The Kalman Filter

This paper provides a gentle introduction to the Kalman filter, a numerical method that can be used for sensor fusion or for calculation of trajectories. First, we consider the Kalman filter for a one-dimensional system. The main idea is that the Kalman filter is simply a linear weighted average of two sensor values. Then, we show that the general case has a similar structure and that the mathematical formulation is quite similar. 1. An example of data filtering The Kalman filter is widely used in aeronautics and engineering for two main purposes: for combining measurements of the same variables but from different sensors, and for combining an inexact forecast of a system’s state with an inexact measurement of the state. The Kalman filter has also applications in statistics and function approximation. When dealing with a time series of data points x1, x2, . . . , xn, a forecaster computes the best guess for the point xn+1. A smoother looks back at the data, and computes the best possible xi taking into account the points before and after xi. A filter provides a correction for xn+1 taking into account all the points x1, x2, . . . , xn and an inexact measurement of xn+1. An example of a filter is the following: Assume that we have a system whose one-dimensional state we can measure at successive steps. The readings of the measurements are x1, x2, . . . , xn. Our task is to compute the average μn of the time series given n points. The solution is μn = 1 n n