# Converting an FIR Filter Model to a State Space Model for Kalman Filtering

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:

Thanks!

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