Editing Discrete sine transform (section)

==Definition==
Formally, the discrete sine transform is a [[linear]], invertible [[function (mathematics)|function]] ''F'' : '''R'''<sup>''N''</sup> {{mono|->}} '''R'''<sup>''N''</sup> (where '''R''' denotes the set of [[real number]]s), or equivalently an ''N'' &times; ''N'' [[square matrix]].  There are several variants of the DST with slightly modified definitions. The ''N'' real numbers ''x''<sub>0</sub>,...,''x''<sub>''N'' − 1</sub> are transformed into the ''N'' real numbers ''X''<sub>0</sub>,...,''X''<sub>''N'' − 1</sub> according to one of the formulas:

===DST-I===
[[File:Discrete sine transform.svg|thumb|upright=1.35|Discrete sine transform (https://www.desmos.com/calculator/r0od93dfgp).]]

:<math>\begin{align}X_k &= \sum_{n=0}^{N-1} x_n \sin \left[\frac \pi {N+1} (n+1) (k+1) \right] & k &= 0, \dots, N-1\\
X_{k-1} &= \sum_{n=1}^{N} x_{n-1} \sin \left[\frac {\pi nk} {N+1}\right] & k &= 1, \dots, N\end{align}</math>

The DST-I matrix is [[orthogonal matrix|orthogonal]] (up to a scale factor).

A DST-I is exactly equivalent to a DFT of a real sequence that is odd around the zero-th and middle points, scaled by 1/2. For example, a DST-I of ''N''=3 real numbers (''a'',''b'',''c'') is exactly equivalent to a DFT of eight real numbers (0,''a'',''b'',''c'',0,&minus;''c'',&minus;''b'',&minus;''a'') (odd symmetry), scaled by 1/2. (In contrast, DST types II–IV involve a half-sample shift in the equivalent DFT.) This is the reason for the ''N''&nbsp;+&nbsp;1 in the denominator of the sine function: the equivalent DFT has 2(''N''+1) points and has 2π/2(''N''+1) in its sinusoid frequency, so the DST-I has π/(''N''+1) in its frequency.

Thus, the DST-I corresponds to the boundary conditions: ''x''<sub>''n''</sub> is odd around ''n''&nbsp;=&nbsp;−1 and odd around ''n''=''N''; similarly for ''X''<sub>''k''</sub>.

===DST-II===
<math display="block">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</math>

Some authors further multiply the ''X''<sub>''N'' − 1</sub> term by 1/{{radic|2}} (see below for the corresponding change in DST-III). This makes the DST-II matrix [[orthogonal matrix|orthogonal]] (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted input.

The DST-II implies the boundary conditions: ''x''<sub>''n''</sub> is odd around ''n''&nbsp;=&nbsp;−1/2 and odd around ''n''&nbsp;=&nbsp;''N''&nbsp;−&nbsp;1/2; ''X''<sub>''k''</sub> is odd around ''k''&nbsp;=&nbsp;−1 and even around ''k''&nbsp;=&nbsp;''N''&nbsp;−&nbsp;1.

===DST-III===
<math display=block>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</math>

Some authors further multiply the ''x''<sub>''N'' − 1</sub> term by {{radic|2}} (see above for the corresponding change in DST-II). This makes the DST-III matrix [[orthogonal matrix|orthogonal]] (up to a scale factor), but breaks the direct correspondence with a real-odd DFT of half-shifted output.

The DST-III implies the boundary conditions: ''x''<sub>''n''</sub> is odd around ''n''&nbsp;=&nbsp;−1 and even around ''n''&nbsp;=&nbsp;''N''&nbsp;−&nbsp;1; ''X''<sub>''k''</sub> is odd around ''k''&nbsp;=&nbsp;−1/2 and odd around ''k''&nbsp;=&nbsp;''N''&nbsp;−&nbsp;1/2.

===DST-IV===
<math display="block">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</math>

The DST-IV matrix is [[orthogonal matrix|orthogonal]] (up to a scale factor).

The DST-IV implies the boundary conditions: ''x''<sub>''n''</sub> is odd around ''n''&nbsp;=&nbsp;−1/2 and even around ''n''&nbsp;=&nbsp;''N''&nbsp;−&nbsp;1/2; similarly for ''X''<sub>''k''</sub>.

===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''&nbsp;+&nbsp;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).

As for the [[discrete Fourier transform|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 <math display="inline">\sqrt{2/N}</math> so that the inverse does not require any additional multiplicative factor.