# scipy.fft.dst

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

  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 `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.

      New in version 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 \(n=-1\) and \(n=N\).

  \[y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right)\]

  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.

  `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 \(n=-1/2\) and \(n=N-1/2\); the output is odd around \(k=-1\) and even around \(k=N-1\)

  \[y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right)\]

  If `orthogonalize=True`, `y[0]` is divided \(\sqrt{2}\) 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 \(n=-1\) and even around \(n=N-1\)

  \[y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( \frac{\pi(2k+1)(n+1)}{2N}\right)\]

  If `orthogonalize=True`, `x[0]` is multiplied by \(\sqrt{2}\) 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 \(2N\). 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 \(n=-0.5\) and even around \(n=N-0.5\)

  \[y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)\]

  `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 \(2N\). The orthonormalized DST-IV is exactly its own inverse.

  References

  [1]

  Wikipedia, “Discrete sine transform”, https://en.wikipedia.org/wiki/Discrete_sine_transform