Problem $1$: The seismic sensor captures ground vibrations, and the amplifier boosts the signal. However, lowfrequency noise from environmental factors $($e$.$g$.,$ wind or traffic$)$ interferes with the detection of significant seismic activity. To address this, you are tasked with designing a Butterworth high$-$pass filter.

Tasks:

a$)$ Design a second$-$order Butterworth high$-$pass filter to yield $\backslash (|\backslash$underline$\{\backslash$underline$\{$H$\}($j $\backslash$infty$)\}|=1\backslash )($or $0$ dB $),$ a cutoff frequency of $\backslash (\backslash$mathrm$\{$f$\}\_\{0\}=21\backslash$mathrm$\{$kHz$\}\backslash ),$ and $\backslash (\backslash$mathrm$\{$Q$\}=0\mathrm{.}93\backslash )$

b$)$ Calculate the filter's order and provide the transfer function.

c$)$ Simulate the filter's frequency response in a circuit simulation software and verify that it meets the requirements.

d$)$ Discuss how the filter impacts the overall system performance and its importance in seismic monitoring.