Discrete sine transform

In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample.

The DST is related to the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions. See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition.^[1] Both the DCT and the DST were described by Nasir Ahmed, T. Natarajan, and K.R. Rao in 1974.^[2]^[3] The type-I DST (DST-I) was later described by Anil K. Jain in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.^[4]

Applications[edit]

DSTs are widely employed in solving partial differential equations by spectral methods, where the different variants of the DST correspond to slightly different odd/even boundary conditions at the two ends of the array.

Computation[edit]

Although the direct application of these formulas would require O(N²) operations, it is possible to compute the same thing with only O(N log N) complexity by factorizing the computation similar to the fast Fourier transform (FFT). (One can also compute DSTs via FFTs combined with O(N) pre- and post-processing steps.)

A DST-III or DST-IV can be computed from a DCT-III or DCT-IV (see discrete cosine transform), respectively, by reversing the order of the inputs and flipping the sign of every other output, and vice versa for DST-II from DCT-II. In this way it follows that types II–IV of the DST require exactly the same number of arithmetic operations (additions and multiplications) as the corresponding DCT types.

Generalizations[edit]

A family of transforms composed of sine and sine hyperbolic functions exists; these transforms are made based on the natural vibration of thin square plates with different boundary conditions.^[5]

S. A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," IEEE Trans. Signal Process. SP-42, 1038–1051 (1994).

Matteo Frigo and : FFTW, FFTW Home Page. A free (GPL) C library that can compute fast DSTs (types I–IV) in one or more dimensions, of arbitrary size. Also M. Frigo and S. G. Johnson, "The Design and Implementation of FFTW3," Proceedings of the IEEE 93 (2), 216–231 (2005).

Steven G. Johnson

Takuya Ooura: General Purpose FFT Package, . Free C & FORTRAN libraries for computing fast DSTs in one, two or three dimensions, power of 2 sizes.

FFT Package 1-dim / 2-dim

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), , Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the original on 2011-08-11, retrieved 2011-08-13.

"Section 12.4.1. Sine Transform"

R. Chivukula and Y. Reznik, "," Proc. SPIE Vol. 8135, 2011.

Fast Computing of Discrete Cosine and Sine Transforms of Types VI and VII

[[Category:Discrete transfo rms]]