$1.$ Butterworth filter design $(4$ points$)$

We would like to design a fifth$-$order Butterworth highpass filter with cutoff frequency $\backslash (10\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$($a$)(1$ point$)$ Determine the locations of the poles.

$($b$)(1$ point$)$ Determine the transfer function $\backslash (\backslash$mathbf$\{$H$\}(\backslash$mathbf$\{$s$\})\backslash ).$

$($c$)(1$ point$)$ We implement this filter using first$-$order RC filters and$/$or second$-$order SallenKey filters. Draw the circuit diagram.

$($d$)(1$ point$)$ Suppose that we use $\backslash (10-\backslash$mathrm$\{$k$\}\backslash$Omega $\backslash )$ resistors. Determine the capacitors in all the stages of the filter.

$2.$ Butterworth filter design $(6$ points$)$

We would like to design a Butterworth lowpass filter that meets the specifications of $\backslash (|\backslash$mathbf$\{$H$\}(\backslash$omega$)|=\backslash )0\mathrm{.}9$ at $10$ Hz and $\backslash (|\backslash$mathbf$\{$H$\}(\backslash$omega$)|=0\mathrm{.}1\backslash )$ at $28$ Hz $.$

$($a$)(2$ points$)$ Determine the order and the cutoff frequency of the filter.

$($b$)(1$ point$)$ Determine the locations of the poles.

$($c$)(1$ point$)$ Determine the transfer function $\backslash (\backslash$mathbf$\{$H$\}(\backslash$mathrm$\{$s$\})\backslash ).$

$($d$)(1$ point$)$ We implement this filter using first$-$order RC filters and$/$or second$-$order SallenKey filters. Draw the circuit diagram.

$($e$)(1$ point$)$ Suppose that we use $\backslash (10-\backslash$mathrm$\{$k$\}\backslash$Omega $\backslash )$ resistors. Determine the capacitors in all the stages of the filter.