Vous êtes un professionnel et vous avez besoin d'une formation ? RAG (Retrieval-Augmented Generation)
et Fine Tuning d'un LLM Voir le programme détaillé

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, orthogonalize=None) 

Description

help(scipy.fft.dct)

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.
orthogonalize : bool, optional
    Whether to use the orthogonalized DCT variant (see Notes).
    Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.

    .. versionadded:: 1.8.0

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)``.

.. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct
             correspondence with the direct Fourier transform. To recover
             it you must specify ``orthogonalize=False``.

For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 1, 2 and 3 means the transform definition is
modified to give orthogonality of the DCT matrix (see below).

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.

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 ``orthogonalize=True``, ``x[0]`` and ``x[N-1]`` are multiplied by a
scaling factor of :math:`\sqrt{2}`, and ``y[0]`` and ``y[N-1]`` are divided
by :math:`\sqrt{2}`. When combined with ``norm="ortho"``, this makes the
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).

.. 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 ``orthogonalize=True``, ``y[0]`` is divided by :math:`\sqrt{2}` which,
when combined with ``norm="ortho"``, 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)

If ``orthogonalize=True``, ``x[0]`` terms are multiplied by
:math:`\sqrt{2}` which, when combined with ``norm="ortho"``, makes the
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).

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)

``orthogonalize`` has no effect here, as the DCT-IV matrix is already
orthogonal up to a scale factor of ``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
>>> import numpy as np
>>> 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.])

Vous êtes un professionnel et vous avez besoin d'une formation ? Programmation Python
Les compléments Voir le programme détaillé