I was just studying an old circuit analysis textbook that was describing how to design a Butterworth filter, and that seemed easy enough.. then, I started to wonder if I can take this analog filter and convert it into a digital filter. Its not really an exercise in the textbook, i was just curious how to convert the analog filter into a digital filter just for fun without the heavy DSP theory.
So I was tried to taking a toy Butterworth filter to do just that. For example, Let's suppose I had an analog filter:
$$H_a(j\Omega) = \frac{1}{(1+j\Omega)(2+j\Omega)}$$
and i wanted to convert this into a digital filter with say a sampling period $T=200\pi$ rad/sec, and neglecting the effects of aliasing, using this formula:
$$H(e^{j\omega}) = H_a(j\Omega)\Big|_{\Omega = \omega/T}$$
What would $H(z)$ and $h[n]$ look like for the digital filter?