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【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
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\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)⟵a⋅f1+b⋅f2⟵f(t±t0)⟵f(t±t0)⟵e∓jω0tf(t)⟵f1(t)⋆f2(t)⟵f1(t)⋅f2(t)⟵f(n)(t)⟵∫−∞tf(x)dx⟵(−jt)nf(t)⟵πf(0)δ(t)+−jtf(t)⟵e−αtε(t)⟵e−α∣t∣⟵gτ(t)⟵1⟵δ⟵δ′⟵δ(n)⟵ε(t)⟵sgn(t)⟵↓R(τ)⟵∫−∞∞f(t)f(t−τ)dt⟵↓R(τ)⟵T→∞lim[T1∫−2T2Tf(t)f(t−τ)dt]⟵ejω0t⟵e−jω0t⟵cos(ω0t)⟵sin(ω0t)⟵fT(t)⟵δT(t)⋆f0(t)⟵δT(t)⋆f0(t)⟵n=−∞∑∞FnejnΩt⟵⟶F(jω)⟶2πf(−ω)⟶∣α∣1F(jαω)⟶a⋅F1+b⋅F2⟶e±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)dx⟶α+jω1⟶α2+ω22α⟶τSa⎩⎧2ωτ⎭⎫⟶2πδ(ω)⟶1⟶jω⟶(jω)n⟶πδ(ω)+jω1⟶jω2⟶E(ω)↓⟶∣F(jω)∣2⟶P(ω)↓⟶T→∞limT∣FT(jω)∣2⟶2πδ(ω−ω0)⟶2πδ(ω+ω0)⟶π[δ(ω+ω0)+δ(ω−ω0)]⟶jπ[δ(ω+ω0)−δ(ω−ω0)]⟶FT(jω)⟶ΩδΩ(ω)F0(jω)⟶Ωn=−∞∑∞F0(jnΩ)δ(ω−nΩ)⟶2πn=−∞∑∞Fnδ(ω−nΩ)
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