Rational Spectral Density Research Articles

Rational spectral densities and strong mixing

This article is concerned with stationary random fields with rational spectral densities. The dichotomy between the one-parameter and multiparameter cases is explored, particularly in terms of a strong mixing condition. The rich variety of behavior exhibited by the multiparameter case is demonstrated.

An efficient method for the evaluation of the controllability and observability Gramians of 2D digital filters and systems

An efficient and general method for the evaluation of the Gramians of causal, stable, 2D recursive digital filters and systems is proposed. The method is based on a two-stage extension of the Astrom-Jury-Agniel algorithm which was originally used for the evaluation of the scalar loss function of a stationary random process with rational spectral density. The new method is compared with other known methods for the evaluation of 2D Gramians with respect to accuracy and computational efficiency. The results obtained show that the new method leads to an accurate evaluation of the 2D Gramians and, furthermore, it requires only a small fraction of the computation required by existing methods. >

Spectral factorisation and prediction of multivariate processes with time-dependent rational spectral density matrices

AbstractThis paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.

Markovian representation and prediction of stochastic processes with time-varying rational spectrum

Abstract This paper solves the spectral factorization problem for a class of processes with time-dependent rational spectral density. As a consequence, the prediction problem for these processes is solved based on the integral operator method. In addition, the connection between processes with time-dependent rational spectral density and time-dependent ARMA processes is discussed.

Exact maximum likelihood time delay estimation for short observation intervals

An exact solution is presented to the problem of maximum likelihood time delay estimation for a Gaussian source signal observed at two different locations in the presence of additive, spatially uncorrelated Gaussian white noise. The solution is valid for arbitrarily small observation intervals; that is, the assumption T>> tau /sub c/, mod d mod made in the derivation of the conventional asymptotic maximum likelihood (AML) time delay estimator (where tau /sub c/ is the correlation time of the various random processes involved and d is the differential time delay) is relaxed. The resulting exact maximum likelihood (EML) instrumentation is shown to consist of a finite-time delay-and-sum beamformer, followed by a quadratic postprocessor based on the eigenvalues and eigenfunctions of a one-dimensional integral equation with nonconstant weight. The solution of this integral equation is obtained for the case of stationary signals with rational power spectral densities. Finally, the performance of the EML and AML estimators is compared by means of computer simulations. >

Optimal Filtering of a Stationary Process with Respect to Measured Discrepancies

It is known that an optimal filter in the linear-quadratic filtering and extrapolation problem for stationary processes with rational spectral densities has a rational transfer function. It is proved below that if only the errors of previous forecasts are measured rather than the values of one of the initial stationary processes, then the transfer function of the optimal filter need not be rational and it may contain interior singular functions in a circle. Standard methods of reducing of the filtering problem to minimization of a quadratic function on a linear manifold [1]-[3] turn out to be inapplicable. This is related to the essential nonlinearity of the set of all admissible transfer functions of the formative filters. The factorization of functions in Hardy spaces [4] is used for the solution. The problem is reduced to minimizing a nonquadratic function on a linear manifold, which is accomplished by

A Two-Dimensional Statistical Model of an Uneven Road Surface.

This paper describes a mathematical model of an uneven road surface where the statistical characteristics of the roughness are isotropic. A spectral density matrix is obtained in consideration of a correlation of the roughness between the right and the left tracks. By the spectral factorization of the rational spectral density matrix, which results from the approximation of the cross-spectral density function, the transfer function matrix is formulated, and the minimal realization of the shaping filter is obtained analytically. The results of the numerical simulation using the shaping filter are in good agreement with the analytical ones.

Non-full-rank causal approximations of full-rank multivariate stationary processes with rational spectrum

Let Y = ( Y n ) n ϵ ζ be a q-variate wide-sense stationary process with full-rank rational spectral density. We characterize the q-variate stationary processes ( Y ̂ n) n ϵ ζ , causally subordinated to Y, whose spectral densities have rank p < q almost everywhere, and for which E ∥ Y n − Y ̂ n ∥ 2 is minimum.

Adaptive Control of Time Varying Stochastic Systems

The stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system is considered.The class of B-bounded stochastic processes is introduced. It is demonstrated that any i.i .d process whose square possesses a moment generating function is B-bounded, and that this class of stochastic processes is closed under finite dimensional linear transformations. Hence a stationary Gaussian process with rational spectral density is always Bounded.If the parameter and disturbance processes entering the system are B-bounded, and the parameter variation is small in a statistical sense, then closed loop stability in all L2 sense will be guaranteed.By strengthening the hypotheses on the disturbance and parameter processes a Markovian state process may be constructed, and in this case it is shown that loss functions on the joint input-output-parameter estimate process converge to their expectation with respect to an invariant probability at a geometric rate.

On a class of random field models which allows long range dependence

SUMMARY In this note it is shown how a convenient class of random field models with rational spectral densities can be augmented to incorporate the possibility of long range dependence.

A two-dimensional statistical model of an uneven road surface.

This paper describes a mathematical model of an uneven road surface where the statistical characteristics of the roughness are isotropic. A spectral density matrix is obtained in consideration of a correlation of the roughness between the right and the left track. By factorization of the rational spectral density matrix, which results from the approximation of the cross spectral density, the transfer function matrix is formulated, and the minimal realization of the shaping filter is obtained analytically. The results of the numerical simulation using the shaping filter are in good agreement with analytical ones.

Explicit extrapolation of stationary processes with generalized rational spectral density

Explicit extrapolation of stationary processes with generalized rational spectral density

Performance of discrete-time predictors of continuous-time stationary processes

The asymptotic performance of linear predictors of continuous-time stationary processes is studied from observations at n sampling instants on a fixed observation interval. Both optimal and simpler choices of predictor coefficients are considered, using uniform sampling as well as nonuniform sampling tailored to the statistics of the process under prediction. The focus is on stationary processes with rational spectral densities and the asymptotic performance for cases with and without a quadratic-mean derivatives is obtained. The analytical results are supplemented by numerical examples depicting small- and large-sample-size performance. >

Rejection of multiple narrow-band interference in both BPSK and QPSK DS spread-spectrum systems

The performance of direct-sequence spread-spectrum systems using a suppression filter in the presence of multiple narrow band interference with rational spectral densities is analyzed. Both BPSK and QPSK systems are considered, and analytical expressions for both the means-square error of the filter output and the performance improvement are established. When the bandwidths of the narrow band components of the interference are all small, approximate expressions are obtained and used to provide further insight into the behavior of the system. Results are presented for the limiting case when the bandwidths approach aero (i.e. when multiple narrow band interference becomes multiple sinusoidal interference).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimation and Model Identification for Continuous Spatial Processes

SUMMARY Formal parameter estimation and model identification procedures for continuous domain spatial processes are introduced. The processes are assumed to be adequately described by a linear model with residuals that follow a second-order stationary Gaussian random field and data are assumed to consist of noisy observations of the process at arbitrary sampling locations. A general class of two-dimensional rational spectral density functions with elliptic contours is used to model the spatial covariance function. An iterative estimation procedure alleviates many of the computational difficulties of conventional maximum likelihood estimation for non-lattice data. The procedure is applied to several generated data sets and to an actual ground-water data set.

Computation of photoelectron counting distributions by numerical contour integration II Light with rational spectral density

For a mixture of an arbitrary number of sinusoidal coherent components and chaotic light with a rational spectral density having simple poles, the probability-generating function h(z) of the number of photoelectrons ejected by the light in an interval is expressed in terms of certain finite determinants. It figures in the integrands of contour integrals in the complex plane for the cumulative and complementary cumulative distributions of the number of photoelectron counts. When these are evaluated by numerical integration, h(z) can be computed at each point on the contour by standard computer routines for solving algebraic equations and evaluating determinants, permitting precise and efficient computation of the photocount distributions.

Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration

The cumulative distribution of the filtered output of a quadratic rectifier whose input is either narrow-band Gaussian noise or Gaussian noise with a low-pass spectral density is to be computed by numerical quadrature of a Laplace inversion integral along a contour in the complex plane chosen to economize the number of steps. The integrand contains the moment-generating function (mgf) of the output. It is expressed in terms of the Fredholm determinant and the resolvent kernel associated with an integral equation involving the autocovariance function of the input and the impulse response of the output filter. A special case is the power of a mean-zero Gaussian process averaged over a finite interval, and when this process has a rational spectral density, the mgf can be expressed as the ratio of certain finite determinants. By this method distributions are calculated for low-pass noise with RLC and second- and fourth-order Chebyshev spectral densities. For rational input spectral densities but arbitrary positive output filtering and an arbitrary additive input signal, the mgf can be calculated by integrating differential equations of the Kalman-Bucy type.

Nonlinear oscillations in dynamical systems subjected to random actions with rational spectral densities

Nonlinear oscillations in dynamical systems subjected to random actions with rational spectral densities

On Sufficient Statistics of Gaussian Processes with Rational Spectral Density Function

On Sufficient Statistics of Gaussian Processes with Rational Spectral Density Function

Characterization and estimation of two-dimensional ARMA models

A class of finite-order two-dimensional autoregressive moving average (ARMA) is introduced that can represent any process with rational spectral density. In this model the driving noise is correlated and need not be Gaussian. Currently known classes of ARMA models or AR models are shown to be subsets of the above class. The three definitions of Markov property are discussed, and the class of ARMA models are precisely stated which have the noncausal and semicausal Markov property without imposing any specific boundary conditions. Next two approaches are considered to estimate the parameters of a model to fit a given image. The first method uses only the empirical correlations and involves the solution of linear equations. The second method is the likelihood approach. Since the exact likelihood function is difficult to compute, we resort to approximations suggested by the toroidal models. Numerical experiments compare the quality of the two estimation schemes. Finally the problem of synthesizing a texture obeying an ARMA model is considered.