# 离散正弦变换

## 定義

### DST-I

$X_k = \sum_{n=0}^{N-1} x_n \sin \left[\frac{\pi}{N+1} (n+1) (k+1) \right] \quad \quad k = 0, \dots, N-1$

$N=3$的實數abc的DST-I變換等價於8點實數0abc0(-c)(-b)(-a)（奇對稱）的DFT轉換，再除2（而DST-II~DST-IV等價於DFT有半個取樣的位移）。

### DST-II

$X_k = \sum_{n=0}^{N-1} x_n \sin \left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) (k+1)\right] \quad \quad k = 0, \dots, N-1$

### DST-III

$X_k = \frac{(-1)^k}{2} x_{N-1} + \sum_{n=0}^{N-2} x_n \sin \left[\frac{\pi}{N} (n+1) \left(k+\frac{1}{2}\right) \right] \quad \quad k = 0, \dots, N-1$

### DST-IV

$X_k = \sum_{n=0}^{N-1} x_n \sin \left[\frac{\pi}{N} \left(n+\frac{1}{2}\right) \left(k+\frac{1}{2}\right) \right] \quad \quad k = 0, \dots, N-1$

## 反變換

DST-I的反變換是把DST-I乘以$\frac{1}{N+1}$。 DST-IV的反變換是把DST-IV乘以$\frac{2}{N}$。 DST-II的反變換是把DST-III乘以$\frac{2}{N}$，反之亦然。

## 參考資料

• 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).