Recursive filtering for two-dimensional random fields (Corresp.)

A class of recursive filtering problems for random fields with a two-dimensional parameter is considered. After a brief introduction of two-parameter stochastic calculus, a class of Markovian random fields generated by stochastic integral equations is defined and considered. It is then shown that the problem of estimating such a Markovian field in additive white Gaussian noise can be reduced to a recursive formalism. If the random field is itself Gaussian, the recursive formalism reduces to a finite set of stochastic integral equations involving the conditional mean and covariance.     

基于随机微分的射频噪声干扰技术研究

根据随机微分与射频噪声干扰信号处理的内在联系,将随机微分引入到雷达噪声干扰信号处理领域,对射频噪声干扰信号进行了系统地分析。首先建立了射频噪声干扰信号通过雷达中频滤波器后所满足的福克尔一普朗克方程,然后利用群移傅立叶变换（Motion—Group Fourier Transform,MGFT）将此偏微分方程化成了齐次线性微分方程组,最后得到了射频噪声干扰信号通过雷达中频滤波器后的概率密度函数。     

The innovations approach to detection and estimation theory

Given a stochastic process, its innovations process will be defined as a white Gaussian noise process obtained from the original process by a causal and causally invertible transformation. The significance of such a representation, when it exists, is that statistical inference problems based on observation of the original process can be replaced by simpler problems based on white noise observations. Seven applications to linear and nonlinear least-squares estimation. Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described. The major new results are summarized in seven theorems. Some powerful mathematical tools will be introduced, but emphasis will be placed on the considerable physical significance of the results.     

Optimal Filtering for Incompletely Measured Polynomial Systems with Multiplicative Noise

In this paper, the optimal filtering problem for polynomial system states with polynomial multiplicative noise over linear observations with an arbitrary, not necessarily invertible, observation matrix is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and the error variance. Thus, the Ito differentials for the optimal estimate and error variance corresponding to the stated filtering problem are first derived. A transformation of the observation equation is introduced to reduce the original problem to the previously solved one with an invertible observation matrix. The procedure for obtaining a closed system of the filtering equations for any polynomial state with polynomial multiplicative noise over linear observations is then established, which yields the explicit closed form of the filtering equations in the particular cases of linear and bilinear state equations. In an example, the performance of the designed optimal filter is verified against those of the optimal filter for a quadratic state with a state-independent noise and a conventional extended Kalman–Bucy filter. The authors thank the Mexican National Science and Technology Council (CONACyT) for financial support under Grants No. 55584 and 52953.     

On the Extension of Multidimensional Speckle Noise Model From Single-Look to Multilook SAR Imagery

Speckle noise represents one of the major problems when synthetic aperture radar (SAR) data are considered. Despite the fact that speckle is caused by the scattering process itself, it must be considered as a noise source due to the complexity of the scattering process. The presence of speckle makes data interpretation difficult, but it also affects the quantitative retrieval of physical parameters. In the case of one-dimensional SAR systems, speckle is completely determined by a multiplicative noise component. Nevertheless, for multidimensional SAR systems, speckle results from the combination of multiplicative and additive noise components. This model has been first developed for single-look data. The objective of this paper is to extend the single-look data model to define a multilook multidimensional speckle noise model. The asymptotic analysis of this extension, for a large number of averaged samples, is also considered to assess the model properties. Details and validation of the multilook multidimensional speckle noise model are provided both theoretically and by means of experimental SAR data acquired by the experimental synthetic aperture radar system, operated by the German Aerospace Center     

Estimation and detection of signals in multiplicative noise (Corresp.)

We consider a class of matrix signal processes that are received in the presence of multiplicative observation noise. By examining the differential version of the observation, we are able to derive finite-dimensional optimal detection-estimation equations that involve a linear filter with gain computed on-line using the incoming observations. An example involving the detection of an actuator failure on a rotating rigid body is considered.     

Introduction to fractional linear systems. Part 2. Discrete-timecase

For pt.I see ibid., vol.147, no.1, p.62 (2000). In the paper, the class of discrete linear systems is enlarged with the inclusion of discrete-time fractional linear systems. These are systems described by fractional difference equations and fractional frequency responses. It is shown how to compute the impulse response and transfer function. Fractal signals are introduced as output of special linear systems: fractional differaccumulators, systems that can be considered as having fractional poles or zeros. The concept of fractional differaccumulation is discussed generalising the notions of fractal and 1/f noise, and introducing two kinds of fractional differaccumulated stochastic process: hyperbolic, resulting from fractional accumulation (similar to the continuous-time case), and parabolic noise, resulting from fractional differencing     

Synthesis of linear optimum distributed parameter systems with time delays

A variational calculus approach is used to study quadratic optimization of linear time-delayed distributed parameter systems with distributed and boundary control function. The canonical equations are derived for the necessary condition of optimality. Then the Riccati equations are obtained and their computational solution is discussed.     

Stochastic differential equations: an approach to the generation ofcontinuous non-Gaussian processes

The generation of continuous random processes with jointly specified probability density and covariation functions is considered. The proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system, excited by white Gaussian noise and described by a system of a first-order stochastic differential equations (SDE). The authors explore how the statistical characteristics of the equation's solution depends on the form of its operator and on the intensity of the input noise. Some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non-Gaussian continuous processes are considered. The approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its Markov model as well as in simulation of continuous correlated processes with the known probability density function     

Study of an integral equation arising in detection theory

This paper considers the solution of a Fredholm equation occurring in detection theory problems. A solution procedure, based on solving differential equations with nonmixed boundary conditions, is described for the case when the kernel of the integral equation is known to be the output covariance of a linear finite-dimensional system excited by white noise. Solutions with discontinuities are considered.     

适合处理乘性噪声估计卫星姿态的非线性迭代滤波算法

针对带乘性噪声的星敏感器/陀螺非线性卫星姿态确定系统,提出了一种迭代MEKF(Iterative Multiplicative Extended Kalman Filter)姿态估计滤波算法.通过对带乘性噪声的非线性卫星姿态确定系统的状态方程和测量方程进行二次线性化迭代,并基于线性最小方差准则和投影公式,导出了姿态状态...     

Criterion for the stability of all moments of a particular stochastic system

It is shown that, for stochastic systems with multiplicative noise, all moments may be asymptotically stable for some values of the noise intensities. For a particular stochastic system, a criterion is derived.     

Performance/dependability modelling using stochastic reward models: transient behaviour

Stochastic Reward Models (SRMs) are commonly used for evaluating combined performance and dependability of fault-tolerant systems. An SRM is composed of a stochastic process, describing the evolution of the system, and a superimposed reward structure, reflecting different performance levels of the system. Evaluating combined transient performance/dependability measures using SRMs leads either to differential or integral equations. This paper discusses the use of SRMs for modelling performance/dependability of fault-tolerant systems and proposes an approach for the numerical solution of integral equations which commonly arise. A bound for the error due to the numerical approximation is obtained. As an example, an n-unit parallel system is analysed numerically in transient state.     

H<Subscript>∞</Subscript> Deconvolution Filters for Stochastic Systems with Interval Uncertainties

This paper is concerned with the robust H_∞ deconvolution filtering problem for continuous- and discrete-time stochastic systems with interval uncertainties. The matrices of the system describing the signal transmissions are assumed to be uncertain within given intervals, and the stochastic perturbation is in the form of multiplicative Gaussian white noise with constant variance. The purpose of the addressed problem is to design a robust H_∞ deconvolution filter such that the input signal distorted by the transmission channel could recover to a specified extent γ. By using stochastic analysis techniques and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the asymptotical stability of the filtering error system. Then the filter parameters are characterized in terms of the solution to linear matrix inequalities, which can be easily solved by using available software packages. Two simulation examples are exploited to demonstrate the effectiveness of the proposed design procedures, respectively, for continuous- and discrete-time systems.     

Perturbation-Based Stochastic Modeling of Nonlinear Circuits

This paper presents a general model for a nonlinear circuit, in which, the circuit parameters (e.g. resistance and capacitance) are subject to random fluctuations due to noise, which vary with time. The fluctuating amplitudes of these parameters are assumed to be Ornstein–Uhlenbeck (O.U.) processes and not the white noise owing to temporal correlations. The nonlinear circuit is represented by a system of nonlinear differential equations depending upon a set of parameters that fluctuate slowly with time. To model these fluctuations, we use the theory of Ito’s stochastic differential equations (SDEs). Then the driving force of the circuit dynamics is in accordance with the general perturbation theory decomposed into the sum of a strong linear component and a weak nonlinear component by the introduction of a small perturbation parameter. The circuit states are expanded in the powers of this small perturbation parameter and recursive solutions to the various approximates obtained. Finally, the approximate expressions for the output states are obtained as stochastic integrals with respect to Brownian motion processes. The proposed method is applied to a half-wave rectifier circuit which is built out of a diode, a resistor and a capacitor. The diode is represented by nonlinear voltage–current equation, and resistance and capacitance are subject to random fluctuations due to noise, which vary slowly with time. The results, obtained using the proposed method, are compared with those obtained via the conventional perturbation-based deterministic differential equations model for a nonlinear circuit. Hence, the noise process component, present at the output, is obtained.     

The extended MENT algorithm: a maximum entropy type algorithm usingprior knowledge for computerized tomography

Unlike MENT (maximum entropy algorithm), the extended MENT algorithm can process prior information and deal with incomplete projections or limited angle data. The reconstruction problem is formulated for solving linear systems involving the Fredholm integral equation. To develop the extended MENT algorithm, maximum entropy is substituted by a more general optimization criterion, that of minimizing the discriminatory function. The a priori knowledge of the shape of the object is easily incorporated in the algorithm by using the discriminatory function. Useful mathematical properties that make the discriminatory function attractive are derived. The sensitivity of the minimum discriminatory solution is derived to determine the characteristics of the noise in the reconstructed images. The extended MENT algorithm is developed for a parallel geometry, and its convergence properties are given. Its image processing performance is better than that for other maximum entropy algorithms such as multiplicative algebraic reconstruction techniques (MART) or more standard methods such as ART and the convolution backprojection     

Adaptive Volterra filters for active control of nonlinear noiseprocesses

This paper presents a Volterra filtered-X least mean square (LMS) algorithm for feedforward active noise control. The research has demonstrated that linear active noise control (ANC) systems can be successfully applied to reduce the broadband noise and narrowband noise, specifically, such linear ANC systems are very efficient in reduction of low-frequency noise. However, in some situations, the noise that comes from a dynamic system may he a nonlinear and deterministic noise process rather than a stochastic, white, or tonal noise process, and the primary noise at the canceling point may exhibit nonlinear distortion. Furthermore, the secondary path estimate in the ANC system, which denotes the transfer function between the secondary source (secondary speaker) and the error microphone, may have nonminimum phase, and hence, the causality constraint is violated. If such situations exist, the linear ANC system will suffer performance degradation. An implementation of a Volterra filtered-X LMS (VFXLMS) algorithm based on a multichannel structure is described for feedforward active noise control. Numerical simulation results show that the developed algorithm achieves performance improvement over the standard filtered-X LMS algorithm for the following two situations: (1) the reference noise is a nonlinear noise process, and at the same time, the secondary path estimate is of nonminimum phase; (2) the primary path exhibits the nonlinear behavior. In addition, the developed VFXLMS algorithm can also be employed as an alternative in the case where the standard filtered-X LMS algorithm does not perform well     

Nonlinear prediction of a class of random processes

This paper is concerned with the minimum mean-squared error (MMSE) nonlinear prediction of a class of random processes. A class of random processes is defined by the property that its MMSE zero-memory predictor is represented by a finite sum of separable terms. Sufficient conditions for the existence of such processes are also considered. The nonlinear predictor is restricted to be composed of a linear filter in parallel with a zero-memory nonlinearity (ZNL) preceded by a variable delay, The optimum predictor is shown to be the solution of linear integral equations with the same kernel as for the optimum linear predictor. The first step of the derivation also yields a simpler scheme which ,only requires the addition of a ZNL to the optimum linear predictor. The improvements in the MMSE of the two nonlinear systems over the linear case are compared and illustrated by a numerical example.