# 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.

New in version 1.8.0.

Returns:
dstndarray of reals

The transformed input array.

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