Consider the FIR filter with the following input$-$output relationship $($note the missing coefficient$)$:

y$[$n$]=$ x$[$n$]+3$x$[$n $1]+3$x$[$n $3]+$ x$[$n $4],$ n in Z

$($i$)$ Determine the response y$($i$)[]$ of the filter to each of the input signals given by the equations

below $($valid for all n in Z$).$

x$(1)[$n$]=(3/4)$n $(2$ pts$.)$

x$(2)[$n$]=(4/3)$n $(2$ pts$.)$

x$(3)[$n$]=1+3$n $(2$ pts$.)$

x$(4)[$n$]=$ cos$($n$($$/6)+2\mathrm{.}5)(3$ pts$.)$

x$(5)[$n$]=2$n$$ cos$($n$/6)(4$ pts$.)$

$($ii$)(5$ pts$.)$ The filter above is connected in series $($cascade$)$ with a filter having input$-$output

relationship

y$[$n$]=$ x$[$n$]2$x$[$n $1]+$ x$[$n $2],$ n in Z

Determine the system function H$($z$)$ of the two$-$filter cascade.

$($iii$)(2$ pts$.)$ Write out the input$-$output relationship of the two$-$filter cascade.