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Module « scipy.fft »

Fonction dst - module scipy.fft

Signature de la fonction dst

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

Description

help(scipy.fft.dst)

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 : {"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 DST variant (see Notes).
    Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.

    .. versionadded:: 1.8.0

Returns
-------
dst : ndarray of reals
    The transformed input array.

See Also
--------
idst : Inverse DST

Notes
-----
.. warning:: For ``type in {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 `dst` and `idst` are scaled by the same
overall factor in both directions. By default, the transform is also
orthogonalized which for types 2 and 3 means the transform definition is
modified to give orthogonality of the DST matrix (see below).

For ``norm="backward"``, there is no scaling on the `dst` and the `idst` is
scaled by ``1/N`` where ``N`` is the "logical" size of the DST.

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="backward"``. DST-I assumes the input is odd around :math:`n=-1` and
:math:`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 :math:`2(N+1)`.
The orthonormalized DST-I is exactly its own inverse.

``orthogonalize`` has no effect here, as the DST-I matrix is already
orthogonal up to a scale factor of ``2N``.

**Type II**

There are several definitions of the DST-II; we use the following for
``norm="backward"``. DST-II assumes the input is odd around :math:`n=-1/2` and
:math:`n=N-1/2`; the output is odd around :math:`k=-1` and even around :math:`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 ``orthogonalize=True``, ``y[-1]`` is divided :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 of the DST-III, we use the following (for
``norm="backward"``). DST-III assumes the input is odd around :math:`n=-1` and
even around :math:`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)

If ``orthogonalize=True``, ``x[-1]`` is 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) DST-III is the inverse of the (unnormalized) DST-II, up
to a factor :math:`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="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and
even around :math:`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)

``orthogonalize`` has no effect here, as the DST-IV matrix is already
orthogonal up to a scale factor of ``2N``.

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

References
----------
.. [1] Wikipedia, "Discrete sine transform",
       https://en.wikipedia.org/wiki/Discrete_sine_transform

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