def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
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 idct 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
-------
idct : ndarray of real
The transformed input array.
See Also
--------
dct : Forward DCT
Notes
-----
For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to
MATLAB ``idct(x)``.
'The' IDCT is the IDCT of type 2, which is the same as DCT of type 3.
IDCT of type 1 is the DCT of type 1, IDCT of type 2 is the DCT of type
3, and IDCT of type 3 is the DCT of type 2. IDCT of type 4 is the DCT
of type 4. For the definition of these types, see `dct`.
Examples
--------
The Type 1 DCT is equivalent to the DFT for real, even-symmetrical
inputs. The output is also real and even-symmetrical. Half of the IFFT
input is used to generate half of the IFFT output:
>>> from scipy.fftpack import ifft, idct
>>> import numpy as np
>>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real
array([ 4., 3., 5., 10., 5., 3.])
>>> idct(np.array([ 30., -8., 6., -2.]), 1) / 6
array([ 4., 3., 5., 10.])
"""
type = _inverse_typemap[type]
return _pocketfft.dct(x, type, n, axis, norm, overwrite_x)
def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
r"""
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 : {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 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
>>> 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.])
"""
return _pocketfft.dct(x, type, n, axis, norm, overwrite_x)
