def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
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.
Returns
-------
idst : ndarray of real
The transformed input array.
See Also
--------
dst : Forward DST
Notes
-----
'The' IDST is the IDST of type 2, which is the same as DST of type 3.
IDST of type 1 is the DST of type 1, IDST of type 2 is the DST of type
3, and IDST of type 3 is the DST of type 2. For the definition of these
types, see `dst`.
.. versionadded:: 0.11.0
"""
type = _inverse_typemap[type]
return _pocketfft.dst(x, type, n, axis, norm, overwrite_x)
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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.
Returns
-------
dst : ndarray of reals
The transformed input array.
See Also
--------
idst : Inverse DST
Notes
-----
For a single dimension array ``x``.
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.
.. versionadded:: 0.11.0
**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.
.. versionadded:: 1.2.0
Support for DST-IV.
References
----------
.. [1] Wikipedia, "Discrete sine transform",
https://en.wikipedia.org/wiki/Discrete_sine_transform
"""
return _pocketfft.dst(x, type, n, axis, norm, overwrite_x)
