# Kalman filtering in one sweep?

I was posed the question:

"With all the computation power available today (as opposed to at the time of the Apollo Program) why can we not simply implement the Kalman Filter in one sweep, instead of fusing the available measurements sequentially?".

I can see the point made, but I am unsure how the Kalman Filter can actually be implemented at once, and thought it made a good question to pose here.

Suppose $$N$$ measurements are available, and each conform to the assumptions of the Kalman Filter, including Gaussian white noise. What expression, if any, can be used to fuse all measurements at once to arrive at the same estimate, as would be arrived at if filtering is done sequentially?

My own thought is that perhaps the measurements can be weighted according to their variance, and an average could then be computed.

Note, I should mention we are dealing with a static Kalman Filter and the simple question of fusing all measurements available at a particular time.

• I am not sure I understand your meaning of fusing. Do you mean the continuous time numerics instead of that of the discrete one?
– A.Γ.
Commented Aug 20, 2019 at 14:01
• By fusion of two estimates I mean the assignment of weights to each of these estimate, to create a weighted average of the two. That provides a third estimate, which can then be fused with fourth, and so on. The question is, can this be done at once and still obtain the final outcome that the Kalman Filter would give. So, I imagine a given set of estimates are available (i.e. estimates from different "sensors"). Commented Aug 20, 2019 at 15:22
• In your question you used the word "estimates" but you probably meant "measurements", didn't you? So you are wondering, if it would be possible to fuse N measurements at once instead of fuse them recursevely, right? Commented Sep 2, 2019 at 15:39
• Anton, yes. That is correct. My mistake. I will change the wording. Commented Sep 2, 2019 at 17:56

"My own thought is that perhaps the measurements can be weighted according to their variance, and an average could then be computed."

You can. For instance, if you have:

$$x_1 (\sigma_1^2) , x_2 (\sigma_2^2), ..., x_n (\sigma_n^2)$$

You could say that:

$$\sigma_{sum} = \sum_{i=1}^n \sigma_i$$

$$x_{sum} = ( \sigma_1^2 / \sigma_{sum} ) x_1 + ( \sigma_2^2 / \sigma_{sum} ) x_2 + ... + ( \sigma_n^2 / \sigma_{sum} ) x_n$$

You do this for each of the components in your state vector, and you would have your "fused" observation, which you could update your Kalman model with.

Here is the associated Wikipedia article which describes this method, based on the Central Limit Theorem:

https://en.wikipedia.org/wiki/Sensor_fusion

Look for the part Example calculation.

I am not sure if this equals updating the filter multiple time though, I did not unfurl the whole equations.