Module « scipy.fft »

Fonction fht - module scipy.fft

Signature de la fonction fht

def fht(a, dln, mu, offset=0.0, bias=0.0) 

Description

fht.__doc__

Compute the fast Hankel transform.

    Computes the discrete Hankel transform of a logarithmically spaced periodic
    sequence using the FFTLog algorithm [1]_, [2]_.

    Parameters
    ----------
    a : array_like (..., n)
        Real periodic input array, uniformly logarithmically spaced.  For
        multidimensional input, the transform is performed over the last axis.
    dln : float
        Uniform logarithmic spacing of the input array.
    mu : float
        Order of the Hankel transform, any positive or negative real number.
    offset : float, optional
        Offset of the uniform logarithmic spacing of the output array.
    bias : float, optional
        Exponent of power law bias, any positive or negative real number.

    Returns
    -------
    A : array_like (..., n)
        The transformed output array, which is real, periodic, uniformly
        logarithmically spaced, and of the same shape as the input array.

    See Also
    --------
    ifht : The inverse of `fht`.
    fhtoffset : Return an optimal offset for `fht`.

    Notes
    -----
    This function computes a discrete version of the Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a(r) \, J_\mu(kr) \, k \, dr \;,

    where :math:`J_\mu` is the Bessel function of order :math:`\mu`.  The index
    :math:`\mu` may be any real number, positive or negative.

    The input array `a` is a periodic sequence of length :math:`n`, uniformly
    logarithmically spaced with spacing `dln`,

    .. math::

        a_j = a(r_j) \;, \quad
        r_j = r_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`r_c`.  Note that the central index
    :math:`j_c = (n+1)/2` is half-integral if :math:`n` is even, so that
    :math:`r_c` falls between two input elements.  Similarly, the output
    array `A` is a periodic sequence of length :math:`n`, also uniformly
    logarithmically spaced with spacing `dln`

    .. math::

       A_j = A(k_j) \;, \quad
       k_j = k_c \exp[(j-j_c) \, \mathtt{dln}]

    centred about the point :math:`k_c`.

    The centre points :math:`r_c` and :math:`k_c` of the periodic intervals may
    be chosen arbitrarily, but it would be usual to choose the product
    :math:`k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j` to be unity.  This can be
    changed using the `offset` parameter, which controls the logarithmic offset
    :math:`\log(k_c) = \mathtt{offset} - \log(r_c)` of the output array.
    Choosing an optimal value for `offset` may reduce ringing of the discrete
    Hankel transform.

    If the `bias` parameter is nonzero, this function computes a discrete
    version of the biased Hankel transform

    .. math::

        A(k) = \int_{0}^{\infty} \! a_q(r) \, (kr)^q \, J_\mu(kr) \, k \, dr

    where :math:`q` is the value of `bias`, and a power law bias
    :math:`a_q(r) = a(r) \, (kr)^{-q}` is applied to the input sequence.
    Biasing the transform can help approximate the continuous transform of
    :math:`a(r)` if there is a value :math:`q` such that :math:`a_q(r)` is
    close to a periodic sequence, in which case the resulting :math:`A(k)` will
    be close to the continuous transform.

    References
    ----------
    .. [1] Talman J. D., 1978, J. Comp. Phys., 29, 35
    .. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)