ó
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 m Z m Z m Z d  d l m Z d d d	 d
 d d g Z d „  Z d „  Z d d „ Z d d e d „ Z d „  Z d „  Z d S(   i    (   t   divisiont   print_functiont   absolute_importN(   t   eig(   t   comb(   t   linspacet   pit   exp(   t   convolvet   daubt   qmft   cascadet   morlett   rickert   cwtc         C` sz  t  j } |  d k  r$ t d ƒ ‚ n  |  d k rS d | d ƒ } t  j | | g ƒ S|  d k r¨ | d ƒ d } | d ƒ } | t  j d | d | d | d | g ƒ S|  d k r«d | d ƒ } d | d	 | ƒ d
 d | d	 ƒ | | d	 ƒ d
 } t  j | ƒ } | d ƒ d } t  j d | d | ƒ } t  j | | ƒ } d t  j | ƒ }	 | | t  j | d | |	 d | d |	 d | d |	 d d |	 d g ƒ S|  d k  rj|  d k  rg  t |  ƒ D]# }
 t |  d |
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 ^ q%d d d … } t  j | ƒ d } t  j	 d d g ƒ |  } t  j	 d g ƒ } x‚ t |  d ƒ D]p }
 | |
 } d | | | d ƒ } d d | } | | } t
 | ƒ d k  r| | } n  | d | g } q²W| t  j | ƒ } | t  j | ƒ | d ƒ } | j d d d … St d ƒ ‚ d S(   sT  
    The coefficients for the FIR low-pass filter producing Daubechies wavelets.

    p>=1 gives the order of the zero at f=1/2.
    There are 2p filter coefficients.

    Parameters
    ----------
    p : int
        Order of the zero at f=1/2, can have values from 1 to 34.

    Returns
    -------
    daub : ndarray
        Return

    i   s   p must be at least 1.i   i   i   i   i
   g      ø?i   i   y              ð?i#   t   exactNiÿÿÿÿg      @i   s<   Polynomial factorization does not work well for p too large.(   t   npt   sqrtt
   ValueErrort   arrayt   conjt   realt   rangeR   t   rootst   poly1dt   abst   sumt   c(   t   pR   R   t   ft   tmpt   z1t   z1ct   d0t   a0t   a1t   kt   Pt   yjt   qt   yvalt   partt   const(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR	      sR    	-8.!CH

c         C` sg   t  |  ƒ d } g  t | d ƒ D]" } i d d 6d d 6| d ^ q! } |  d d d … t j | ƒ S(   s‘   
    Return high-pass qmf filter from low-pass

    Parameters
    ----------
    hk : array_like
        Coefficients of high-pass filter.

    i   i    iÿÿÿÿi   N(   t   lenR   R   R   (   t   hkt   NR$   t   asgn(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR
   Q   s    
9i   c   !      C` sj  t  |  ƒ d } | d t j | d ƒ k r< t d ƒ ‚ n  | d k  rW t d ƒ ‚ n  t j d | … d | … f \ } } t j d ƒ } t j |  d f } t |  ƒ } t j | d f } t j d | | d | d ƒ }	 t j d | | d d | d ƒ }
 t j	 d d | | f d	 ƒ } t j
 | |	 d ƒ | d <t j
 | |
 d ƒ | d <t j
 | |	 d ƒ | d <t j
 | |
 d ƒ | d <| | 9} t j d | d | >d
 t ƒd | >} d | } d | } t | d ƒ \ } } t j t j | d ƒ ƒ } t j | d d … | f ƒ } t j | ƒ } | d k  rH| } | } n  i | | d 6} t j | d | d ƒ | d <d | >} | d | d d | … <| d | d | d >d | … <t j | d | d ƒ | d d | … <t j | d | d ƒ | d | d >d | … <d g } xDt d | d ƒ D]/} g  d d g D]# } | D] } d | | f ^ qGq=} d | | >} xÜ | D]Ô } d } x= t | ƒ D]/ } | | d k r”| d | d | >7} q”q”W| | d } t | d ƒ } t j | d | f | ƒ }  |  | | <|  | | | d | … <t j | d | f | ƒ | | | d | … <q{W| } q*W| | | f S(   sð  
    Return (x, phi, psi) at dyadic points ``K/2**J`` from filter coefficients.

    Parameters
    ----------
    hk : array_like
        Coefficients of low-pass filter.
    J : int, optional
        Values will be computed at grid points ``K/2**J``. Default is 7.

    Returns
    -------
    x : ndarray
        The dyadic points ``K/2**J`` for ``K=0...N * (2**J)-1`` where
        ``len(hk) = len(gk) = N+1``.
    phi : ndarray
        The scaling function ``phi(x)`` at `x`:
        ``phi(x) = sum(hk * phi(2x-k))``, where k is from 0 to N.
    psi : ndarray, optional
        The wavelet function ``psi(x)`` at `x`:
        ``phi(x) = sum(gk * phi(2x-k))``, where k is from 0 to N.
        `psi` is only returned if `gk` is not None.

    Notes
    -----
    The algorithm uses the vector cascade algorithm described by Strang and
    Nguyen in "Wavelets and Filter Banks".  It builds a dictionary of values
    and slices for quick reuse.  Then inserts vectors into final vector at the
    end.

    i   i   s   Too many levels.s   Too few levels.Ni   i    iÿÿÿÿt   dt   dtypet   0t   1s   %d%s(   i    i    (   i    i   (   i   i    (   i   i   (   i    i    (   i    i   (   i   i    (   i   i   (   R+   R   t   log2R   t   ogridR   t   r_R
   t   clipt   zerost   taket   aranget   floatR   t   argmint   absoluteR   R   t   dotR   t   int(!   R,   t   JR-   t   nnt   kkt   s2t   thkt   gkt   tgkt   indx1t   indx2t   mt   xt   phit   psit   lamt   vt   indt   smt   bitdict   stept   prevkeyst   levelt   xxt   yyt   newkeyst   fact   keyt   numt   post   pastphit   iit   temp(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR   `   sh     %!%
(



'/	6
1
g      @g      ð?c         C` s|   t  | d t | d t |  ƒ } t d | | ƒ } | rX | t d | d ƒ 8} n  | t d | d ƒ t d 9} | S(   sû  
    Complex Morlet wavelet.

    Parameters
    ----------
    M : int
        Length of the wavelet.
    w : float, optional
        Omega0. Default is 5
    s : float, optional
        Scaling factor, windowed from ``-s*2*pi`` to ``+s*2*pi``. Default is 1.
    complete : bool, optional
        Whether to use the complete or the standard version.

    Returns
    -------
    morlet : (M,) ndarray

    See Also
    --------
    scipy.signal.gausspulse

    Notes
    -----
    The standard version::

        pi**-0.25 * exp(1j*w*x) * exp(-0.5*(x**2))

    This commonly used wavelet is often referred to simply as the
    Morlet wavelet.  Note that this simplified version can cause
    admissibility problems at low values of `w`.

    The complete version::

        pi**-0.25 * (exp(1j*w*x) - exp(-0.5*(w**2))) * exp(-0.5*(x**2))

    This version has a correction
    term to improve admissibility. For `w` greater than 5, the
    correction term is negligible.

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

    The fundamental frequency of this wavelet in Hz is given
    by ``f = 2*s*w*r / M`` where `r` is the sampling rate.
    
    Note: This function was created before `cwt` and is not compatible
    with it.

    i   y              ð?g      à¿g      Ð¿(   R   R   R   (   t   Mt   wt   st   completeRI   t   output(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR   Ê   s    3# c   	      C` sŒ   d t  j d | ƒ t  j d } | d } t  j d |  ƒ |  d d } | d } d | | } t  j | d | ƒ } | | | } | S(   s×  
    Return a Ricker wavelet, also known as the "Mexican hat wavelet".

    It models the function:

        ``A (1 - x^2/a^2) exp(-x^2/2 a^2)``,

    where ``A = 2/sqrt(3a)pi^1/4``.

    Parameters
    ----------
    points : int
        Number of points in `vector`.
        Will be centered around 0.
    a : scalar
        Width parameter of the wavelet.

    Returns
    -------
    vector : (N,) ndarray
        Array of length `points` in shape of ricker curve.

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

    >>> points = 100
    >>> a = 4.0
    >>> vec2 = signal.ricker(points, a)
    >>> print(len(vec2))
    100
    >>> plt.plot(vec2)
    >>> plt.show()

    i   i   g      Ð?i    g      ð?i   (   R   R   R   R9   R   (	   t   pointst   at   At   wsqt   vect   xsqt   modt   gausst   total(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR     s    %"

c         C` s‰   t  j t | ƒ t |  ƒ g ƒ } xa t | ƒ D]S \ } } | t d | t |  ƒ ƒ | ƒ } t |  | d d ƒ| | d d … f <q. W| S(   sV  
    Continuous wavelet transform.

    Performs a continuous wavelet transform on `data`,
    using the `wavelet` function. A CWT performs a convolution
    with `data` using the `wavelet` function, which is characterized
    by a width parameter and length parameter.

    Parameters
    ----------
    data : (N,) ndarray
        data on which to perform the transform.
    wavelet : function
        Wavelet function, which should take 2 arguments.
        The first argument is the number of points that the returned vector
        will have (len(wavelet(length,width)) == length).
        The second is a width parameter, defining the size of the wavelet
        (e.g. standard deviation of a gaussian). See `ricker`, which
        satisfies these requirements.
    widths : (M,) sequence
        Widths to use for transform.

    Returns
    -------
    cwt: (M, N) ndarray
        Will have shape of (len(widths), len(data)).

    Notes
    -----
    ::

        length = min(10 * width[ii], len(data))
        cwt[ii,:] = signal.convolve(data, wavelet(length,
                                    width[ii]), mode='same')

    Examples
    --------
    >>> from scipy import signal
    >>> import matplotlib.pyplot as plt
    >>> t = np.linspace(-1, 1, 200, endpoint=False)
    >>> sig  = np.cos(2 * np.pi * 7 * t) + signal.gausspulse(t - 0.4, fc=2)
    >>> widths = np.arange(1, 31)
    >>> cwtmatr = signal.cwt(sig, signal.ricker, widths)
    >>> plt.imshow(cwtmatr, extent=[-1, 1, 31, 1], cmap='PRGn', aspect='auto',
    ...            vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max())
    >>> plt.show()

    i
   t   modet   sameN(   R   R7   R+   t	   enumeratet   minR   (   t   datat   wavelett   widthsRb   RN   t   widtht   wavelet_data(    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyR   7  s    1!"(   t
   __future__R    R   R   t   numpyR   t
   numpy.dualR   t   scipy.specialR   t   scipyR   R   R   t   scipy.signalR   t   __all__R	   R
   R   t   TrueR   R   R   (    (    (    s4   /tmp/pip-build-7oUkmx/scipy/scipy/signal/wavelets.pyt   <module>   s   	E	j>	/