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. 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{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 factor- f\[\begin{split}f = \begin{cases} \sqrt{\frac{1}{4N}} & \text{if }k = 0, \\ \sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}\end{split}\]- 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