What's the formula behind the coefficients calculation of the Filter Designer?

I'm using the Filter Designer to find coefficients for a Butterworth digital filter on microcontroller. How can I calculate them without the use of the app? Given the sample rate and the cut-off frequency, what's the formula find the values of a0, a1, a2, b0, b1, b2 and the gains for every section?

Answer 1

[Star Strider]

This is covered in all the signal processing texts I’ve ever used. A good discussion is given in DESIGN OF a 5th ORDER BUTTERWORTH LOW-PASS FILTER USING SALLEN & KEY CIRCUIT (link). It first discusses the calculation of the filter coefficients for a lowpass Butterworth design, and then component calculations if you want to implement it in hardware.

Since I get the impression that you want to implement this in software as a discrete filter, the easiest way to do that is with the Bilinear transform (link).

If you want to avoid all of these complications, use the MATLAB buttord (link) and butter (link) functions, (and related functions to transform them into second-order-section or other implementations if you want to do that).

Answer 2

[Honglei Chen]

If you go to File -> Generate MATLAB Code, you will be able to get a script containing functions that used by the app to design the filter. You can then run that script without the app.

HTH

[Andrea Verdecchia]

ok, but i need the formula to find the coefficients.

[Honglei Chen]

You can find those information in reference pages. For example, you will use butter to design Butterworth filter. It's reference page is at

https://www.mathworks.com/help/signal/ref/butter.html

There is an algorithm section pointing to different functions and in those functions, there are reference listed. You can also find similar information online, such as,

https://www.dsprelated.com/freebooks/filters/Butterworth_Lowpass_Design.html

HTH