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# buttap() - Signal Processing

### Syntax

### Example

### Output / Return Value

### Limitations

### Alternatives / See Also

### Reference

[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 isH(s)=z(s)p(s)=k(s−p(1))(s−p(2))⋯(s−p(n))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|H(ω)|2=11+(ω/ω0)2ncorresponding 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 1/2 regardless of the filter order. buttap sets ω0 to 1 for a normalized result.

[z,p,k] = buttap(n)

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