$($Kalman filter$)$ Consider the following state$-$space form

$x[n+1]=Ax[n]+Bu[n]$

$y[n]=Cx[n]+w[n]$

where

$A=\left[\begin{array}{ccccccc}1\mathrm{.}1269& -0\mathrm{.}4940& 0\mathrm{.}1129& 0& 0& 1\mathrm{.}0000& 0\end{array}\right]$

$0$

$0$

$1\mathrm{.}0000$

$B=\left[\begin{array}{c}-0\mathrm{.}3832\end{array}\right]$

$0\mathrm{.}5919$

$0\mathrm{.}5191$

$C=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

$u[n]=N(0,2\mathrm{.}3)$

$w[n]=N(0,1)$

$($a$)$ Generate $10,000$ measurement samples using MATLAB. Draw the samples for the signal $Cx[n]$ using plot function.

$($b$)$ Run Kalman filter to produce the estimate $[n|]$ based on the measurement $y[n].$ Draw the last $100$ samples of $[n|]$ and $Cx[n]$ in the same plot.

$($c$)$ Compute the MSE between $[n|]$ and $Cx[n]$ by

$[i]|$

Describe how MSE changes with respect to the variance of $w[n].$