Discrete sine transformation

The discrete sine transform (DST, English discrete sine transform ) is a real-valued , discrete , linear , orthogonal transformation that, like the imaginary part of the discrete Fourier transform (DFT), generates a time-discrete signal from the time domain (for time signals) or the spatial domain (for spatial Signals) transformed into the frequency domain.

It is closely related to the discrete cosine transform (DCT), but in contrast is based on the odd sine function .

Application of the DST, as well as the DCT, lies in the solution of partial differential equations . With the H.265 video standard , the DST can be used with certain settings. In contrast to the DCT, the DST has in most cases no significant application in the field of signal processing and data compression.

definition

Different periodic continuations in DST-I to DST-IV on a 9 element long example sequence in red.

There are a total of eight different forms of DST, which are referred to in the literature as DST-I to DST-VIII. They differ in the way in which the finite sequence is continued oddly at the beginning of the sequence. The DST-I to DST-IV are, except for a constant factor, equivalent to the real-valued, odd DFT with an even order. The different types of DST each map the real-valued input sequence, from the space or time domain, with N elements x [ n ] to a real-valued output sequence, the spectral range, X [ n ]:

${\displaystyle x[n]=x_{0},\ldots ,x_{N-1}\Rightarrow X[n]=X_{0},\ldots ,X_{N-1}}$

The four most common types of DST are DST-I through DST-IV:

DST-I

The DST-I with respect to its boundary values at the beginning of odd order x _-1 and odd at the end to x _N .

${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N+1}}(n+1)(k+1)\right]\quad \quad k=0,\dots ,N-1}$

DST-II

With regard to its boundary values, DST-II is odd at the beginning by x _−1/2 and odd at the end by x _{N −1/2} .

${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)(k+1)\right]\quad \quad k=0,\dots ,N-1}$

DST-III

The DST-III is odd at the beginning by x ₋₁ and even at the end by x _{N −1} with regard to its boundary _values .

${\displaystyle X_{k}={\frac {(-1)^{k}}{2}}x_{N-1}+\sum _{n=0}^{N-2}x_{n}\sin \left[{\frac {\pi }{N}}(n+1)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}$

DST-IV

The DST-IV is odd at the beginning by x _−1/2 and even at the end by x _{N −1/2} with regard to its boundary _values .

${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\sin \left[{\frac {\pi }{N}}\left(n+{\frac {1}{2}}\right)\left(k+{\frac {1}{2}}\right)\right]\quad \quad k=0,\dots ,N-1}$

Inverse transformation

Like any transformation, the DST also has an inverse transformation. The inverse of the DST-I is the DST-I with a constant factor of ² / _{( N +1)} . The inverse of the DST-IV is the DST-IV with the constant factor ² / _N . The inverse of DST-II is DST-III with a factor of ² / _N and vice versa.

Similar to the DCT, the pre-factors of the DST are not standardized in the literature. For example, some authors introduce an additional factor of in order to avoid the additional factor in inverse surgery. By suitable choice of the constant factor, the transformation matrix can represent an orthogonal matrix . ${\displaystyle {\sqrt {2/N}}}$

literature

• Vladimir Britanak, Patrick C. Yip, KR Rao: Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations . Academic Press, 2007, ISBN 978-0-12-373624-6 .

Web links

• FFTW . An open source C library under the GPL for the calculation of DST-I to DST-IV in one or more dimensions.
• The Design and Implementation of FFTW3 (English; PDF; 342 kB)

Individual evidence

1. ^ SA Martucci: Symmetric convolution and the discrete sine and cosine transforms , in Proceedings of the IEEE in Signal Processing , Edition SP-42 , 1994, pp. 1038-1051.
2. ↑ Martin Fiedler: Video compression process - from MPEG-1 to H.264 and H.265. Retrieved March 10, 2014 .