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 Nth-order digital or analog Butterworth filter and return the filter coefficients.
- Parameters
- Nint
The order of the filter. For ‘bandpass’ and ‘bandstop’ filters, the resulting order of the final second-order sections (‘sos’) matrix is
2*N
, with N the number of biquad sections of the desired system.- Wnarray_like
The critical frequency or frequencies. For lowpass and highpass filters, Wn is a scalar; for bandpass and bandstop filters, Wn is a length-2 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, if fs is not specified, Wn units are normalized from 0 to 1, where 1 is the Nyquist frequency (Wn is thus in half cycles / sample and defined as 2*critical frequencies / fs). If fs is specified, Wn is in the same units as fs.
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’), pole-zero (‘zpk’), or second-order sections (‘sos’). Default is ‘ba’ for backwards compatibility, but ‘sos’ should be used for general-purpose 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
Second-order 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 high-pass filter at 15 Hz to remove the 10 Hz tone, and apply it to the signal. (It’s recommended to use second-order 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 high-pass filter') >>> ax2.axis([0, 1, -2, 2]) >>> ax2.set_xlabel('Time [seconds]') >>> plt.tight_layout() >>> plt.show()