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from ._basic import _dispatch 

from scipy._lib.uarray import Dispatchable 

import numpy as np 

 

__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] 

 

 

@_dispatch 

def dctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, 

workers=None): 

""" 

Return multidimensional Discrete Cosine Transform along the specified axes. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, optional 

Type of the DCT (see Notes). Default type is 2. 

s : int or array_like of ints or None, optional 

The shape of the result. If both `s` and `axes` (see below) are None, 

`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is 

``scipy.take(x.shape, axes, axis=0)``. 

If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. 

If ``s[i] < x.shape[i]``, the ith dimension is truncated to length 

``s[i]``. 

If any element of `s` is -1, the size of the corresponding dimension of 

`x` is used. 

axes : int or array_like of ints or None, optional 

Axes over which the DCT is computed. If not given, the last ``len(s)`` 

axes are used, or all axes if `s` is also not specified. 

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. 

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 

-------- 

idctn : Inverse multidimensional DCT 

 

Notes 

----- 

For full details of the DCT types and normalization modes, as well as 

references, see `dct`. 

 

Examples 

-------- 

>>> from scipy.fft import dctn, idctn 

>>> y = np.random.randn(16, 16) 

>>> np.allclose(y, idctn(dctn(y))) 

True 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def idctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, 

workers=None): 

""" 

Return multidimensional Discrete Cosine Transform along the specified axes. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, optional 

Type of the DCT (see Notes). Default type is 2. 

s : int or array_like of ints or None, optional 

The shape of the result. If both `s` and `axes` (see below) are 

None, `s` is ``x.shape``; if `s` is None but `axes` is 

not None, then `s` is ``scipy.take(x.shape, axes, axis=0)``. 

If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. 

If ``s[i] < x.shape[i]``, the ith dimension is truncated to length 

``s[i]``. 

If any element of `s` is -1, the size of the corresponding dimension of 

`x` is used. 

axes : int or array_like of ints or None, optional 

Axes over which the IDCT is computed. If not given, the last ``len(s)`` 

axes are used, or all axes if `s` is also not specified. 

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. 

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 

-------- 

dctn : multidimensional DCT 

 

Notes 

----- 

For full details of the IDCT types and normalization modes, as well as 

references, see `idct`. 

 

Examples 

-------- 

>>> from scipy.fft import dctn, idctn 

>>> y = np.random.randn(16, 16) 

>>> np.allclose(y, idctn(dctn(y))) 

True 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def dstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, 

workers=None): 

""" 

Return multidimensional Discrete Sine Transform along the specified axes. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, optional 

Type of the DST (see Notes). Default type is 2. 

s : int or array_like of ints or None, optional 

The shape of the result. If both `s` and `axes` (see below) are None, 

`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is 

``scipy.take(x.shape, axes, axis=0)``. 

If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. 

If ``s[i] < x.shape[i]``, the ith dimension is truncated to length 

``s[i]``. 

If any element of `shape` is -1, the size of the corresponding dimension 

of `x` is used. 

axes : int or array_like of ints or None, optional 

Axes over which the DST is computed. If not given, the last ``len(s)`` 

axes are used, or all axes if `s` is also not specified. 

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. 

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 

-------- 

idstn : Inverse multidimensional DST 

 

Notes 

----- 

For full details of the DST types and normalization modes, as well as 

references, see `dst`. 

 

Examples 

-------- 

>>> from scipy.fft import dstn, idstn 

>>> y = np.random.randn(16, 16) 

>>> np.allclose(y, idstn(dstn(y))) 

True 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def idstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False, 

workers=None): 

""" 

Return multidimensional Discrete Sine Transform along the specified axes. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, optional 

Type of the DST (see Notes). Default type is 2. 

s : int or array_like of ints or None, optional 

The shape of the result. If both `s` and `axes` (see below) are None, 

`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is 

``scipy.take(x.shape, axes, axis=0)``. 

If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros. 

If ``s[i] < x.shape[i]``, the ith dimension is truncated to length 

``s[i]``. 

If any element of `s` is -1, the size of the corresponding dimension of 

`x` is used. 

axes : int or array_like of ints or None, optional 

Axes over which the IDST is computed. If not given, the last ``len(s)`` 

axes are used, or all axes if `s` is also not specified. 

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. 

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 

-------- 

dstn : multidimensional DST 

 

Notes 

----- 

For full details of the IDST types and normalization modes, as well as 

references, see `idst`. 

 

Examples 

-------- 

>>> from scipy.fft import dstn, idstn 

>>> y = np.random.randn(16, 16) 

>>> np.allclose(y, idstn(dstn(y))) 

True 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

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

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. 

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=None``, 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='ortho'`` 

both directions are scaled by the same factor ``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=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} 

 

.. 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}} 

 

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.]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, 

workers=None): 

""" 

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. 

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 

------- 

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-II, which is the same as the normalized DCT-III. 

 

The IDCT is equivalent to a normal DCT except for the normalization and 

type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each 

other's inverses. 

 

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.fft import ifft, idct 

>>> 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) 

array([ 4., 3., 5., 10.]) 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

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

r""" 

Return the Discrete Sine Transform of arbitrary type sequence x. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, 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. 

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 

------- 

dst : ndarray of reals 

The transformed input array. 

 

See Also 

-------- 

idst : Inverse DST 

 

Notes 

----- 

For a single dimension array ``x``. 

 

For ``norm=None``, there is no scaling on the `dst` and the `idst` is 

scaled by ``1/N`` where ``N`` is the "logical" size of the DST. For 

``norm='ortho'`` both directions are scaled by the same factor 

``1/sqrt(N)``. 

 

There are, theoretically, 8 types of the DST for different combinations of 

even/odd boundary conditions and boundary off sets [1]_, only the first 

4 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`. 

 

.. math:: 

 

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

 

Note that the DST-I is only supported for input size > 1. 

The (unnormalized) DST-I is its own inverse, up to a factor `2(N+1)`. 

The orthonormalized DST-I is exactly its own inverse. 

 

**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 :math:`k=-1` and even around `k=N-1` 

 

.. math:: 

 

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

 

**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` 

 

.. math:: 

 

y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( 

\frac{\pi(2k+1)(n+1)}{2N}\right) 

 

The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up 

to a factor `2N`. The orthonormalized DST-III is exactly the inverse of the 

orthonormalized DST-II. 

 

**Type IV** 

 

There are several definitions of the DST-IV, we use the following (for 

``norm=None``). DST-IV assumes the input is odd around `n=-0.5` and even 

around `n=N-0.5` 

 

.. math:: 

 

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

 

The (unnormalized) DST-IV is its own inverse, up to a factor `2N`. The 

orthonormalized DST-IV is exactly its own inverse. 

 

References 

---------- 

.. [1] Wikipedia, "Discrete sine transform", 

https://en.wikipedia.org/wiki/Discrete_sine_transform 

 

""" 

return (Dispatchable(x, np.ndarray),) 

 

 

@_dispatch 

def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, 

workers=None): 

""" 

Return the Inverse Discrete Sine Transform of an arbitrary type sequence. 

 

Parameters 

---------- 

x : array_like 

The input array. 

type : {1, 2, 3, 4}, 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 idst 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. 

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 

------- 

idst : ndarray of real 

The transformed input array. 

 

See Also 

-------- 

dst : Forward DST 

 

Notes 

----- 

 

'The' IDST is the IDST-II, which is the same as the normalized DST-III. 

 

The IDST is equivalent to a normal DST except for the normalization and 

type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each 

other's inverses. 

 

""" 

return (Dispatchable(x, np.ndarray),)