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 andsine transforms and their fast algorithms. 2006.
3 Z. Wang and B. Hunt, The Discrete W Transform, Applied Mathematics and Computation,16. 1985.