 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 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=None``)

.. 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}

.. versionadded:: 1.2.0
   Orthonormalization in DCT-I.

.. 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=None``)

.. 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=None``)

.. 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=None``)

.. 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}}

.. versionadded:: 1.2.0
   Support for DCT-IV.

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.fftpack 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.])

)