SciPy

scipy.signal.tf2zpk

scipy.signal.tf2zpk(b, a)[source]

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

Parameters :

b : ndarray

Numerator polynomial.

a : ndarray

Denominator polynomial.

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)} + \cdots + b_M}
{a_0 s^N + a_1 s^{(N-1)} + \cdots + a_N}

or a discrete-time filter of the form:

H(z) = \frac
{b_0 z^M + b_1 z^{(M-1)} + \cdots + b_M}
{a_0 z^N + a_1 z^{(N-1)} + \cdots + 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} + \cdots + b_M z^{-M}}
{a_0 + a_1 z^{-1} + \cdots + 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.

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