Populating the interactive namespace from numpy and matplotlib

Help on function ifft in module scipy.fftpack.basic:

ifft(x, n=None, axis=-1, overwrite_x=False)
    Return discrete inverse Fourier transform of real or complex sequence.
    
    The returned complex array contains ``y(0), y(1),..., y(n-1)`` where
    
    ``y(j) = (x * exp(2*pi*sqrt(-1)*j*np.arange(n)/n)).mean()``.
    
    Parameters
    ----------
    x : array_like
        Transformed data to invert.
    n : int, optional
        Length of the inverse Fourier transform.  If ``n < x.shape[axis]``,
        `x` is truncated.  If ``n > x.shape[axis]``, `x` is zero-padded.
        The default results in ``n = x.shape[axis]``.
    axis : int, optional
        Axis along which the ifft's are computed; the default is over the
        last axis (i.e., ``axis=-1``).
    overwrite_x : bool, optional
        If True, the contents of `x` can be destroyed; the default is False.
    
    Returns
    -------
    ifft : ndarray of floats
        The inverse discrete Fourier transform.
    
    See Also
    --------
    fft : Forward FFT
    
    Notes
    -----
    This function is most efficient when `n` is a power of two, and least
    efficient when `n` is prime.
    
    If the data type of `x` is real, a "real IFFT" algorithm is automatically
    used, which roughly halves the computation time.

Help on function dst in module scipy.fftpack.realtransforms:

dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False)
    Return the Discrete Sine Transform of arbitrary type sequence x.
    
    Parameters
    ----------
    x : array_like
        The input array.
    type : {1, 2, 3}, optional
        Type of the DST (see Notes). Default type is 2.
    n : int, optional
        Length of the transform.  If ``n < x.shape[axis]``, `x` is
        truncated.  If ``n > x.shape[axis]``, `x` is zero-padded. The
        default results in ``n = x.shape[axis]``.
    axis : int, optional
        Axis along which the dst is computed; the default is over the
        last axis (i.e., ``axis=-1``).
    norm : {None, 'ortho'}, optional
        Normalization mode (see Notes). Default is None.
    overwrite_x : bool, optional
        If True, the contents of `x` can be destroyed; the default is False.
    
    Returns
    -------
    dst : ndarray of reals
        The transformed input array.
    
    See Also
    --------
    idst : Inverse DST
    
    Notes
    -----
    For a single dimension array ``x``.
    
    There are theoretically 8 types of the DST for different combinations of
    even/odd boundary conditions and boundary off sets [1]_, only the first
    3 types are implemented in scipy.
    
    **Type I**
    
    There are several definitions of the DST-I; we use the following
    for ``norm=None``.  DST-I assumes the input is odd around n=-1 and n=N. ::
    
                 N-1
      y[k] = 2 * sum x[n]*sin(pi*(k+1)*(n+1)/(N+1))
                 n=0
    
    Only None is supported as normalization mode for DCT-I. Note also that the
    DCT-I is only supported for input size > 1
    The (unnormalized) DCT-I is its own inverse, up to a factor `2(N+1)`.
    
    **Type II**
    
    There are several definitions of the DST-II; we use the following
    for ``norm=None``.  DST-II assumes the input is odd around n=-1/2 and
    n=N-1/2; the output is odd around k=-1 and even around k=N-1 ::
    
                N-1
      y[k] = 2* sum x[n]*sin(pi*(k+1)*(n+0.5)/N), 0 <= k < N.
                n=0
    
    if ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f` ::
    
        f = sqrt(1/(4*N)) if k == 0
        f = sqrt(1/(2*N)) otherwise.
    
    **Type III**
    
    There are several definitions of the DST-III, we use the following
    (for ``norm=None``).  DST-III assumes the input is odd around n=-1
    and even around n=N-1 ::
    
                                 N-2
      y[k] = x[N-1]*(-1)**k + 2* sum x[n]*sin(pi*(k+0.5)*(n+1)/N), 0 <= k < N.
                                 n=0
    
    The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
    to a factor `2N`.  The orthonormalized DST-III is exactly the inverse of
    the orthonormalized DST-II.
    
    .. versionadded:: 0.11.0
    
    References
    ----------
    .. [1] Wikipedia, "Discrete sine transform",
           http://en.wikipedia.org/wiki/Discrete_sine_transform

Help on function dct in module scipy.fftpack.realtransforms:

dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False)
    Return the Discrete Cosine Transform of arbitrary type sequence x.
    
    Parameters
    ----------
    x : array_like
        The input array.
    type : {1, 2, 3}, optional
        Type of the DCT (see Notes). Default type is 2.
    n : int, optional
        Length of the transform.  If ``n < x.shape[axis]``, `x` is
        truncated.  If ``n > x.shape[axis]``, `x` is zero-padded. The
        default results in ``n = x.shape[axis]``.
    axis : int, optional
        Axis along which the dct is computed; the default is over the
        last axis (i.e., ``axis=-1``).
    norm : {None, 'ortho'}, optional
        Normalization mode (see Notes). Default is None.
    overwrite_x : bool, optional
        If True, the contents of `x` can be destroyed; the default is False.
    
    Returns
    -------
    y : ndarray of real
        The transformed input array.
    
    See Also
    --------
    idct : Inverse DCT
    
    Notes
    -----
    For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to
    MATLAB ``dct(x)``.
    
    There are theoretically 8 types of the DCT, only the first 3 types are
    implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the'
    Inverse DCT generally refers to DCT type 3.
    
    **Type I**
    
    There are several definitions of the DCT-I; we use the following
    (for ``norm=None``)::
    
                                         N-2
      y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1))
                                         n=1
    
    Only None is supported as normalization mode for DCT-I. Note also that the
    DCT-I is only supported for input size > 1
    
    **Type II**
    
    There are several definitions of the DCT-II; we use the following
    (for ``norm=None``)::
    
    
                N-1
      y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
                n=0
    
    If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f`::
    
      f = sqrt(1/(4*N)) if k = 0,
      f = sqrt(1/(2*N)) otherwise.
    
    Which makes the corresponding matrix of coefficients orthonormal
    (``OO' = Id``).
    
    **Type III**
    
    There are several definitions, we use the following
    (for ``norm=None``)::
    
                        N-1
      y[k] = x[0] + 2 * sum x[n]*cos(pi*(k+0.5)*n/N), 0 <= k < N.
                        n=1
    
    or, for ``norm='ortho'`` and 0 <= k < N::
    
                                          N-1
      y[k] = x[0] / sqrt(N) + sqrt(2/N) * sum x[n]*cos(pi*(k+0.5)*n/N)
                                          n=1
    
    The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
    to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of
    the orthonormalized DCT-II.
    
    References
    ----------
    .. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J.
           Makhoul, `IEEE Transactions on acoustics, speech and signal
           processing` vol. 28(1), pp. 27-34,
           http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980).
    .. [2] Wikipedia, "Discrete cosine transform",
           http://en.wikipedia.org/wiki/Discrete_cosine_transform
    
    Examples
    --------
    The Type 1 DCT is equivalent to the FFT (though faster) for real,
    even-symmetrical inputs.  The output is also real and even-symmetrical.
    Half of the FFT input is used to generate half of the FFT output:
    
    >>> from scipy.fftpack import fft, dct
    >>> fft(np.array([4., 3., 5., 10., 5., 3.])).real
    array([ 30.,  -8.,   6.,  -2.,   6.,  -8.])
    >>> dct(np.array([4., 3., 5., 10.]), 1)
    array([ 30.,  -8.,   6.,  -2.])