Homework $9($Additional Problems$)$

$1.$ For each of the following pairs of systems, determine the system TF$,$ Hp$($z$)$ and

the impulse response hp$[$n$]$ for the parallel interconnection of the two systems, and

likewise for Hc$($z$)$ and hc$[$n$]$ for the cascade interconnection.

$($i$)$ h$1[$n$]=(2/3)$n u$[$n$]$ and h$2[$n$]=(1/2)$n u$[$n$-1]$

$($ii$)$ h$1[$n$]=$ anu$[$n$]=$ h$2[-$n$]$

$2.$ For each of the following systems, find all the corresponding inverse systems,

HI$($z$).$

Which inverses are stable? Causal? FIR?

$($i$)$ h$[$n$]=(0\mathrm{.}9$n $+0\mathrm{.}5$n$)$ u$[$n$]$

$($ii$)$ h$[$n$]=[$n$]+[$n$-2]$

$($iii$)$ h$[$n$]=(-1)$n u$[$n$]$

$3.$ Draw the direct$-$form and direct form II structures for each of the following

filters:

$($i$)$ H$($z$)=1/(10\mathrm{.}49$z $2)$

$($ii$)$ H$($z$)=(1+2$ z$-1)/(11\mathrm{.}3$z$-1+0\mathrm{.}4$z$-2)$

If each of the above systems are stable, find the output, y$[$n$]$

to the input: x$[$n$]=(-2)$n u$[-$n$-1].$