Editing Discrete sine transform (section)

==Informal overview==
[[Image:DST-symmetries.svg|thumb|right|350px|Illustration of the implicit even/odd extensions of DST input data, for ''N''=9 data points (red dots), for the four most common types of DST (types I–IV).]]

Like any Fourier-related transform, discrete sine transforms (DSTs) express a function or a signal in terms of a sum of [[sinusoid]]s with different [[frequencies]] and [[amplitude]]s. 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 function]]s, while the latter uses both cosines and sines (in the form of [[complex exponential]]s). However, this visible difference is merely a consequence of a deeper distinction: a DST implies different [[boundary condition]]s than the DFT or other related transforms.

The Fourier-related transforms that operate on a function over a finite [[domain of a function|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 <math>f(x)</math> as a sum of sinusoids, you can evaluate that sum at any <math>x</math>, even for <math>x</math> where the original <math>f(x)</math> was not specified. The DFT, like the Fourier series, implies a [[periodic function|periodic]] extension of the original function. A DST, like a [[Sine and cosine transforms|sine transform]], implies an [[even and odd functions|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 (&minus;''c'',&minus;''b'',&minus;''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 (&minus;''c'',&minus;''b'',&minus;''a'',''a'',''b'',''c'')

These choices lead to all the standard variations of DSTs and also [[discrete cosine transform]]s (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 <math>2 \times 2 \times 2 \times 2 = 16</math> 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 equation]]s by [[spectral method]]s, the boundary conditions are directly specified as a part of the problem being solved.