非加性噪声情形下卡尔曼滤波器的推广

在数字信号处理中，得到的信号总是或多或少伴随着噪声。如何去除噪声，恢复真实的信号，是信号处理面临的首要问题。一般情形下我们都假定噪声是加性的，即噪声是不依赖于信号的，此时，卡尔曼滤波器是一种非常简便的降噪方法，它是一个最优化自回归数据处理算法，是用前一个估计值和最近一个观察数据来估计信号的当前值，是用状态方程和递推的方法进行估计的，而且它在均方误差意义下是最优的。本文将噪声推广到一般的乘性噪声的情形，利用卡尔曼滤波的基本思想，同样可以得到均方误差意义下的最优滤波，最后通过一个模拟的例子验证了该方法的有效性。

Extension of Kalman filtering when noise could be non-additive

In digital signal processing,the signals we get are often accompany with noise more or less.How to delete noise and recover the actual signal is the first important problem in signal processing.In ordinary cases,we assume the noise is additive,i.e.,noise is independent of the signal,then Kalman filtering is one simple method to delete the noise.It is an optimal auto-regression data processing algorithm,which uses the past estimate value and the observe value at the present time to estimate the estimate value at the present time.This filtering can minimize the mean standard error.In this article,the author extend the noise to be the ordinary multiplicative noise,by using the method of Kalman filtering,we can also obtain the optimal filtering which minimizes the mean standard error.At last,the author validate the method by a simulated example.