Module « scipy.signal »

Fonction freqz - module scipy.signal

Signature de la fonction freqz

def freqz(b, a=1, worN=512, whole=False, plot=None, fs=6.283185307179586, include_nyquist=False) 

Description

freqz.__doc__

    Compute the frequency response of a digital filter.

    Given the M-order numerator `b` and N-order denominator `a` of a digital
    filter, compute its frequency response::

                 jw                 -jw              -jwM
        jw    B(e  )    b[0] + b[1]e    + ... + b[M]e
     H(e  ) = ------ = -----------------------------------
                 jw                 -jw              -jwN
              A(e  )    a[0] + a[1]e    + ... + a[N]e

    Parameters
    ----------
    b : array_like
        Numerator of a linear filter. If `b` has dimension greater than 1,
        it is assumed that the coefficients are stored in the first dimension,
        and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies
        array must be compatible for broadcasting.
    a : array_like
        Denominator of a linear filter. If `b` has dimension greater than 1,
        it is assumed that the coefficients are stored in the first dimension,
        and ``b.shape[1:]``, ``a.shape[1:]``, and the shape of the frequencies
        array must be compatible for broadcasting.
    worN : {None, int, array_like}, optional
        If a single integer, then compute at that many frequencies (default is
        N=512). This is a convenient alternative to::

            np.linspace(0, fs if whole else fs/2, N, endpoint=include_nyquist)

        Using a number that is fast for FFT computations can result in
        faster computations (see Notes).

        If an array_like, compute the response at the frequencies given.
        These are in the same units as `fs`.
    whole : bool, optional
        Normally, frequencies are computed from 0 to the Nyquist frequency,
        fs/2 (upper-half of unit-circle). If `whole` is True, compute
        frequencies from 0 to fs. Ignored if worN is array_like.
    plot : callable
        A callable that takes two arguments. If given, the return parameters
        `w` and `h` are passed to plot. Useful for plotting the frequency
        response inside `freqz`.
    fs : float, optional
        The sampling frequency of the digital system. Defaults to 2*pi
        radians/sample (so w is from 0 to pi).

        .. versionadded:: 1.2.0
    include_nyquist : bool, optional
        If `whole` is False and `worN` is an integer, setting `include_nyquist` to True
        will include the last frequency (Nyquist frequency) and is otherwise ignored.

        .. versionadded:: 1.5.0

    Returns
    -------
    w : ndarray
        The frequencies at which `h` was computed, in the same units as `fs`.
        By default, `w` is normalized to the range [0, pi) (radians/sample).
    h : ndarray
        The frequency response, as complex numbers.

    See Also
    --------
    freqz_zpk
    sosfreqz

    Notes
    -----
    Using Matplotlib's :func:`matplotlib.pyplot.plot` function as the callable
    for `plot` produces unexpected results, as this plots the real part of the
    complex transfer function, not the magnitude.
    Try ``lambda w, h: plot(w, np.abs(h))``.

    A direct computation via (R)FFT is used to compute the frequency response
    when the following conditions are met:

    1. An integer value is given for `worN`.
    2. `worN` is fast to compute via FFT (i.e.,
       `next_fast_len(worN) <scipy.fft.next_fast_len>` equals `worN`).
    3. The denominator coefficients are a single value (``a.shape[0] == 1``).
    4. `worN` is at least as long as the numerator coefficients
       (``worN >= b.shape[0]``).
    5. If ``b.ndim > 1``, then ``b.shape[-1] == 1``.

    For long FIR filters, the FFT approach can have lower error and be much
    faster than the equivalent direct polynomial calculation.

    Examples
    --------
    >>> from scipy import signal
    >>> b = signal.firwin(80, 0.5, window=('kaiser', 8))
    >>> w, h = signal.freqz(b)

    >>> import matplotlib.pyplot as plt
    >>> fig, ax1 = plt.subplots()
    >>> ax1.set_title('Digital filter frequency response')

    >>> ax1.plot(w, 20 * np.log10(abs(h)), 'b')
    >>> ax1.set_ylabel('Amplitude [dB]', color='b')
    >>> ax1.set_xlabel('Frequency [rad/sample]')

    >>> ax2 = ax1.twinx()
    >>> angles = np.unwrap(np.angle(h))
    >>> ax2.plot(w, angles, 'g')
    >>> ax2.set_ylabel('Angle (radians)', color='g')
    >>> ax2.grid()
    >>> ax2.axis('tight')
    >>> plt.show()

    Broadcasting Examples

    Suppose we have two FIR filters whose coefficients are stored in the
    rows of an array with shape (2, 25). For this demonstration, we'll
    use random data:

    >>> rng = np.random.default_rng()
    >>> b = rng.random((2, 25))

    To compute the frequency response for these two filters with one call
    to `freqz`, we must pass in ``b.T``, because `freqz` expects the first
    axis to hold the coefficients. We must then extend the shape with a
    trivial dimension of length 1 to allow broadcasting with the array
    of frequencies.  That is, we pass in ``b.T[..., np.newaxis]``, which has
    shape (25, 2, 1):

    >>> w, h = signal.freqz(b.T[..., np.newaxis], worN=1024)
    >>> w.shape
    (1024,)
    >>> h.shape
    (2, 1024)

    Now, suppose we have two transfer functions, with the same numerator
    coefficients ``b = [0.5, 0.5]``. The coefficients for the two denominators
    are stored in the first dimension of the 2-D array  `a`::

        a = [   1      1  ]
            [ -0.25, -0.5 ]

    >>> b = np.array([0.5, 0.5])
    >>> a = np.array([[1, 1], [-0.25, -0.5]])

    Only `a` is more than 1-D. To make it compatible for
    broadcasting with the frequencies, we extend it with a trivial dimension
    in the call to `freqz`:

    >>> w, h = signal.freqz(b, a[..., np.newaxis], worN=1024)
    >>> w.shape
    (1024,)
    >>> h.shape
    (2, 1024)