`scipy.signal._signaltools:filtfilt`

source: /scipy/signal/_signaltools.py :4708

Signature

   def     filtfilt (    b    ,    a    ,    x    ,    axis    =  -1   ,    padtype    =  odd   ,    padlen    =  None   ,    method    =  pad   ,    irlen    =  None     )

Summary

Apply a digital filter forward and backward to a signal.

Extended Summary

This function applies a linear digital filter twice, once forward and once backwards. The combined filter has zero phase and a filter order twice that of the original.

The function provides options for handling the edges of the signal.

The function sosfiltfilt (and filter design using output='sos') should be preferred over filtfilt for most filtering tasks, as second-order sections have fewer numerical problems.

Parameters

b : (N,) array_like: The numerator coefficient vector of the filter.
a : (N,) array_like: The denominator coefficient vector of the filter. If a[0] is not 1, then both a and b are normalized by a[0].
x : array_like: The array of data to be filtered.
axis : int, optional: The axis of x to which the filter is applied. Default is -1.
padtype : str or None, optional: Must be 'odd', 'even', 'constant', or None. This determines the type of extension to use for the padded signal to which the filter is applied. If padtype is None, no padding is used. The default is 'odd'.
padlen : int or None, optional: The number of elements by which to extend x at both ends of axis before applying the filter. This value must be less than x.shape[axis] - 1. padlen=0 implies no padding. The default value is 3 * max(len(a), len(b)).
method : str, optional: Determines the method for handling the edges of the signal, either "pad" or "gust". When method is "pad", the signal is padded; the type of padding is determined by padtype and padlen, and irlen is ignored. When method is "gust", Gustafsson's method is used, and padtype and padlen are ignored.
irlen : int or None, optional: When method is "gust", irlen specifies the length of the impulse response of the filter. If irlen is None, no part of the impulse response is ignored. For a long signal, specifying irlen can significantly improve the performance of the filter.

Returns

y : ndarray: The filtered output with the same shape as x.

Notes

When method is "pad", the function pads the data along the given axis in one of three ways: odd, even or constant. The odd and even extensions have the corresponding symmetry about the end point of the data. The constant extension extends the data with the values at the end points. On both the forward and backward passes, the initial condition of the filter is found by using lfilter_zi and scaling it by the end point of the extended data.

When method is "gust", Gustafsson's method ^[1] is used. Initial conditions are chosen for the forward and backward passes so that the forward-backward filter gives the same result as the backward-forward filter.

The option to use Gustaffson's method was added in scipy version 0.16.0.

Array API Standard Support

filtfilt has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ✅                     ✅                   
Dask                  ✅                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

The examples will use several functions from `scipy.signal`.

import numpy as np
from scipy import signal
import matplotlib.pyplot as plt

First we create a one second signal that is the sum of two pure sine waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz.

t = np.linspace(0, 1.0, 2001)
xlow = np.sin(2 * np.pi * 5 * t)
xhigh = np.sin(2 * np.pi * 250 * t)
x = xlow + xhigh

Now create a lowpass Butterworth filter with a cutoff of 0.125 times the Nyquist frequency, or 125 Hz, and apply it to ``x`` with `filtfilt`. The result should be approximately ``xlow``, with no phase shift.

b, a = signal.butter(8, 0.125)
y = signal.filtfilt(b, a, x, padlen=150)
np.abs(y - xlow).max()

We get a fairly clean result for this artificial example because the odd extension is exact, and with the moderately long padding, the filter's transients have dissipated by the time the actual data is reached. In general, transient effects at the edges are unavoidable. The following example demonstrates the option ``method="gust"``. First, create a filter.

b, a = signal.ellip(4, 0.01, 120, 0.125)  # Filter to be applied.

`sig` is a random input signal to be filtered.

rng = np.random.default_rng()
n = 60
sig = rng.standard_normal(n)**3 + 3*rng.standard_normal(n).cumsum()

Apply `filtfilt` to `sig`, once using the Gustafsson method, and once using padding, and plot the results for comparison.

fgust = signal.filtfilt(b, a, sig, method="gust")
fpad = signal.filtfilt(b, a, sig, padlen=50)
plt.plot(sig, 'k-', label='input')
plt.plot(fgust, 'b-', linewidth=4, label='gust')
plt.plot(fpad, 'c-', linewidth=1.5, label='pad')
plt.legend(loc='best')
plt.show()

The `irlen` argument can be used to improve the performance of Gustafsson's method. Estimate the impulse response length of the filter.

z, p, k = signal.tf2zpk(b, a)
eps = 1e-9
r = np.max(np.abs(p))
approx_impulse_len = int(np.ceil(np.log(eps) / np.log(r)))
approx_impulse_len

Apply the filter to a longer signal, with and without the `irlen` argument. The difference between `y1` and `y2` is small. For long signals, using `irlen` gives a significant performance improvement.

x = rng.standard_normal(4000)
y1 = signal.filtfilt(b, a, x, method='gust')
y2 = signal.filtfilt(b, a, x, method='gust', irlen=approx_impulse_len)
print(np.max(np.abs(y1 - y2)))

Aliases

scipy.signal.filtfilt