xrscipy.fftpack.dst

xrscipy.fftpack.dst(x, coord, type=2, n=None, norm=None)

dst(x, coord, type=2, n=None, norm=None)

Parameters:

  • obj (xarray object) – The input array.

  • coord (string) – Coordinate along which the dst is computed. The coordinate must be evenly spaced.

  • 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].

  • norm ({None, 'ortho'}, optional) – Normalization mode (see Notes). Default is None.

Returns:

dst – The transformed input array.

Return type:

ndarray of reals

See also

idst

Inverse DST

numpy.fftpack.dst

scipy.fft.dst : Original scipy implementation

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