# 離散哈特利轉換

## 定義

${\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\left[\cos \left({\frac {2\pi }{N}}nk\right)+\sin \left({\frac {2\pi }{N}}nk\right)\right]\quad \quad k=0,\dots ,N-1}$

## 性質

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

## 一般化離散哈特利轉換

### 型態一

${\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\left[\cos \left({\frac {2\pi }{N}}nk\right)+\sin \left({\frac {2\pi }{N}}nk\right)\right]\quad \quad k=0,\dots ,N-1}$

### 型態二

${\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\left[\cos \left({\frac {2\pi }{N}}(n+{\frac {1}{2}})k\right)+\sin \left({\frac {2\pi }{N}}(n+{\frac {1}{2}})k\right)\right]\quad \quad k=0,\dots ,N-1}$

### 型態三

${\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\left[\cos \left({\frac {2\pi }{N}}n(k+{\frac {1}{2}})\right)+\sin \left({\frac {2\pi }{N}}n(k+{\frac {1}{2}})\right)\right]\quad \quad k=0,\dots ,N-1}$

### 型態四

${\displaystyle H_{k}=\sum _{n=0}^{N-1}x_{n}\left[\cos \left({\frac {2\pi }{N}}(n+{\frac {1}{2}})(k+{\frac {1}{2}})\right)+\sin \left({\frac {2\pi }{N}}(n+{\frac {1}{2}})(k+{\frac {1}{2}})\right)\right]\quad \quad k=0,\dots ,N-1}$

## 整數離散哈特利轉換

${\displaystyle HI_{P}={\begin{bmatrix}1&1&1&1&1&1&1&1\\8&10&8&0&-8&-10&-8&0\\1&1&-1&-1&1&1&-1&-1\\5&0&-5&8&-5&0&5&-8\\1&-1&1&-1&1&-1&1&-1\\5&-8&5&0&-5&8&-5&0\\1&-1&-1&1&1&-1&-1&1\\8&0&-8&-10&-8&0&8&10\\\end{bmatrix}}}$

## 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