Check Bilinear transformation for the formulas!!
Butterworth design
- Find epsilon and lambda
- Find the actual frequencies (same as the normal frequencies unless T not equal to 1)
- Find order N
- Round off order N upwards
- Find omegaC cutoff frequency
- Plug N and omegaC into H(s) transfer function formula
- Find Ha(s) using sigma formula
- Take H(s) denominator and find roots for that. Get a complex number for S
- Replace H(s) denominator with both versions of the root multiplied
- Use partial fractions to get the numerator coefficients
- Get C1 and C2 in J terms
- Plug all of this stuff into the Ha(s) formula and get the analog transfer function fully
- Now plug all of these values into the H(z) formula and get the discrete transfer function