As part of my effort to understand how a high-pass filter can be implemented, I'm reading this section of HandWiki.
My question here is: How do the 2 simple formula presented there filter out at a cut-off at 8kHz? Do we just cleverly set the $a_{1,2}$ and $b_{0,1,2}$ coefficents? Do we not need another (set of) components?
Do we just cleverly set the $a_{1,2}$ and $b_{0,1,2}$ coefficents?
Yes.
The coefficients determine entirely what he filter does: shape (lowpass, highpass, bandpass, peaking EQ, shelve, etc.), steepness, crossover frequency, Q, gain, etc.
Calculating the coefficients is called "filter design" and many textbooks have been written on the topic. It's mathematically challenging and typically requires university level math/engineering skills. An example for an introductionary course is https://ccrma.stanford.edu/~jos/filters/
This being said: most scientific programming languages (Matlab/Octave, Python, etc) have built-in libraries: you can input your filter requirements (including the cutoff frequency) and the filter coefficients pop out.
