Module « scipy.signal »

Fonction lfilter_zi - module scipy.signal

Signature de la fonction lfilter_zi

def lfilter_zi(b, a) 

Description

lfilter_zi.__doc__

    Construct initial conditions for lfilter for step response steady-state.

    Compute an initial state `zi` for the `lfilter` function that corresponds
    to the steady state of the step response.

    A typical use of this function is to set the initial state so that the
    output of the filter starts at the same value as the first element of
    the signal to be filtered.

    Parameters
    ----------
    b, a : array_like (1-D)
        The IIR filter coefficients. See `lfilter` for more
        information.

    Returns
    -------
    zi : 1-D ndarray
        The initial state for the filter.

    See Also
    --------
    lfilter, lfiltic, filtfilt

    Notes
    -----
    A linear filter with order m has a state space representation (A, B, C, D),
    for which the output y of the filter can be expressed as::

        z(n+1) = A*z(n) + B*x(n)
        y(n)   = C*z(n) + D*x(n)

    where z(n) is a vector of length m, A has shape (m, m), B has shape
    (m, 1), C has shape (1, m) and D has shape (1, 1) (assuming x(n) is
    a scalar).  lfilter_zi solves::

        zi = A*zi + B

    In other words, it finds the initial condition for which the response
    to an input of all ones is a constant.

    Given the filter coefficients `a` and `b`, the state space matrices
    for the transposed direct form II implementation of the linear filter,
    which is the implementation used by scipy.signal.lfilter, are::

        A = scipy.linalg.companion(a).T
        B = b[1:] - a[1:]*b[0]

    assuming `a[0]` is 1.0; if `a[0]` is not 1, `a` and `b` are first
    divided by a[0].

    Examples
    --------
    The following code creates a lowpass Butterworth filter. Then it
    applies that filter to an array whose values are all 1.0; the
    output is also all 1.0, as expected for a lowpass filter.  If the
    `zi` argument of `lfilter` had not been given, the output would have
    shown the transient signal.

    >>> from numpy import array, ones
    >>> from scipy.signal import lfilter, lfilter_zi, butter
    >>> b, a = butter(5, 0.25)
    >>> zi = lfilter_zi(b, a)
    >>> y, zo = lfilter(b, a, ones(10), zi=zi)
    >>> y
    array([1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.])

    Another example:

    >>> x = array([0.5, 0.5, 0.5, 0.0, 0.0, 0.0, 0.0])
    >>> y, zf = lfilter(b, a, x, zi=zi*x[0])
    >>> y
    array([ 0.5       ,  0.5       ,  0.5       ,  0.49836039,  0.48610528,
        0.44399389,  0.35505241])

    Note that the `zi` argument to `lfilter` was computed using
    `lfilter_zi` and scaled by `x[0]`.  Then the output `y` has no
    transient until the input drops from 0.5 to 0.0.