Linear Filtering

Abstract In this chapter, we consider a very special filtering model. Namely, the signal is a Gaussian process and the observation function is linear. The corresponding filtering theory is called Kalman–Bucy filtering.

We will derive the Kalman–Bucy filter as a special case of the non-linear filter introduced in the previous chapters. After the filtering equations are established and the discrete-time approximation is studied, we will investigate the long time stability of the linear filter in Sections 9.4 and 9.5. The results in the last two sections can be regarded as a standard to compare with when we study the stability for the non-linear filter in Chapter 10.

The material in Section 9.4 is very technical and can be skipped in the first reading.