# 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

Q: How do you deal with time correlated noise (eg, GPS jumps on Alice)?

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.

For Alice, the most correct approach would be to model the noise as something other than a Gaussian process (in which case the theory we have studied doesn't directly apply). However, we can also take data from the sensor and develop the correlation function numerically, then determine the linear system that best models the correlation.