# 傅利葉分析

## 經典傅利葉分析啲變體

${\displaystyle {\bar {x}}(t)=\sum _{k=-\infty }^{+\infty }\!X[k]\;e^{i{\frac {2\pi k}{T_{0}}}t}}$  ${\displaystyle x_{n}={\frac {1}{N}}\sum _{k=0}^{N-1}X_{k}\;e^{i{\frac {2\pi }{N}}kn},\quad n=0,\dots ,N-1.}$
${\displaystyle X[k]={\frac {1}{T_{0}}}\int _{T_{0}}{\bar {x}}(t)\;e^{-i{\frac {2\pi k}{T_{0}}}t}\,dt}$  ${\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\;e^{-i{\frac {2\pi }{N}}kn},\quad k=0,\dots ,N-1.}$

${\displaystyle x(t)=\int _{-\infty }^{\infty }X(f)\ e^{i2\pi ft}\,df}$  ${\displaystyle x[n]=T_{s}\int _{1/T_{s}}{\bar {X}}(f)\ e^{i2\pi fnT_{s}}\ df}$
${\displaystyle X(f)=\int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\,dt}$  ${\displaystyle {\bar {X}}(f)=\sum _{n=-\infty }^{+\infty }x[n]\ e^{-i2\pi fnT_{s}}}$
• ${\displaystyle {\bar {x}}(t)}$ ${\displaystyle {\bar {X}}(f)}$  都係週期函數，週期分別係 ${\displaystyle T_{0}}$ ${\displaystyle f_{s}=1/T_{s}}$
• ${\displaystyle x[n]}$ ${\displaystyle X[k]}$  都係無限序列，間隔分別係 ${\displaystyle T_{s}}$ ${\displaystyle f_{0}=1/T_{0}}$
• ${\displaystyle x_{n}}$ ${\displaystyle X_{k}}$  都係有限序列，序列長度都係 ${\displaystyle N}$

## 廣義傅利葉分析

{\displaystyle {\begin{aligned}{\mathcal {F}}(f)\colon {\widehat {G}}&\rightarrow {\mathbb {C} },\\{\mathcal {F}}(f)(\chi )&=\int _{G}f(x){\overline {\chi (x)}}\mathrm {d} \lambda (x)\end{aligned}}}