連續傅里葉變換

正變換：
X ( ω ) = ∫ ? ∞ ∞ x ( t ) e ? i ω t d t X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i\omega t}dt X(ω)=∫?∞∞?x(t)e?iωtdt
逆變換：
x ( t ) = 1 2 π ∫ ? ∞ ∞ X ( ω ) e i ω t d ω x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{i\omega t}d\omega x(t)=2π1?∫?∞∞?X(ω)eiωtdω

∫ ? ∞ ∞ ∣ x ( t ) ∣ 2 d t = ∫ ? ∞ ∞ x ( t ) x ˉ ( t ) d t = 1 2 π ∫ ? ∞ ∞ x ( t ) [ ∫ ? ∞ ∞ X ( ω ) e i ω t d ω ] ￣ d t = 1 2 π ∫ ? ∞ ∞ x ( t ) [ ∫ ? ∞ ∞ X ￣ ( ω ) e ? i ω t d ω ] d t = 1 2 π ∫ ? ∞ ∞ [ ∫ ? ∞ ∞ x ( t ) e ? i ω t d t ] X ￣ ( ω ) d ω = 1 2 π ∫ ? ∞ ∞ X ( ω ) X ￣ ( ω ) d ω = 1 2 π ∫ ? ∞ ∞ ∣ X ( ω ) ∣ 2 d ω \begin{aligned} &\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) |^2d\omega \end{aligned} ======?∫?∞∞?∣x(t)∣2dt∫?∞∞?x(t)xˉ(t)dt2π1?∫?∞∞?x(t)[∫?∞∞?X(ω)eiωtdω]?dt2π1?∫?∞∞?x(t)[∫?∞∞?X(ω)e?iωtdω]dt2π1?∫?∞∞?[∫?∞∞?x(t)e?iωtdt]X(ω)dω2π1?∫?∞∞?X(ω)X(ω)dω2π1?∫?∞∞?∣X(ω)∣2dω?

離散傅里葉變換

正變換:
X k = 1 N ∑ n = 0 N ? 1 x [ n ] e ? i ω 0 k n X_k = \frac{1}{N} \sum_{n=0}^{N-1} x[n]e^{-i\omega_0 kn} Xk?=N1?n=0∑N?1?x[n]e?iω0?kn
逆變換：
x [ n ] = ∑ k = 0 N ? 1 X k e i ω 0 k n x[n] = \sum_{k=0}^{N-1} X_k e^{i\omega_0 kn} x[n]=k=0∑N?1?Xk?eiω0?kn

∑ n = 0 N ? 1 ∣ x [ n ] ∣ 2 = ∑ n = 0 N ? 1 x [ n ] x [ n ] ￣ = ∑ n = 0 N ? 1 x [ n ] ∑ k = 0 N ? 1 X k e i ω 0 k n ￣ = ∑ n = 0 N ? 1 x [ n ] ∑ k = 0 N ? 1 X k ￣ e ? i ω 0 k n = N ∑ k = 0 N ? 1 [ 1 N ∑ n = 0 N ? 1 x [ n ] e ? i ω 0 k n ] X k ￣ = N ∑ k = 0 N ? 1 X k X k ￣ = N ∑ k = 0 N ? 1 ∣ X k ∣ 2 \begin{aligned} &\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{aligned} ======?n=0∑N?1?∣x[n]∣2n=0∑N?1?x[n]x[n]?n=0∑N?1?x[n]k=0∑N?1?Xk?eiω0?kn?n=0∑N?1?x[n]k=0∑N?1?Xk??e?iω0?knNk=0∑N?1?[N1?n=0∑N?1?x[n]e?iω0?kn]Xk??Nk=0∑N?1?Xk?Xk??Nk=0∑N?1?∣Xk?∣2?