Butterworth filter prototype
[z,p,k] = buttap(n)
[z,p,k] = buttap(n) returns the poles and gain of an order n Butterworth analog lowpass filter prototype. The function returns the poles in the length n column vector p and the gain in scalar k. z is an empty matrix because there are no zeros. The transfer function is
Butterworth filters are characterized by a magnitude response that is maximally flat in the passband and monotonic overall. In the lowpass case, the first 2n-1 derivatives of the squared magnitude response are zero at ω = 0. The squared magnitude response function is
corresponding to a transfer function with poles equally spaced around a circle in the left half plane. The magnitude response at the cutoff angular frequency ω0 is always regardless of the filter order. buttap sets ω0 to 1 for a normalized result.
Design a 9th-order Butterworth analog lowpass filter. Display its magnitude and phase responses.
[z,p,k] = buttap(9); % Butterworth filter prototype [num,den] = zp2tf(z,p,k); % Convert to transfer function form freqs(num,den) % Frequency response of analog filter
 Parks, T. W., and C. S. Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987, chap. 7.