# 正交

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

## 各种正交概念

### 正交变换

${\displaystyle \langle Tx,Ty\rangle =\langle x,y\rangle .}$

## 正交函数集

${\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)g(x)w(x)\,dx.}$

${\displaystyle \int _{a}^{b}f(x)g(x)w(x)\,dx=0.}$

${\displaystyle ||f||_{w}={\sqrt {\langle f,f\rangle _{w}}}}$

${\displaystyle \langle f_{i},f_{j}\rangle =\int _{-\infty }^{\infty }f_{i}(x)f_{j}(x)w(x)\,dx=||f_{i}||^{2}\delta _{i,j}=||f_{j}||^{2}\delta _{i,j}}$

${\displaystyle \delta _{i,j}={\begin{cases}1&\mathrm {if} \ i=j\\0&\mathrm {if} \ i\neq j\end{cases}}}$

${\displaystyle \langle f_{i},f_{j}\rangle =\int _{-\infty }^{\infty }f_{i}(x)f_{j}(x)w(x)\,dx=\delta _{i,j}}$

## 訊號處理中的正交轉換的例子及優點

• Discrete Fourier transform
• Discrete cosine, sine, Hartley transforms
• Walsh Transform, Haar Transform
• discrete Legendre transform
• discrete orthogonal polynomial transforms
• Hahn, Meixner, Krawtchouk, Charlier

1. 可以完整了解訊號的完整度
2. 訊號彼此不影響
3. 確保近似誤差最小化
4. 正轉換跟反轉換的架構是相似的

## 註釋

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