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.

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.

References

[1]

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