""" Real spectrum transforms (DCT, DST, MDCT) """ __all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] from scipy.fft import _pocketfft from ._helper import _good_shape _inverse_typemap = {1: 1, 2: 3, 3: 2, 4: 4} def dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): """ 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. shape : int or array_like of ints or None, optional The shape of the result. If both `shape` and `axes` (see below) are None, `shape` is ``x.shape``; if `shape` is None but `axes` is not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the ith dimension is truncated to length ``shape[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 along which the DCT is computed. The default is over all axes. 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 -------- idctn : Inverse multidimensional DCT Notes ----- For full details of the DCT types and normalization modes, as well as references, see `dct`. Examples -------- >>> import numpy as np >>> from scipy.fftpack import dctn, idctn >>> rng = np.random.default_rng() >>> y = rng.standard_normal((16, 16)) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True """ shape = _good_shape(x, shape, axes) return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x) def idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): """ 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. shape : int or array_like of ints or None, optional The shape of the result. If both `shape` and `axes` (see below) are None, `shape` is ``x.shape``; if `shape` is None but `axes` is not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the ith dimension is truncated to length ``shape[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 along which the IDCT is computed. The default is over all axes. 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 -------- dctn : multidimensional DCT Notes ----- For full details of the IDCT types and normalization modes, as well as references, see `idct`. Examples -------- >>> import numpy as np >>> from scipy.fftpack import dctn, idctn >>> rng = np.random.default_rng() >>> y = rng.standard_normal((16, 16)) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True """ type = _inverse_typemap[type] shape = _good_shape(x, shape, axes) return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_x) def dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): """ 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. shape : int or array_like of ints or None, optional The shape of the result. If both `shape` and `axes` (see below) are None, `shape` is ``x.shape``; if `shape` is None but `axes` is not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the ith dimension is truncated to length ``shape[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 along which the DCT is computed. The default is over all axes. 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 -------- idstn : Inverse multidimensional DST Notes ----- For full details of the DST types and normalization modes, as well as references, see `dst`. Examples -------- >>> import numpy as np >>> from scipy.fftpack import dstn, idstn >>> rng = np.random.default_rng() >>> y = rng.standard_normal((16, 16)) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True """ shape = _good_shape(x, shape, axes) return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x) def idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False): """ 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. shape : int or array_like of ints or None, optional The shape of the result. If both `shape` and `axes` (see below) are None, `shape` is ``x.shape``; if `shape` is None but `axes` is not None, then `shape` is ``numpy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the ith dimension is truncated to length ``shape[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 along which the IDST is computed. The default is over all axes. 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 -------- dstn : multidimensional DST Notes ----- For full details of the IDST types and normalization modes, as well as references, see `idst`. Examples -------- >>> import numpy as np >>> from scipy.fftpack import dstn, idstn >>> rng = np.random.default_rng() >>> y = rng.standard_normal((16, 16)) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True """ type = _inverse_typemap[type] shape = _good_shape(x, shape, axes) return _pocketfft.dstn(x, type, shape, axes, 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 >>> import numpy as np >>> 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 >>> 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 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)