Although the Kalman filter is often presented within a continuous-time context [1]-[4], the original derivation was provided in discrete time [5]. In practice, controllers and observers are invariably implemented digitally; thus discrete-time algorithms deserve special attention. This article focuses on the real-time implementation of the discrete-time Kalman filter and its relation to the discrete-time Kalman predictor. These algorithms differ in subtle ways, as highlighted in this article. Equations for the discrete-time Kalman filter and predictor are given in [6]. Unfortunately, the covariance matrix becomes indefinite after a few steps, which shows that those equations are erroneous. The present article corrects and extends the results from [6].

中文翻译：

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最佳预测器和最佳滤波器的实时实现：准确性与延迟[讲义]

尽管卡尔曼滤波器通常在连续时间上下文中出现[1]-[4]，但原始推导是在离散时间中提供的[5]。实际上，控制器和观察者总是以数字方式实现的。因此，离散时间算法值得特别注意。本文重点介绍离散时间卡尔曼滤波器的实时实现及其与离散时间卡尔曼预测器的关系。这些算法在微妙的方式上有所不同，如本文所述。离散时间卡尔曼滤波器和预测器的方程式在[6]中给出。不幸的是，经过几步后，协方差矩阵变得不确定，这表明这些方程是错误的。本文更正并扩展了[6]中的结果。