discrete sine transform

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

1 Definition

The orthonormal variants of the DST, where xn is the original vector of N real numbers, Ck is the transformed vector of N real numbers and δ is the Kronecker delta, are defined by the following equations:

1.1 DST-I

SkI = pn=0N-1xnsinπ(n+1)(k+1)N+1  k=0,1,2,,N-1
p = 2N+1

The DST-I is its own inverse.

1.2 DST-II

SkII = pkn=0N-1xnsinπ(n+12)(k+1)N  k=0,1,2,,N-1
pk = 2-δk,0N

The inverse of DST-II is DST-III.


SkIII = pn=0N-1xnqnsinπ(n+1)(k+12)N  k=0,1,2,,N-1
p = 2N
qn = 11+δn,0

The inverse of DST-III is DST-II.

1.4 DST-IV

SkIV = pn=0N-1xnsinπ(n+12)(k+12)N  k=0,1,2,,N-1
p = 2N

The DST-IV is its own inverse.

1.5 DST-V

SkV = pn=0N-1xnsinπ(n+1)(k+1)N+12  k=0,1,2,,N-1
p = 2N+12

The DST-V is its own inverse.

1.6 DST-VI

SkVI = pn=0N-1xnsinπ(n+12)(k+1)N+12  k=0,1,2,,N-1
p = 2N+12

The inverse of DST-VI is DST-VII.


SkVII = pn=0N-1xnsinπ(n+1)(k+12)N+12  k=0,1,2,,N-1
p = 2N+12

The inverse of DST-VII is DST-VI.


SkVIII = pkn=0N-1xnqnsinπ(n+12)(k+12)N-12  k=0,1,2,,N-1
pk = 2-δk,N-1N-12
qn = 11+δn,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 N1×N2 matrix is given by

Sk1,k2II = pk1pk2n1=0N1-1n2=0N2-1xn1,n2sinπ(n1+12)(k1+1)N1sinπ(n2+12)(k2+1)N2


  • 1 Xuancheng Shao, Steven G. Johnson. Type-II/III DCT/DST algorithmsMathworldPlanetmath 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