On the generation of the baseband and narrowband non-Gaussianprocesses

One of the widely used classifications of random processes implies their separation from the baseband and narrowband ones. Exponent and exponentially decaying cosine waves represent the most popular approximations of the observed processes' autocovariation functions (ACF). We consider the generation of such processes with any specified marginal probability density function (PDF). The approach is based on the representation of the process considered to be a stationary solution of certain stochastic differential equations (SDE) with the white Gaussian noise in the right-hand side. It removes the known Pearson restrictions on the kind of available process' PDF and, in the part related to the generation of the processes with strictly exponential ACF, is very close to that considered by Haddad (1970). Concerning the generation of the narrowband non-Gaussian processes with the exact exponential envelope of its ACF, synthesis of the corresponding SDE may be considered to be principally novel. Several examples of nonlinear dynamical systems generating baseband and narrowband stationary continuous processes with the given PDF are considered. Some aspects of the SDE numerical simulation are presented. The generation method proposed may be useful in communication and signal processing applications where a proper interference simulation appears to be really important     

Modeling and analysis of stochastic differential equations driven by point processes

The modeling and analysis of nonlinear systems described by differential equations driven by point process noise are considered. The stochastic calculus of McShane is generalized to include such differential equations, and a more general canonical extension is defined. It is proved that this canonical extension possesses the same desirable properties for point process noise that it does for the noise processes, such as Brownian motion, considered by McShane. In addition, a new stochastic integral with respect to a point process is defined; this alternative integral obeys the rules of ordinary calculus. As a special case of the analysis of such systems, linear systems with multiplicative point process noise are investigated. The consistency of the canonical extension is studied by means of the product integral. Finally, moment equations and criteria for the stochastic stability of linear systems with multiplicative Poisson noise are derived.     

Discrete simulation of colored noise and stochastic processes and1/fα power law noise generation

This paper discusses techniques for generating digital sequences of noise which simulate processes with certain known properties or describing equations. Part I of the paper presents a review of stochastic processes and spectral estimation (with some new results) and a tutorial on simulating continuous noise processes with a known autospectral density or autocorrelation function. In defining these techniques for computer generating sequences, it also defines the necessary accuracy criteria. These methods are compared to some of the common techniques for noise generation and the problems, or advantages, of each are discussed. Finally, Part I presents results on simulating stochastic differential equations. A Runge-Kutta (RK) method is presented for numerically solving these equations. Part II of the paper discusses power law, or 1/f^α, noises. Such noise processes occur frequently in nature and, in many cases, with nonintegral values for α. A review of 1/f noises in devices and systems is followed by a discussion of the most common continuous 1/f noise models. The paper then presents a new digital model for power law noises. This model allows for very accurate and efficient computer generation of 1/f^α noises for any α. Many of the statistical properties of this model are discussed and compared to the previous continuous models. Lastly, a number of approximate techniques for generating power law noises are presented for rapid or real time simulation     

无色散光纤信道的非线性演化

在单模光纤中由于非线性效应和拉曼增益效应的共同作用,导致光子在各向同性介质中传输时满足非线性薛定谔方程。利用随机微分方程研究了长距离光纤通信中噪声对光纤信道的影响,给出了光纤信道的动力学机理模型。首先在非线性薛定谔方程的基础上引入噪声项,然后利用It公式将其整理成极坐标系下标准的随机微分方程组,最后利用福克尔-普朗克(Fokker-Planck)方程得到了光脉冲在光纤信道中的概率密度函数,精细地研究了光纤信道的非线性演化规律。即在加入噪声项的情况下,分析了光纤通信的传输性能指标,得到了概率密度函数。     

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。     

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.     

Traffic models for wireless communication networks

Introduces a deterministic fluid model and two stochastic traffic models for wireless networks. The setting is a highway with multiple entrances and exits. Vehicles are classified as calling or noncalling, depending upon whether or not they have calls in progress. The main interest is in the calling vehicles; but noncalling vehicles are important because they can become calling vehicles if they initiate (place or receive) a call. The deterministic model ignores the behavior of individual vehicles and treats them as a continuous fluid, whereas the stochastic traffic models consider the random behavior of each vehicle. However, all three models use the same two coupled partial differential equations (PDEs) or ordinary differential equations (ODEs) to describe the evolution of the system. The call density and call handoff rate (or their expected values in the stochastic models) are readily computable by solving these equations. Since no capacity constraints are imposed in the models, these computed quantities can be regarded as offered traffic loads. The models complement each other, because the fluid model can be extended to include additional features such as capacity constraints and the interdependence between velocity and vehicular density, while the stochastic traffic model can provide probability distributions. Numerical examples are presented to illustrate how the models can be used to investigate various aspects of time and space dynamics in wireless networks     

有色噪声的扩散过程模型

研究使用基于随机微分方程的扩散过程模型产生有色噪声.首先给出Markov扩散过程的平稳分布,该分布给出了扩散过程模型中的漂移系数、扩散系数和有色噪声概率密度分布之间的关系;选择扩散过程模型中的扩散系数为x的一次幂,其系数决定了所生成噪声的相关特性;数值算法使用Milstein高阶法.以三元独立积构造的G-分布雷达杂波为例进行仿真分析,验证所提出方法的准确性和有效性.     

量测随机延迟下带厚尾噪声的鲁棒Student's t随机容积卡尔曼滤波器

针对量测随机延迟下带厚尾过程噪声和量测噪声的非线性状态估计问题，本文通过充分考虑量测一步随机延迟特性及过程噪声和量测噪声的"厚尾"特性，推导了一种新的鲁棒Student's t滤波器框架，并采用随机Student's t-球面相径容积规则近似计算Student's t权值积分，从而设计了一种新的鲁棒Student's t随机容积滤波器.首先，采用一组服从伯努利分布的随机序列来描述系统中可能存在的量测一步随机延迟现象，并采用Student's t分布刻画过程噪声和量测噪声中存在的"厚尾"特性；其次，从理论上证明了当自由度参数趋于无穷以及随机延迟概率为零时，该鲁棒Student's t滤波器就自动地降为标准的非线性高斯近似滤波器；最后，采用随机Student's t-球面相径容积规则给出了一种新的鲁棒Student's t随机容积滤波器，并通过协同转弯模型验证了该滤波器的有效性和优越性.     

Moment characterization of phase noise in coherent optical systems

The statistical characterization of the phase noise introduced by a semiconductor laser in a coherent optical transmission system is a key problem in the system performance evaluation. The authors consider the moment characterization, of the complex random process. Starting from the implicit representation of the probability density function through the Fokker-Planck equation, the authors obtain closed form analytical expressions for the moments of the filtered phase noise both in stationary and nonstationary conditions. Then the use of the moments for the computation of probability densities through orthogonal polynomial series expansion and maximum entropy approach is considered in application examples     

一种新的脉冲压缩雷达干扰处理方法

本文通过在雷达干扰信号处理领域中应用随机微分,系统地对噪声调频干扰信号进行了分析。首先建立了噪声调频干扰信号通过脉冲压缩雷达中频滤波器后所满足的福克尔-普朗克方程,再通过群移傅立叶变换(Motion-Group Fourier Transform,MGFT)对偏微分方程组进行了转换,将其转化成了变系数齐次线性微分方程组,结合Peano-Baker级数法得到了该方程组的解,并得到了其概率密度函数。     

Optimal estimation for discrete time jump processes

Optimum estimates of nonobservable random variables or random processes which influence the rate functions of a discrete time jump process (D) are derived. The approach used is based on the {em a posteriori} probability of a nonobservable event expressed in terms of the {em a priori} probability of that event and of the sample function probability of the DTJP. Thus a general representation is obtained for optimum estimates, and recursive equations are derived for minimum mean-squared error (MMSE) estimates. In general, MMSE estimates are nonlinear functions of the observations. The problem of estimating the rate of a DTJP when the rate is a random variable with a beta probability density function and the jump amplitudes are binomially distributed is considered. It is shown that the MMSE estimates ale linear. The class of beta density functions is rather rich and explains why there are insignificant differences between optimum unconstrained and linear MMSE estimates in a variety of problems.     

Solving phase-noise Fokker-Planck equations using the motion-group Fourier transform

A quantity of importance in coherent optical communications is the probability density of a filtered signal in the presence of phase noise (PN). The Fokker-Planck(FP) approach has been recognized as a rigorous way to describe these statistical properties. However, computational difficulties in solving these FP equation shave prevented their widespread application. In this paper, we present a new and simple computational solution method based on techniques from noncommutative harmonic analysis on motion groups. This proposed method can easily solve all the PN FP equations with any kind of intermediate frequency filter. We also present a new derivation of PN FP equations from the viewpoint of stochastic processes.     

Phase-Locked Loop Operation in the Presence of Impulsive and Guassian Noise

A first-order phase-locked loop with detuning is considered in the presence of white Gaussian noise and random amplitude impulsive noise with Poisson times. The stochastic equation for the phase error density is of infinite order, but when the stationary mod-2π phase density is represented by a Fourier series, a linear second-order difference equation is the Fourier coefficients results. The difference equation is solved numerically, and the phase error density is generated from the Fourier series. This method uses no approximations and is valid for any impulsive amplitude probability density.     

Segmentation of stochastic images with a stochastic random walker method

We present an extension of the random walker segmentation to images with uncertain gray values. Such gray-value uncertainty may result from noise or other imaging artifacts or more general from measurement errors in the image acquisition process. The purpose is to quantify the influence of the gray-value uncertainty onto the result when using random walker segmentation. In random walker segmentation, a weighted graph is built from the image, where the edge weights depend on the image gradient between the pixels. For given seed regions, the probability is evaluated for a random walk on this graph starting at a pixel to end in one of the seed regions. Here, we extend this method to images with uncertain gray values. To this end, we consider the pixel values to be random variables (RVs), thus introducing the notion of stochastic images. We end up with stochastic weights for the graph in random walker segmentation and a stochastic partial differential equation (PDE) that has to be solved. We discretize the RVs and the stochastic PDE by the method of generalized polynomial chaos, combining the recent developments in numerical methods for the discretization of stochastic PDEs and an interactive segmentation algorithm. The resulting algorithm allows for the detection of regions where the segmentation result is highly influenced by the uncertain pixel values. Thus, it gives a reliability estimate for the resulting segmentation, and it furthermore allows determining the probability density function of the segmented object volume.     

Wavelet decomposition of harmonizable random processes

The discrete wavelet decomposition of second-order harmonizable random processes is considered. The deterministic wavelet decomposition of a complex exponential function is examined, where its pointwise and bounded convergence to the function is proved. This result is then used for establishing the stochastic wavelet decomposition of harmonizable processes. The similarities and differences between the wavelet decompositions of general harmonizable processes and a subclass of processes having no spectral mass at zero frequency, e.g., those that are wide-sense stationary and have continuous power spectral densities, are also investigated. The relationships between the harmonization of a process and that of its wavelet decomposition are examined. Finally, certain linear operations such as addition, differentiation, and linear filtering on stochastic wavelet decompositions are considered. It is shown that certain linear operations can be performed term by term with the decomposition     

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.     

CMOS LC-oscillator phase-noise analysis using nonlinear models

In this paper, a second-order stochastic differential equation is used as a tool for the analysis of phase noise in a submicron CMOS LC oscillator. A cross-coupled topology typical of integrated CMOS designs is considered. Nonlinear limiting and mobility degradation effects in the circuit are modeled and used to predict the statistics of the random amplitude and phase deviations in terms of design variables. Assuming Gaussian noise disturbances and describing the phase noise as a random diffusion process, the average phase-noise power spectrum is derived and its accuracy verified with measurement and simulation results. Calculations for phase noise arising from stationary tank noise, nonstationary channel thermal noise, and flicker noise are discussed. The analysis is used to emphasize the fundamental power/performance tradeoff associated with compensation of tank losses via adjustments in the power supply and device size.     

Estimation of multilevel digital signals in the presence ofarbitrary impulsive interference

We first study the a posteriori probability density function of the state of a discrete-time system given the measurement data. By applying the Bayesian law to the state and measurement equations of the stochastic system, the a posteriori density is obtained in closed-form and computed recursively for arbitrary i.i.d. state noise and any discrete-type measurement noise (or multilevel digital signal). Then, our effort concentrates on the estimation of impulsive noise which interferes the multilevel signal of interest. By considering the L^p -metric performance criterion, where 0相似文献   

Optimal nonlinear filtering for independent increment processes--Part II

The main result obtained in Part I [1]--which is contained in Theorem1of Part I--is generalized and extended in several ways. First, a new basic stochastic integro-partial differential equation for the conditional probability density function for the state of a nonlinear dynamic system with disturbance noise given noisy nonlinear measurements of the state is derived under the less restrictive assumption that the disturbance noise is an arbitrary independent increment process with an infinitely divisible distribution, and the measurement noise is a Gaussian independent increment process with an infinitely divisible distribution. It is then shown that, under proper restrictions, this basic equation reduces to either the Fokker-Planck equation for diffusion processes or the Kolmogorov-Feller equation for jump processes. Also, it is shown that this basic equation contains the earlier results of Kushner [2] and Wonhamt [3] as special cases. Next, it is shown how the result represented by this basic equation can be easily extended to include the case where the disturbance noise is a Markov process of the type initially assumed for the state of the dynamic system. Finally, it is shown that, in contrast to results obtained by Bryson and Johansen [4] and Cox [5] for the linear, Gaussian case, it is not generally possible to extend the result represented by this basic equation to include either the case where the measurement noise is a Markov process of the type initially assumed for the state of the dynamic system, or the case where the covariance matrix of the Gaussian measurement noise is singular. However, some incomplete results indicating when such extensions might be possible are given.