scipy.signal.lfilter¶

scipy.signal.
lfilter
(b, a, x, axis=1, zi=None)[source]¶ Filter data along onedimension with an IIR or FIR filter.
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).
 Parameters
 barray_like
The numerator coefficient vector in a 1D sequence.
 aarray_like
The denominator coefficient vector in a 1D sequence. If
a[0]
is not 1, then both a and b are normalized bya[0]
. xarray_like
An Ndimensional input array.
 axisint, optional
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.
 ziarray_like, optional
Initial conditions for the filter delays. It is a vector (or array of vectors for an Ndimensional input) of length
max(len(a), len(b))  1
. If zi is None or is not given then initial rest is assumed. Seelfiltic
for more information.
 Returns
 yarray
The output of the digital filter.
 zfarray, optional
If zi is None, this is not returned, otherwise, zf holds the final filter delay values.
See also
lfiltic
Construct initial conditions for
lfilter
.lfilter_zi
Compute initial state (steady state of step response) for
lfilter
.filtfilt
A forwardbackward filter, to obtain a filter with linear phase.
savgol_filter
A SavitzkyGolay filter.
sosfilt
Filter data using cascaded secondorder sections.
sosfiltfilt
A forwardbackward filter using secondorder sections.
Notes
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[n1] + ... + b[M]*x[nM]  a[1]*y[n1]  ...  a[N]*y[nN]
where M is the degree of the numerator, N is the degree of the denominator, and 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][n1] d[0][n] = b[1] * x[n]  a[1] * y[n] + d[1][n1] d[1][n] = b[2] * x[n]  a[2] * y[n] + d[2][n1] ... d[N2][n] = b[N1]*x[n]  a[N1]*y[n] + d[N1][n1] d[N1][n] = b[N] * x[n]  a[N] * y[n]
where d are the state variables.
The rational transfer function describing this filter in the ztransform 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
Examples
Generate a noisy signal to be filtered:
>>> from scipy import signal >>> import matplotlib.pyplot as plt >>> t = np.linspace(1, 1, 201) >>> x = (np.sin(2*np.pi*0.75*t*(1t) + 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 + np.random.randn(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()