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Fonction kaiser - module numpy

Signature de la fonction kaiser

def kaiser(M, beta) 

Description

kaiser.__doc__

    Return the Kaiser window.

    The Kaiser window is a taper formed by using a Bessel function.

    Parameters
    ----------
    M : int
        Number of points in the output window. If zero or less, an
        empty array is returned.
    beta : float
        Shape parameter for window.

    Returns
    -------
    out : array
        The window, with the maximum value normalized to one (the value
        one appears only if the number of samples is odd).

    See Also
    --------
    bartlett, blackman, hamming, hanning

    Notes
    -----
    The Kaiser window is defined as

    .. math::  w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}}
               \right)/I_0(\beta)

    with

    .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2},

    where :math:`I_0` is the modified zeroth-order Bessel function.

    The Kaiser was named for Jim Kaiser, who discovered a simple
    approximation to the DPSS window based on Bessel functions.  The Kaiser
    window is a very good approximation to the Digital Prolate Spheroidal
    Sequence, or Slepian window, which is the transform which maximizes the
    energy in the main lobe of the window relative to total energy.

    The Kaiser can approximate many other windows by varying the beta
    parameter.

    ====  =======================
    beta  Window shape
    ====  =======================
    0     Rectangular
    5     Similar to a Hamming
    6     Similar to a Hanning
    8.6   Similar to a Blackman
    ====  =======================

    A beta value of 14 is probably a good starting point. Note that as beta
    gets large, the window narrows, and so the number of samples needs to be
    large enough to sample the increasingly narrow spike, otherwise NaNs will
    get returned.

    Most references to the Kaiser window come from the signal processing
    literature, where it is used as one of many windowing functions for
    smoothing values.  It is also known as an apodization (which means
    "removing the foot", i.e. smoothing discontinuities at the beginning
    and end of the sampled signal) or tapering function.

    References
    ----------
    .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by
           digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285.
           John Wiley and Sons, New York, (1966).
    .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The
           University of Alberta Press, 1975, pp. 177-178.
    .. [3] Wikipedia, "Window function",
           https://en.wikipedia.org/wiki/Window_function

    Examples
    --------
    >>> import matplotlib.pyplot as plt
    >>> np.kaiser(12, 14)
     array([7.72686684e-06, 3.46009194e-03, 4.65200189e-02, # may vary
            2.29737120e-01, 5.99885316e-01, 9.45674898e-01,
            9.45674898e-01, 5.99885316e-01, 2.29737120e-01,
            4.65200189e-02, 3.46009194e-03, 7.72686684e-06])


    Plot the window and the frequency response:

    >>> from numpy.fft import fft, fftshift
    >>> window = np.kaiser(51, 14)
    >>> plt.plot(window)
    [<matplotlib.lines.Line2D object at 0x...>]
    >>> plt.title("Kaiser window")
    Text(0.5, 1.0, 'Kaiser window')
    >>> plt.ylabel("Amplitude")
    Text(0, 0.5, 'Amplitude')
    >>> plt.xlabel("Sample")
    Text(0.5, 0, 'Sample')
    >>> plt.show()

    >>> plt.figure()
    <Figure size 640x480 with 0 Axes>
    >>> A = fft(window, 2048) / 25.5
    >>> mag = np.abs(fftshift(A))
    >>> freq = np.linspace(-0.5, 0.5, len(A))
    >>> response = 20 * np.log10(mag)
    >>> response = np.clip(response, -100, 100)
    >>> plt.plot(freq, response)
    [<matplotlib.lines.Line2D object at 0x...>]
    >>> plt.title("Frequency response of Kaiser window")
    Text(0.5, 1.0, 'Frequency response of Kaiser window')
    >>> plt.ylabel("Magnitude [dB]")
    Text(0, 0.5, 'Magnitude [dB]')
    >>> plt.xlabel("Normalized frequency [cycles per sample]")
    Text(0.5, 0, 'Normalized frequency [cycles per sample]')
    >>> plt.axis('tight')
    (-0.5, 0.5, -100.0, ...) # may vary
    >>> plt.show()