# discrete sine transform

The are a family of transforms closely related to the discrete cosine transform and the discrete Fourier transform. The set of variants of the DST was first introduced by Wang and Hunt [3].

## 1 Definition

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:

### 1.1 DST-I

 $\displaystyle S^{I}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi(n+1)(k+1)}{N+1}\quad\quad k=% 0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N+1}}$

The DST-I is its own inverse.

### 1.2 DST-II

 $\displaystyle S^{II}_{k}$ $\displaystyle=$ $\displaystyle p_{k}\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi\left(n+\frac{1}{2}\right% )(k+1)}{N}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p_{k}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2-\delta_{k,0}}{N}}$

The inverse of DST-II is DST-III.

### 1.3 DST-III

 $\displaystyle S^{III}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}q_{n}\sin\frac{\pi(n+1)\left(k+\frac{1}{2}% \right)}{N}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N}}$ $\displaystyle q_{n}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1}{1+\delta_{n,0}}}$

The inverse of DST-III is DST-II.

### 1.4 DST-IV

 $\displaystyle S^{IV}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi\left(n+\frac{1}{2}\right)% \left(k+\frac{1}{2}\right)}{N}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N}}$

The DST-IV is its own inverse.

### 1.5 DST-V

 $\displaystyle S^{V}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi(n+1)(k+1)}{N+\frac{1}{2}}% \quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N+\frac{1}{2}}}$

The DST-V is its own inverse.

### 1.6 DST-VI

 $\displaystyle S^{VI}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi\left(n+\frac{1}{2}\right)(k+% 1)}{N+\frac{1}{2}}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N+\frac{1}{2}}}$

The inverse of DST-VI is DST-VII.

### 1.7 DST-VII

 $\displaystyle S^{VII}_{k}$ $\displaystyle=$ $\displaystyle p\sum_{n=0}^{N-1}x_{n}\sin\frac{\pi(n+1)\left(k+\frac{1}{2}% \right)}{N+\frac{1}{2}}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2}{N+\frac{1}{2}}}$

The inverse of DST-VII is DST-VI.

### 1.8 DST-VIII

 $\displaystyle S^{VIII}_{k}$ $\displaystyle=$ $\displaystyle p_{k}\sum_{n=0}^{N-1}x_{n}q_{n}\sin\frac{\pi\left(n+\frac{1}{2}% \right)\left(k+\frac{1}{2}\right)}{N-\frac{1}{2}}\quad\quad k=0,1,2,\dots,N-1$ $\displaystyle p_{k}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2-\delta_{k,N-1}}{N-\frac{1}{2}}}$ $\displaystyle q_{n}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1}{1+\delta_{n,N-1}}}$

The DST-VIII is its own inverse.

## 2 Two-dimensional DST

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

 $\displaystyle S^{II}_{k_{1},k_{2}}$ $\displaystyle=$ $\displaystyle p_{k_{1}}p_{k_{2}}\sum_{n_{1}=0}^{N_{1}-1}\sum_{n_{2}=0}^{N_{2}-% 1}x_{n_{1},n_{2}}\sin\frac{\pi\left(n_{1}+\frac{1}{2}\right)(k_{1}+1)}{N_{1}}% \sin\frac{\pi\left(n_{2}+\frac{1}{2}\right)(k_{2}+1)}{N_{2}}$

## References

• 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.
 Title discrete sine transform Canonical name DiscreteSineTransform Date of creation 2013-03-22 17:23:45 Last modified on 2013-03-22 17:23:45 Owner stitch (17269) Last modified by stitch (17269) Numerical id 7 Author stitch (17269) Entry type Definition Classification msc 42-00 Classification msc 65T50 Synonym DST Synonym discrete trigonometric transforms Related topic DiscreteCosineTransform Related topic DiscreteFourierTransform2 Related topic DiscreteFourierTransform Defines DST-I Defines DST-II Defines DST-III Defines DST-IV Defines DST-V Defines DST-VI Defines DST-VII Defines DST-VII Defines DST-VIII