`scipy.signal._filter_design:tf2zpk`

source: build-install/usr/lib/python3.14/site-packages/scipy/signal/_filter_design.py:1171

Signature

def tf2zpk(b, a)

Summary

Return zero, pole, gain (z, p, k) representation from a numerator, denominator representation of a linear filter.

Parameters

b : array_like, shape (M,): Numerator polynomial coefficients. Must be 1-D.
a : array_like, shape (N,): Denominator polynomial coefficients. Must be 1-D.

Returns

z : ndarray: Zeros of the transfer function.
p : ndarray: Poles of the transfer function.
k : float: System gain.

Notes

If some values of b are too close to 0, they are removed. In that case, a BadCoefficients warning is emitted.

The b and a arrays are interpreted as coefficients for positive, descending powers of the transfer function variable. So the inputs $b = [b_{0}, b_{1}, ..., b_{M}]$ and $a = [a_{0}, a_{1}, ..., a_{N}]$ can represent an analog filter of the form:

H (s) = \frac{b _{0} s ^{M} + b _{1} s ^{(M - 1)} + \dots + b _{M}}{a _{0} s ^{N} + a _{1} s ^{(N - 1)} + \dots + a _{N}}

or a discrete-time filter of the form:

H (z) = \frac{b _{0} z ^{M} + b _{1} z ^{(M - 1)} + \dots + b _{M}}{a _{0} z ^{N} + a _{1} z ^{(N - 1)} + \dots + a _{N}}

This "positive powers" form is found more commonly in controls engineering. If M and N are equal (which is true for all filters generated by the bilinear transform), then this happens to be equivalent to the "negative powers" discrete-time form preferred in DSP:

H (z) = \frac{b _{0} + b _{1} z ^{- 1} + \dots + b _{M} z ^{- M}}{a _{0} + a _{1} z ^{- 1} + \dots + a _{N} z ^{- N}}

Although this is true for common filters, remember that this is not true in the general case. If M and N are not equal, the discrete-time transfer function coefficients must first be converted to the "positive powers" form before finding the poles and zeros.

Examples

Find the zeroes, poles and gain of a filter with the transfer function .. math:: H(s) = \frac{3s^2}{s^2 + 5s + 13}

from scipy.signal import tf2zpk

✓

tf2zpk([3, 0, 0], [1, 5, 13])

✗

(array([0., 0.]), array([-2.5+2.6j, -2.5-2.6j]), np.float64(3.0))

Aliases

scipy.signal.tf2zpk