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. 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{“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(). See- fftfor more details.
- orthogonalizebool, optional
- Whether to use the orthogonalized DST variant (see Notes). Defaults to - Truewhen- norm="ortho"and- Falseotherwise.- Added 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 specify- orthogonalize=False.- For - norm="ortho"both the- dstand- idstare 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- dstand the- idstis scaled by- 1/Nwhere- Nis 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. - orthogonalizehas 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[-1]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[-1]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 \(n=N-0.5\)\[y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)\]- orthogonalizehas 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. - References [1]- Wikipedia, “Discrete sine transform”, https://en.wikipedia.org/wiki/Discrete_sine_transform