在數學 中，帕塞瓦爾定理 （或稱帕塞瓦爾等式 ），經常指「傅里葉轉換 是么正算符 」這一結論；簡而言之，就是說函數平方的和（或積分）等於其傅里葉轉換式平方之和（或者積分）。這個定理產生於法國數學家馬克-安托萬·帕塞瓦爾 （Marc-Antoine Parseval ）在1799年所得到的一個有關級數 的定理，該定理隨後被應用於傅里葉級數 。它也被稱為瑞利能量定理 或瑞利恆等式 ，以物理學家瑞利 命名。
雖說帕塞瓦爾定理 這一術語常用來描述任何 傅里葉轉換的么正性，尤其是在物理學 和工程學 上，但這種屬性最一般的形式還是稱為普朗歇爾定理 而不是帕塞瓦爾定理 才更合適。
該定理是勾股定理 在希爾伯特空間 或更廣泛的內積空間 中的推廣，或者說勾股定理是帕塞瓦爾定理在定義了內積的二維歐氏空間中的特例。
在一般的歐氏平面幾何中，勾股定理說明直角三角形的兩個直角邊之長度的平方加起來等於斜邊的平方。從另一種角度來看，若在平面上定義了一個直角坐標系xOy（單位向量分別是
(
e
x
,
e
y
)
{\displaystyle (e_{x},e_{y})}
對於一個有限維的歐幾里得空間
R
n
{\displaystyle \mathbb {R} ^{n}}
(
e
1
,
e
2
,
⋯
,
e
n
)
{\displaystyle (e_{1},e_{2},\cdots ,e_{n})}
v
=
(
v
1
,
v
2
,
⋯
,
v
n
)
{\displaystyle v=(v_{1},v_{2},\cdots ,v_{n})}
長度 的平方等於它在各個基向量上的投影的長度的平方之和：
‖
v
‖
2
=
v
1
2
+
v
2
2
+
⋯
+
v
n
2
{\displaystyle \left\|v\right\|^{2}=v_{1}^{2}+v_{2}^{2}+\cdots +v_{n}^{2}}
在一般的希爾伯特空間之中，也有類似的等式。設
H
{\displaystyle {\mathcal {H}}}
內積 ：
⟨
⋅
,
⋅
⟩
{\displaystyle \left\langle \cdot ,\cdot \right\rangle }
希爾伯特空間 。考慮
H
{\displaystyle {\mathcal {H}}}
規範正交 基 ：
(
e
1
,
e
2
,
⋯
,
e
n
,
⋯
)
{\displaystyle (e_{1},e_{2},\cdots ,e_{n},\cdots )}
H
{\displaystyle {\mathcal {H}}}
範數 的平方都等於它在各個基向量上的投影的平方之和。
∑
k
|
⟨
x
,
e
k
⟩
|
2
=
‖
x
‖
2
{\displaystyle \sum _{k}\left|\left\langle x,e_{k}\right\rangle \right|^{2}=\left\|x\right\|^{2}}
假定A (x )和B (x )都是平方可積 的（參照勒貝格測度 ）複變函數，且定義在R 上周期為2π的區間上，分別寫成傅里葉級數的形式:
A
(
x
)
=
∑
n
=
−
∞
∞
a
n
e
i
n
x
{\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}
和
B
(
x
)
=
∑
n
=
−
∞
∞
b
n
e
i
n
x
.
{\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}.}
然後
∑
n
=
−
∞
∞
a
n
b
n
¯
=
1
2
π
∫
−
π
π
A
(
x
)
B
(
x
)
¯
d
x
,
{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,dx,}
這裏的i 是虛數單位 而上劃線（horizontal bars）表示復共軛 運算。
一般地， 給定一個交換的拓撲群 G 和它的Pontryagin對偶 G^, 帕塞瓦爾定理 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 物理學 和 工程學 中, 帕塞瓦爾定理通常描述如下:
∫
−
∞
∞
|
x
(
t
)
|
2
d
t
=
∫
−
∞
∞
|
X
(
f
)
|
2
d
f
{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df}
其中
X
(
f
)
=
F
{
x
(
t
)
}
{\displaystyle X(f)={\mathcal {F}}\{x(t)\}}
x (t ) 的連續傅立葉變換 （以歸一化酉形式），而f 代表x 的頻率分量（非角頻率 ）
帕塞瓦爾定理的此表達形式解釋了波形x (t )依時間域t 累積的總能量 與該波形的傅立葉變換X (f )在頻率域f 累積的總能量相等。
對於離散時間 信號 ，該理論表達式變換為：
∑
n
=
−
∞
∞
|
x
[
n
]
|
2
=
1
2
π
∫
−
π
π
|
X
(
e
i
ω
)
|
2
d
ω
{\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 為x 的離散時間傅立葉變換 (DTFT)，而
ω
{\displaystyle \omega }
x 的角頻率 （度 每樣本）。
此外，對於離散傅立葉變換 (DFT)，表達式變換為：
∑
n
=
0
N
−
1
|
x
[
n
]
|
2
=
1
N
∑
k
=
0
N
−
1
|
X
[
k
]
|
2
{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}
其中，X [k ]為x [n ]的DFT變換，變換前後樣本長度皆為N 。
∫
−
∞
∞
|
x
(
t
)
|
2
d
t
{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}dt}
=
∫
−
∞
∞
x
(
t
)
x
∗
(
t
)
d
t
{\displaystyle =\int _{-\infty }^{\infty }x(t)x^{*}(t)dt}
=
∫
−
∞
∞
x
(
t
)
[
∫
−
∞
∞
X
∗
(
f
)
e
−
j
2
π
f
t
d
f
]
d
t
{\displaystyle =\int _{-\infty }^{\infty }x(t)[\int _{-\infty }^{\infty }X^{*}(f)e^{-j2{\pi }ft}df]dt}
=
∫
−
∞
∞
X
∗
(
f
)
[
∫
−
∞
∞
x
(
t
)
e
−
j
2
π
f
t
d
t
]
d
f
{\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)[\int _{-\infty }^{\infty }x(t)e^{-j2{\pi }ft}dt]df}
=
∫
−
∞
∞
X
∗
(
f
)
X
(
f
)
d
f
{\displaystyle =\int _{-\infty }^{\infty }X^{*}(f)X(f)df}
=
∫
−
∞
∞
|
X
(
f
)
|
2
d
f
{\displaystyle =\int _{-\infty }^{\infty }|X(f)|^{2}df}
其中，
x
∗
(
t
)
{\displaystyle x^{*}(t)}
x
(
t
)
{\displaystyle x(t)}
∑
n
=
−
∞
∞
|
x
[
n
]
|
2
{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}}
=
∑
n
=
−
∞
∞
x
[
n
]
x
∗
[
n
]
{\displaystyle =\sum _{n=-\infty }^{\infty }x[n]x^{*}[n]}
=
∑
n
=
−
∞
∞
x
[
n
]
[
1
2
π
∫
0
2
π
X
∗
(
e
j
ω
)
e
−
j
ω
n
d
ω
]
{\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 ]}
=
1
2
π
∫
0
2
π
[
∑
n
=
−
∞
∞
x
[
n
]
e
−
j
ω
n
]
X
∗
(
e
j
ω
)
d
ω
{\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 }
=
1
2
π
∫
0
2
π
X
(
e
j
ω
)
X
∗
(
e
j
ω
)
d
ω
{\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}X(e^{j\omega })X^{*}(e^{j\omega })d\omega }
=
1
2
π
∫
0
2
π
|
X
(
e
j
ω
)
|
2
d
ω
{\displaystyle ={\frac {1}{2{\pi }}}\int _{0}^{2{\pi }}|X(e^{j\omega })|^{2}d\omega }
其中，
x
∗
[
n
]
{\displaystyle x^{*}[n]}
x
[
n
]
{\displaystyle x[n]}
令x(t)是周期為
T
0
=
1
f
0
{\displaystyle T_{0}={\frac {1}{f_{0}}}}
c
n
{\displaystyle c_{n}}
連續時間傅立葉級數 。
c
n
=
1
T
0
∫
0
T
0
x
(
t
)
e
−
j
2
π
n
f
0
t
d
t
{\displaystyle c_{n}={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x(t)e^{-j2\pi {n}{f_{0}}t}dt}
∑
n
=
−
∞
∞
|
c
n
|
2
{\displaystyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}}
=
∑
n
=
−
∞
∞
c
n
c
n
∗
{\displaystyle =\sum _{n=-\infty }^{\infty }c_{n}{c_{n}}^{*}}
=
∑
n
=
−
∞
∞
c
n
[
1
T
0
∫
0
T
0
x
∗
(
t
)
e
j
2
π
n
f
0
t
d
t
]
{\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]}
=
1
T
0
∫
0
T
0
x
∗
(
t
)
[
∑
n
=
−
∞
∞
c
n
e
j
2
π
n
f
0
t
]
d
t
{\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}
=
1
T
0
∫
0
T
0
x
∗
(
t
)
x
(
t
)
d
t
{\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}x^{*}(t)x(t)dt}
=
1
T
0
∫
0
T
0
|
x
(
t
)
|
2
d
t
{\displaystyle ={\frac {1}{T_{0}}}\int _{0}^{T_{0}}|x(t)|^{2}dt}
x[n]是長度為N的離散時間信號，
a
k
{\displaystyle a_{k}}
a
k
=
1
N
∑
n
=
0
N
−
1
x
[
n
]
e
−
j
ω
0
k
n
{\displaystyle a_{k}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]e^{-j\omega _{0}kn}}
其中
ω
0
{\displaystyle \omega _{0}}
ω
0
=
2
π
N
{\displaystyle \omega _{0}={\frac {2\pi }{N}}}
∑
k
=
0
N
−
1
|
a
k
|
2
{\displaystyle \sum _{k=0}^{N-1}|a_{k}|^{2}}
=
∑
k
=
0
N
−
1
a
k
a
k
∗
{\displaystyle =\sum _{k=0}^{N-1}a_{k}{a_{k}}^{*}}
=
∑
k
=
0
N
−
1
a
k
[
1
N
∑
n
=
0
N
−
1
x
∗
[
n
]
e
j
ω
0
k
n
]
{\displaystyle =\sum _{k=0}^{N-1}a_{k}[{\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]e^{j\omega _{0}kn}]}
=
1
N
∑
n
=
0
N
−
1
x
∗
[
n
]
[
∑
k
=
0
N
−
1
a
k
e
j
ω
0
k
n
]
{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n][\sum _{k=0}^{N-1}a_{k}e^{j\omega _{0}kn}]}
=
1
N
∑
n
=
0
N
−
1
x
∗
[
n
]
x
[
n
]
{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}x^{*}[n]x[n]}
=
1
N
∑
n
=
0
N
−
1
|
x
[
n
]
|
2
{\displaystyle ={\frac {1}{N}}\sum _{n=0}^{N-1}|x[n]|^{2}}
令
x
[
n
]
{\displaystyle x[n]}
x
[
n
]
=
0
{\displaystyle x[n]=0}
n
<
0
{\displaystyle n<0}
n
>
N
−
1
{\displaystyle n>N-1}
其DFT為
X
[
k
]
{\displaystyle X[k]}
X
[
k
]
=
0
{\displaystyle X[k]=0}
k
<
0
{\displaystyle k<0}
k
>
N
−
1
{\displaystyle k>N-1}
設
W
N
=
e
j
2
π
n
{\displaystyle W_{N}=e^{j{\frac {2\pi }{n}}}}
∑
n
=
0
N
−
1
|
x
[
n
]
|
2
{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}}
=
∑
n
=
0
N
−
1
x
[
n
]
x
∗
[
n
]
{\displaystyle =\sum _{n=0}^{N-1}x[n]x^{*}[n]}
=
∑
n
=
0
N
−
1
x
[
n
]
[
1
N
∑
k
=
0
N
−
1
X
∗
[
k
]
W
N
−
k
n
]
{\displaystyle =\sum _{n=0}^{N-1}x[n][{\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]{W_{N}}^{-kn}]}
=
1
N
∑
k
=
0
N
−
1
X
∗
[
k
]
[
∑
n
=
0
N
−
1
x
[
n
]
W
N
−
k
n
]
{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k][\sum _{n=0}^{N-1}x[n]{W_{N}}^{-kn}]}
=
1
N
∑
k
=
0
N
−
1
X
∗
[
k
]
X
[
k
]
{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]X[k]}
=
1
N
∑
k
=
0
N
−
1
|
X
[
k
]
|
2
{\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)
![{\displaystyle X[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44d31b5739397a3a3a58892804677840e85bb2a)
![{\displaystyle X[k]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5140be2ae2adc3eb67548c794a4c6566e408833)
![{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1020af4636e13010651483af0ef6ab83f7e43a74)
![{\displaystyle =\sum _{n=0}^{N-1}x[n]x^{*}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d1bfb74439fadd6af1a8a9e52c1b15cb83b9ea)
![{\displaystyle =\sum _{n=0}^{N-1}x[n][{\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]{W_{N}}^{-kn}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b270a16665931435af5d33d4a3434368ed7ed5f8)
![{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k][\sum _{n=0}^{N-1}x[n]{W_{N}}^{-kn}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d15a6e9563561102676f65258e92aa847710d8f6)
![{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}X^{*}[k]X[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad69c76456241728ccfb85d24f110f51ac67e2a1)
![{\displaystyle ={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adad34b1c31e0300f4772bcd70ac795554610b18)