I want to try and determine the true value of a quantity $\alpha[k]$ from observations of a related quantity $\vartheta[k]$ using a Kalman filter. The observations are of the following FIR filter form:

$$ \alpha[k] = \frac{1}{h} \sum_{n=0}^{N} c_{n} \vartheta[k - nh] + w[k]$$

where $w[k] \sim \mathcal{N}(0, \sigma^{2})$ and the $c_{n}$ are known ahead of time, and so is $h$.

I want to put this into a state space model in order to process it with a Kalman filter assuming additive process noise. To start the conversion, I can put it in transfer function form:

$$ \alpha[k] = \frac{1}{h} \sum_{n=0}^{N} c_{n} z^{-n} \vartheta[k] $$ $$ \frac{\alpha[k]}{\vartheta[k]} = \sum_{n=0}^{N} \tilde{c}_{n} z^{-n}$$

where $\tilde{c}_{n} = \frac{c_{n}}{h}$.

This is where I get stuck. I am not sure what to do -- which canonical form should I use? Should I even use one? Where does the noise term go anyway?

This answer gives the link to all the forms, and a general idea:

However my model is slightly different. Other answers that may be helpful, but are confusing to me:


  • $\begingroup$ Have a look at Simon Haykin_Adaptive Filter Theory, where he shows Kalman filters for FIR process equations. Derivation is due Goddard. $\endgroup$
    – Fat32
    Oct 17 '19 at 21:33

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