The Nth order Butterworth filter In general the magnitude response of the Nth order Butterworth filter with cut-off frequency Ωc is given by
With the normalized frequency variable defined as r = W / Wc , the MATLAB segment below plots the magnitude response for 1st, 3rd, and 10th order filters, that is, N = 1, 3, and 10, respectively. Note that as the filter order increases the response becomes flatter on either side of the cut-off; and the transition (cut-off) becomes sharper.
r = 0: 0.1: 3;
H1 = 1./sqrt(1.+ r .^2); H3 = 1./sqrt(1.+ r .^6); H10 = 1./sqrt(1.+ r .^20);
plot (r, H1, r, H3, 'r', r, H10, 'k');
legend ('1st Order', '3rd Order', '10th Order');
xlabel ('Normalized frequency, r'), ylabel('|H(r)|'); grid; title ('Magnitude')
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