$(20$ points$)$

$($a$)$ Design an analog lowpass filter with a cut$-$off frequency of $12ra\frac{d}{sec}$ by starting with an analogue prototype first$-$order lowpass filter with cut$-$off frequency of $1ra\frac{d}{sec}.$ Show the system transfer function $H(s).$

$($b$)$ Design an IIR digital filter $H(z)$ that corresponds to the above $H(s)$ by using the bilinear transform method without prewarping with $T_s=0\mathrm{.}01$ second. Show the system transfer function $H(z)$ and find its corresponding digital cut$-$off frequency ${}_c$ approximately.

$($c$)$ What is the desired $($true$)$ digital cut$-$off frequency ${}_c$ for the above $H(z)?$ Prewarp the analog cut$-$off frequency ${}_c$ corresponding to the true ${}_c$ and find $H(z)$ using the bilinear transform method with $T_s=0\mathrm{.}01$ second. please don't use ai