Signal Processing Toolbox™

buttap - Butterworth analog lowpass filter prototype

Syntax

[z,p,k] = buttap(n)

Description

[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.

Algorithm

z = [];
p = exp(sqrt(-1)*(pi*(1:2:2*n-1)/(2*n)+pi/2)).';
k = real(prod(-p));

References

[1] Parks, T.W., and C.S. Burrus. Digital Filter Design. New York: John Wiley & Sons, 1987. Chapter7.

See Also

besselap, butter, cheb1ap, cheb2ap, ellipap

buffer

butter