# scipy.fftpack.dst

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

  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.

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

  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 \(k=-1\) 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 `norm='ortho'`, `y[k]` is multiplied by a scaling factor `f`

  \[\begin{split}f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}\end{split}\]

  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

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

  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.

  New in version 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

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

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

  New in version 1.2.0: Support for DST-IV.

  References

  [1]

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