Calculate Coefficients of 2nd Order Butterworth Low Pass Filter

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

For those wondering where those magical formulas from the other answers come from, here's a derivation following this example.

Starting with the transfer function for the Butterworth filter

G(s) = wc^2 / (s^2 + s*sqrt(2)*wc + wc^2)

where wc is the cutoff frequency, apply the bilinear z-transform, i.e. substitute s = 2/T*(1-z^-1)/(1+z^-1):

G(z) = wc^2 / ((2/T*(1-z^-1)/(1+z^-1))^2 + (2/T*(1-z^-1)/(1+z^-1))*sqrt(2)*wc + wc^2)

T is the sampling period [s].

The cutoff frequency needs to be pre-warped to compensate for the nonlinear relation between analog and digital frequency introduced by the z-transform:

wc = 2/T * tan(wd*T/2)

where wd is the desired cutoff frequency [rad/s].

Let C = tan(wd*T/2), for convenience, so that wc = 2/T*C.

Substituting this into the equation, the 2/T factors drop out:

G(z) = C^2 / ((1-z^-1)/(1+z^-1))^2 + (1-z^-1)/(1+z^-1)*sqrt(2)*C + C^2)

Multiply the numerator and denominator by (1+z^-1)^2 and expand, which yields:

G(z) = C^2*(1 + 2*z^-1 + z^-2) / (1 + sqrt(2)*C + C^2 + 2*(C^2-1)*z^-1 + (1-sqrt(2)*C+C^2)*z^-2')

Now, divide both numerator and denominator by the constant term from the denominator. For convenience, let D = 1 + sqrt(2)*C + C^2:

G(z) = C^2/D*(1 + 2*z^-1 + z^-2) / (1 + 2*(C^2-1)/D*z^-1 + (1-sqrt(2)*C+C^2)/D*z^-2')

This form is equivalent to the one we are looking for:

G(z) = (b0 + b1*z^-1 + b2*z^-1) / (1 + a1*z^-1 +a2*z^-2)

So we get the coefficients by equating them:

a0 = 1

a1 = 2*(C^2-1)/D

a2 = (1-sqrt(2)*C+C^2)/D

b0 = C^2/D

b1 = 2*b0

b2 = b0

where, again, D = 1 + sqrt(2)*C + C^2, C = tan(wd*T/2), wd is the desired cutoff frequency [rad/s], T is the sampling period [s].

You can use this link to get the coefficients of n-order Butterworth filter with specific sample rate and cut of frequency. In order to test the result. You can use MATLAB to obtain the coefficients and compare with program's output

http://www.exstrom.com/journal/sigproc

fnorm = f_cutoff/(f_sample_rate/2); % normalized cut off freq, http://www.exstrom.com/journal/sigproc
% Low pass Butterworth filter of order N
[b1, a1] = butter(nth_order, fnorm,'low');