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)