discrete sine transform

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

δ

is the Kronecker delta, are defined by the following equations:

1.1

DST-I

S

k

I

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

)

⁢

(

k

+

1

)

N

+

1

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

+

1

The DST-I is its own inverse.

1.2

DST-II

S

k

I

⁢

I

=

p

k

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

2

)

⁢

(

k

+

1

)

N

k

=

0

,

1

,

2

,

…

,

N

-

1

p

k

=

2

-

δ

k

,

0

N

The inverse of DST-II is

DST-III

.

1.3

DST-III

S

k

I

⁢

I

⁢

I

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

q

n

⁢

sin

⁡

π

⁢

(

n

+

1

)

⁢

(

k

+

1

2

)

N

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

q

n

=

1

1

+

δ

n

,

0

The inverse of DST-III is DST-II.

1.4

DST-IV

S

k

I

⁢

V

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

2

)

⁢

(

k

+

1

2

)

N

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

The DST-IV is its own inverse.

1.5

DST-V

S

k

V

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

)

⁢

(

k

+

1

)

N

+

1

2

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

+

1

2

The DST-V is its own inverse.

1.6

DST-VI

S

k

V

⁢

I

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

2

)

⁢

(

k

+

1

)

N

+

1

2

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

+

1

2

The inverse of DST-VI is DST-VII.

1.7

DST-VII

S

k

V

⁢

I

⁢

I

=

p

⁢

∑

n

=

0

N

-

1

x

n

⁢

sin

⁡

π

⁢

(

n

+

1

)

⁢

(

k

+

1

2

)

N

+

1

2

k

=

0

,

1

,

2

,

…

,

N

-

1

p

=

2

N

+

1

2

The inverse of DST-VII is DST-VI.

1.8

DST-VIII

S

k

V

⁢

I

⁢

I

⁢

I

=

p

k

⁢

∑

n

=

0

N

-

1

x

n

⁢

q

n

⁢

sin

⁡

π

⁢

(

n

+

1

2

)

⁢

(

k

+

1

2

)

N

-

1

2

k

=

0

,

1

,

2

,

…

,

N

-

1

p

k

=

2

-

δ

k

,

N

-

1

N

-

1

2

q

n

=

1

1

+

δ

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

×

N

2

matrix is given by

S

k

1

,

k

2

I

⁢

I

=

p

k

1

⁢

p

k

2

⁢

∑

n

1

=

0

N

1

-

1

∑

n

2

=

0

N

2

-

1

x

n

1

,

n

2

⁢

sin

⁡

π

⁢

(

n

1

+

1

2

)

⁢

(

k

1

+

1

)

N

1

⁢

sin

⁡

π

⁢

(

n

2

+

1

2

)

⁢

(

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

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