polyspectral analysis of mixed processes and coupled harmonics polyspectral analysis of processes with mixed spectra is considered and scaled polyperiodograms are introduced to clarify issues related to stationarity ergodicity and suppression of additive stationary noise in harmonic retrieval problems spectral and polyspectral approaches are capable of retrieving un coupled harmonics not only when the harmonics have constant amplitudes but also when they are observed in nonzero mean multiplicative noise fourier series polyspectra and asymptotic properties of scaled polyperiodograms provide general tools for higher order analysis of time series with mixed spectra a single record phase coupling detector is derived to obviate the assumption of independent multiple records required by existing methods the novelties are illustrated by simulation examples

asymptotic relative efficiencies of multistage tests corresp asymptotic relative efficiencies of two stage and multistage tests of a simple null hypothesis versus a simple location alternative are considered observations are taken in groups not necessarily of the same size and a test statistic is computed for comparison with thresholds in a two stage test if the test statistic of the first group of observations exceeds an upper threshold the alternative is accepted and if it crosses a lower threshold the null hypothesis is accepted otherwise a second group of samples is taken and a second test is performed two different classes of two stage tests are considered one of them computes the test statistic in the second stage from observations in the second group alone while the other uses both the first and the second groups of observations it is shown that these tests are asymptotically more efficient than fixed sample size tests but are less efficient than sequential probability ratio tests with proper choices of parameters the improvement over fixed sample size tests can be significant especially when the error probabilities are small however the complexity of two stage tests is comparable to that of fixed sample size tests making their use desirable the efficiency of k stage tests k 2 is also investigated with the conclusion that the behavior of a two stage test can be enhanced by adding stages up to a point beyond which the effect of adding stages diminishes

statistical tests for presence of cyclostationarity the presence of kth order cyclostationarity is defined in terms of nonvanishing cyclic cumulants or polyspectra relying upon the asymptotic normality and consistency of kth order cyclic statistics asymptotically optimal spl chi sup 2 tests are developed to detect the presence of cycles in the kth order cyclic cumulants or polyspectra without assuming any specific distribution on the data constant false alarm rate tests are derived in both time and frequency domain and yield consistent estimates of possible cycles present in the kth order cyclic statistics explicit algorithms for k spl les 4 are discussed existing approaches are rather empirical and deal only with k spl les 2 case simulation results are presented to confirm the performance of the given tests

locally optimum detection of signals in a generalized observation model the random signal case locally optimum detectors for weak random signals are derived for a generalized observation model incorporating signal dependent and multiplicative noise it is shown that the locally optimum random signal detectors in the generalized observation model are interesting generalizations of those that would be obtained in the purely additive noisy signal model examples of explicit results for the locally optimum detector test statistics are given for some typical cases both asymptotic and finite sample size performances of the locally optimum detectors are considered and compared with those of other standard detector structures

sequential cfar detectors using a dead zone limiter the performances of some proposed sequential constant false alarm rate cfar detectors are evaluated the observations are passed through a dead zone limiter the output of which is 1 0 or 1 depending on whether the input is less than c between c and c or greater than c where c is a constant the test statistic is the sum of the outputs the test is performed on a reduced set of data those with absolute value larger than c with the test statistic being the sum of the signs of the reduced set of data both constant and linear boundaries are considered numerical results show a significant reduction of the average number of observations needed to achieve the same false alarm and detection probabilities as compared to a fixed sample size cfar detector using the same kind of test statistic

persistent stability of a chaotic system abstract we report that trajectories of a one dimensional model for inertial particles in a random velocity field can remain stable for a surprisingly long time despite the fact that the system is chaotic we provide a detailed quantitative description of this effect by developing the large deviation theory for fluctuations of the finite time lyapunov exponent of this system specifically the determination of the entropy function for the distribution reduces to the analysis of a schrodinger equation which is tackled by semi classical methods the system has generic instability properties and we consider the broader implications of our observation of long term stability in chaotic systems

performance analysis of blind carrier frequency offset estimators for noncircular transmissions through frequency selective channels this paper deals with the problem of blind estimation of the carrier frequency offset of a linearly modulated noncircular transmission through an unknown frequency selective channel a frequency estimator is developed based on the unique conjugate cyclic frequency of the received signal which is equal to twice the frequency offset consistency and asymptotic normality of the frequency estimator together with a closed form expression for its asymptotic variance are also established the closed form expression of the asymptotic variance enables analysis of the performance of the proposed frequency offset estimator as a function of the number of estimated cyclic correlation coefficients used it is shown that optimum is obtained if the number of correlation coefficients taken into account coincides with the degree of the channel numerical simulations are provided and confirm the conclusion of the theoretical asymptotic analysis

asymptotic analysis of blind cyclic correlation based symbol rate estimators this paper considers the problem of blind symbol rate estimation of signals linearly modulated by a sequence of unknown symbols oversampling the received signal generates cyclostationary statistics that are exploited to devise symbol rate estimators by maximizing in the cyclic domain a possibly weighted sum of modulus squares of cyclic correlation estimates although quite natural the asymptotic large sample performance of this estimator has not been studied rigorously the consistency and asymptotic normality of this symbol rate estimator is established when the number of samples n converges to infinity it is shown that this estimator exhibits a fast convergence rate proportional to n sup 3 2 and it admits a simple closed form expression for its asymptotic variance this asymptotic expression enables performance analysis of the rate estimator as a function of the number of estimated cyclic correlation coefficients and the weighting matrix a justification for the high performance of the unweighted estimator in high signal to noise scenarios is also provided

two stage wilcoxon detectors using conditional tests the idea of the wilcoxon detector using a conditional test is extended to two stage detectors which constitute a class of sequential detectors three different two stage detectors are considered using the wilcoxon statistic with conditional tests the three detectors differ in the usage of the samples of the first stage for computing the test statistic in the second stage in computing the test statistic in the second stage the first detector disregards all the samples of the first stage the second detector uses the value of the test statistic of the first stage and the third detector uses all the samples of the first stage the asymptotic relative efficiency improves from the first to the second detector comparisons of these detectors with fixed sample size fss detectors show that they are better than both the fss linear detector and the fss conditional wilcoxon detector under the same false alarm and detection probabilities

fourth order spectra of gaussian amplitude modulated sinusoids propagating underwater signals are sometimes amplitude modulated by the medium or other physical effects since this is a multiplicative phenomenon the resulting spectrum is a convolution of the spectra of the modulation and the desired signal a new method based on the spectrum of a special case of the fourth order cumulant is proposed to extract the desired signal it is shown that the special case of the fourth order cumulant is independent of the covariance of a gaussian modulating function and therefore it can extract the desired signal even when the modulating process is gaussian white noise convergence equations are derived which show that the fourth order cumulant and its special case will converge to their asymptotic forms as the data length increases the spectrum of the special case of the fourth order cumulant is also derived at the output of a low pass filter to demonstrate these theoretical results an experiment was conducted a sinusoid was modulated by white gaussian noise and transmitted through the water and received on an omnidirectional hydrophone the second order spectrum the spectrum of the special case of the fourth order moment and the spectrum of the special case of the fourth order cumulant were estimated from the filtered data the experiment corroborated the theoretical results by showing that the second order spectrum could not extract the sinusoidal frequency but the spectrum of the special case of the fourth order moment and the spectrum of the special case of the fourth order cumulant could simulations are included which further corroborate the theoretical results

power law distributions of particle concentration in free surface flows particles floating on the surface of a turbulent incompressible fluid accumulate along string like structures while leaving large regions of the flow domain empty this is reflected experimentally by a very peaked probability distribution function of c r the coarse grained particle concentration at scale r around c r 0 with a power law decay over two decades of c r pi c r proportional variant c r beta r the positive exponent beta r decreases with scale in the inertial range and stays approximately constant in the dissipative range thus indicating a qualitative difference between the dissipative and the inertial ranges of scales also visible in the first moment of c r

harmonics in multiplicative and additive noise performance analysis of cyclic estimators multiplicative noise causes smearing of spectral lines and thus hampers frequency estimation relying on conventional spectral analysis in contrast cyclic mean and correlation statistics have proved to be useful for harmonic retrieval in the presence of multiplicative and additive noise of arbitrary color and distribution performance analysis of cyclic estimators is carried through both for nonzero and zero mean multiplicative noises cyclic estimators are shown to be asymptotically equivalent to certain nonlinear least squares estimators and are also compared with the maximum likelihood ones large sample variance expressions of the cyclic estimators are derived and compared with the corresponding cramer rao bounds when the noises are white gaussian it is demonstrated that previously well established results on constant amplitude harmonics are special cases of the present analysis simulations not only validate the large sample performance analysis but also provide concrete examples regarding relative statistical efficiency of the cyclic estimators

performance analysis of cyclic statistics for the estimation of harmonics in multiplicative and additive noise the problem of estimating the frequencies of harmonics in multiplicative and additive noise is addressed the cyclic mean cm can be used if the multiplicative noise has nonzero mean the cyclic variance cv can be used whether or not the multiplicative noise has zero mean this paper answers the following question under what conditions should we use the cv instead of the cm the criteria used are the ease of detection and the accuracy of estimation the cv is preferable to the cm if the coherent to noncoherent harmonic power ratio is less than a threshold that depends on the first four cumulants when the noises are colored this threshold becomes frequency dependent third and fourth order cyclic statistics are also studied and it is shown that they will always be outperformed either by the cm or cv when the multiplicative noise is symmetric

detection of multiplicative noise in stationary random processes using second and higher order statistics this paper addresses the problem of detecting the presence of colored multiplicative noise when the information process can be modeled as a parametric arma process for the case of zero mean multiplicative noise a cumulant based suboptimal detector is studied this detector tests the nullity of a specific cumulant slice a second detector is developed when the multiplicative noise is nonzero mean this detector consists of filtering the data by an estimated ar filter cumulants of the residual data are then shown to be well suited to the detection problem theoretical expressions for the asymptotic probability of detection are given simulation derived finite sample roc curves are shown for different sets of model parameters

a modified sequential detection procedure a modified version of sequential testing is proposed and applied to discrete time signal detection the resulting detection procedure exhibits simplicity in structure and in analysis and retains most of the optimal features of sequential detection also two simple and efficient truncation schemes are suggested for the proposed detection procedure which is unrealizable without truncation

analytic behavior of the lms adaptive line enhancer for sinusoids corrupted by multiplicative and additive noise the least mean squares adaptive line enhancer lms ale has been widely used for the enhancement of coherent sinusoids in additive wideband noise this paper studies the behavior of the lms ale when applied to the enhancement of sinusoids that have been corrupted by both colored multiplicative and white additive noise the multiplicative noise decorrelates the sinusoid spreads its power spectrum and acts as an additional corrupting noise closed form expressions are derived for the optimum wiener filter ale output snr as a function of the residual coherent sine wave power the noncoherent sine wave power spectrum and the background additive white noise when the coherent to noncoherent sine wave power ratio is sufficiently small it is shown that a nonlinear e g square law transformation of the ale input results in a larger optimum ale output snr

harmonics in multiplicative and additive noise parameter estimation using cyclic statistics the problem of concern here is parameter estimation of harmonics in the presence of multiplicative and additive noise cyclic statistics are employed to estimate the frequencies and phases after which the time series is demodulated and cumulants of the noise processes are estimated the latter are then supplied to linear or nonlinear cumulant based algorithms to identify arma model parameters for the noises cyclic statistics and higher order spectra based approaches are shown to yield the same frequency estimates simulation examples illustrate the algorithms

sinusoidal signals with random amplitude least squares estimators and their statistical analysis the asymptotic properties of constrained and unconstrained least squares estimates of the parameters of a random amplitude sinusoid are analyzed an explicit formula for the asymptotic covariance matrix of the estimation errors is derived for both the constrained and unconstrained estimators accuracy aspects are investigated with the following main results for a certain weighting matrix which is shown to be the same for the constrained and unconstrained methods the estimation errors achieve their lower bounds it is proven that in the optimal case the constrained method always outperforms the unconstrained method it is also proven that the accuracy of the optimal estimators improves as the number of least squares equations increases a formula for the sample length needed for the asymptotic theory to hold is derived and its dependence on the lowpass modulating sequence is stressed simulations provide illustrations of the difference between the constrained and unconstrained estimators as well as the difference between the optimal and basic estimates the influence of the number of least squares equations and the characteristics of the lowpass envelope on the estimation accuracy is also investigated

a matrix contraction process we consider a stochastic process in which independent identically distributed random matrices are multiplied and where the lyapunov exponent of the product is positive we continue multiplying the random matrices as long as the norm of the product is less than unity if the norm is greater than unity we reset the matrix to a multiple of the identity and then continue the multiplication we address the problem of determining the probability density function of the norm we argue that in the limit as where and are two real parameters our motivation for analysing this matrix contraction process is that it serves as a model for describing the fine structure of strange attractors where a dense concentration of trajectories results from the differential of the flow being contracting in some region we exhibit a matrix product model for the differential of the flow in a random velocity field and show that there is a phase transition with the parameter changing abruptly from to as a parameter of the flow field model is varied

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

asymptotic theory of mixed time averages and kth order cyclic moment and cumulant statistics we generalize parzen s 1961 analysis of asymptotically stationary processes to mixtures of deterministic stationary nonstationary and generally complex time series under certain mixing conditions expressed in terms of joint cumulant summability we show that time averages of such mixtures converge in the mean square sense to their ensemble averages we additionally show that sample averages of arbitrary orders are jointly complex normal and provide their covariance expressions these conclusions provide us with statistical tools that treat random and deterministic signals on a common framework and are helpful in defining generalized moments and cumulants of mixed processes as an important consequence we develop consistent and asymptotically normal estimators for time varying and cyclic moments and cumulants of kth order cyclostationary processes and provide computable variance expressions some examples are considered to illustrate the salient features of the analysis