Part I: Warm$-$up$.$

See Lecture notes on $$System Identification$$ for suggestions regarding useful Matlab commands.

$($a$)$ Write, by hand, a Toeplitz matrix multiply to implement the convolution x$[$n$]$h$[$n$]=$ y$[$n$]$ for n $=0,1,...,5,$

where the finite$-$duration input and output are given by

x$[$n$]=\{1,2,1,1\}$ and y$[$n$]=\{3,8,2,3,1,1\}$ h is unknown.

$($Pro tip: ensure that your array sizes are conformal: m $\backslash$times n$,$ n $\backslash$times $1,$ m $\backslash$times $1$ for some integers m$,$ n$.)$

$($b$)$ Use Matlab to construct the matrix$-$vector equations and solve least$-$squares for h$.($Report h and your

Matlab commands.$)$

$($c$)$ Write, by hand, a Toepliz matrix multiply to implement equations corresponding to the LTI system

y$[$n$]=$ b$0$x$[$n$]$ a$1$y$[$n $1]$ a$2$y$[$n $2],$ n $=0,1,...,5$

where

x$[$n$]=\{1,2,1,1\}$ and y$[$n$]=\{1,1\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}125,1\mathrm{.}4375,1\mathrm{.}25\}$ b$0,$ a$1,$ a$2$ are unknown

$.$

$($d$)$ Use Matlab to construct the matrix$-$vector equations and solve least$-$squares for b$0,$ a$1,$ a$2.($Report b$0,$ a$1,$ a$2$

and your Matlab commands.$)$