帕塞瓦尔定理

在数学中，帕塞瓦尔定理经常指“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Marc-Antoine Parseval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，以物理学家约翰·斯特拉特，第三代瑞利男爵命名。

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Plancherel theorem而不是帕塞瓦尔定理才更合适。

帕塞瓦尔定理的陈述

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:

$A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$

和

$B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}.$

然后

$\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, dx,$

这里的i是虚数单位而上划线（horizontal bars）表示复共轭运算。

More generally, given an abelian topological group G with Pontryagin dual G^, Parseval's theorem 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.

物理学和工程学上使用的记号

In physics and engineering, Parseval's theorem is often written as:

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

where $X(f) = \mathcal{F} \{ x(t) \}$ represents the continuous Fourier transform (in normalized, unitary form) of x(t) and f represents the frequency component (not angular frequency) of x.

The interpretation of this form of the theorem is that the total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f.

For discrete time signals, the theorem becomes:

$\sum_{n=-\infty}^\infty | x[n] |^2 = \frac{1}{2\pi} \int_{-\pi}^\pi | X(e^{i\phi}) |^2 d\phi$

where X is the discrete-time Fourier transform (DTFT) of x and Φ represents the angular frequency (in radians per sample) of x.

Alternatively, for the discrete Fourier transform (DFT), the relation becomes:

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

where X[k] is the DFT of x[n], both of length N.

参见

• 帕塞瓦尔恒等式
• Plancherel's theorem
• Parseval–Gutzmer formula

参考链接

• 傅立葉級數,單維彰