# Evaluation of Jacobian for Extended Kalman Filter

For the non-additive noise case, $$x_k = f(x_{k-1}, u_{k-1}, \xi_{k-1}) \\ y_k = h(x_k, \nu_k)$$

the EKF takes into account the jacobian wrt to the noise terms $L_{k-1} = \frac{\partial f}{\partial \xi} |_{\hat{x}_{k-1|k-1}, u_{k-1}}$ and $M_{k} = \frac{\partial h}{\partial \nu} |_{\hat{x}_{k|k-1}}$

I know that jacobian wrt to state: $A = \frac{\partial f}{\partial x}$ and $H = \frac{\partial h}{\partial x}$ are evaluated at means of noises i.e. at $\xi = 0, \nu = 0$ resp.

But I'm not clear where to evaluate $L, M$ ? The above expression for $L_{k-1} , M_k$ tells only the $x, u$ but not the noise terms. Should $\xi = 0, \nu = 0$ or some random sample?

## 1 Answer

Yes, you should use 0 for both. As the EKF assumes that noises have 0 mean, I would evaluate them there. If you look up the Wikipedia page, it also suggests the same.

If you would use a single sample, then

1. You would need a random number generator (if you use the algorithm on an embedded system, you might want to avoid this).
2. Most importantly, this would introduce high variance - you can use multiple samples, but then using the mean could be a compromise.