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In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but with one additional property: If the input consists of only real numbers, so will the output. 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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
This is a list of linear transformations of functions related to the Fourier transform. ...
In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...
In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ...
Sphere symmetry group o. ...
A related transform is 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. 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ...
Applications
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. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
Definition Formally, the discrete sine transform is a linear, invertible function F : RN -> RN (where R denotes the set of real numbers), or equivalently an N × N square matrix. There are several variants of the DST with slightly modified definitions. The N real numbers x0, ...., xN-1 are transformed into the N real numbers X0, ..., XN-1 according to one of the formulas: The word linear comes from the Latin word linearis, which means created by lines. ...
Partial plot of a function f. ...
In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...
For the square matrix section, see square matrix. ...
DST-I ![X_k = sum_{n=0}^{N-1} x_n sin left[frac{pi}{N+1} (n+1) (k+1) right]](http://upload.wikimedia.org/math/1/0/2/1020a00c5307056be1e2cdd80d785e1a.png) The DST-I matrix is orthogonal (up to a scale factor). In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
A DST-I of N=3 real numbers abc is exactly equivalent to a DFT of eight real numbers 0abc0(-c)(-b)(-a) (odd symmetry), here divided by two. (In contrast, DST types II-IV involve a half-sample shift in the equivalent DFT.)
Thus, the DST-I corresponds to the boundary conditions: xn is odd around n=-1 and odd around n=N; similarly for Xk.
DST-II ![X_k = sum_{n=0}^{N-1} x_n sin left[frac{pi}{N} left(n+frac{1}{2}right) (k+1)right]](http://upload.wikimedia.org/math/d/d/f/ddfe13f85ad148265a330d235caecff6.png) Some authors further multiply the XN-1 term by 1/√2 (see below for the corresponding change in DST-III). This makes the DST-II matrix orthogonal (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted input. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
The DST-II implies the boundary conditions: xn is odd around n=-1/2 and odd around n=n-1/2; Xk is odd around k=-1 and even around k=N-1.
DST-III ![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]](http://upload.wikimedia.org/math/7/c/d/7cd955e451dc075546357ff04aada162.png) Some authors further multiply the xN-1 term by √2 (see above for the corresponding change in DST-II). This makes the DST-III matrix orthogonal (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted output. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
The DST-III implies the boundary conditions: xn is odd around n=-1 and even around n=N-1; Xk is odd around k=-1/2 and odd around k=N-1/2.
DST-IV ![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]](http://upload.wikimedia.org/math/9/0/f/90ffc4dbc009348df4a928e0dd57c828.png) The DST-IV matrix is orthogonal (up to a scale factor). In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
The DST-IV implies the boundary conditions: xn is odd around n=-1/2 and even around n=N-1/2; similarly for Xk.
DST V-VIII DST types I-IV are equivalent to real-odd DFTs of even order. In principle, there are actually four additional types of discrete sine transform (Martucci, 1994), corresponding to real-odd DFTs of logically odd order, which have factors of N+1/2 in the denominators of the sine arguments. However, these variants seem to be rarely used in practice.
Inverse transforms The inverse of DST-I is DST-I multiplied by 2/(N+1). The inverse of DST-IV is DST-IV multiplied by 2/N. The inverse of DST-II is DST-III multiplied by 2/N (and vice versa).
Like for the DFT, the normalization factor in front of these transform definitions is merely a convention and differs between treatments. For example, some authors multiply the transforms by  so that the inverse does not require any additional multiplicative factor. In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...
Computation 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 fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ...
A DST-II or DST-IV can be computed from a DCT-II 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-III from DCT-III. 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. 2-D DCT compared to the DFT The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. ...
References - S. A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," IEEE Trans. Sig. Processing SP-42, 1038-1051 (1994).
- Matteo Frigo and Steven G. Johnson: FFTW, http://www.fftw.org/. 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).
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