Editing Discrete sine transform (section)

===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.