589 lines
18 KiB
Python
589 lines
18 KiB
Python
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"""
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Real spectrum transforms (DCT, DST, MDCT)
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"""
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__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn']
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from scipy.fft import _pocketfft
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from .helper import _good_shape
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_inverse_typemap = {1: 1, 2: 3, 3: 2, 4: 4}
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def dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Cosine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3, 4}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : int or array_like of ints or None, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the ith dimension is truncated to
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length ``shape[i]``.
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If any element of `shape` is -1, the size of the corresponding
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dimension of `x` is used.
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axes : int or array_like of ints or None, optional
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Axes along which the DCT is computed.
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The default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idctn : Inverse multidimensional DCT
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Notes
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-----
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For full details of the DCT types and normalization modes, as well as
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references, see `dct`.
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Examples
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--------
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>>> from scipy.fftpack import dctn, idctn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
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True
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"""
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shape = _good_shape(x, shape, axes)
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return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x)
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def idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Cosine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3, 4}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : int or array_like of ints or None, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the ith dimension is truncated to
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length ``shape[i]``.
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If any element of `shape` is -1, the size of the corresponding
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dimension of `x` is used.
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axes : int or array_like of ints or None, optional
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Axes along which the IDCT is computed.
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The default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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dctn : multidimensional DCT
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Notes
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-----
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For full details of the IDCT types and normalization modes, as well as
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references, see `idct`.
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Examples
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--------
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>>> from scipy.fftpack import dctn, idctn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
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True
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"""
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type = _inverse_typemap[type]
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shape = _good_shape(x, shape, axes)
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return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x)
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def dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Sine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3, 4}, optional
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Type of the DST (see Notes). Default type is 2.
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shape : int or array_like of ints or None, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the ith dimension is truncated to
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length ``shape[i]``.
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If any element of `shape` is -1, the size of the corresponding
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dimension of `x` is used.
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axes : int or array_like of ints or None, optional
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Axes along which the DCT is computed.
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The default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idstn : Inverse multidimensional DST
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Notes
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-----
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For full details of the DST types and normalization modes, as well as
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references, see `dst`.
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Examples
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--------
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>>> from scipy.fftpack import dstn, idstn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
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True
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"""
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shape = _good_shape(x, shape, axes)
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return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x)
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def idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Sine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3, 4}, optional
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Type of the DST (see Notes). Default type is 2.
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shape : int or array_like of ints or None, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the ith dimension is truncated to
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length ``shape[i]``.
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If any element of `shape` is -1, the size of the corresponding
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dimension of `x` is used.
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axes : int or array_like of ints or None, optional
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Axes along which the IDST is computed.
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The default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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dstn : multidimensional DST
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Notes
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-----
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For full details of the IDST types and normalization modes, as well as
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references, see `idst`.
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Examples
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--------
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>>> from scipy.fftpack import dstn, idstn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
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True
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"""
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type = _inverse_typemap[type]
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shape = _good_shape(x, shape, axes)
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return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x)
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def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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r"""
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Return the Discrete Cosine Transform of arbitrary type sequence x.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3, 4}, optional
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Type of the DCT (see Notes). Default type is 2.
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n : int, optional
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Length of the transform. If ``n < x.shape[axis]``, `x` is
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truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
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default results in ``n = x.shape[axis]``.
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axis : int, optional
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Axis along which the dct is computed; the default is over the
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last axis (i.e., ``axis=-1``).
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idct : Inverse DCT
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Notes
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-----
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For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to
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MATLAB ``dct(x)``.
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There are, theoretically, 8 types of the DCT, only the first 4 types are
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implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the'
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Inverse DCT generally refers to DCT type 3.
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**Type I**
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There are several definitions of the DCT-I; we use the following
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(for ``norm=None``)
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.. math::
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y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left(
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\frac{\pi k n}{N-1} \right)
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If ``norm='ortho'``, ``x[0]`` and ``x[N-1]`` are multiplied by a scaling
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factor of :math:`\sqrt{2}`, and ``y[k]`` is multiplied by a scaling factor
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``f``
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.. math::
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f = \begin{cases}
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\frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\
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\frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases}
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.. versionadded:: 1.2.0
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Orthonormalization in DCT-I.
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.. note::
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The DCT-I is only supported for input size > 1.
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**Type II**
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There are several definitions of the DCT-II; we use the following
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(for ``norm=None``)
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.. math::
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y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)
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If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``
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.. math::
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f = \begin{cases}
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\sqrt{\frac{1}{4N}} & \text{if }k=0, \\
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\sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}
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which makes the corresponding matrix of coefficients orthonormal
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(``O @ O.T = np.eye(N)``).
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**Type III**
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There are several definitions, we use the following (for ``norm=None``)
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.. math::
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y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)
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or, for ``norm='ortho'``
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.. math::
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y_k = \frac{x_0}{\sqrt{N}} + \sqrt{\frac{2}{N}} \sum_{n=1}^{N-1} x_n
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\cos\left(\frac{\pi(2k+1)n}{2N}\right)
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The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
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to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of
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the orthonormalized DCT-II.
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**Type IV**
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There are several definitions of the DCT-IV; we use the following
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(for ``norm=None``)
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.. math::
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y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)
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If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``
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.. math::
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f = \frac{1}{\sqrt{2N}}
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.. versionadded:: 1.2.0
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Support for DCT-IV.
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References
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----------
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.. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J.
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Makhoul, `IEEE Transactions on acoustics, speech and signal
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processing` vol. 28(1), pp. 27-34,
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:doi:`10.1109/TASSP.1980.1163351` (1980).
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.. [2] Wikipedia, "Discrete cosine transform",
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https://en.wikipedia.org/wiki/Discrete_cosine_transform
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Examples
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--------
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The Type 1 DCT is equivalent to the FFT (though faster) for real,
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even-symmetrical inputs. The output is also real and even-symmetrical.
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Half of the FFT input is used to generate half of the FFT output:
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>>> 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)
|
||
|
|
|
||
|
|
|
||
|
|
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
|
||
|
|
>>> 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 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)
|
||
|
|
|
||
|
|
|
||
|
|
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)
|