Coherence function of interrelated periodically nonstationary random processes

A coherence function characterizing the correlation between harmonic components of two signals that are described by periodically correlated random processes has been proposed. Such function is shown to be invariant with regard to linear transformations of signals.Aformula for coherence function is concretized for the amplitude- and phase-modulated signals.     

Spectral analysis of stationary components of periodically correlated random processes

The properties of stationary components of periodically correlated random processes separated by using zonal filtering have been considered. The properties of estimates of their correlation and spectral characteristics were investigated. The latter were built by using the Blackman-Tukey method.     

Coherent estimates of correlation characteristics of interconnected periodically correlated random processes

Brief analysis of correlation and spectral characteristics that describe interconnections between two periodically correlated random processes is conducted. Properties of coherent estimates of signals?? mutual correlation function and estimates of mutual correlation components are studied. Expressions for biases and dispersions of estimates are specified for amplitude modulated signals.     

Least squares method in the statistic analysis of periodically correlated random processes

The properties of least-squares estimates of mathematical expectations and correlation function of periodically correlated random processes (mathematical model of stochastic oscillations) have been investigated. The formulas defining the statistical characteristics of estimates were analyzed. In addition, examples were presented for illustrating the analysis of modulated signals.     

On the spectral coherence of nonstationary processes

It is shown that for a single path of a nonstationary process, reliable (i.e., low-variance) measurements of the degree of coherence between two spectral bands with substantial separation must be smaller than 1/2 unless the nonstationarity is of a special type. That is, the nonstationarity must be either of a known form (prior to measurement) or of a periodic nature, which is known as cyclostationarity. It is concluded that spectral correlation measurements on single time series are of highly questionable value for detection and characterization of nonstationarity, except in the special cases of cyclostationarity     

Properties of characteristics estimators of periodically correlated random processes in preliminary determination of the period of correlation

The coherent estimators of probabilistic characteristics of periodically correlated random processes with unknown period have been investigated. It is shown that these estimators are asymptotically unbiased and consistent. In a first approximation formulas were obtained for the bias and dispersion of estimators defining the impact of the preliminary determination of the period on the value of estimation error.     

Discrete estimations of cross-correlation components of periodically correlated random signals

Properties of cross-correlation components estimations of two periodically correlated random signals are analyzed. The conditions of absence of the first and the second genus aliasing effects are obtained. The formulas for estimations variance which allow to select a valid sampling step in dependences on realization length and signal properties are derived. The results are concretized for amplitude modulated signals.     

Synthesis of correlated multichannel random processes

This paper describes a method for synthesizing multichannel autoregressive (AR) random processes. The procedure allows for the variation of temporal and cross-channel correlation subject to specific constraints for correlation functions. The resulting synthesized processes provide a “fit” in a minimum mean squared error (MMSE) sense to the process correlation functions specified in terms of their temporal and cross-channel correlation parameters. Computer simulation results are presented showing the case of a two-channel AR process with various values of temporal and cross-channel correlation. A method is also suggested to synthesize a more general class of Gaussian processes with unconstrained quadrature components     

Component correlation analysis of vectorial periodically non-stationary random processes

Properties of linear and quadratic invariants of correlation tensor-function of vectorial periodically non-stationary random processes are considered. We analyze the properties of their estimations, given in the form of trigonometric polynomials, whose Fourier coefficients can be obtained using averaging on time of realization parts. Formula for estimations bias and variance, which allow to calculate systematic and mean square errors of processing in dependence on processing parameters and process parameters, are derived.     

A single-record estimator for correlation functions of nonstationary random processes

The need for estimation of auto- and cross-correlation functions of nonstationary random processes arises in many problems of communication, signal processing, system identification and control. The practical evaluation of such correlation functions is hindered by the fact that ensemble averages are extremely burdensome to obtain. This paper proposes a time-average estimator that yields unbiased and consistent estimates of those correlation functions by use of a single record of the processes involved. It is expected that this estimator will be applicable in most cases of practical interest since the conditions for its validity are fairly weak.     

A sampling theorem for nonstationary random processes (Corresp.)

The well-known sampling theorem for wide-sense stationary random processes is generalized to the class of nonstationary random processes.     

Line tracking in nonstationary processes

Semistationary processes as defined by Priestley possess an evolutionary spectrum, the discrete component of which is used to define nonstationary processes possessing a modulated line spectrum, i.e. lines modulated by complex-valued functions of time which can be interpreted as amplitude and phase modulation. It is shown (1) that the derivative of an appropriately modified version of the phase modulation of a line exists, is continuous and can therefore be defined as an instantaneous frequency, (2) that the modulation functions can be estimated using complex demodulation. Using the estimated phase modulation, estimates of the instantaneous frequency are obtained by using a linear spline regression with unknown nodes, (3) that the estimate of a modulated line spectrum is unique up to an error ? depending on the possible time-frequency resolution, (4) that if the underlying process possesses a modulated line spectrum and if all the modulation functions are band-limited, then the process can be exactly reconstructed from its sampled version (i.q.m.).The successful application of the estimation procedure is shown by an example dealing with the problem of detecting of instabilities and transient behaviour of the internal clock of organisms which plays a role for instance, if organisms have to be adapted to new time conditions (intercontinental flights, shiftwork, seasonal changes, etc.).     

Levinson filter for nonstationary processes

The linear optimal filter theory given by Levinson for stationary processes has been extended to nonstationary processes. A fast algorithm to reduce computation time is given. The modified Levinson filter is compared with the Kalman filter.     

非平稳确定性信号与非平稳随机信号统一分类法的探讨

根据信号通过线性系统可产生新信号的理论,对非平稳确定性信号与非平稳随机信号的统一分类方法进行了深入研究,分别给出格林函数描述的线性时变系统法、调制函数作用于边界稳定线性系统的自激振荡正弦波法、随机输入线性时不变系统与边界稳定的与不稳定的线性系统组合法。这3种分类法便于对非平稳确定性信号与非平稳随机信号进行统一分类研究,并具有比较全面与系统性好的优点。     

Cross spectral analysis of nonstationary processes

Consideration is given to the generalization of stationary cross spectral analysis methods to a class of nonstationary processes, specifically, the class of semistationary finite energy processes possessing sample functions that are of finite energy almost surely. A new quantity called the time-frequency coherence (TFC) is defined, and it is demonstrated that its properties are analogous to those possessed by the stationary coherence function. The problem of estimating the TFC by using elements of L. Cohen's (1966) class of joint time-frequency representations is investigated. It is shown that the only admissible estimators are those based on the class of time-frequency smoothed periodograms. Thus, the familiar procedure of segmentation and (smoothed) short-time Fourier analysis cannot be improved upon (within the framework considered) by the use of the higher-resolution nonparametric time-frequency methods. Procedures for selection of the appropriate estimators and a possible application are suggested     

Spectral representation of a periodic nonstationary random process

This paper deals with the periodic nonstationary process, the mean value and the correlation function of which are invariant under shift by a multiple of a certain period. The spectral representation is derived by making use of Loève's harmonizability theorem on a second-order nonstationary process. The process is represented as a sum of infinite stationary processes among which covariances exist. Each stationary process has a nonoverlapping frequency band of equal width, the center of which corresponds to a harmonic of the fundamental frequency determined by the period. The correlation function, dependent on two points, is represented in terms of a matrix-valued spectral density that is hermitian and nonnegative definite. The representations in other possible forms are also given. Finally some properties, special processes, and examples produced by a certain stationary random sequence are discussed.     

Behaviour of automata in a nonstationary random environment

The behaviour of automata operating in a nonstationary random environment is investigated. The automaton is assumed to be specified in terms of its associated connection matrix, and the nonstationary random environment is characterised by superimposing a time-varying component on a stationary one.     

Parsimonious correlated nonstationary models for real baseband UWB data

Few ultrawide-band (UWB) models directly fit original baseband UWB data to simultaneously account for their correlation structure, non-Gaussianity, and nonstationarity. The difficulty arises from the fact that no relevant result is available, even in multivariate statistical analysis. It also arises from the attempt to pursue the details of the mechanism that is imagined to be responsible for the generation of UWB data. The consequence is the modeling complexity, which makes it difficult to handle the received signal correlation on the basis of a single realization. Accordingly, various partial characterization is used in the literature instead by virtue of second-order statistics (such as power delay profile), nonparametric characteristics (such as zero-crossing rate), or their combination. In this paper, we take a different philosophy, which believes that the information in the received UWB data itself, as long as fully exploited, plus some simple physical intuition should suffice for the model identification and its parameter estimation. A received UWB signal is decomposed into three factor processes and each is parsimoniously parameterized. The application of the new model to data regeneration and receiver design is illustrated by using the real UWB data acquired by Intel and the TimeDomain Corporation.