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discrete sine transform
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(Definition)
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The discrete sine transforms (DST) are a family of similar transforms closely related to the discrete cosine transform and the discrete Fourier transform. The complete set of variants of the DST was first introduced by Wang and Hunt [3].
The orthonormal variants of the DST, where $x_n$ is the original vector of $N$ real numbers, $C_k$ is the transformed vector of $N$ real numbers and $\delta$ is the Kronecker delta, are defined by the following equations:
\begin{eqnarray*} S^{I}_k&=&p \sum _{n=0}^{N-1} x_n \sin \frac{\pi (n+1) (k+1)}{N+1} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N+1}} \end{eqnarray*}The DST-I is its own inverse.
\begin{eqnarray*} S^{II}_k&=&p_k \sum _{n=0}^{N-1} x_n \sin \frac{\pi \left(n+\frac{1}{2}\right) (k+1)}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,0}}{N}} \end{eqnarray*}The inverse of DST-II is DST-III.
\begin{eqnarray*} S^{III}_k&=&p \sum _{n=0}^{N-1} x_n q_n \sin \frac{\pi (n+1) \left(k+\frac{1}{2}\right)}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,0}}} \end{eqnarray*}The inverse of DST-III is DST-II.
\begin{eqnarray*} S^{IV}_k&=&p \sum _{n=0}^{N-1} x_n \sin \frac{\pi \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)}{N} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N}} \end{eqnarray*}The DST-IV is its own inverse.
\begin{eqnarray*} S^V_k&=&p \sum _{n=0}^{N-1} x_n \sin \frac{\pi (n+1) (k+1)}{N+\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N+\frac{1}{2}}} \end{eqnarray*}The DST-V is its own inverse.
\begin{eqnarray*} S^{VI}_k&=&p \sum _{n=0}^{N-1} x_n \sin \frac{\pi \left(n+\frac{1}{2}\right) (k+1)}{N+\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N+\frac{1}{2}}} \end{eqnarray*}The inverse of DST-VI is DST-VII.
\begin{eqnarray*} S^{VII}_k&=&p \sum _{n=0}^{N-1} x_n \sin \frac{\pi (n+1) \left(k+\frac{1}{2}\right)}{N+\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p&=&\sqrt{\frac{2}{N+\frac{1}{2}}} \end{eqnarray*}The inverse of DST-VII is DST-VI.
\begin{eqnarray*} S^{VIII}_k&=&p_k \sum _{n=0}^{N-1} x_n q_n \sin \frac{\pi \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right)}{N-\frac{1}{2}} \quad \quad k=0, 1, 2, \dots, N-1\\ p_k&=&\sqrt{\frac{2-\delta _{k,N-1}}{N-\frac{1}{2}}}\\ q_n&=&\sqrt{\frac{1}{1+\delta _{n,N-1}}} \end{eqnarray*}The DST-VIII is its own inverse.
The DST in two dimensions is simply the one-dimensional transform computed in each row and each column. For example, the DST-II of a $N_1\times N_2$ matrix is given by
\begin{eqnarray*} S^{II}_{k_1,k_2}&=&p_{k_1}p_{k_2}\sum _{n_1=0}^{N_1-1}\sum _{n_2=0}^{N_2-1} x_{n_1,n_2} \sin \frac{\pi \left(n_1+\frac{1}{2}\right) (k_1+1)}{N_1} \sin \frac{\pi \left(n_2+\frac{1}{2}\right) (k_2+1)}{N_2} \end{eqnarray*}
- 1
- Xuancheng Shao, Steven G. Johnson. Type-II/III DCT/DST algorithms with reduced number of arithmetic operations. 2007.
- 2
- Markus Päuschel, José M. F. Mouray. The algebraic approach to the discrete cosine and sine transforms and their fast algorithms. 2006.
- 3
- Z. Wang and B. Hunt, The Discrete W Transform, Applied Mathematics and Computation, 16. 1985.
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"discrete sine transform" is owned by stitch.
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See Also: discrete cosine transform, discrete Fourier transform
| Other names: |
DST, discrete trigonometric transforms |
| Also defines: |
DST-I, DST-II, DST-III, DST-IV, DST-V, DST-VI, DST-VII, DST-VII, DST-VIII |
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Cross-references: matrix, column, row, dimensions, inverse, equations, Kronecker delta, real numbers, vector, orthonormal, discrete Fourier transform, Transforms
This is version 4 of discrete sine transform, born on 2007-07-12, modified 2007-08-22.
Object id is 9764, canonical name is DiscreteSineTransform.
Accessed 6179 times total.
Classification:
| AMS MSC: | 42-00 (Fourier analysis :: General reference works ) | | | 65T50 (Numerical analysis :: Numerical methods in Fourier analysis :: Discrete and fast Fourier transforms) |
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Pending Errata and Addenda
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