scipy.fft.dst¶

scipy.fft.
dst
(x, type=2, n=None, axis= 1, norm=None, overwrite_x=False, workers=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 zeropadded. 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.
 Returns
 dstndarray of reals
The transformed input array.
See also
idst
Inverse DST
Notes
For a single dimension array
x
.For
norm="backward"
, there is no scaling on thedst
and theidst
is scaled by1/N
whereN
is the “logical” size of the DST. Fornorm='ortho'
both directions are scaled by the same factor1/sqrt(N)
.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 DSTI; we use the following for
norm="backward"
. DSTI assumes the input is odd around \(n=1\) and \(n=N\).\[y_k = 2 \sum_{n=0}^{N1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right)\]Note that the DSTI is only supported for input size > 1. The (unnormalized) DSTI is its own inverse, up to a factor \(2(N+1)\). The orthonormalized DSTI is exactly its own inverse.
Type II
There are several definitions of the DSTII; we use the following for
norm="backward"
. DSTII assumes the input is odd around \(n=1/2\) and \(n=N1/2\); the output is odd around \(k=1\) and even around \(k=N1\)\[y_k = 2 \sum_{n=0}^{N1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right)\]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 DSTIII, we use the following (for
norm="backward"
). DSTIII assumes the input is odd around \(n=1\) and even around \(n=N1\)\[y_k = (1)^k x_{N1} + 2 \sum_{n=0}^{N2} x_n \sin\left( \frac{\pi(2k+1)(n+1)}{2N}\right)\]The (unnormalized) DSTIII is the inverse of the (unnormalized) DSTII, up to a factor \(2N\). The orthonormalized DSTIII is exactly the inverse of the orthonormalized DSTII.
Type IV
There are several definitions of the DSTIV, we use the following (for
norm="backward"
). DSTIV assumes the input is odd around \(n=0.5\) and even around \(n=N0.5\)\[y_k = 2 \sum_{n=0}^{N1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)\]The (unnormalized) DSTIV is its own inverse, up to a factor \(2N\). The orthonormalized DSTIV is exactly its own inverse.
References
 1
Wikipedia, “Discrete sine transform”, https://en.wikipedia.org/wiki/Discrete_sine_transform