lfilter(b, a, x, axis=-1, zi=None)
Filter a data sequence, x, using a digital filter. This works for many fundamental data types (including Object type). The filter is a direct form II transposed implementation of the standard difference equation (see Notes).
The function sosfilt
(and filter design using output='sos'
) should be preferred over lfilter
for most filtering tasks, as second-order sections have fewer numerical problems.
The filter function is implemented as a direct II transposed structure. This means that the filter implements:
a[0]*y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[M]*x[n-M]
- a[1]*y[n-1] - ... - a[N]*y[n-N]
where :None:None:`M` is the degree of the numerator, :None:None:`N` is the degree of the denominator, and :None:None:`n` is the sample number. It is implemented using the following difference equations (assuming M = N):
a[0]*y[n] = b[0] * x[n] + d[0][n-1] d[0][n] = b[1] * x[n] - a[1] * y[n] + d[1][n-1] d[1][n] = b[2] * x[n] - a[2] * y[n] + d[2][n-1] ... d[N-2][n] = b[N-1]*x[n] - a[N-1]*y[n] + d[N-1][n-1] d[N-1][n] = b[N] * x[n] - a[N] * y[n]
where d
are the state variables.
The rational transfer function describing this filter in the z-transform domain is:
-1 -M
b[0] + b[1]z + ... + b[M] z
Y(z) = -------------------------------- X(z)
-1 -N
a[0] + a[1]z + ... + a[N] z
The numerator coefficient vector in a 1-D sequence.
The denominator coefficient vector in a 1-D sequence. If a[0]
is not 1, then both a and b are normalized by a[0]
.
An N-dimensional input array.
The axis of the input data array along which to apply the linear filter. The filter is applied to each subarray along this axis. Default is -1.
Initial conditions for the filter delays. It is a vector (or array of vectors for an N-dimensional input) of length max(len(a), len(b)) - 1
. If :None:None:`zi` is None or is not given then initial rest is assumed. See lfiltic
for more information.
The output of the digital filter.
If :None:None:`zi` is None, this is not returned, otherwise, :None:None:`zf` holds the final filter delay values.
Filter data along one-dimension with an IIR or FIR filter.
filtfilt
A forward-backward filter, to obtain a filter with linear phase.
lfilter_zi
Compute initial state (steady state of step response) for :None:None:`lfilter`.
lfiltic
Construct initial conditions for :None:None:`lfilter`.
savgol_filter
A Savitzky-Golay filter.
sosfilt
Filter data using cascaded second-order sections.
sosfiltfilt
A forward-backward filter using second-order sections.
Generate a noisy signal to be filtered:
>>> from scipy import signal
... import matplotlib.pyplot as plt
... rng = np.random.default_rng()
... t = np.linspace(-1, 1, 201)
... x = (np.sin(2*np.pi*0.75*t*(1-t) + 2.1) +
... 0.1*np.sin(2*np.pi*1.25*t + 1) +
... 0.18*np.cos(2*np.pi*3.85*t))
... xn = x + rng.standard_normal(len(t)) * 0.08
Create an order 3 lowpass butterworth filter:
>>> b, a = signal.butter(3, 0.05)
Apply the filter to xn. Use lfilter_zi to choose the initial condition of the filter:
>>> zi = signal.lfilter_zi(b, a)
... z, _ = signal.lfilter(b, a, xn, zi=zi*xn[0])
Apply the filter again, to have a result filtered at an order the same as filtfilt:
>>> z2, _ = signal.lfilter(b, a, z, zi=zi*z[0])
Use filtfilt to apply the filter:
>>> y = signal.filtfilt(b, a, xn)
Plot the original signal and the various filtered versions:
>>> plt.figure
... plt.plot(t, xn, 'b', alpha=0.75)
... plt.plot(t, z, 'r--', t, z2, 'r', t, y, 'k')
... plt.legend(('noisy signal', 'lfilter, once', 'lfilter, twice',
... 'filtfilt'), loc='best')
... plt.grid(True)
... plt.show()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.signal._spectral_py.coherence
scipy.signal._signaltools.lfilter_zi
scipy.signal._signaltools.lfilter
scipy.signal._spectral_py.csd
scipy.signal._waveforms.unit_impulse
scipy.signal._signaltools.filtfilt
scipy.signal._signaltools.lfiltic
scipy.signal._signaltools.decimate
scipy.signal._signaltools.sosfilt
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