常用函数的傅里叶变换汇总

  • 快速链接:
    • 【总目录】信号与系统 学习笔记 Signals and Systems with Python
    • (1) 简介 Intro
    • (2) 傅里叶 Fourier
      • 常用函数的傅里叶变换汇总
    • (3) LTI 系统 与 滤波器

常用函数的傅里叶变换汇总

f ( t ) ⟵ ⟶ F ( j ω ) F ( j t ) ⟵ ⟶ 2 π f ( − ω ) f ( α t ) ⟵ ⟶ 1 ∣ α ∣ F ( j ω α ) a ⋅ f 1 + b ⋅ f 2 ⟵ ⟶ a ⋅ F 1 + b ⋅ F 2 f ( t ± t 0 ) ⟵ ⟶ e ± j ω t 0 F ( j ω ) f ( t ± t 0 ) ⟵ ⟶ ∣ F ( j ω ) ∣ e j [ φ ( ω ) ± ω t 0 ] e ∓ j ω 0 t f ( t ) ⟵ ⟶ F [ j ( ω ± ω 0 ) ] f 1 ( t ) ⋆ f 2 ( t ) ⟵ ⟶ F 1 ( j ω ) ⋅ F 2 ( j ω ) f 1 ( t ) ⋅ f 2 ( t ) ⟵ ⟶ 1 2 π F 1 ( j ω ) ⋆ F 2 ( j ω ) f ( n ) ( t ) ⟵ ⟶ ( j ω ) n F ( j ω ) ∫ − ∞ t f ( x ) d x ⟵ ⟶ π F ( 0 ) δ ( ω ) + F ( j ω ) j ω ( − j t ) n f ( t ) ⟵ ⟶ F ( n ) ( j ω ) π f ( 0 ) δ ( t ) + f ( t ) − j t ⟵ ⟶ ∫ − ∞ ω F ( j x ) d x e − α t ε ( t ) ⟵ ⟶ 1 α + j ω e − α ∣ t ∣ ⟵ ⟶ 2 α α 2 + ω 2 g τ ( t ) ⟵ ⟶ τ Sa ⟮ ω τ 2 ⟯ 1 ⟵ ⟶ 2 π δ ( ω ) δ ⟵ ⟶ 1 δ ′ ⟵ ⟶ j ω δ ( n ) ⟵ ⟶ ( j ω ) n ε ( t ) ⟵ ⟶ π δ ( ω ) + 1 j ω sgn ( t ) ⟵ ⟶ 2 j ω ↓ R ( τ ) ⟵ ⟶ E ( ω ) ↓ ∫ − ∞ ∞ f ( t ) f ( t − τ ) d t ⟵ ⟶ ∣ F ( j ω ) ∣ 2 ↓ R ( τ ) ⟵ ⟶ P ( ω ) ↓ lim ⁡ T → ∞ [ 1 T ∫ − T 2 T 2 f ( t ) f ( t − τ ) d t ] ⟵ ⟶ lim ⁡ T → ∞ ∣ F T ( j ω ) ∣ 2 T e j ω 0 t ⟵ ⟶ 2 π δ ( ω − ω 0 ) e − j ω 0 t ⟵ ⟶ 2 π δ ( ω + ω 0 ) cos ⁡ ( ω 0 t ) ⟵ ⟶ π [ δ ( ω + ω 0 ) + δ ( ω − ω 0 ) ] sin ⁡ ( ω 0 t ) ⟵ ⟶ j π [ δ ( ω + ω 0 ) − δ ( ω − ω 0 ) ] f T ( t ) ⟵ ⟶ F T ( j ω ) δ T ( t ) ⋆ f 0 ( t ) ⟵ ⟶ Ω δ Ω ( ω ) F 0 ( j ω ) δ T ( t ) ⋆ f 0 ( t ) ⟵ ⟶ Ω ∑ n = − ∞ ∞ F 0 ( j n Ω ) δ ( ω − n Ω ) ∑ n = − ∞ ∞ F n e j n Ω t ⟵ ⟶ 2 π ∑ n = − ∞ ∞ F n δ ( ω − n Ω ) \begin{aligned} \displaystyle f({\color{blue}t}) \longleftarrow& \longrightarrow F({\color{blue}j\omega}) \\ F(j t) \longleftarrow& \longrightarrow {\color{blue}2\pi }f(-\omega)\\ f({\color{blue}\alpha} t) \longleftarrow& \longrightarrow {\color{blue}\frac{1}{\lvert \alpha \rvert}}F(j\frac{\omega}{{\color{blue}\alpha}})\\ {\color{blue}a}\cdot f_1 + {\color{blue}b}\cdot f_2 \longleftarrow& \longrightarrow {\color{blue}a}\cdot F_1 + {\color{blue}b}\cdot F_2 \\ f(t {\color{blue}\pm t_0}) \longleftarrow& \longrightarrow {\color{blue}e^{\pm j \omega t_0}}F(j\omega)\\ f(t {\color{blue}\pm t_0}) \longleftarrow& \longrightarrow \lvert F(j\omega)\rvert {\color{blue}e^{j[\varphi(\omega)\pm \omega t_0]}}\\ {\color{blue}e^{\mp j\omega_0 t}}f(t)\longleftarrow& \longrightarrow F\big[j(\omega{\color{blue}\pm\omega_0})\big]\\ f_1(t) {\color{blue}\star} f_2(t) \longleftarrow& \longrightarrow F_1(j\omega){\color{blue}\cdot} F_2(j\omega)\\ f_1(t){\color{blue}\cdot} f_2(t) \longleftarrow& \longrightarrow {\color{blue}\frac{1}{2\pi}}F_1(j\omega){\color{blue}\star} F_2(j\omega)\\ f^{{\color{blue}(n)}} (t) \longleftarrow& \longrightarrow {\color{blue}(j\omega)^n} F(j\omega)\\ \int^{t}_{-\infty} f(x) dx \longleftarrow& \longrightarrow \pi F(0)\delta(\omega) + \frac{F(j\omega)}{j\omega}\\ {\color{blue}(-jt)^n} f (t) \longleftarrow& \longrightarrow F^{{\color{blue}(n)}}(j\omega)\\ \pi f(0)\delta(t) + \frac{f(t)}{{\color{red}-}jt} \longleftarrow& \longrightarrow \int^{\omega}_{-\infty}F(jx)dx\\ e^{-\alpha t} \varepsilon(t)\longleftarrow& \longrightarrow \frac{1}{\alpha + j\omega}\\ e^{-\alpha \lvert t\rvert} \longleftarrow& \longrightarrow \frac{2\alpha}{\alpha^2 + \omega^2} \\ g_{\color{blue}\tau}(t) \longleftarrow& \longrightarrow {\color{blue}\tau} \text{Sa} \Big\lgroup \displaystyle \frac{\omega{\color{blue}\tau}}{2} \Big\rgroup\\ {\color{red}1} \longleftarrow& \longrightarrow {\color{blue}2\pi}\delta{(\omega)}\\ {\color{red}\delta} \longleftarrow& \longrightarrow 1 \\ \delta^{\color{blue}\prime} \longleftarrow& \longrightarrow {\color{blue}j\omega} \\ \delta^{{\color{blue}(n)}} \longleftarrow& \longrightarrow (j\omega)^{\color{blue}n} \\ {\color{red}\varepsilon}(t)\longleftarrow& \longrightarrow \pi \delta(\omega) + \frac{1}{j\omega}\\ {\color{blue}\text{sgn}}(t)\longleftarrow& \longrightarrow \frac{2}{j\omega}\\ \downarrow R(\tau) \longleftarrow& \longrightarrow {\color{red}E}(\omega) \downarrow \\ {\int^{\infty}_{-\infty}f(t)f(t-\tau)dt} \longleftarrow& \longrightarrow \lvert F(j\omega) \rvert ^2\\ \downarrow R(\tau) \longleftarrow& \longrightarrow {\color{red}P}(\omega)\downarrow \\ \lim_{T\to\infty} \big[ \frac{1}{T} \int^{\frac{T}{2}}_{-\frac{T}{2}} f(t)f(t-\tau)dt \big] \longleftarrow& \longrightarrow \lim_{T\to\infty} \frac{\lvert F_T(j\omega)\rvert ^2}{T}\\ e^{j{\color{blue}\omega_0} t} \longleftarrow& \longrightarrow 2\pi \delta (\omega {\color{blue}- \omega_0}) \\ e^{-j\omega_0 t} \longleftarrow& \longrightarrow 2\pi \delta (\omega + \omega_0) \\ {\color{blue}\cos} ( \omega_0 t )\longleftarrow& \longrightarrow \pi \big[ \delta(\omega + \omega_0) {\color{blue}+} \delta(\omega-\omega_0)\big] \\ {\color{blue}\sin} (\omega_0 t) \longleftarrow& \longrightarrow {\color{blue}j}\pi \big[ \delta(\omega + \omega_0){\color{blue} -} \delta(\omega-\omega_0)\big] \\ f_{\color{blue}T}(t) \longleftarrow& \longrightarrow F_{\color{blue}T}(j\omega)\\ \delta_T(t) \star f_0(t) \longleftarrow& \longrightarrow \Omega \delta_\Omega(\omega) F_0(j\omega)\\ \delta_T(t) \star f_0(t) \longleftarrow& \longrightarrow \Omega \sum_{n=-\infty}^{\infty} F_0(jn\Omega) \delta (\omega- n\Omega)\\ \sum_{n=-\infty}^{\infty} F_n e^{jn\Omega t} \longleftarrow& \longrightarrow 2\pi \sum_{n=-\infty}^{\infty} F_n \delta (\omega- n\Omega) \\ \end{aligned} f(t)F(jt)f(αt)af1+bf2f(t±t0)f(t±t0)ejω0tf(t)f1(t)f2(t)f1(t)f2(t)f(n)(t)tf(x)dx(jt)nf(t)πf(0)δ(t)+jtf(t)eαtε(t)eαtgτ(t)1δδδ(n)ε(t)sgn(t)R(τ)f(t)f(tτ)dtR(τ)Tlim[T12T2Tf(t)f(tτ)dt]ejω0tejω0tcos(ω0t)sin(ω0t)fT(t)δT(t)f0(t)δT(t)f0(t)n=FnejnΩtF(jω)2πf(ω)α1F(jαω)aF1+bF2e±jωt0F(jω)F(jω)ej[φ(ω)±ωt0]F[j(ω±ω0)]F1(jω)F2(jω)2π1F1(jω)F2(jω)(jω)nF(jω)πF(0)δ(ω)+jωF(jω)F(n)(jω)ωF(jx)dxα+jω1α2+ω22ατSa2ωτ2πδ(ω)1jω(jω)nπδ(ω)+jω1jω2E(ω)F(jω)2P(ω)TlimTFT(jω)22πδ(ωω0)2πδ(ω+ω0)π[δ(ω+ω0)+δ(ωω0)]jπ[δ(ω+ω0)δ(ωω0)]FT(jω)ΩδΩ(ω)F0(jω)Ωn=F0(jnΩ)δ(ωnΩ)2πn=Fnδ(ωnΩ)

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