scipy.signal.

bilinear

scipy.signal.bilinear(b, a, fs=1.0)

Calculate a digital IIR filter from an analog transfer function by utilizing the bilinear transform.

Parameters:

barray_like

Coefficients of the numerator polynomial of the analog transfer function in form of a complex- or real-valued 1d array.

aarray_like

Coefficients of the denominator polynomial of the analog transfer function in form of a complex- or real-valued 1d array.

fsfloat

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

Returns:

betandarray

Coefficients of the numerator polynomial of the digital transfer function in form of a complex- or real-valued 1d array.

alphandarray

Coefficients of the denominator polynomial of the digital transfer function in form of a complex- or real-valued 1d array.

See also

lp2lp, lp2hp, lp2bp, lp2bs, bilinear_zpk

Notes

The parameters \(b = [b_0, \ldots, b_Q]\) and \(a = [a_0, \ldots, a_P]\) are 1d arrays of length \(Q+1\) and \(P+1\). They define the analog transfer function

\[H_a(s) = \frac{b_0 s^Q + b_1 s^{Q-1} + \cdots + b_Q}{ a_0 s^P + a_1 s^{P-1} + \cdots + a_P}\ .\]

The bilinear transform [1] is applied by substituting

\[s = \kappa \frac{z-1}{z+1}\ , \qquad \kappa := 2 f_s\ ,\]

into \(H_a(s)\), with \(f_s\) being the sampling rate. This results in the digital transfer function in the \(z\)-domain

\[H_d(z) = \frac{b_0 \left(\kappa \frac{z-1}{z+1}\right)^Q + b_1 \left(\kappa \frac{z-1}{z+1}\right)^{Q-1} + \cdots + b_Q}{ a_0 \left(\kappa \frac{z-1}{z+1}\right)^P + a_1 \left(\kappa \frac{z-1}{z+1}\right)^{P-1} + \cdots + a_P}\ .\]

This expression can be simplified by multiplying numerator and denominator by \((z+1)^N\), with \(N=\max(P, Q)\). This allows \(H_d(z)\) to be reformulated as

\[\begin{split}& & \frac{b_0 \big(\kappa (z-1)\big)^Q (z+1)^{N-Q} + b_1 \big(\kappa (z-1)\big)^{Q-1} (z+1)^{N-Q+1} + \cdots + b_Q(z+1)^N}{ a_0 \big(\kappa (z-1)\big)^P (z+1)^{N-P} + a_1 \big(\kappa (z-1)\big)^{P-1} (z+1)^{N-P+1} + \cdots + a_P(z+1)^N}\\ &=:& \frac{\beta_0 + \beta_1 z^{-1} + \cdots + \beta_N z^{-N}}{ \alpha_0 + \alpha_1 z^{-1} + \cdots + \alpha_N z^{-N}}\ .\end{split}\]

This is the equation implemented to perform the bilinear transform. Note that for large \(f_s\), \(\kappa^Q\) or \(\kappa^P\) can cause a numeric overflow for sufficiently large \(P\) or \(Q\).

References

[1] (1,2)

“Bilinear Transform”, Wikipedia, https://en.wikipedia.org/wiki/Bilinear_transform

Examples

Try it in your browser!

The following example shows the frequency response of an analog bandpass filter and the corresponding digital filter derived by utilitzing the bilinear transform:

>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> import numpy as np
...
>>> fs = 100  # sampling frequency
>>> om_c = 2 * np.pi * np.array([7, 13])  # corner frequencies
>>> bb_s, aa_s = signal.butter(4, om_c, btype='bandpass', analog=True, output='ba')
>>> bb_z, aa_z = signal.bilinear(bb_s, aa_s, fs)
...
>>> w_z, H_z = signal.freqz(bb_z, aa_z)  # frequency response of digitial filter
>>> w_s, H_s = signal.freqs(bb_s, aa_s, worN=w_z*fs)  # analog filter response
...
>>> f_z, f_s = w_z * fs / (2*np.pi), w_s / (2*np.pi)
>>> Hz_dB, Hs_dB = (20*np.log10(np.abs(H_).clip(1e-10)) for H_ in (H_z, H_s))
>>> fg0, ax0 = plt.subplots()
>>> ax0.set_title("Frequency Response of 4-th order Bandpass Filter")
>>> ax0.set(xlabel='Frequency $f$ in Hertz', ylabel='Magnitude in dB',
...         xlim=[f_z[1], fs/2], ylim=[-200, 2])
>>> ax0.semilogx(f_z, Hz_dB, alpha=.5, label=r'$|H_z(e^{j 2 \pi f})|$')
>>> ax0.semilogx(f_s, Hs_dB, alpha=.5, label=r'$|H_s(j 2 \pi f)|$')
>>> ax0.legend()
>>> ax0.grid(which='both', axis='x')
>>> ax0.grid(which='major', axis='y')
>>> plt.show()

The difference in the higher frequencies shown in the plot is caused by an effect called “frequency warping”. [1] describes a method called “pre-warping” to reduce those deviations.

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