$($Problem $1$: $30\%)$

The transfer function of a third$-$order Butterworth prototype filter is

$H_p(p)=\frac{1}{p^3+2p^2+2p+1}.$

$($a$)$ Then, determine the circuit components of Ladder$-$type.

$($b$)$ Design a third$-$order Butterworth low$-$pass filter with a cutoff frequency of $3$ kHz and a $1k$ load.

$($c$)$ Design a third$-$order Butterworth band$-$pass filter with cutoff frequencies of $1$ kHz and $4$ kHz and $1k$ load. Use the results obtained in $($b$).$