# scipy.signal.bilinear_zpk#

scipy.signal.bilinear_zpk(z, p, k, fs)[source]#

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 Tustinâ€™s method, which substitutes (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.

Notes

New in version 1.1.0.

Examples

>>> 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('Magnitude [dB]')
>>> plt.grid(True)