wavelets.py 文件源码

def time(self, t, s=1.0):
        """
        Return a DOG wavelet,

        When m = 2, this is also known as the "Mexican hat", "Marr"
        or "Ricker" wavelet.

        It models the function::

            ``A d^m/dx^m exp(-x^2 / 2)``,

        where ``A = (-1)^(m+1) / (gamma(m + 1/2))^.5``
        and   ``x = t / s``.

        Note that the energy of the return wavelet is not normalised
        according to s.

        Parameters
        ----------
        t : float
            Time. If s is not specified, this can be used as the
            non-dimensional time t/s.
        s : scalar
            Width parameter of the wavelet.

        Returns
        -------
        float : value of the ricker wavelet at the given time


        Notes
        -----
        The derivative of the gaussian has a polynomial representation:

        from http://en.wikipedia.org/wiki/Gaussian_function:

        "Mathematically, the derivatives of the Gaussian function can be
        represented using Hermite functions. The n-th derivative of the
        Gaussian is the Gaussian function itself multiplied by the n-th
        Hermite polynomial, up to scale."

        http://en.wikipedia.org/wiki/Hermite_polynomial

        Here, we want the 'probabilists' Hermite polynomial (He_n),
        which is computed by scipy.special.hermitenorm

        """
        x = t / s
        m = self.m

        # compute the hermite polynomial (used to evaluate the
        # derivative of a gaussian)
        He_n = scipy.special.hermitenorm(m)
        gamma = scipy.special.gamma

        const = (-1) ** (m + 1) / gamma(m + 0.5) ** .5
        function = He_n(x) * np.exp(-x ** 2 / 2)

        return const * function