帕塞瓦尔定理（Parseval）

连续傅里叶变换

正变换：
$$X(\omega )={\int }_{-\infty }^{\infty }x(t)e^{-i\omega t}dt$$
逆变换：
$$x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )e^{i\omega t}d\omega$$

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$$\begin{array}{rl}& {\int }_{-\infty }^{\infty }|x(t){|}^{2}dt\\ & \\ =& {\int }_{-\infty }^{\infty }x(t)\bar{x}(t)dt\\ & \\ =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }x(t)\overline{\left[{\int }_{-\infty }^{\infty }X(\omega )e^{i\omega t}d\omega \right]}dt\\ & \\ =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }x(t)\left[{\int }_{-\infty }^{\infty }\overline{X}(\omega )e^{-i\omega t}d\omega \right]dt\\ & \\ =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }\left[{\int }_{-\infty }^{\infty }x(t)e^{-i\omega t}dt\right]\overline{X}(\omega )d\omega \\ & \\ =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )\overline{X}(\omega )d\omega \\ & \\ =& \frac{1}{2\pi }{\int }_{-\infty }^{\infty }|X(\omega ){|}^{2}d\omega \end{array}$$

离散傅里叶变换

正变换:
$$X_k=\frac{1}{N}\sum _{n=0}^{N-1}x[n]e^{-i{\omega }_{0}kn}$$
逆变换：
$$x[n]=\sum _{k=0}^{N-1}{X}_k{e}^{i{\omega }_{0}kn}$$

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$$\begin{array}{rl}& \sum _{n=0}^{N-1}{\left|x[n]\right|}^2\\ & \\ =& \sum _{n=0}^{N-1}x[n]\overline{x[n]}\\ & \\ =& \sum _{n=0}^{N-1}x[n]\overline{\sum _{k=0}^{N-1}{X}_k{e}^{i{\omega }_{0}kn}}\\ & \\ =& \sum _{n=0}^{N-1}x[n]\sum _{k=0}^{N-1}\overline{X_k}{e}^{-i{\omega }_{0}kn}\\ & \\ =& N\sum _{k=0}^{N-1}\left[\frac{1}{N}\sum _{n=0}^{N-1}x[n]e^{-i{\omega }_{0}kn}\right]\overline{X_k}\\ & \\ =& N\sum _{k=0}^{N-1}{X}_k\overline{X_k}\\ & \\ =& N\sum _{k=0}^{N-1}|X_k{|}^2\end{array}$$