scipy.signal.lfilter

scipy.signal.lfilter(b, a, x, axis=-1, zi=None)

Filter data along one-dimension 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:

b : array_like

The numerator coefficient vector in a 1-D sequence.

a : array_like

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].

x : array_like

An N-dimensional input array.

axis : int, 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.

zi : array_like, optional

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 zi is None or is not given then initial rest is assumed. See lfiltic for more information.

Returns:

y : array

The output of the digital filter.

zf : array, 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 forward-backward filter, to obtain a filter with linear phase.

savgol_filter

A Savitzky-Golay filter.

sosfilt

Filter data using cascaded second-order sections.

sosfiltfilt

A forward-backward filter using second-order 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[n-1] + ... + b[M]*x[n-M]
                      - a[1]*y[n-1] - ... - a[N]*y[n-N]

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][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

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*(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 + 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()