I'm trying to learn more about high order filter design, with my goal being to design an 8th order Butterworth lowpass filter in spice, then in the real world. I've been reading a filter book I have and I was trying to follow this tutorial here.
I was doing ok up until this point:
From the normalised low pass Butterworth Polynomials table above, the coefficient for a third-order filter is given as \$(1+s)(1+s+s^2)\$ and this gives us a gain of \$3-A = 1\$, or \$A = 2\$. As \$A = 1 + (R_f/R_1)\$, choosing a value for both the feedback resistor \$R_f\$ and resistor \$R1\$ gives us values of \$1\$kΩ and \$1\$kΩ respectively, ( \$1\$kΩ\$/1\$kΩ\$ + 1 = 2\$ ).
I don't understand how they went from \$(1+s)(1+s+s^2)\$, and all of a sudden they know the gain from that which you use to calculate your resistors. There's no more explanation, this basically reads to me as if you start with this polynomial table, then a wizard appears, and now you know your gain.
So how did they go from the polynomial to the gain? The 8th order equation is even larger:
Does that imply that I will have different gain settings for each stage?