fundamental limitations in passive time delay estimation part i narrow band systems time delay estimation of a noise like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar sonar applications a new method is presented to analyze the mean square error performance of delay estimation schemes based on a modified improved version of the ziv zakai lower bound zzlb this technique is shown to yield the tightest results on the attainable system performance for a wide range of signal to noise ratio snr conditions for delay estimation using narrow band ambiguity prone signals the fundamental result of this study is illustrated in fig 3 the entire domain of snr is divided into several disjoint segments indicating several distinct modes of operation if the available snr does not exceed snr 1 signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation as a result the attainable mean square error bar epsilon 2 is bounded only by the a priori parameter domain if snr 1 2 the modified zzlb coincides with the barankin bound in this regime differential delay observations are subject to ambiguities if snr snr 3 the modified zzlb coincides with the cramer rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved the transition from the ambiguity dominated mode of operation to the ambiguity free mode of operation starts at snr 2 and ends at snr 3 this is the threshold phenomenon in time delay estimation the various deflection points snr i and the various segments of the bound fig 3 are given as functions of such important system parameters as time bandwidth product wt signal bandwidth to center frequency ratio w 0 and the number of half wavelengths of the signal center frequency contained in the spacing between receivers with this information the composite bound illustrated in fig 3 provides the most complete characterization of the attainable system performance under any prespecified snr conditions

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

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

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

time delay estimation using the cross bispectrum the cross bispectrum phase can be effectively used to estimate the time required for a nongaussian signal to propagate between a pair of spatially separated sensors in the presence of highly correlated gaussian noise the authors present a consistent estimator of the phase of the cross bispectrum derive the exact distribution of the phase of a complex gaussian sample bispectrum and show that in most cases the exact distribution can be approximated by a gaussian distribution using this gaussian approximation the authors derive the variance of the time delay estimate computed from the sample cross bispectrum of a signal in additive correlated noise these results allow the performance of time delay estimators based on the cross bispectrum phase to be quantified as a function of the sample size the skewness of the signal the signal to noise ratio snr and the noise correlation

exploitation of cyclostationarity for identifying the volterra kernels of nonlinear systems a class of random time series inputs for nonlinear time invariant systems that permit the analytical specification of a set of operators on the input that are orthonormal over all time to the volterra operators for all orders and all lag sets is introduced the time series in this class are cyclostationary and complex valued the orthonormal operators are used to obtain an input output type of cross correlation formula for identifying the individual volterra kernels of arbitrary order for a nonlinear system of possibly infinite order and possibly infinite memory the real parts of the complex valued inputs in this class comprise a class of real valued inputs for which the same sets of specified operators apply however the orthogonality for different orders holds for these real inputs only for volterra operators of order less than the order of the specified operator thus these real inputs can be used to identify volterra kernels only for finite order systems frequency domain counterparts of the time domain methods that can utilize an fft algorithm are developed

signal reconstruction from the phase of the bispectrum the authors present a simple procedure the bispectrum signal reconstruction bsr algorithm to recover the fourier phase of a signal from the phase of its bispectrum by simple analogy a procedure that recovers the fourier magnitude of a signal from the magnitude of its bispectrum is also presented in addition the authors propose an iterative scheme the bicepstrum iterative reconstruction algorithm bira for the reconstruction of a finite impulse response fir sequence from only the phase of its bispectrum and they demonstrate how some a priori information on the energy of the cepstra coefficients can improve significantly the convergence rate of the algorithm both schemes are based on the key observation that the differences of the bispectrum coefficients contain all the information concerning the fourier phase of the signal whereas their sums contain the fourier magnitude information

blind equalization of digital communication channels using high order moments the authors describe algorithms for blind equalization of digital communication channels of the quadrature amplitude modulation qam type these algorithms are based on the fourth order statistical moments of the received data sequence the first of the two is a linear least squares type algorithm the second algorithm is of the nonlinear least squares type the algorithms use the fourth order statistical moments of the symbol sequence to explicitly estimate the channel impulse response the estimated impulse response is used in turn to construct a linear mean square error equalizer the performance of this equalizer is not optimal in any sense but it is adequate for channels with mild intersymbol interference or when the number of data points available for estimating the channel response is very large

non criticality criteria for abelian sandpile models with sources and sinks we prove that the abelian sandpile model on a random binary and binomial tree as introduced in redig ruszel and saada j stat phys 147 653 677 2012 is not critical for all branching probabilities p 1 by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps we obtain some more information about the exponential tail of the avalanche radius next we study the sandpile model on zd with some additional dissipative sites we provide examples and sufficient conditions for non criticality we also make a connection with the parabolic anderson model finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non criticality in the presence of a finite number of sources using a connection with the homogeneous pinning model

a comment on the finite memory of stochastic processes it is shown that a proposed concept of finite memory for a zero mean strictly stationary stochastic process results in a stochastic process of random variables each of which is almost surely equal to zero

estimation of frequency delay of arrival fdoa using fourth order statistics in unknown correlated gaussian noise sources a new family of nonparametric and parametric methods based on fourth order statistics for the estimation of the frequency delay of arrival fdoa between two sensor signal measurements corrupted by correlated gaussian noise sources in an unknown way is presented the new family of fdoa estimation methods utilizes the fourth order cumulants or 1 d fourier transforms of sliced fourth order cumulants of the two signal measurements it is demonstrated that the new family of fdoa estimation methods suppresses the effect due to correlated gaussian measurement noises and outperforms existing second order statistics based fdoa estimation methods using either cross ambiguity function or music algorithm various simulation results are presented for different types of signals different lengths of data and different signal to noise ratios

time delay estimation with unknown spatially correlated gaussian noise the problem of estimating the difference in arrival times of a non gaussian signal at two spatially separated sensors is considered the signal is assumed to be corrupted by spatially correlated gaussian noises of unknown cross correlation the author proposes and analyzes two new classes of methods for time delay estimation based on higher order statistics the proposed methods are conceptually very similar to the traditional cross correlation based techniques in that the proposed criteria peak at a lag value equalizing true delay since the proposed methods are based upon higher order cumulant statistics of the data they result in estimators that remain unbiased in the presence of gaussian noise

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

strongly consistent coefficient estimate for errors in variables models for the single input single output siso errors in variables system it is assumed that the system input is an arma process and that the driven noise of the system input and the observation noise are jointly gaussian the two dimensional observation made on system input and output is represented as a two dimensional 2d arma system of minimum phase driven by a sequence of 2d i i d gaussian random vectors innovation representation it is shown that the resulting arma system is identifiable i e its coefficients are uniquely defined under reasonable conditions recursive algorithms are proposed for estimating coefficients of the arma representation including those contained in the original siso system the estimates are proved to be convergent to the true values with probability one and the convergence rate is derived as well the theoretical results are justified by numerical simulation

damping estimates from experimental non linear time series this paper seeks to illustrate the utility of the lyapunov spectrum in estimating the damping of an experimental non linear system a mechanical model of duffing s equation operating in the chaotic regime is used to generate a single observable using standard techniques from non linear time series analysis the complete lyapunov spectrum is estimated the sum of these exponents may via the divergence theorem be related directly to the coefficient of viscous damping estimations are performed in this manner for both a three and four dimensional response and results are compared to estimates taken from two linear based techniques the indication is that use of the lyapunov spectrum to obtain quantitative damping estimates is a comparable alternative to methods requiring transient data or detailed knowledge of the dynamics

time delay estimation in unknown gaussian spatially correlated noise a novel class of methods that estimate the difference in arrival time between signals corrupted by spatially correlated gaussian noise sources of unknown cross correlation is presented the methods are based on the idea of comparing the similarities between the two sensor measurements in higher order spectrum domains bispectrum rather than in the cross correlation domain it is demonstrated that estimation techniques based on higher order cumulants suppress the effect of correlated gaussian noise sources and therefore exhibit improved performance over generalized cross correlation methods results are reported for different types of signals lengths of data records and signal to noise ratios

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

the abelian sandpile model on an infinite tree we consider the standard abelian sandpile process on the bethe lattice we show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume markov process exhibiting features of self organized criticality

common pitfalls in the application of stationary process theory to time sampled and modulated signals the common practice of applying the theory of stationary stochastic processes to a cyclostationary process by introducing random phase s into the probabilistic model in order to stationarize the process can lead to erroneous results such as incorrect formulas for power spectral density this is illustrated by showing that commonly used formulas for signals that have undergone frequency conversion or time sampling can be incorrect the source of error is shown to be inappropriate phase randomization procedures the correct procedure is described and corrected formulas are given the problem is further illustrated by showing that commonly used resolution and reliability mean and variance formulas for spectrum analyzers must be corrected for cyclostationary signals it is explained that all corrections to formulas reflect the effects of spectral correlation these effects are inappropriately averaged out by inappropriate phase randomization procedures it is further explained that these inappropriate procedures destroy the important property of ergodicity of the probabilistic model

nonparametric cyclic polyspectral analysis of am signals and processes with missing observations by viewing discrete time amplitude modulated signals and processes with missing observations as cyclostationary signals nonparametric mean square sense consistent and asymptotically normal single record estimators are developed for their kth order cumulants and polyspectra along with the asymptotic covariances the proposed estimation schemes use cyclic cumulants and polyspectra and are theoretically insensitive to any additive stationary noise in addition schemes of order k or 3 convey complete phase information and are insensitive to additive cyclostationary gaussian noise of unknown covariance the conventional approaches cannot recover mixed phase linear processes are susceptible to additive noise and are a special case of the proposed schemes simulations demonstrate superior performance of the proposed algorithms

stochastic system identification with noisy input output measurements using polyspectra two new classes of parametric frequency domain approaches are proposed for estimation of the parameters of scalar linear errors in variables models i e linear systems where measurements of both input and output of the system are noise contaminated the first approach consists of linear estimators where using the bispectrum or the integrated polyspectrum of the input and the cross bispectrum or the integrated cross polyspectrum the system transfer function is first estimated at a number of frequencies exceeding one half the number of unknown parameters the estimated transfer function is then used to estimate the unknown parameters using an overdetermined linear system of equations in the second class of approaches quadratic transfer function matching criteria are optimized by using the results of the linear estimators as initial guesses both classes of the parameter estimators are shown to be consistent in any measurement noise that has symmetric probability density function when the bispectral approaches are used the proposed parameter estimators are shown to be consistent in gaussian measurement noise when trispectral approaches are used

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

a unifying maximum likelihood view of cumulant and polyspectral measures for non gaussian signal classification and estimation classification and estimation of non gaussian signals observed in additive gaussian noise of unknown covariance are addressed using cumulants or polyspectra by integrating ideas from pattern recognition and model identification asymptotically optimum maximum likelihood classifiers and arma autoregressive moving average parameter estimators are derived without knowledge of the data distribution identifiability of noncausal and nonminimum phase arma models is established using a finite number of cumulant or polyspectral lags of any order greater than two a unifying view of cumulant and polyspectral discriminant measures utilizes these lags and provides a common framework for development and performance analysis of novel and existing estimation and classification algorithms tentative order determination and model validation tests for non gaussian arma processes are described briefly illustrative simulations are also presented

a test for the absence of aliasing or local white noise in locally stationary wavelet time series aliasing is often overlooked in time series analysis but can seriously distort the spectrum autocovariance and their estimates we show that dyadic subsampling of a locally stationary wavelet process which can cause aliasing results in a process that is the sum of asymptotic white noise and another locally stationary wavelet process with a modified spectrum we develop a test for the absence of aliasing in a locally stationary wavelet series at a fixed location and illustrate it on simulated data and a wind energy time series a useful by product is a new test for local white noise the tests are robust to model misspecification in that it is unnecessary for the analysis and synthesis wavelets to be identical hence in principle the tests work irrespective of which wavelet is used to analyze the time series though in practice there is a tradeoff between increasing statistical power and time localization of the test

the complex cepstrum and bicepstrum analytic performance evaluation in the presence of gaussian noise an analytic performance evaluation of the complex cepstrum power cepstrum bicepstrum and power bicepstrum is presented by providing approximate expressions of the bias and variance of the cepstrum parameters due to the presence of gaussian noise the approximations are based on the asymptotic properties of bispectrum estimates and they are valid for moderate signal to noise ratios snrs the assumed model consists of a deterministic signal e g multiple echoes in additive white gaussian noise and finite length data the validity of the analytic expressions is verified with monte carlo simulations performance comparisons between bicepstrum and cepstrum methods suggest that the improved performance of the bicepstrum is a function of snr a omega m and n

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

nonparametric bispectrum based time delay estimators for multiple sensor data when estimating time delays of signals from sensor outputs the bispectrum has advantages when the signal has a skewed distribution the authors provide nonparametric estimation approaches which exploit the two dimensional nature of the bispectrum when dealing with observations from more than a pair of sensors the new algorithms have computational advantages and in some cases show improved performance compared to the conventional nonparametric bispectrum based time delay estimation approach

maximum likelihood time delay estimation in non gaussian noise in a non gaussian noise environment it is theoretically possible to design a delay estimator that performs significantly better than the conventional linear correlator we study the maximum likelihood estimator for passive time delay in non gaussian noise we show that the form of the best estimator depends strongly on signal to noise ratio snr and the estimator optimal at low snr can fail catastrophically at high values of snr the paper uses simulations to examine this sensitivity problem and proposes an ad hoc instrumentation that is reasonably robust over a wide range of snr values

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

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

bearing estimation in the bispectrum domain a new array processing method is presented for bearing estimation based on the cross bispectrum of the array output data the method is based on the asymptotic distribution of cross bispectrum estimates and uses maximum likelihood theory it is demonstrated that when the noise additive sources are spatially correlated gaussian with unknown cross spectral matrix csm the cross bispectrum method provides better bearing estimates than the stochastic maximum likelihood method with known csm analytical studies and simulations are given to document the performance of the new method

the bispectrum of sampled data part 1 detection of the sampling jitter the effect of timing error jitter on the power spectrum second order spectrum of sampled data has been investigated previously the current increasing interest in higher order spectra calls for investigation of the effect of sam pling jitter on the discrete higher than order two spectra in this paper we present results concerning the bispectrum third order spectrum we derive expressions for the bispectrum of sampled data under the assumption that the timing errors are independently identically distributed i i d random variables it is shown that while the discrete bispectrum of a uniformly sampled third order stationary signal is zero in a triangle that is a proper subset of the principal domain it differs from zero in the presence of jitter exploiting this effect we propose a test for detection of sampling jitter using the bispectrum namely given estimates of the discrete bispectrum of an un known band limited signal we can decide whether jitter is present or not the performance of the proposed detector is analyzed and its dependence on the jitter variance signal skewness and observation time is shown

suboptimal detection of non gaussian signals by third order spectral analysis the use of higher order spectra hos is proposed for improving detection performance in non gaussian signals the method comprises two stages first the higher order spectra of the received signal are estimated using conventional spectral estimation techniques then a maximum likelihood ratio test lrt is performed in the higher order spectra domain it is shown that under certain low signal to noise ratio snr conditions the hos based method performs much better than the conventional energy one the required processor is derived and its performance is analyzed while the method is demonstrated using the third order spectrum called bispectrum it can easily be extended to higher order analysis e g trispectrum etc

adaptive multistage beamforming using cyclic higher order statistics chos a new blind beamforming algorithm for signals that exhibit higher order cyclostationarity is presented exploiting some previous theoretical developments we show how cyclic cumulants of the received signals can be used to obtain the weights of the beamformer that perform blind extraction the method is based on a spatial interpretation of a deconvolution procedure known as the super exponential algorithm the basic block processing algorithm is made fully adaptive using an adaptive urv scheme and applied to a typical mobile communications scenario where several cochannel interferers corrupt the signals of interest

detection of non gaussian signals in non gaussian noise using the bispectrum the problem of detecting a non gaussian time series in the presence of additive gaussian or non gaussian noise is cast into a classical hypothesis testing framework using the sample bispectrum as the test statistic the power of the test is demonstrated as a function of signal to noise ratio the degree of skewness of the signal and processing parameters the results are compared to the power of a classical energy detection test it is concluded that the bispectrum can be used effectively to detect non gaussian signals in the presence of interfering noise and that it may perform better depending on the degree of non gaussianity than energy detection

identification of systems with cyclostationary input and correlated input output measurement noise an alternative approach to wiener s optimum filtering theory for stationary random time series that is applicable when the system input exhibits cyclostationarity is presented this alternative approach can in principle eliminate performance degradation due to input output corruption this method of system identification is called the speccorr spectral correlation ratio method since the cross spectra in the ratio are actually spectral correlation functions an application to interference tolerant time difference of arrival estimation is described and illustrated by simulation results showing its effectiveness for system identification with severely corrupted input output measurements a comparison of wiener s method to the speccorr method shows the superior performance of the latter

time delay estimation via cross correlation in the presence of large estimation errors the estimate of the difference in time of arrival of a common random signal received at two sensors each of which also receives uncorrelated noise is examined for both small and large estimation errors it is shown that as the post integration signal to noise ratio decreases the correlator exhibits a thresholding effect that is the probability of a large error an anomalous estimate increases rapidly approximate theoretical results for the probability of an anomaly are presented and are verified experimentally the variance of the time delay estimate is examined for both a gated mode in which the time delay corresponding to the correlation peak closest to the true time delay is used as the estimate of time delay and an ungated mode in which the time delay corresponding to the largest peak over the full range of the correlator delay times is used as the estimate the observed variance for both modes is compared with the theoretical variance based on a small error analysis for the gated modes the signal to noise ratio below which the observed variance begins to differ significantly from the small error theory can be reliably predicted from a linearity criterion it is shown however that the expected variance for the ungated mode can depart from the small error theory at a higher signal to noise ratio than for the gated modes thus the variance due to anomalies can be the most important factor in determining the region of applicability of the small error analysis

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

mean field avalanche size exponent for sandpiles on galton watson trees we show that in abelian sandpiles on infinite galton watson trees the probability that the total avalanche has more than t topplings decays as t 1 2 we prove both quenched and annealed bounds under suitable moment conditions our proofs are based on an analysis of the conductance martingale of morris probab theory relat fields 125 259 265 2003 that was previously used by lyons et al electron j probab 13 58 1702 1725 2008 to study uniform spanning forests on mathbb z d d ge 3 and other transient graphs

how to extract lyapunov exponents from short and noisy time series in this correspondence we discuss an algorithm for the extraction of lyapunov spectra from short and noisy time series initially we briefly describe an existing algorithm and include a method for improving the robustness to noise of this algorithm the problem of short time series is then addressed and a novel concatenation technique is presented and combined with the extraction algorithm previously utilized results of applying this extended algorithm to chaotic time series are presented and the effects of noise and data length are discussed

linear time variant transformations of generalized almost cyclostationary signals i theory and method the problem of linear time variant filtering is addressed in the fraction of time fot probability framework the adopted approach which is an alternative to the classical stochastic one provides a statistical characterization of the system in terms of time averages of functions of time rather than ensemble averages of stochastic processes thus it is particularly useful when stochastic systems transform ergodic input signals into nonergodic output signals as it happens with several channel models encountered in practice the analysis is carried out with reference to the wide class of the generalized almost cyclostationary signals which includes as a special case the class of almost cyclostationary signals in this paper systems are classified as deterministic or random in the fot probability framework moreover the new concept of expectation in the fot probability framework of the impulse response function of a system is introduced for the linear time variant systems the higher order system characterization in the time domain is provided in terms of the system temporal moment function which is the kernel of the operator that transforms the additive sinewave components contained in the input lag product into the additive sinewave components contained in the output lag product moreover the higher order characterization in the frequency domain is also provided and input output relationships are derived in terms of temporal and spectral moment and cumulant functions developments and examples of application of the theory introduced here are presented in part ii of this two part paper

signal interception a unifying theoretical framework for feature detection the unifying framework of the spectral correlation theory of cyclostationary signals is used to present a broad treatment of weak random signal detection for interception purposes the relationships among a variety of previously proposed ad hoc detectors optimum detectors and newly proposed detectors are established the spectral correlation plane approach to the interception problem is put forth as especially promising for detection classification and estimation in particularly difficult environments involving unknown and changing noise levels and interference activity a fundamental drawback of the popular radiometric methods in such environments is explained

passive time delay estimation in non gaussian noise this article deals with the structure of the maximum likelihood ml estimator for time delay with arbitrary signal and noise statistics at high signal to noise ratios snrs the ml estimation performs a nonlinear operation on the delayed difference of the two received waveshapes the required nonlinearity depends only on the noise statistics at low snr a closed form simple expression for the ml which depends only on the noise statistics and on the second order statistics of the signal is provided with statistically independent noise processes the estimator correlates two vectors generated by separate nonlinear operations on the two received waveshapes

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 cross bicepstrum definition properties and application for simultaneous reconstruction of three nonminimum phase signals the recovery of three signals from their cross bispectrum or the identification of the impulse responses of three parallel linear time invariant lti systems from the cross bispectrum of their system outputs by computing the complex cepstrum of the cross bispectrum as long as the signals or impulse responses have no zeros on the unit circle is discussed it is shown that the three signals can be separated completely and approximately recovered in the cross bicepstrum domain except for their magnitude and linear phase factors the computation of the cross bicepstrum can be seen as a method for the simultaneous computation of the ordinary complex cepstra of three nonminimum phase signals without the need for phase unwrapping both least squares and fast fourier transform fft based methods for computing the bicepstral coefficients are presented simulation examples of signal reconstruction in gaussian white and nongaussian colored noise and of system identification are included the results are extended to nth order cross spectra and the factorization problem for these spectra is discussed

the abelian sandpile model on a random binary tree we study the abelian sandpile model on a random binary tree using a transfer matrix approach introduced by dhar and majumdar we prove exponential decay of correlations and in a small supercritical region i e where the branching process survives with positive probability exponential decay of avalanche sizes this shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes our main technical tools are 1 a recursion for the ratio between the numbers of weakly and strongly allowed configurations which is proved to have a well defined stochastic solution 2 quenched and annealed estimates of the eigenvalues of a product of n random transfer matrices

direction finding algorithms based on high order statistics two direction finding algorithms are presented for nongaussian signals which are based on the fourth order cumulants of the data received by the array the first algorithm is similar to music while the second is asymptotically minimum variance in a certain sense the first algorithm requires singular value decomposition of the cumulant matrix while the second is based on nonlinear minimization of a certain cost function the performance of the minimum variance algorithm can be assessed by analytical means at least for the case of discrete probability distributions of the source signals and spatially uncorrelated gaussian noise the numerical experiments performed seem to confirm the insensitivity of these algorithms to the gaussian noise parameters

on time delay estimation with unknown spatially correlated gaussian noise using fourth order cumulants and cross cumulants the problem of estimating the difference in arrival times of a linear non gaussian signal at two spatially separated sensors is considered the signal is assumed to be corrupted by spatially correlated gaussian noises of unknown cross correlation a parameter estimation approach is adopted and the fourth order cumulant statistics of the noisy measurements are exploited to obtain the time delay estimate specific and mild sufficient conditions are given for time delay identifiability via a persistence of excitation condition illustrative monte carlo simulation examples are also presented

the higher order theory of generalized almost cyclostationary time series in this paper the class of generalized almost cyclostationary gacs time series is introduced time series belonging to this class are characterized by multivariate statistical functions that are almost periodic functions of time whose fourier series expansions can exhibit coefficients and frequencies depending on the lag shifts of the time series moreover the union over all the lag shifts of the lag dependent frequency sets is not necessarily countable almost cyclostationary acs time series turn out to be the subclass of gacs time series for which the frequencies do not depend on the lag shifts and the union of the above mentioned sets is countable the higher order characterization of gacs time series in the strict and wide sense is provided it is shown that the characterization in terms of cyclic moment and cumulant functions is inadequate for those gacs time series that are not acs then generalized cyclic moment and cumulant functions in both the time and frequency domains are introduced finally the problem of estimating the introduced generalized cyclic statistics is addressed and two examples of gacs time series are considered

on the principal domain of the discrete bispectrum of a stationary signal the paper presents a simplifying yet general approach to determining the symmetry structure of a bispectrum the principal domain pd of the bispectrum and its region of positive support are derived in a way that illuminates the controversy surrounding the triangle in the pd which is called the outer triangle ot by hinich and wolinsky 1988 where the bispectrum is zero for a stationary random sampled process that is not aliased the basic statistical issues of testing for nonzero bispectral structure are reviewed

abelian sandpile model on the bethe lattice the authors study bak tang and wiesenfeld s abelian sandpile model 1987 of self organised criticality on the bethe lattice exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained they also determine the probability distribution of the number of distinct sites that topple at least once the number of toplings at the origin and the total number of toplings in an avalanche the probability that an avalanche consists of more than n toplings varies as n 1 2 for large n the probability that its duration exceeds t decreases as 1 t for large t these exponents are the same as for the critical percolation clusters in mean field theory

complex ambiguity functions using nonstationary higher order cumulant estimates the complex ambiguity function based on second order statistics caf sos has been used to simultaneously estimate the frequency delay of arrival fdoa and time delay of arrival tdoa between two signal measurements its performance however is sensitive to the correlation between two additive noise sources when the noise sources are assumed to be gaussian we develop a new complex ambiguity function based on higher order statistics caf hos that reduces the unknown noise correlation effect the new caf hos algorithm utilizes nonstationary higher order cross cumulant estimates and their fourier transform in fact we suggest a nonstationary estimate of fourth order cross cumulants and obtain the analytical expressions for its mean value and variance we compare the analytical expressions with results obtained by monte carlo runs also we compare the performance of the new complex ambiguity function based on fourth order statistics caf fos against the caf sos algorithm using different gaussian noise sources different signals of interest different signal to noise ratios and different lengths of data

cyclic wiener filtering theory and method conventional time and space filtering of stationary random signals which amounts to forming linear combinations of time translates and space translates exploits the temporal and spatial coherence of the signals by including frequency translates as well the spectral coherence that is characteristic of cyclostationary signals can also be exploited some of the theoretical concepts underlying this generalized type of filtering called frequency shift fresh filtering are developed the theory of optimum fresh filtering which is a generalization of wiener filtering called cyclic wiener filtering is summarized and the theory is illustrated with specific examples of separating temporally and spectrally overlapping communications signals including am bpsk and qpsk the structures and performances of optimum fresh filters are presented and adaptive adjustment of the weights in these structures is discussed also specific results on the number of digital qam signals that can be separated as a function of excess bandwidth are obtained

a new method for adaptive time delay estimation for non gaussian signals a novel adaptive scheme for time delay estimation is introduced for signal environments where the signal is non gaussian and the additive noise sources are spatially correlated gaussian with unknown power spectrum characteristics this scheme is based on parametric modeling between two sensor measurements and employs higher order statistics third or fourth order of the data it is demonstrated by means of extensive simulations that this scheme works well for both stationary and nonstationary cases as expected it outperforms the cross correlation based gradient method for time delay adaptation in spatially correlated gaussian noises the proposed scheme is compared to the overdetermined recursive instrumental variable method and is shown to exhibit substantially less computational complexity

the cumulant theory of cyclostationary time series i foundation the problem of characterizing the sine wave components in the output of a polynomial nonlinear system with a cyclostationary random time series input is investigated the concept of a pure nth order sine wave is introduced and it is shown that pure nth order sine wave strengths in the output time series are given by scaled fourier coefficients of the polyperiodic temporal cumulant of the input time series the higher order moments and cumulants of narrowband spectral components of time series are defined and then idealized to the case of infinitesimal bandwidth such spectral moments and cumulants are shown to be characterized by the fourier transforms of the temporal moments and cumulants of the time series it is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts to put the contributions of the paper in perspective a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided

analysis of self noise in a fourth power clock regenerator a method for computing the power spectra of the in phase and quadrature components of self noise in a fourth power clock recovery scheme is presented this method is shown to be effective if the symbol waveform is time limited or can be so approximated the method applies to baseband signaling schemes such as pulse amplitude modulation and to modulated carrier schemes such as quadrature amplitude modulation comparison is made with the squaring type recovery technique using cosine rolloff nyquist pulses on the binary phase shift keyed mode at several excess bandwidth factors alpha it is shown that when alpha is less than 0 4 the fourth power circuit gives less time jitter than the squarer for a noiseless channel the method for self noise spectrum calculation is verified by computer simulation a prefilter that completely eliminates time jitter in the fourth power clock recovery circuit is described

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

a computationally efficient and interference tolerant nonparametric algorithm for lti system identification based on higher order cyclostationarity a new nonparametric algorithm for the identification of linear time invariant systems is proposed the method is based on the cyclic correlations of the input and output signals with a nonlinear transformation of the input signal consequently although it exploits the higher order cyclostationarity properties of the input and output signals its computational complexity is comparable with that of methods based on second order statistics the proposed estimator of the system transfer function is inherently immune to the presence of noise and interference on both input and output signal measurements and turns out to be asymptotically unbiased and consistent moreover bias and variance of the estimate exhibit a rate of convergence to zero equal to that of estimates based on second order statistics finally simulation results show that the proposed algorithm significantly outperforms in terms of both bias and variance of the estimates several nonparametric identification algorithms previously presented in the literature

image motion estimation algorithms using cumulants a class of algorithms is presented that estimates the displacement vector from two successive image frames consisting of signal plus noise in the model the signals are assumed to be either non gaussian or quasistationary deterministic and via a consistency result for cumulant estimators the authors unify the stochastic and deterministic signal viewpoints the noise sources are assumed to be gaussian perhaps spatially and temporally correlated and of unknown covariance viewing image motion estimation as a 2d time delay estimation problem the displacement vector of a moving object is estimated by solving linear equations involving third order auto cumulants and cross cumulants additionally a block matching algorithm is developed that follows from a cumulant error optimality criterion finally the displacement vector for each pel is estimated using a recursive algorithm that minimizes a mean 2d fourth order cumulant criterion simulation results are presented and discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

blind identification of simo systems and simultaneous estimation of multiple time delays from hos based inverse filter criteria higher order statistics based inverse filter criteria ifc have been effectively used for blind equalization of single input multiple output simo systems recently chi and chen reported a relationship between the unknown simo system and the optimum equalizer designed by the ifc for finite signal to noise ratio snr in this paper based on this relationship an iterative fast fourier transform fft based nonparametric blind system identification bsi algorithm and an fft based multiple time delay estimation mtde algorithm are proposed with a given set of non gaussian measurements the proposed bsi algorithm allows the unknown simo system to have common subchannel zeros and its performance estimation accuracy is superior to that of the conventional ifc based methods the proposed mtde algorithm can simultaneously estimate all the p 1 time delays with respect to a reference sensor with space diversity of sensors exploited therefore its performance estimation accuracy is robust to the nonuniform distribution of snrs of p spl ges 2 sensors due to channel fading some simulation results are presented to support the efficacy of the proposed bsi algorithm and mtde algorithm