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. Ifn > x.shape[axis]
, x is zero-padded. The default results inn = 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()
. Seefft
for more details.- orthogonalizebool, optional
Whether to use the orthogonalized DST variant (see Notes). Defaults to
True
whennorm="ortho"
andFalse
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 specifyorthogonalize=False
.For
norm="ortho"
both thedst
andidst
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 thedst
and theidst
is scaled by1/N
whereN
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 of2N
.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 withnorm="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 withnorm="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 of2N
.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