I am performing some research about digital filtering for my BS Degree thesis in Electronic and TLC Engineering.

I found myself quite stucked with the project since I can't easily figure out some important concept about the parameters of the 2nd order filters;

to give an idea of what I am talking about, the whole project is related to a digital signal generated by a ROM beeing filtered through a second order filter and its implementation on a FPGA with the use of some external components to set some parameters (I am going to use some potentiometers to set, for instance, the values of frequency and amplitude for the input wave; at this moment I want to set up the regulation of Q factor and the cutoff frequency for the filter. This last point is driving me mad, since I cannot find a correlation between the choice of some poles/zeros and those parameters)

I am going to attach a couple images reporting what I was basically searching for, this is a paper from a NASA scientist I found online. I am just wondering where these calculation come from, I tried to google them but with no success. My question is, do you have some paper or link that explain how to get these results, starting from why do you factor out 1/2 from the general equation arriving to the constraints given for certain values. I want to take those results as certain and correct, of course, but I sholud further document them in my paperwork, as you can imagine.

How can you derive beta, theta, tau and gamma, as in these 2 screen?

How can you derive tau in this third slide?

and finally, how do you get these constraints about gamma and beta?

Thanks in advance for your help