# CDS 110b: Sensor Fusion

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In this lecture we show how the Kalman filter can be used for sensor fusion and explore some variations on the basic Kalman filter, including the extended Kalman filter.

## Lecture Outline

1. Sensor fusion using Kalman filters
2. The extended Kalman filter
3. Parameter estimation using EKF

## Lecture Materials

Correlated noise can be put into the Kalman filtering framework by using a (linear) filter to give a correlated noise source with a given correlation function (or spectral density). Suppose that $$H(s)$$ is a transfer function that filters Gaussian white noise and provides the desired correlation. Let $$(A_f, B_f, C_f)$$ be a state space representation for the filter. Then the entire system can be written as
 $$\left[\begin{matrix} x \\ z \end{matrix}\right] = \left[\begin{matrix} A & F C_f \\ 0 & A_f \end{matrix}\right] \left[\begin{matrix} x \\ z \end{matrix}\right] + \left[\begin{matrix} B \\ 0 \end{matrix}\right] u + \left[\begin{matrix} 0 \\ B_f \end{matrix}\right] v$$ $$y = \left[\begin{matrix} C & 0 \end{matrix}\right] \left[\begin{matrix} x \\ z \end{matrix}\right] + w$$
This system takes a Guassian white noise input $$v$$, filters it to give the desired spectrum, and uses it to drive the system.