# NCS: Kalman Filtering

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In this lecture, we study the Kalman filter for discrete-time linear systems. In particular, we see under what assumptions and in what senses the Kalman filter is an optimal estimator. To prove the results we use some results about conditional expectations and Gaussian probabiliy distributions. We show that the filter contains one prediction step and one correcter step that takes the most recent measurement into account. How the filter deals with sensor fusion is discussed and an example is used to illustrate the results.

## Lecture Materials

## Reading

An Introduction to the Kalman Filter, G. Welch and G. Bishop. A brief introduction to the Kalman filter in discrete time. No proofs are given, but it is a good first read.

Wikipedia: Kalman Filter A webpage that gives a proof and some applications.

A New Approach to Linear Filtering and Prediction Problem, R.E. Kalman.

*Transactions of the ASME*, Series D, 1960. A classical paper. Still very readable. It uses different notation than the lecture, and present a different and more general proof.

## Additional Resources

The Kalman Filter, G. Welch and G. Bishop. A webpage with many links on Kalman filter.

Optimal Filtering, B.D.O Anderson and J.B. Moore. Dover Books on Engineering, 2005. A reissue of a book from 1979. It contains a detailed mathematical presentation of filtering problems and the Kalman filter. A very good book.