# 帕塞瓦尔定理

## 帕塞瓦尔定理的陈述

${\displaystyle \left\|v\right\|^{2}=v_{1}^{2}+v_{2}^{2}+\cdots +v_{n}^{2}}$

${\displaystyle \sum _{k}\left|\left\langle x,e_{k}\right\rangle \right|^{2}=\left\|x\right\|^{2}}$

${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}$

${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}.}$

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,dx,}$

## 物理学和工程学上使用的记号

${\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df}$

${\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X(e^{i\omega })|^{2}d\omega }$

${\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}$

## 證明

### 连续傅立叶变换(CTFT)的帕塞瓦爾定理

${\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}dt}$

${\displaystyle =\int _{-\infty }^{\infty }x(t)x^{*}(t)dt}$

${\displaystyle =\int _{-\infty }^{\infty }x(t)[\int _{-\infty }^{\infty }X^{*}(f)e^{-j2{\pi }ft}df]dt}$

${\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)[\int _{-\infty }^{\infty }x(t)e^{-j2{\pi }ft}dt]df}$

${\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)X(f)df}$

${\displaystyle =\int _{-\infty }^{\infty }|X(f)|^{2}df}$

### 离散时间傅立叶变换(DTFT)的帕塞瓦爾定理

${\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}}$

${\displaystyle =\sum _{n=-\infty }^{\infty }x[n]x^{*}[n]}$

${\displaystyle =\sum _{n=-\infty }^{\infty }x[n][{\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}X^{*}(e^{j\omega })e^{-j{\omega }n}d\omega ]}$

${\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}[\sum _{n=-\infty }^{\infty }x[n]e^{-j{\omega }n}]X^{*}(e^{j\omega })d\omega }$

${\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}X(e^{j\omega })X^{*}(e^{j\omega })d\omega }$

${\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}|X(e^{j\omega })|^{2}d\omega }$

### 連續時間傅立葉級數(CTFS)的帕塞瓦爾定理

${\displaystyle c_{n}}$ 是其連續時間傅立葉級數${\displaystyle c_{n}={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x(t)e^{-j2\pi {n}{f_{0}}t}dt}$

${\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}}$

${\displaystyle =\sum _{n=-\infty }^{\infty }c_{n}{c_{n}}^{*}}$

${\displaystyle =\sum _{n=-\infty }^{\infty }c_{n}[{\frac {1}{T_{0}}}\int _{0}^{T_{0}}x^{*}(t)e^{j2\pi {n}{f_{0}}t}dt]}$

${\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x^{*}(t)[\sum _{n=-\infty }^{\infty }c_{n}e^{j2\pi {n}{f_{0}}t}]dt}$

${\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x^{*}(t)x(t)dt}$

${\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}|x(t)|^{2}dt}$

### 离散时间傅里叶级数(DTFS)的帕塞瓦爾定理

x[n]是長度為N的離散時間信號，${\displaystyle a_{k}}$ 為其離散時間傅立葉級數，亦即${\displaystyle a_{k}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]e^{-j\omega _{0}kn}}$

${\displaystyle \sum _{k=0}^{N-1}|a_{k}|^{2}}$

${\displaystyle =\sum _{k=0}^{N-1}a_{k}{a_{k}}^{*}}$

${\displaystyle =\sum _{k=0}^{N-1}a_{k}[{\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]e^{j\omega _{0}kn}]}$

${\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n][\sum _{k=0}^{N-1}a_{k}e^{j\omega _{0}kn}]}$

${\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]x[n]}$

${\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}|x[n]|^{2}}$

### 离散傅立叶变换(DFT)的帕塞瓦爾定理

${\displaystyle x[n]}$ 為一長度是N點的離散時間信號，僅在0≤n≤N-1有值，${\displaystyle x[n]=0}$  for ${\displaystyle n<0}$  or ${\displaystyle n>N-1}$

${\displaystyle W_{N}=e^{j{\frac {2\pi }{n}}}}$

${\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}}$

${\displaystyle =\sum _{n=0}^{N-1}x[n]x^{*}[n]}$

${\displaystyle =\sum _{n=0}^{N-1}x[n][{\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]{W_{N}}^{-kn}]}$

${\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k][\sum _{n=0}^{N-1}x[n]{W_{N}}^{-kn}]}$

${\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]X[k]}$

${\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}$