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Cauchy Estimation for Linear Scalar Systems

机译：线性标量系统的柯西估计

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摘要

An estimation paradigm is presented for scalar discrete linear systems entailing additive process and measurement noises that have Cauchy probability density functions (pdf). For systems with Gaussian noises, the Kalman filter has been the main estimation paradigm. However, many practical system uncertainties that have impulsive character, such as radar glint, are better described by stable non-Gaussian densities, for example, the Cauchy pdf. Although the Cauchy pdf does not have a well defined mean and does have an infinite second moment, the conditional density of a Cauchy random variable, given its linear measurements with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. For a single measurement, simple expressions are obtained for the conditional mean and variance, by deriving closed form expressions for the infinite integrals associated with the minimum variance estimation problem. To alleviate the complexity of the multi-stage estimator, the conditional pdf is represented in a special factored form. A recursion scheme is then developed based on this factored form and closed form integrations, allowing for the propagation of the conditional mean and variance over an arbitrary number of time stages. In simulations, the performance of the newly developed scalar discrete-time Cauchy estimator is significantly superior to a Kalman filter in the presence of Cauchy noise, whereas the Cauchy estimator deteriorates only slightly compared to the Kalman filter in the presence of Gaussian noise. Remarkably, this new recursive Cauchy conditional mean estimator has parameters that are generated by linear difference equations with stochastic coefficients, providing computational efficiency.
机译：提出了一种标量离散线性系统的估计范例，该系统需要加法过程和测量噪声，这些噪声具有柯西概率密度函数（pdf）。对于具有高斯噪声的系统，卡尔曼滤波器已成为主要的估计范例。但是，许多具有脉冲特性的实际系统不确定性（例如雷达闪烁）可以通过稳定的非高斯密度更好地描述，例如Cauchy pdf。尽管Cauchy pdf没有明确定义的均值并且确实具有无限的第二矩，但考虑到Cauchy随机变量的线性测量以及附加的Cauchy噪声，其条件密度具有条件均值和有限条件方差，两者均为测量功能。对于单次测量，通过推导与最小方差估计问题关联的无限积分的闭式表达式，可以获得条件均值和方差的简单表达式。为了减轻多阶段估算器的复杂性，条件pdf以特殊的分解形式表示。然后，基于此因式形式和闭合形式积分来开发递归方案，从而允许在任意数量的时间段内传播条件均值和方差。在仿真中，在存在柯西噪声的情况下，新开发的标量离散时间柯西估计器的性能明显优于卡尔曼滤波器，而在存在高斯噪声的情况下，柯西估计器的性能仅稍低于卡尔曼滤波器。值得注意的是，这种新的递归柯西条件均值估计器具有由具有随机系数的线性差分方程生成的参数，从而提高了计算效率。

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