Apply a digital filter forward and backward to a signal.
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 ``a0`` is not 1, then both `a` and `b` are normalized by ``a0``. 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.shapeaxis - 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`.
See Also -------- sosfiltfilt, lfilter_zi, lfilter, lfiltic, savgol_filter, sosfilt
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.
References ---------- .. 1 F. Gustaffson, 'Determining the initial states in forward-backward filtering', Transactions on Signal Processing, Vol. 46, pp. 988-992, 1996.
Examples -------- The examples will use several functions from `scipy.signal`.
>>> 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() 9.1086182074789912e-06
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. >>> np.random.seed(123456)
`sig` is a random input signal to be filtered.
>>> n = 60 >>> sig = np.random.randn(n)**3 + 3*np.random.randn(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 137
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 = np.random.randn(5000) >>> 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))) 1.80056858312e-10