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

nonlinear boundary value problems on semi infinite intervals abstract a boundary value problem approach to solving nonlinear differential equations on the half line is discussed several examples are presented to illustrate the ideas

sampling of generalized almost cyclostationary signals in this paper the problem of sampling a continuous time generalized almost cyclostationary gacs signal is addressed the class of such nonstationary signals includes as a special case the almost cyclostationary acs signals acs signals filtered by some linear time variant channels are further examples it is shown that the discrete time signal constituted by the samples of a gacs signal is a discrete time acs signal thus discrete time acs signals can arise not only from the sampling of continuous time acs signals but from the sampling of a wider class of nonstationary signals as well namely the continuous time gacs signals in the paper relationships between generalized cyclic statistics of a continuous time gacs signal and cyclic statistics of the discrete time acs signal constituted by its samples are derived the problem of aliasing in the domain of the cycle frequencies is considered and a condition assuring that the cyclic temporal moment function of the discrete time signal can be obtained by sampling that of the continuous time signal is determined finally it is shown that starting from the sampled signal the gacs or acs nature of the continuous time signal can be conjectured provided that the analysis parameters such as the sampling period padding factor and data record length are properly chosen

asymptotic performance analysis of cyclic detectors the paper deals with the analytical evaluation of the asymptotic detection and false alarm probabilities of multicycle and single cycle detectors operating in additive white gaussian noise with unknown and nonrandom spectral level which are based on the cyclostationarity properties of the signal to be intercepted the receiver operating characteristics are derived by using the asymptotic complex normality and the covariance expression of the sample average estimator of the cyclic covariance in an observed discrete time series a numerical example for interception of a binary phase shift keying signal is considered

high order moments and gaussianity of single carrier and ofdm signals this paper analyzes the high order moments of linearly modulated single carrier as well as orthogonal frequency division multiplexing ofdm signals assuming that the modulated symbols that form resulting continuous signal waveform are statistically independent for a single carrier case with the ideal low pass filter used for pulse shaping closed form expressions of the moments with up to the eighth order are derived the proposed approach is then extended to gaussianity assessment of ofdm signals we show how the moments of ofdm signals can approach to those of gaussian distribution as the number of subcarriers increases for a given modulation constraint on each subcarrier also reported is the peculiar nonstationary behavior of the fourth order moments in the case of bpsk modulated ofdm signals

cyclic statistic estimators with uncertain cycle frequencies for almost cyclostationary processes sufficient conditions are derived such that estimates of the second order cyclic probabilistic functions with estimated cycle frequencies are mean square consistent and asymptotically complex normal cyclic conjugate autocorrelation functions and cyclic conjugate spectra are considered under the derived conditions asymptotically as the data record length approaches infinity the estimates of cyclic statistics with estimated cycle frequencies have the same complex normal distribution as the case of exactly known cycle frequencies the results are applied to the detection of a moving cyclostationary source in the presence of strong doppler effect and for low values of snr

stochastic time frequency analysis using the analytic signal why the complementary distribution matters we challenge the perception that we live in a proper world where complex random signals can always be assumed to be proper also called circularly symmetric rather we stress the fact that the analytic signal constructed from a nonstationary real signal is in general improper which means that its complementary correlation function is nonzero we explore the consequences of this finding in the context of stochastic time frequency analysis in cohen s class there the analytic signal plays a prominent role because it reduces interference terms however the usual time frequency representation tfr based on the analytic signal gives only an incomplete signal description it must be augmented by a complementary tfr whose properties we develop in detail we show why it is still advantageous to use the pair of standard and complementary tfrs of the analytic signal rather than the tfr of the corresponding real signal

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

nonparametric time series analysis for periodically correlated processes correlation functions of continuous time periodically correlated processes can be represented by a fourier series with coefficient functions it is shown that the usual estimator for stationary covariances formed from a single sample path of the process can be simply modified to provide a consistent in quadratic mean estimator for any of the coefficient functions resulting from the aforementioned representation it is shown that if the process is gaussian and b sub k tau is a fourier integral with respect to a density function g sub k lambda a two dimensional periodogram formed from a single sample function can be smoothed along a line of constant difference frequency to provide a consistent estimator for g sub k lambda this natural extension of the well known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes

detection and source location of weak cyclostationary signals simplifications of the maximum likelihood receiver the problem of dual receiver interception of low snr signals which includes detection of the presence of a particular signal type and location of its source is considered in particular source location based on time difference of arrival tdoa measurements is examined and the low snr maximum likelihood receiver for joint detection and tdoa estimation is taken as a starting point by explicitly revealing the way in which this receiver exploits the spectral correlation properties of the cyclostationary signal several partial implementations with optimality properties of their own are proposed these greatly simplified implementations which require only a one dimensional search over the tdoa parameter are shown by simulation to perform competitively with the relatively complicated maximum likelihood receiver which requires a two or three dimensional search for phase shift keyed psk signals

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

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

discrete time estimation of second order statistics of generalized almost cyclostationary processes in this paper the problem of estimating second order statistics of continuous time generalized almost cyclostationary gacs processes is addressed gacs processes in the wide sense have autocorrelation function almost periodic in time whose generalized fourier series expansion has both frequencies and coefficients that depend on the lag shifts almost cyclostationary acs processes are obtained as a special case when the frequencies do not depend on the lag shifts acs processes filtered by doppler channels and communications signals with time varying parameters are further examples it is shown that continuous time gacs processes do not have a discrete time counterpart the discrete time cyclic cross correlogram of the discrete time acs processes obtained by uniformly sampling gacs processes is considered as estimator of samples of the continuous time cyclic cross correlation function the asymptotic performance analysis is carried out by resorting to the hybrid cyclic cross correlogram which is partially continuous time and partially discrete time its mean square consistency and asymptotic complex normality as the number of data samples approaches infinity and the sampling period approaches zero are proved under mild conditions on the regularity of the fourier series coefficients and the finite or practically finite memory of the processes expressed in terms of summability of cumulants it is shown that the asymptotic properties of the hybrid cyclic cross correlogram are coincident with those of the continuous time cyclic cross correlogram hence discrete time estimation does not give rise to any loss in asymptotic performance with respect to continuous time estimation

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

bounded stabilisation of stochastic port hamiltonian systems this paper proposes a stochastic bounded stabilisation method for a class of stochastic port hamiltonian systems both full actuated and underactuated mechanical systems in the presence of noise are considered in this class the proposed method gives conditions for the controller gain and design parameters under which the state remains bounded in probability the bounded region and achieving probability are both assignable and a stochastic lyapunov function is explicitly provided based on a hamiltonian structure although many conventional stabilisation methods assume that the noise vanishes at the origin the proposed method is applicable to systems under persistent disturbances

blind identification of second order volterra systems with complex random inputs using higher order cumulants in this correspondence closed form expressions for the blind identification of linear quadratic volterra systems are established the system is excited by a complex valued random sequence and the output cumulants of order up to 4 are employed it is assumed that the memory of the linear part is greater than or equal to the memory of the quadratic part cumulant based formulas are developed demonstrating that the system is uniquely identifiable an svd based variant with improved performance is also derived simulations and comparisons with existing techniques are presented

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

compressive temporal higher order cyclostationary statistics the application of nonlinear transformations to a cyclostationary signal for the purpose of revealing hidden periodicities has proven to be useful for applications requiring signal selectivity and noise tolerance the fact that the hidden periodicities referred to as cyclic moments are often compressible in the fourier domain motivates the use of compressive sensing cs as an efficient acquisition protocol for capturing such signals in this paper we consider the class of temporal higher order cyclostationary statistics thocs estimators when cs is used to acquire the cyclostationary signal assuming compressible cyclic moments in the fourier domain we develop a theoretical framework for estimating thocs using the low rate nonuniform sampling protocol from cs and illustrate the performance of this framework using simulated data

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

central limit theorem in the functional approach the central limit theorem is proved within the framework of the functional approach for signal analysis in this framework a signal is modeled as a single function of time rather than a stochastic process distribution function expectation and all the familiar probabilistic parameters are built starting from this single function of time by resorting to the concept of relative measure furthermore the concept of independence among functions of time can be introduced in the paper it is shown that if a sequence of independent signals fulfills some mild regularity assumptions then the asymptotic distribution of the appropriately scaled average of such signals has a limiting normal distribution the approach is shown to be useful when only one realization of a signal is available and no ensemble of realizations is observed or exists the obtained results also allow one to rigorously justify stochastic models for signals and channels that up to now have been derived starting from a deterministic description of phenomena and for which the inferred stochastic model is built invoking a not proved ergodicity property an application to the statistical characterization of the output signal of a multipath doppler channel is 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

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

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

blind wideband spatio temporal filtering based on higher order cyclostationarity properties in this paper a new method for blind minimum mean square error mmse spatio temporal signal filtering is proposed to obtain a blind version of the mmse filter the proposed method utilizes the signal selectivity property of the higher order cyclostationary statistics hocs exhibited by the signal of interest an accurate estimate of the transfer function of the mmse filter is obtained also in the presence of strong frequency overlapped interfering signals which would render standard stationarity based techniques ineffective the method is useful when the desired signal cannot be accurately extracted by exploiting its second order cyclostationarity properties because they are too weak and or they are shared by the interference and noise signals as well the performance analysis which is carried out by computer simulations in the case of quadrature amplitude modulated qam modulated signals with partially overlapping bands substantiates the effectiveness of the proposed method

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

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