dst#
- scipy.fftpack.dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False)[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{None, ‘ortho’}, optional
Normalization mode (see Notes). Default is None.
- overwrite_xbool, optional
If True, the contents of x can be destroyed; the default is False.
- Returns:
- dstndarray of reals
The transformed input array.
See also
idst
Inverse DST
Notes
For a single dimension array
x
.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=None
. 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.Type II
There are several definitions of the DST-II; we use the following for
norm=None
. 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
norm='ortho'
,y[k]
is multiplied by a scaling factorf
\[\begin{split}f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}\end{split}\]Type III
There are several definitions of the DST-III, we use the following (for
norm=None
). DST-III assumes the input is odd around n=-1 and even around n=N-1\[y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left( \frac{\pi(2k+1)(n+1)}{2N}\right)\]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.Added in version 0.11.0.
Type IV
There are several definitions of the DST-IV, we use the following (for
norm=None
). DST-IV assumes the input is odd around n=-0.5 and even around n=N-0.5\[y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)\]The (unnormalized) DST-IV is its own inverse, up to a factor
2N
. The orthonormalized DST-IV is exactly its own inverse.Added in version 1.2.0: Support for DST-IV.
References
[1]Wikipedia, “Discrete sine transform”, https://en.wikipedia.org/wiki/Discrete_sine_transform