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Fonction bessel - module scipy.signal

Signature de la fonction bessel

def bessel(N, Wn, btype='low', analog=False, output='ba', norm='phase', fs=None) 

Description

bessel.__doc__

    Bessel/Thomson digital and analog filter design.

    Design an Nth-order digital or analog Bessel filter and return the
    filter coefficients.

    Parameters
    ----------
    N : int
        The order of the filter.
    Wn : array_like
        A scalar or length-2 sequence giving the critical frequencies (defined
        by the `norm` parameter).
        For analog filters, `Wn` is an angular frequency (e.g., rad/s).

        For digital filters, `Wn` are in the same units as `fs`.  By default,
        `fs` is 2 half-cycles/sample, so these are normalized from 0 to 1,
        where 1 is the Nyquist frequency. (`Wn` is thus in
        half-cycles / sample.)
    btype : {'lowpass', 'highpass', 'bandpass', 'bandstop'}, optional
        The type of filter.  Default is 'lowpass'.
    analog : bool, optional
        When True, return an analog filter, otherwise a digital filter is
        returned. (See Notes.)
    output : {'ba', 'zpk', 'sos'}, optional
        Type of output:  numerator/denominator ('ba'), pole-zero ('zpk'), or
        second-order sections ('sos'). Default is 'ba'.
    norm : {'phase', 'delay', 'mag'}, optional
        Critical frequency normalization:

        ``phase``
            The filter is normalized such that the phase response reaches its
            midpoint at angular (e.g. rad/s) frequency `Wn`. This happens for
            both low-pass and high-pass filters, so this is the
            "phase-matched" case.

            The magnitude response asymptotes are the same as a Butterworth
            filter of the same order with a cutoff of `Wn`.

            This is the default, and matches MATLAB's implementation.

        ``delay``
            The filter is normalized such that the group delay in the passband
            is 1/`Wn` (e.g., seconds). This is the "natural" type obtained by
            solving Bessel polynomials.

        ``mag``
            The filter is normalized such that the gain magnitude is -3 dB at
            angular frequency `Wn`.

        .. versionadded:: 0.18.0
    fs : float, optional
        The sampling frequency of the digital system.

        .. versionadded:: 1.2.0

    Returns
    -------
    b, a : ndarray, ndarray
        Numerator (`b`) and denominator (`a`) polynomials of the IIR filter.
        Only returned if ``output='ba'``.
    z, p, k : ndarray, ndarray, float
        Zeros, poles, and system gain of the IIR filter transfer
        function.  Only returned if ``output='zpk'``.
    sos : ndarray
        Second-order sections representation of the IIR filter.
        Only returned if ``output=='sos'``.

    Notes
    -----
    Also known as a Thomson filter, the analog Bessel filter has maximally
    flat group delay and maximally linear phase response, with very little
    ringing in the step response. [1]_

    The Bessel is inherently an analog filter. This function generates digital
    Bessel filters using the bilinear transform, which does not preserve the
    phase response of the analog filter. As such, it is only approximately
    correct at frequencies below about fs/4. To get maximally-flat group
    delay at higher frequencies, the analog Bessel filter must be transformed
    using phase-preserving techniques.

    See `besselap` for implementation details and references.

    The ``'sos'`` output parameter was added in 0.16.0.

    Examples
    --------
    Plot the phase-normalized frequency response, showing the relationship
    to the Butterworth's cutoff frequency (green):

    >>> from scipy import signal
    >>> import matplotlib.pyplot as plt

    >>> b, a = signal.butter(4, 100, 'low', analog=True)
    >>> w, h = signal.freqs(b, a)
    >>> plt.semilogx(w, 20 * np.log10(np.abs(h)), color='silver', ls='dashed')
    >>> b, a = signal.bessel(4, 100, 'low', analog=True, norm='phase')
    >>> w, h = signal.freqs(b, a)
    >>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
    >>> plt.title('Bessel filter magnitude response (with Butterworth)')
    >>> plt.xlabel('Frequency [radians / second]')
    >>> plt.ylabel('Amplitude [dB]')
    >>> plt.margins(0, 0.1)
    >>> plt.grid(which='both', axis='both')
    >>> plt.axvline(100, color='green')  # cutoff frequency
    >>> plt.show()

    and the phase midpoint:

    >>> plt.figure()
    >>> plt.semilogx(w, np.unwrap(np.angle(h)))
    >>> plt.axvline(100, color='green')  # cutoff frequency
    >>> plt.axhline(-np.pi, color='red')  # phase midpoint
    >>> plt.title('Bessel filter phase response')
    >>> plt.xlabel('Frequency [radians / second]')
    >>> plt.ylabel('Phase [radians]')
    >>> plt.margins(0, 0.1)
    >>> plt.grid(which='both', axis='both')
    >>> plt.show()

    Plot the magnitude-normalized frequency response, showing the -3 dB cutoff:

    >>> b, a = signal.bessel(3, 10, 'low', analog=True, norm='mag')
    >>> w, h = signal.freqs(b, a)
    >>> plt.semilogx(w, 20 * np.log10(np.abs(h)))
    >>> plt.axhline(-3, color='red')  # -3 dB magnitude
    >>> plt.axvline(10, color='green')  # cutoff frequency
    >>> plt.title('Magnitude-normalized Bessel filter frequency response')
    >>> plt.xlabel('Frequency [radians / second]')
    >>> plt.ylabel('Amplitude [dB]')
    >>> plt.margins(0, 0.1)
    >>> plt.grid(which='both', axis='both')
    >>> plt.show()

    Plot the delay-normalized filter, showing the maximally-flat group delay
    at 0.1 seconds:

    >>> b, a = signal.bessel(5, 1/0.1, 'low', analog=True, norm='delay')
    >>> w, h = signal.freqs(b, a)
    >>> plt.figure()
    >>> plt.semilogx(w[1:], -np.diff(np.unwrap(np.angle(h)))/np.diff(w))
    >>> plt.axhline(0.1, color='red')  # 0.1 seconds group delay
    >>> plt.title('Bessel filter group delay')
    >>> plt.xlabel('Frequency [radians / second]')
    >>> plt.ylabel('Group delay [seconds]')
    >>> plt.margins(0, 0.1)
    >>> plt.grid(which='both', axis='both')
    >>> plt.show()

    References
    ----------
    .. [1] Thomson, W.E., "Delay Networks having Maximally Flat Frequency
           Characteristics", Proceedings of the Institution of Electrical
           Engineers, Part III, November 1949, Vol. 96, No. 44, pp. 487-490.