在数学 中，帕塞瓦尔定理 （或称帕塞瓦尔等式 ），经常指“傅里叶转换 是幺正算符 ”这一结论；简而言之，就是说函数平方的和（或积分）等于其傅里叶转换式平方之和（或者积分）。这个定理产生于马克-安托万·帕塞瓦尔 在1799年所得到的一个有关级数 的定理，该定理随后被应用于傅里叶级数 。它也被称为瑞利能量定理 或瑞利恒等式 ，以物理学家瑞利 命名。
虽说帕塞瓦尔定理 这一术语常用来描述任何 傅里叶转换的幺正性，尤其是在物理学 和工程学 上，但这种属性最一般的形式还是称为普朗歇尔定理 而不是帕塞瓦尔定理 才更合适。
该定理是勾股定理 在希尔伯特空间 或更广泛的内积空间 中的推广，或者说勾股定理是帕塞瓦尔定理在定义了内积的二维欧氏空间中的特例。
在一般的欧氏平面几何中，勾股定理说明直角三角形的两个直角边之长度的平方加起来等于斜边的平方。从另一种角度来看，若在平面上定义了一个直角坐标系xOy（单位向量分别是
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{\displaystyle (e_{x},e_{y})}
对于一个有限维的欧几里得空间
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{\displaystyle \mathbb {R} ^{n}}
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{\displaystyle (e_{1},e_{2},\cdots ,e_{n})}
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{\displaystyle v=(v_{1},v_{2},\cdots ,v_{n})}
长度 的平方等于它在各个基向量上的投影的长度的平方之和：
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{\displaystyle \left\|v\right\|^{2}=v_{1}^{2}+v_{2}^{2}+\cdots +v_{n}^{2}}
在一般的希尔伯特空间之中，也有类似的等式。设
H
{\displaystyle {\mathcal {H}}}
内积 ：
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{\displaystyle \left\langle \cdot ,\cdot \right\rangle }
希尔伯特空间 。考虑
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{\displaystyle {\mathcal {H}}}
规范正交 基 ：
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{\displaystyle (e_{1},e_{2},\cdots ,e_{n},\cdots )}
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{\displaystyle {\mathcal {H}}}
范数 的平方都等于它在各个基向量上的投影的平方之和。
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{\displaystyle \sum _{k}\left|\left\langle x,e_{k}\right\rangle \right|^{2}=\left\|x\right\|^{2}}
假定A (x )和B (x )都是平方可积 的（参照勒贝格测度 ）复变函数，且定义在R 上周期为2π的区间上，分别写成傅里叶级数的形式:
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{\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}
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{\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}.}
然后
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{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,dx,}
这里的i 是虚数单位 而上划线（horizontal bars）表示复共轭 运算。
一般地， 给定一个交换的拓扑群 G 和它的Pontryagin对偶 G^, 帕塞瓦尔定理 says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line R, G^ is also R and the unitary transform is the Fourier transform on the real line. When G is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete-time Fourier transform in applied contexts.
在 物理学 和 工程学 中, 帕塞瓦尔定理通常描述如下:
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{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df}
其中
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{\displaystyle X(f)={\mathcal {F}}\{x(t)\}}
x (t ) 的连续傅立叶变换 （以归一化酉形式），而f 代表x 的频率分量（非角频率 ）
帕塞瓦尔定理的此表达形式解释了波形x (t )依时间域t 累积的总能量 与该波形的傅立叶变换X (f )在频率域f 累积的总能量相等。
对于离散时间 信号 ，该理论表达式变换为：
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{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X(e^{i\omega })|^{2}d\omega }
其中，X 为x 的离散时间傅立叶变换 (DTFT)，而
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{\displaystyle \omega }
x 的角频率 （度 每样本）。
此外，对于离散傅立叶变换 (DFT)，表达式变换为：
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{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}
其中，X [k ]为x [n ]的DFT变换，变换前后样本长度皆为N 。
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{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}dt}
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{\displaystyle =\int _{-\infty }^{\infty }x(t)x^{*}(t)dt}
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{\displaystyle =\int _{-\infty }^{\infty }x(t)[\int _{-\infty }^{\infty }X^{*}(f)e^{-j2{\pi }ft}df]dt}
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{\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)[\int _{-\infty }^{\infty }x(t)e^{-j2{\pi }ft}dt]df}
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{\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)X(f)df}
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{\displaystyle =\int _{-\infty }^{\infty }|X(f)|^{2}df}
其中，
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{\displaystyle x^{*}(t)}
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{\displaystyle x(t)}
编辑
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{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}}
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{\displaystyle =\sum _{n=-\infty }^{\infty }x[n]x^{*}[n]}
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{\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 ]}
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{\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 }
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{\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}X(e^{j\omega })X^{*}(e^{j\omega })d\omega }
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{\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}|X(e^{j\omega })|^{2}d\omega }
其中，
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{\displaystyle x^{*}[n]}
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{\displaystyle x[n]}
编辑
令x(t)是周期为
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{\displaystyle T_{0}={\frac {1}{f_{0}}}}
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{\displaystyle c_{n}}
连续时间傅立叶级数 。
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{\displaystyle c_{n}={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x(t)e^{-j2\pi {n}{f_{0}}t}dt}
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{\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}}
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{\displaystyle =\sum _{n=-\infty }^{\infty }c_{n}{c_{n}}^{*}}
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{\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]}
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{\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}
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{\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x^{*}(t)x(t)dt}
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{\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}|x(t)|^{2}dt}
编辑
x[n]是长度为N的离散时间信号，
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{\displaystyle a_{k}}
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{\displaystyle a_{k}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]e^{-j\omega _{0}kn}}
其中
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{\displaystyle \omega _{0}}
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{\displaystyle \omega _{0}={\frac {2\pi }{N}}}
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{\displaystyle \sum _{k=0}^{N-1}|a_{k}|^{2}}
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{\displaystyle =\sum _{k=0}^{N-1}a_{k}{a_{k}}^{*}}
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{\displaystyle =\sum _{k=0}^{N-1}a_{k}[{\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]e^{j\omega _{0}kn}]}
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{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n][\sum _{k=0}^{N-1}a_{k}e^{j\omega _{0}kn}]}
=
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N
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∗
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x
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{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]x[n]}
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{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}|x[n]|^{2}}
令
x
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{\displaystyle x[n]}
x
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{\displaystyle x[n]=0}
n
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{\displaystyle n<0}
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{\displaystyle n>N-1}
其DFT为
X
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{\displaystyle X[k]}
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{\displaystyle X[k]=0}
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{\displaystyle k<0}
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{\displaystyle k>N-1}
设
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{\displaystyle W_{N}=e^{j{\frac {2\pi }{n}}}}
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{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}}
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{\displaystyle =\sum _{n=0}^{N-1}x[n]x^{*}[n]}
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{\displaystyle =\sum _{n=0}^{N-1}x[n][{\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]{W_{N}}^{-kn}]}
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{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k][\sum _{n=0}^{N-1}x[n]{W_{N}}^{-kn}]}
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{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]X[k]}
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{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}
![{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X(e^{i\omega })|^{2}d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f43bed86b88bdcb38b1e86f93a95224c6de57f)
![{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd01ef048644b8295d7e2332488c2013ec04b742)
![{\displaystyle =\int _{-\infty }^{\infty }x(t)[\int _{-\infty }^{\infty }X^{*}(f)e^{-j2{\pi }ft}df]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12e340c1f8556ec0445e92f5719c0db921fc2d28)
![{\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)[\int _{-\infty }^{\infty }x(t)e^{-j2{\pi }ft}dt]df}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce92b8a25e6823db2737b059091a393e0d6adca)
![{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8569380eb2fc3e53f3375fd0d19fd760f4f10ee5)
![{\displaystyle =\sum _{n=-\infty }^{\infty }x[n]x^{*}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2677306e1d943b99829e06c127d9d588212ed2)
![{\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 ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f91295edc68fab0ca28e6d8fa8ff41e278171b)
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c7246206f049dcb274a612ed7759bc59ad09c5)
![{\displaystyle x^{*}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ebfe6cd83983535242e2e7090ec8afd92fda490)
![{\displaystyle x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e251c81a2a3450a6d3ae18f21627ab35e688ed2)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37a966069a1e2acbeb8d9ece38927285a38e049)
![{\displaystyle a_{k}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]e^{-j\omega _{0}kn}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d3636e7b741822eb39b3a991c72909eccb17d0d)
![{\displaystyle =\sum _{k=0}^{N-1}a_{k}[{\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]e^{j\omega _{0}kn}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b1e79fd12de200a7bfa21d80524fae785ea221)
![{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n][\sum _{k=0}^{N-1}a_{k}e^{j\omega _{0}kn}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64e38a474d8902c6976bdd4c906e5d1c3258b7b0)
![{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697f1f19d740c8dc4168c96a3763025dfac5d1c9)
![{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}|x[n]|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9350e830a6bcf94f066210d528321adf8e88de77)
![{\displaystyle x[n]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f8ad33743130063213bccb0c3dc06396e8814c)
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![{\displaystyle X[k]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5140be2ae2adc3eb67548c794a4c6566e408833)
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