discrete signals - Doubt in state space representation

$y$ is scalar observations and so C will be a 1x2 matrix.

I want to represent the following model as a state space representation so as to estimate the hidden states from the noisy observations $y$ using Kalman filter.

The state space model :

x(t+1) = Ax(t) + w(t)

y(t) = Cx(t) + v(t)

w(t) = N(0,Q)


v(t) = N(0,R)

$w(t)$ is a pseudo-random binary signal that excites/ drives ; and $v(t) = N(0,\sigma^2_v)$ is the measurement noise.

1. The model is an FIR (MA) filter

$$x(t) = h_1 \epsilon(t-1) + h_2 \epsilon(t-2) + \epsilon(t)$$ $$y(t) = x(t) + v(t)$$ $$ y(t) = h^T \epsilon(t) + v(t)$$

(In vector form)

where $\epsilon(t) = w(t)$.

2. The other model is an IIR (AR) filter $$x(t) = ax(t-1) + bx(t-2)+ w(t)$$

The state space representation:

$$x(t+1) = a^Tx(t) + w(t)$$

$$y(t) = h^Tx(t) + v(t)$$

How do I represent these as state space so as to apply Kalman Filter?

There are several ways to represent time series models. This is how I proceeded, but unsure because the output of the log-likelihood is a matrix of 2 by 2 with off diagonal elements being infinity and the diagonal elements are same positive values. So, the dimension and the value of log-likelihood is incorrect, I should get negative instead of positive values.

1. FIR :

Re-writing the above model as:

$$x(t+1) = h_1 \epsilon(t) + h_2 \epsilon(t-1) + \epsilon(t+1)$$ $$y(t) = Cx(t)+v(t)$$

State Space :

$ \left[ \begin{array}{c} x(t+1) \\ x(t)\\ x(t-1) \end{array} \right] $ = $ \left[ \begin{array}{ccc} 1 & h_1 & h_2 \\ 0 & 1 & h_1 \\ 0 & 0 & 1\end{array} \right] $ $\times$ $ \left[ \begin{array}{c} e(t+1)\\ e(t) \\ e(t-1)\end{array} \right] $

$ y(t)$ = $\left[ \begin{array}{c} 1 \hskip 5 pt 0 \hskip 5 pt 0\end{array} \right] $ $\times$ $ \left[ \begin{array}{c} e(t+1)\\ e(t) \\ e(t-1)\end{array} \right] $ + $v(t)$

2. 
   

   IIR (AR model)

   
   

   $ \left[ \begin{array}{c} x(t+1) \\ x(t) \end{array} \right] $ = $ \left[ \begin{array}{cc} a & b \\ 1 & 0 \end{array} \right] $ $\times$ $ \left[ \begin{array}{c} x(t)\\ x(t-1)\end{array} \right] $ + $\left[ \begin{array}{c} 1\\ 0\end{array} \right] $ $\times$ $\left[ \begin{array}{c} w(t+1)\\ w(t)\end{array} \right] $

   
   
   

   $ y(t)$ = $\left[ \begin{array}{cc} 1 \hskip 5 pt 0\end{array} \right]$ $\times$ $ \left[ \begin{array}{c} x(t)\\ x(t-1)\end{array} \right] $ + $v(t)$

   
   

Answer

Your FIR state space representation seems to be doing too much.

The way I would write it is to have the process noise is $\epsilon(t)$ as your input, and your state as two time-delayed copies of it:

$$x(t+1) = \left[ \begin{array}{c} \epsilon(t+1)\\ \epsilon(t) \\ \epsilon(t-1) \end{array} \right] = \left[ \begin{array}{ccc} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array} \right] x(t) + \left[\begin{array}{c} 1\\ 0\\ 0 \end{array} \right] \epsilon(t + 1) $$

then your output equation is just: $$ y(t) = \left[ \begin{array}{ccc} 1 & h_1 & h_2 \end{array} \right] x(t) + v(t) $$

Your IIR representation is too confused for me to make head or tail out of. Can you clarify that you have the right information there?

Other questions: