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