scipy.signal.

bilinear_zpk

scipy.signal.bilinear_zpk(z, p, k, fs)

Return a digital IIR filter from an analog one using a bilinear transform.

Transform a set of poles and zeros from the analog s-plane to the digital z-plane using Tustin’s method, which substitutes 2*fs*(z-1) / (z+1) for s, maintaining the shape of the frequency response.

Parameters:

zarray_like

Zeros of the analog filter transfer function.

parray_like

Poles of the analog filter transfer function.

kfloat

System gain of the analog filter transfer function.

fsfloat

Sample rate, as ordinary frequency (e.g., hertz). No prewarping is done in this function.

Returns:

zndarray

Zeros of the transformed digital filter transfer function.

pndarray

Poles of the transformed digital filter transfer function.

kfloat

System gain of the transformed digital filter.

See also

lp2lp_zpk, lp2hp_zpk, lp2bp_zpk, lp2bs_zpk

bilinear

Notes

Added in version 1.1.0.

Array API Standard Support

bilinear_zpk has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	⚠️ no JIT	⛔
Dask	⚠️ computes graph	n/a

See Support for the array API standard for more information.

Examples

Try it in your browser!

>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt

>>> fs = 100
>>> bf = 2 * np.pi * np.array([7, 13])
>>> filts = signal.lti(*signal.butter(4, bf, btype='bandpass', analog=True,
...                                   output='zpk'))
>>> filtz = signal.lti(*signal.bilinear_zpk(filts.zeros, filts.poles,
...                                         filts.gain, fs))
>>> wz, hz = signal.freqz_zpk(filtz.zeros, filtz.poles, filtz.gain)
>>> ws, hs = signal.freqs_zpk(filts.zeros, filts.poles, filts.gain,
...                           worN=fs*wz)
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hz).clip(1e-15)),
...              label=r'$|H_z(e^{j \omega})|$')
>>> plt.semilogx(wz*fs/(2*np.pi), 20*np.log10(np.abs(hs).clip(1e-15)),
...              label=r'$|H(j \omega)|$')
>>> plt.legend()
>>> plt.xlabel('Frequency [Hz]')
>>> plt.ylabel('Amplitude [dB]')
>>> plt.grid(True)

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