# 離散哈特利轉換

## 定義

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

## 性質

$\begin{matrix} Z_k & = & \left[ X_k \left( Y_k + Y_{N-k} \right) + X_{N-k} \left( Y_k - Y_{N-k} \right) \right] / 2 \\ Z_{N-k} & = & \left[ X_{N-k} \left( Y_k + Y_{N-k} \right) - X_k \left( Y_k - Y_{N-k} \right) \right] / 2 \end{matrix}$

## 一般化離散哈特利轉換

### 型態一

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

### 型態二

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

### 型態三

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

### 型態四

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

## 整數離散哈特利轉換

$HI_P=\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 8 & 10 & 8 & 0 & -8 & -10 & -8 & 0\\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\ 5 & 0 & -5 & 8 & -5 & 0 & 5 & -8\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\ 5 & -8 & 5 & 0 & -5 & 8 & -5 & 0\\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\ 8 & 0 & -8 & -10 & -8 & 0 & 8 & 10\\ \end{bmatrix}$

## 參考

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• R. V. L. Hartley, "A more symmetrical Fourier analysis applied to transmission problems," Proc. IRE 30, 144–150 (1942).
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• Neng-Chung Hu et al., "Generalized Discrete Hartley Transforms," IEEE Trans. Signal Processing, vol. 42, No. 12, Dec. 1992
• Guoan Bi et al., "Fast Algorithms for Generalized Discrete Hartley Transform of Composite Sequence Lengths," IEEE Trans. Circuits and System-II vol. 49, No. 9, Sept. 2000
• Soo-Chang Pei and Jian-Jiun Ding, "The Integer Transforms Analogous to Discrete Trigonometric Transforms," IEEE Trans. on Signal Processing vol.48, No. 12, Dec. 2000