Module « scipy.fft »

Fonction dct - module scipy.fft

Signature de la fonction dct

def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None) 

Description

dct.__doc__

Return the Discrete Cosine Transform of arbitrary type sequence x.

    Parameters
    ----------
    x : array_like
        The input array.
    type : {1, 2, 3, 4}, 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 : {"backward", "ortho", "forward"}, optional
        Normalization mode (see Notes). Default is "backward".
    overwrite_x : bool, optional
        If True, the contents of `x` can be destroyed; the default is False.
    workers : int, optional
        Maximum number of workers to use for parallel computation. If negative,
        the value wraps around from ``os.cpu_count()``.
        See :func:`~scipy.fft.fft` for more details.

    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)``.

    For ``norm="backward"``, there is no scaling on `dct` and the `idct` is
    scaled by ``1/N`` where ``N`` is the "logical" size of the DCT. For
    ``norm="forward"`` the ``1/N`` normalization is applied to the forward
    `dct` instead and the `idct` is unnormalized. For ``norm='ortho'`` both
    directions are scaled by the same factor of ``1/sqrt(N)``.

    There are, theoretically, 8 types of the DCT, only the first 4 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="backward"``)

    .. math::

       y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left(
       \frac{\pi k n}{N-1} \right)

    If ``norm='ortho'``, ``x[0]`` and ``x[N-1]`` are multiplied by a scaling
    factor of :math:`\sqrt{2}`, and ``y[k]`` is multiplied by a scaling factor
    ``f``

    .. math::

        f = \begin{cases}
         \frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\
         \frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases}

    .. note::
       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="backward"``)

    .. math::

       y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)

    If ``norm="ortho"``, ``y[k]`` is multiplied by a scaling factor ``f``

    .. math::
       f = \begin{cases}
       \sqrt{\frac{1}{4N}} & \text{if }k=0, \\
       \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}

    which makes the corresponding matrix of coefficients orthonormal
    (``O @ O.T = np.eye(N)``).

    **Type III**

    There are several definitions, we use the following (for
    ``norm="backward"``)

    .. math::

       y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)

    or, for ``norm="ortho"``

    .. math::

       y_k = \frac{x_0}{\sqrt{N}} + \sqrt{\frac{2}{N}} \sum_{n=1}^{N-1} x_n
       \cos\left(\frac{\pi(2k+1)n}{2N}\right)

    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.

    **Type IV**

    There are several definitions of the DCT-IV; we use the following
    (for ``norm="backward"``)

    .. math::

       y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)

    If ``norm="ortho"``, ``y[k]`` is multiplied by a scaling factor ``f``

    .. math::

        f = \frac{1}{\sqrt{2N}}

    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,
           :doi:`10.1109/TASSP.1980.1163351` (1980).
    .. [2] Wikipedia, "Discrete cosine transform",
           https://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.fft 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.])