corresponding aliased wiener filter and provide a technique for determining a closed form for the necessary power spectral density functions we conclude with an example where both signal and noise are modeled using a second order innovations representation

high rate interpolation of random signals from nonideal samples we address the problem of reconstructing a random signal from samples of its filtered version using a given interpolation kernel in order to reduce the mean squared error mse when using a nonoptimal kernel we propose a high rate interpolation scheme in which the interpolation grid is finer than the sampling grid a digital correction system that processes the samples prior to their multiplication with the shifts of the interpolation kernel is developed this system is constructed such that the reconstructed signal is the linear minimum mse lmmse estimate of the original signal given its samples an analytic expression for the mse as a function of the interpolation rate is provided which leads to an explicit condition such that the optimal mse is achieved with the given nonoptimal kernel simulations confirm the reduction in mse with respect to a system with equal sampling and reconstruction rates

polynomial estimation of the amplitude of a signal the problem of estimating the amplitude of a signal is addressed using higher order statistics the probability distribution of the noise is assumed to be unknown so that the maximum likelihood estimator cannot be calculated the estimator is taken as a polynomial of the observation the coefficients of which are determined so that the estimate is unbiased with minimum variance this method generalizes the linear approach and the estimate variance is reduced the ease of linear quadratic estimation is detailed and numerical examples are presented

on estimation of a class of nonlinear systems by the kernel regression estimate the estimation of a multiple input single output discrete hammerstein system is studied such a system contains a nonlinear memoryless subsystem followed by a dynamic linear subsystem the impulse response of the dynamic linear subsystem is obtained by the correlation method the main results concern the estimation of the nonlinear memoryless subsystem no conditions are imposed on the functional form of the nonlinear subsystem and the nonlinearity is recovered using the kernel regression estimate the distribution free pointwise and global convergence of the estimate is demonstrated that is no conditions are imposed on the input distribution and convergence is proven for virtually all nonlinearities the rates of pointwise as well as global convergence are obtained for all input distributions and for lipschitz type nonlinearities

identification of the coefficients in a non linear time series of the quadratic type abstract this paper presents a method for identifying the structure of the quadratic model x t e t r r 1 s s 0 a r s e t s e t r s where the e t are indepen dent identically distributed zero mean random variables this finite parameter model is a simplified second order volterra expansion of a stable non linear filter the standard deviation of e t is not estimable from observations of the x t it is shown that the m m n th third order cumulant called the bicovariance of x t is cub to a m n times the square of the variance of e t the large sample distribution of the sample bicovariance is used to determine which quadratic coefficients are significantly different from zero the method is illustrated using artificial and real data daily stock price series for several securities

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

an interior point method for fractional programs with convex constraints we present an interior point method for a class of fractional programs with convex constraints the proposed algorithm converges at a polynomial rate similarly as in the case of a convex problem even though fractional programs are only pseudo convex here the rate of convergence is measured in terms of the area of two dimensional convex setsck containing the origin and certain projections of the optimal points and the area ofck is reduced by a constant factorc 1 at each iteration the factorc depends only on the self concordance parameter of a barrier function associated with the feasible set we present an outline of a practical implementation of the proposed method and we report results of some preliminary numerical experiments

arma systems excited by non gaussian processes are not always identifiable it is a popular myth that the zeros of an arma autoregressive moving average process excited by an unobservable non gaussian process can be resolved from noisy output measurements a counterexample is given

optimal linear quadratic systems for detection and estimation the problem of linear quadratic systems for detection has long been solved by assuming the deflection criterion and gaussian noise it is shown here that the gaussian assumption can be removed and a complete solution is presented for an arbitrary probability distribution with finite fourth order moments the optimal solution can always be obtained by solving a linear system of equations some properties of the optimal systems are developed for particular examples of nongaussian noise it is shown that there is a strong relationship between linear quadratic optimal detection and optimal estimation which extends results known for the purely linear case

blind identification of linear subsystems of lti zmnl lti models with cyclostationary inputs discrete time nonlinear models consisting of two linear time invariant lti filters separated by a finite order zero memory nonlinearity zmnl of the polynomial type the lti zmnl lti model are appropriate in a large number of practical applications we discuss some approaches to the problem of blind identification of such nonlinear models it is shown that for an nth order nonlinearity the possibly non minimum phase finite memory linear subsystems of lti zmnl and lti zmnl lti models can be identified using the n 1th order cyclic statistics of the output sequence alone provided the input is cyclostationary and satisfies certain conditions the coefficients of the zmnl are not needed for identification of the linear subsystems and are not estimated it is shown that the theory presented leads to analytically simple identification algorithms that possess several noise and interference suppression characteristics

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

deconvolution bandwidth and the trispectrum abstract in the largest application area of time series analysis geophysical exploration the underlying innovations sequence is of primary interest and must be estimated this sequence is estimated by deconvolving the non gaussian noninvertible time series this involves estimation of a phase shift correction from the time series which can be carried out by maximizing the kurtosis of the series unfortunately the method is hampered by the fact that the time series is typically deficient in power in certain bands of frequencies band limited the consequences of this can be analyzed by studying the trispectrum the third of the polyspectra of the series this reveals two important results first we are able to easily appreciate why for certain types of band limitation kurtosis cannot be used to determine a phase correction second by looking at the inner and outer subvolumes of the support volume for the discrete parameter trispectrum we see that for the standard linear model the trispectrum is no

nonparametric estimation of nonlinear dynamic systems using semirecursive regression estimates abstract in the paper we apply semirecursive nonparametric estimation techniques for the estimation of dynamic systems we investigate in detail the estimation of hammerstein dynamic systems using kernel and partitioning regression estimates in particular we propose estimation algorithms and study their convergence for all distributions of the data

polyspectrum modeling using linear or quadratic filters the polyspectrum modeling problem using linear or quadratic filters is investigated in the linear case it is shown that if the output pth order cumulant function of a filter driven by a white noise is of finite extent then the filter necessarily has a finite extent impulse response it is proved that every factorable polyspectrum with a non gaussian white noise can also be modeled with a quadratic filter driven by a gaussian white noise it is shown that if the quadratic filter has a finite extent impulse response then the output pth order cumulant function is of finite extent and if the output pth order cumulant function of a quadratic filter is of finite extent then the impulse response may or may not be of finite extent it is shown that there exist finite and infinite extent pth order cumulant functions that are not factorable but can be modeled with quadratic filters

blind identification of volterra hammerstein systems this paper is concerned with the blind identification of volterra hammerstein systems two identification scenarios are covered the first scenario assumes that although the input is not available the statistics of the input are a priori known this case appears in communication applications where the input statistics of the transmitter are known to the receiver the second scenario assumes that the input statistics are unknown in the case of known input statistics the input is stationary higher order white noise with arbitrary probability density function under the scenario of unknown input statistics the input is restricted to gaussian white process new cumulant based identification methods are described for the above scenarios the problem is converted into a linear multivariable form and the output cumulants are calculated using kronecker products first initial conditions are determined by a linear system of equations these correspond to the boundary values of the volterra kernels the remaining kernel coefficients can be determined under both identification schemes from a possibly overdetermined system of linear equations

the distortion rate function of cyclostationary gaussian processes a general expression for the quadratic distortion rate function drf of cyclostationary gaussian processes in terms of their spectral properties is derived this expression can be seen as the result of orthogonalization over the different components in the polyphase decomposition of the process we use this expression to derive in a closed form the drf of several cyclostationary processes arising in practice we first consider the drf of a combined sampling and source coding problem it is known that the optimal coding strategy for this problem involves source coding applied to a signal with the same structure as one resulting from pulse amplitude modulation pam since a pam modulated signal is cyclostationary our drf expression can be used to solve for the minimal distortion in the combined sampling and source coding problem we also analyze in more detail the drf of a source with the same structure as a pam modulated signal and show that it is obtained by reverse waterfilling over an expression that depends on the energy of the pulse and the baseband process modulated to obtain the pam signal this result is then used to explore the effect of the symbol rate in pam on the drf of its output in addition we also study the drf of sources with an amplitude modulation structure and show that the drf of a narrow band gaussian stationary process modulated by either a deterministic or a random phase sine wave equals the drf of the baseband process

higher order spectrum factorization in one and two dimensions with applications in signal modeling and nonminimum phase system identification the problem of modeling a given higher order spectrum as that of the output of a linear time invariant lti system driven by a higher order white random signal is discussed this can be posed as a higher order spectrum factorization problem a theorem is provided to discuss the existence of such a factorization a fast algorithm for efficient implementation of the higher order spectrum factorization if it exists is then proposed the results are extended to two dimensions straightforwardly the relationship of the higher order spectrum factorization problem to the well known power spectrum factorization is also discussed as applications the problems of identification of nonminimum phase lti systems given the higher order statistics of both input and output processes and phase reconstruction are presented experimental results are provided for the one dimensional case

quadratic digital filters for image processing a class of nonlinear two dimensional filters is presented a matrix description is developed for such operators and some symmetry conditions are imposed which simplify and render more effective the filter structure a strategy for the design of filters capable of enhancing images having reduced contrast and degraded by noise is proposed

volterra filtering and higher order whiteness some properties of volterra filtering are established finite order finite horizon volterra filtering is investigated as well as its asymptotic properties next the concepts of volterra unpredictability and uninterpolability lead to generalizations of the notion of white noise to higher orders these generalizations are introduced and relations are established between them

detecting a transient signal by bispectral analysis a method for detecting a transient waveform of unknown shape in additive stationary noise is presented the method uses a statistic computed from the sample bispectrum of a sampled record of the signal plus noise of length t n tau where 1 tau is the sampling rate the key result underlying the method is that the bispectrum of the noise is zero in a triangle that is a proper subset of the principal domain triangle it is shown that the probability of detecting the signal is high if 3 16 sup 1 2 n sup 5 6 is larger than rho sup 1 where rho is the energy signal to noise ratio the result implies that the bispectrum based test may detect a weak signal of unknown form which evades detection by other methods

the finite criss cross method for hyperbolic programming in this paper the finite criss cross method is generalized to solve hyperbolic fractional linear programming problems just as in the case of linear or quadratic programming the criss cross method can be initialized with any not necessarily feasible basic solution it is known that if the feasible region of the problem is unbounded then some of the known algorithms fail to solve the problem our criss cross algorithm does not have such drawback finiteness of the procedure is proved under the usual mild assumptions some small numerical examples illustrate the main features of the algorithm and show that our method generates different iterates than other earlier published methods

global identification of nonlinear hammerstein systems by recursive kernel approach abstract in this paper we discuss identification of hammerstein systems by the recursive kernel regression estimate we propose the identification algorithm and study its global convergence and the rate of convergence

on the identifiability of non gaussian arma models using cumulants a fixed set of output cumulants of order greater than two guarantees unique identification of known order causal arma autoregressive moving average models which are driven by unobservable non gaussian i i d noise the models are allowed to be non minimum phase and their outputs may be corrupted by additive colored gaussian noise of unknown covariance the arma parameters can be estimated either by means of linear equations and closed form expressions or by minimizing quadratic cumulant matching criteria the latter approach requires computation of cumulants in terms of the arma parameters which is carried out in the state space using kronecker products

polyspectra of ordered signals polyspectra are related to fourier transforms of moment or cumulant functions of any order of random signals they play an important role in many problems of signal analysis and processing however there are only a few statistical models giving explicitly the expression of polyspectra ordered signals are signals for which the explicit expression of the moment functions requires that the time instants appearing in these moments are put in an increasing order there are many examples of such signals the best known being the random telegraph signal constructed from a poisson process some of these examples are presented and analyzed the origin of the ordering structure is related with the point that real time is an oriented variable making a difference between past and future this especially appears in markov processes the calculation of polyspectra is difficult because ordering is not adapted to fourier analysis by an appropriate grouping of various terms the explicit expression of spectral moment functions is obtained it shows in particular that many ordered signals present a normal density on the normal manifolds of the frequency domain and another contribution on the stationary manifold that is explicitly calculated the analysis of the structure of this expression allows us to discuss some relationships with normal distribution central limit theorem and time reversibility

on estimating noncausal nonminimum phase arma models of non gaussian processes the authors address the problem of estimating the parameters of non gaussian arma autoregressive moving average processes using only the cumulants of the noisy observation the measurement noise is allowed to be colored gaussian or independent and identically non gaussian distributed the arma model is not restricted to be causal or minimum phase and may even contain all pass factors the unique parameter estimates of both the ma and ar parts are obtained by linear equations the structure of the proposed algorithm facilitates asymptotic performance evaluation of the parameter estimators and model order selection using cumulant statistics the method is computationally simple and can be viewed as the least squares solution to a quadratic model fitting of a sampled cumulant sequence identifiability issues are addressed simulations are presented to illustrate the proposed algorithm

an interior point method for generalized linear fractional programming we develop an interior point polynomial time algorithm for a generalized linear fractional problem the latter problem can be regarded as a nonpolyhedral extension of the usual linear fractional programming typical example which is of interest for control theory is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables

singular value decomposition based ma order determination of non gaussian arma models singular value decomposition svd based moving average ma order determination of non gaussian processes using higher order statistics is addressed it is shown that the ma order determination of autoregressive moving average arma models is equivalent to the rank determination of a certain error matrix and a svd approach is proposed its simplified form is applied to pure ma models to improve the robustness of the order selection a combination of the svd and the product of diagonal entries pode test is proposed some interesting applications of the two svd approaches are presented and simulations verify their performance

on tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty we present efficiently verifiable sufficient conditions for the validity of specific np hard semi infinite systems of linear matrix inequalities lmis arising from lmis with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor in particular we prove that given an n times n interval matrix cal u rho a mid a ij a ij leq rho c ij one can build a computable lower bound accurate within the factor pi over 2 on the supremum of those rho for which all instances of cal u rho share a common quadratic lyapunov function we then obtain a similar result for the problem of quadratic lyapunov stability synthesis finally we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube

frequency analysis and synthesis of a class of nonlinear filters nonlinear filters are analyzed from a spectral point of view the equivalent frequency behavior of a filter is described by its linear part which represents the linear nonrecursive filter that best matches in the mean square sense the behavior of the filter the class considered includes all the order statistic filters l filters such as median midrange alpha trim and some new order statistic filters ll filters defined by the authors to generalize l filters and fir filters the formulation leads to a technique for imposing on the nonlinear filters a given spectral behavior a simulation of the design for a low pass ll filter is given

global convergence of the recursive kernel regression estimates with applications in classification and nonlinear system estimation an improved exponential bound on the l sub 1 error for the recursive kernel regression estimates is derived it is shown using the martingale device that weak strong and complete l sub 1 consistencies are equivalent consequently the conditions on a certain smoothing sequence are necessary and sufficient for strong l sub 1 consistency of the recursive kernel regression estimate the rates of global convergence are also given obtained results are applied to recursive classification rules and to nonlinear time series estimation

generalized fractional programming algorithms and numerical experimentation abstract several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case these algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case the most efficient algorithm is obtained by marrying the newton approach within the dinkelbach approach for fractional programming