Noise Whiteness hypothesis in Kalman filtering

[Unconscious]

In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise. I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where such hypothesis is crucial.
Could you help me showing where the mathematical proof fails if I remove such hypothesis?
Thanks.

 

[jedishrfu]

Your answer might be found in this thesis on the Kalman filter:

https://people.duke.edu/~hpgavin/SystemID/References/Balut-KalmanFilter-PhD-NEU-2011.pdf

Around page 137, they mention modifying the white noise hypothesis.

 

[Unconscious]

Thank you for the link, but unfortunately I have not found an answer in that pages.
Anyway, I try to share with you my doubt more precisely.

Let's consider the formulas in Fig. 1 in this article:
https://arxiv.org/pdf/1712.01406.pdf

These two formulas have been obtained without assumptions on the process noise. Then, again without assumptions on the process noise, the author arrives at formula (5), in which appears the probability density $p(x_k | x_{k-1})$. Taking into account the first equation of the system (1), now he says that if we suppose the gaussianity and the whiteness of the process noise, we can then write such probability density as $\mathcal{N}(f(x_{k-1}),Q)$.
It is exactly in this point that my doubt arises: it seems to me that I could have written $p(x_k | x_{k-1})=\mathcal{N}(f(x_{k-1}),Q)$ even if I had only assumed gaussianity of the process noise, without whiteness, because $x_k$ is conditioned only by $x_{k-1}$ and not also by $q_{k-1},q_{k-2},...$.

 

[Office_Shredder]

By definition, white noise is noise that is uncorrelated from sample to sample.

Suppose the actual noise of your sensor was totally correlated - on the first sample, you pick a gaussian, and that's the noise the entire time.

Let's say the first pick is a positive noise.

Then a better algorithm would be able to learn that the noise picked is positive (e.g. it notices that 2/3 of the measurements are higher than its estimate), and could adjust to start making lower estimates given the measurements.

On each step, in some sense you could say ok, I don't know anything about the noise, so since I'm ignorant about it it's a gaussian as far as I can tell, and this kind of works (you can use a Kalman filter in a variety of real world situations that don't fit the theoretical framework and it works pretty well still). But it's not the best estimate you can form if you *are* informed about the noise.This is just a modeling question. If you make 8 samples and all of them return higher measurements than you expect, do you chalk it up to bad luck, bad measuring, or a bad understanding of what the process is? The assumption of the noise being white means the Kalman filter says it's bad luck.

 

[Unconscious]

This is clear to me, thank you.
What is not clear to me is the pure math, in particular the step highlighted in my last post.
So, are you saying to me that if the noise is not white, then I can’t correctly write the equality $p(x_k|x_{k-1})=\mathcal{N}(f(x_{k-1}),Q)$? In other words, this equality would be a logic error?