`scipy.fft._realtransforms:dst`

source: build-install/usr/lib/python3.14/site-packages/scipy/fft/_realtransforms.py:506

Signature

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

Summary

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.
versionadded 1.8.0

Returns

dst : ndarray of reals: The transformed input array.

Notes

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 n = 0 \sum N - 1 x_{n} sin (\frac{π ( k + 1 ) ( n + 1 )}{N + 1})

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 n = 0 \sum N - 1 x_{n} sin (\frac{π ( k + 1 ) ( 2 n + 1 )}{2 N})

If orthogonalize=True, y[-1] is divided $2$ 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 n = 0 \sum N - 2 x_{n} sin (\frac{π ( 2 k + 1 ) ( n + 1 )}{2 N})

If orthogonalize=True, x[-1] is multiplied by $2$ 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 $2 N$ . 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 $n = - 1/2$ and even around $n = N - 1/2$

y_{k} = 2 n = 0 \sum N - 1 x_{n} sin (\frac{π ( 2 k + 1 ) ( 2 n + 1 )}{4 N})

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 $2 N$ . The orthonormalized DST-IV is exactly its own inverse.

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

Aliases

scipy.fft.dst

Signature

Summary

Parameters

Returns

Notes

Examples

See also

Aliases