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I am studying the Kalman filter and its basic implementation, and it was asked to use the filter to estimate a signal observed in noise $$y(n) = x(n) + v(n)$$ where $v(n) \sim \mathcal{N}(0, \sigma^2)$ and $x(n)$ is modeled as an ARMA(2,2) process $$x(n) = b_0 u(n) + b_1 u(n-1) - a_1 x(n-1) - a_2 x(n-2).$$

The exercise is conceptually intersting, but I was wondering if there is an direct application for this case. I mean, which processes, in real life, can be modeled as an ARMA(2,2) and in which context the Kalman filter is used, not as a predictor, for a signal like $s(n)$?

Thanks!

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ARMA models are useful when you need to model a Signal plus Noise situation where the signal is an AR process and the noise models sensor noise. The overall model is an ARMA model.

See HL Van Trees, Detection, Estimation, and Modulation Theory, vol 4 Array Processing. He gives an example of a Spatial AR process sensed by noisy sensors. The overall model is ARMA

ARMA model are also useful in situations where you have strong nulls n your spectrum like multi path and you have a noise like signal

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  • $\begingroup$ Would you be able to elaborate on "ARMA model are also useful in situations where you have strong nulls n your spectrum like multi path and you have a noise like signal"? $\endgroup$ – Brian Feb 27 at 22:17
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My friend.... there are simply too much applications for ARMAs because.... they are second order systems models in discrete space. Not in every workprocess, mainly in analysis studies or design stages. Remember that most all data acquisition systems works on "discrete space" - did you realize that? :)

If you are in engineering or physics or whatever analysis project, almost every simple process can be modeled as a linear dynamic second order system.

The Kalman Filter is essentially an estimator, de-noiser, filter, predictor, etc. whatever utility you can gave to this method for "discarding" the noise. On this regard, the Kalman Filter is exploited when there is a "noisy state" which is desired to be recovered.

For applications of the Kalman Filter: https://en.wikipedia.org/wiki/Kalman_filter#Applications

Best regards.

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