Calculate Coefficients of 2nd Order Butterworth Low Pass Filter

Question

With the,

Sampling Freq: 10kHz
Cut-off Freq: 1kHz

How do I actually calculate the coefficients for the difference equation below?

I know the difference equation will be in this form, but do not know how to actually work out and come up with the numbers for the coefficients b0, b1, b2, a1, a2

y(n)  =  b0.x(n) + b1.x(n-1) + b2.x(n-2) + a1.y(n-1) + a2.y(n-2)

I will eventually be implementing this LPF in C++ but I need to know how to actually calculate the coefficients first before I can get anywhere with it

Answer 1

Here you go. ff is the frequency ratio, 0.1 in your case:

    const double ita =1.0/ tan(M_PI*ff);
    const double q=sqrt(2.0);
    b0 = 1.0 / (1.0 + q*ita + ita*ita);
    b1= 2*b0;
    b2= b0;
    a1 = 2.0 * (ita*ita - 1.0) * b0;
    a2 = -(1.0 - q*ita + ita*ita) * b0;

and the result is:

b0=0.0674553
b1=0.134911
b2=0.0674553
a1=1.14298
a2=-0.412802