Module « scipy.signal »

Fonction lfilter - module scipy.signal

Signature de la fonction lfilter

def lfilter(b, a, x, axis=-1, zi=None) 

Description

lfilter.__doc__

    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).

    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.

    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
    >>> 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()