scipy.fft.

dst#

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

Return the Discrete Sine Transform of arbitrary type sequence x.

Parameters:

xarray_like: The input array.
type{1, 2, 3, 4}, optional: Type of the DST (see Notes). Default type is 2.
nint, 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].
axisint, 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_xbool, optional: If True, the contents of x can be destroyed; the default is False.
workersint, 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.
orthogonalizebool, optional: Whether to use the orthogonalized DST variant (see Notes). Defaults to True when norm="ortho" and False otherwise.

Added in version 1.8.0.

Returns:

dstndarray 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[-1] 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[-1] 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.

Array API Standard Support

dst has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	⛔	⛔
Dask	⚠️ computes graph	n/a

See Support for the array API standard for more information.

References

[1]

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

Examples

Compute the DST of a simple 1D array:

>>> import numpy as np
>>> from scipy.fft import dst
>>> x = np.array([1, -1, 1, -1])
>>> dst(x, type=2)
array([0., 0., 0., 8.])

This computes the Discrete Sine Transform (DST) of type-II for the input array. The output contains the transformed values corresponding to the given input sequence