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 , 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.

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 ~scipy.fft.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

Returns

dst : ndarray of reals

The transformed input array.

Return the Discrete Sine Transform of arbitrary type sequence x.

See Also

idst

Inverse DST

Examples

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.fft._realtransforms.dstn scipy.fft._realtransforms.dst scipy.fft._realtransforms.idst

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