scipy.signal.butter¶

scipy.signal.
butter
(N, Wn, btype='low', analog=False, output='ba', fs=None)[source]¶ Butterworth digital and analog filter design.
Design an Nthorder digital or analog Butterworth filter and return the filter coefficients.
 Parameters
 Nint
The order of the filter.
 Wnarray_like
The critical frequency or frequencies. For lowpass and highpass filters, Wn is a scalar; for bandpass and bandstop filters, Wn is a length2 sequence.
For a Butterworth filter, this is the point at which the gain drops to 1/sqrt(2) that of the passband (the “3 dB point”).
For digital filters, Wn are in the same units as fs. By default, fs is 2 halfcycles/sample, so these are normalized from 0 to 1, where 1 is the Nyquist frequency. (Wn is thus in halfcycles / sample.)
For analog filters, Wn is an angular frequency (e.g. rad/s).
 btype{‘lowpass’, ‘highpass’, ‘bandpass’, ‘bandstop’}, optional
The type of filter. Default is ‘lowpass’.
 analogbool, optional
When True, return an analog filter, otherwise a digital filter is returned.
 output{‘ba’, ‘zpk’, ‘sos’}, optional
Type of output: numerator/denominator (‘ba’), polezero (‘zpk’), or secondorder sections (‘sos’). Default is ‘ba’ for backwards compatibility, but ‘sos’ should be used for generalpurpose filtering.
 fsfloat, optional
The sampling frequency of the digital system.
New in version 1.2.0.
 Returns
 b, andarray, ndarray
Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if
output='ba'
. z, p, kndarray, ndarray, float
Zeros, poles, and system gain of the IIR filter transfer function. Only returned if
output='zpk'
. sosndarray
Secondorder sections representation of the IIR filter. Only returned if
output=='sos'
.
Notes
The Butterworth filter has maximally flat frequency response in the passband.
The
'sos'
output parameter was added in 0.16.0.If the transfer function form
[b, a]
is requested, numerical problems can occur since the conversion between roots and the polynomial coefficients is a numerically sensitive operation, even for N >= 4. It is recommended to work with the SOS representation.Examples
Design an analog filter and plot its frequency response, showing the critical points:
>>> from scipy import signal >>> import matplotlib.pyplot as plt
>>> b, a = signal.butter(4, 100, 'low', analog=True) >>> w, h = signal.freqs(b, a) >>> plt.semilogx(w, 20 * np.log10(abs(h))) >>> plt.title('Butterworth filter frequency response') >>> plt.xlabel('Frequency [radians / second]') >>> plt.ylabel('Amplitude [dB]') >>> plt.margins(0, 0.1) >>> plt.grid(which='both', axis='both') >>> plt.axvline(100, color='green') # cutoff frequency >>> plt.show()
Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz
>>> t = np.linspace(0, 1, 1000, False) # 1 second >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t) >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True) >>> ax1.plot(t, sig) >>> ax1.set_title('10 Hz and 20 Hz sinusoids') >>> ax1.axis([0, 1, 2, 2])
Design a digital highpass filter at 15 Hz to remove the 10 Hz tone, and apply it to the signal. (It’s recommended to use secondorder sections format when filtering, to avoid numerical error with transfer function (
ba
) format):>>> sos = signal.butter(10, 15, 'hp', fs=1000, output='sos') >>> filtered = signal.sosfilt(sos, sig) >>> ax2.plot(t, filtered) >>> ax2.set_title('After 15 Hz highpass filter') >>> ax2.axis([0, 1, 2, 2]) >>> ax2.set_xlabel('Time [seconds]') >>> plt.tight_layout() >>> plt.show()