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. 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. 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. 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. Like any Fourier-related transform, discrete sine transforms (DSTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the discrete Fourier transform (DFT), a DST operates on a function at a finite number of discrete data points. The obvious distinction between a DST and a DFT is that the former uses only sine functions, while the latter uses both cosines and sines (in the form of complex exponentials). However, this visible difference is merely a consequence of a deeper distinction: a DST implies different boundary conditions than the DFT or other related transforms. The Fourier-related transforms that operate on a function over a finite domain, such as the DFT or DST or a Fourier series, can be thought of as implicitly defining an extension of that function outside the domain. That is, once you write a function f ( x ) {displaystyle f(x)} as a sum of sinusoids, you can evaluate that sum at any x {displaystyle x} , even for x {displaystyle x} where the original f ( x ) {displaystyle f(x)} was not specified. The DFT, like the Fourier series, implies a periodic extension of the original function. A DST, like a sine transform, implies an odd extension of the original function. However, because DSTs operate on finite, discrete sequences, two issues arise that do not apply for the continuous sine transform. First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain (i.e. the min-n and max-n boundaries in the definitions below, respectively). Second, one has to specify around what point the function is even or odd. In particular, consider a sequence (a,b,c) of three equally spaced data points, and say that we specify an odd left boundary. There are two sensible possibilities: either the data is odd about the point prior to a, in which case the odd extension is (−c,−b,−a,0,a,b,c), or the data is odd about the point halfway between a and the previous point, in which case the odd extension is (−c,−b,−a,a,b,c) These choices lead to all the standard variations of DSTs and also discrete cosine transforms (DCTs). Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 {displaystyle 2 imes 2 imes 2 imes 2=16} possibilities. Half of these possibilities, those where the left boundary is odd, correspond to the 8 types of DST; the other half are the 8 types of DCT. These different boundary conditions strongly affect the applications of the transform, and lead to uniquely useful properties for the various DCT types. Most directly, when using Fourier-related transforms to solve partial differential equations by spectral methods, the boundary conditions are directly specified as a part of the problem being solved. 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: