# Kalman, state-space model

1. Aug 19, 2008

### mr.t

1. The problem statement, all variables and given/known data
A time discrete stocastic signal is described by
$$s(k) = w(k-1) + aw(k-2)$$, |a|<1
and w(n) is white gaussian noise with $$m_w = 0, \sigma_w^2 = 1$$. It is observed under influence of white noise:
$$y(k) = s(k) + v(k)$$
where v(n) is white gaussian noise with $$m_v = 0, \sigma_v^2=1$$. v(n) and w(n) are independant.

Problem: Find the space-state model:
$$x(k+1) = Ax(k) + Bw(k) y(k) = Cx(k) + v(k)$$

By using the state:
$$x(k) = \bmatrix s(k) \\ w(k-1) \endbmatrix$$

2. Relevant equations
(given above)

3. The attempt at a solution
I have only solved these problems when there is a AR-part. As this is an ARMA(0,2) I have no clue and need help. If its just an MA-part, then the whole A-matrix is zero? And how should I use the fact that Im suppose to use the specified states? How does that affect the state-space model?

Thanks!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 20, 2008

### mr.t

Just want to let you guys know that i've solved it. (pretty sure at least :tongue2:)