傅里叶变换和拉普拉斯变换公式总结

　

傅里叶变换公式

因为傅里叶变换之类的很常用，时间长了不用总会忘记，所以一次性罗列出来权当总结好了。主要参考《信号与线性系统分析》(吴大正)，也有的部分参考了复变函数。

δ" role="presentation" style="position: relative;">δ$\delta$-函数相关运算

n" role="presentation" style="position: relative;">n$n$阶导数的尺度变换

(56)δ(n)(at)=1|a|1anδ(n)(t)" role="presentation">δ(n)(at)=1|a|1anδ(n)(t)(56)$$\begin{array}{}\text{(56)}& {\delta }^{(n)}(at)=\frac{1}{|a|}\frac{1}{a^n}{\delta }^{(n)}(t)\end{array}$$

一阶导数和函数的乘积

(57)f(t)δ′(t−t0)=f(t0)δ′(t−t0)−f′(t0)δ(t−t0)" role="presentation">f(t)δ′(t−t0)=f(t0)δ′(t−t0)−f′(t0)δ(t−t0)(57)$$\begin{array}{}\text{(57)}& f(t){\delta }^{\prime }(t-t_0)=f(t_0){\delta }^{\prime }(t-t_0)-f^{\prime }(t_0)\delta (t-t_0)\end{array}$$

n" role="presentation" style="position: relative;">n$n$阶导数和函数的乘积

(58)f(t)δ(n)(t−t0)=∑i=0n(−1)i(ni)f(i)(t0)δ(n−i)(t−t0)" role="presentation">f(t)δ(n)(t−t0)=∑i=0n(−1)i(ni)f(i)(t0)δ(n−i)(t−t0)(58)$$\begin{array}{}\text{(58)}& f(t){\delta }^{(n)}(t-t_0)=\sum _{i=0}^n(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})f^{(i)}(t_0){\delta }^{(n-i)}(t-t_0)\end{array}$$

傅里叶级数和傅里叶变换

傅里叶级数

(4)f(x)=a02+∑n=1∞(ancosnπLx+bnsinnπLx)" role="presentation">f(x)=a02+∑n=1∞(ancosnπLx+bnsinnπLx)(4)$$\begin{array}{}\text{(4)}& f(x)=\frac{a_0}{2}+\sum _{n=1}^{\mathrm{\infty }}\left(a_{n}cos\frac{n\pi }{L}x+b_{n}sin\frac{n\pi }{L}x\right)\end{array}$$

(5)an=1L∫−LLf(x)cosnπLxdx" role="presentation">an=1L∫L−Lf(x)cosnπLxdx(5)$$\begin{array}{}\text{(5)}& a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)cos\frac{n\pi }{L}xdx\end{array}$$

(6)bn=1L∫−LLf(x)sinnπLxdx" role="presentation">bn=1L∫L−Lf(x)sinnπLxdx(6)$$\begin{array}{}\text{(6)}& b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)sin\frac{n\pi }{L}xdx\end{array}$$

半幅傅里叶级数

(7)ϕ(x)=∑n=1∞CnsinnπxL" role="presentation">ϕ(x)=∑n=1∞CnsinnπxL(7)$$\begin{array}{}\text{(7)}& \varphi (x)=\sum _{n=1}^{\mathrm{\infty }}{C}_{n}sin\frac{n\pi x}{L}\end{array}$$

(8)Cn=2L∫0Lϕ(x)sinnπxLdx" role="presentation">Cn=2L∫L0ϕ(x)sinnπxLdx(8)$$\begin{array}{}\text{(8)}& C_n=\frac{2}{L}{\int }_0^L\varphi (x)sin\frac{n\pi x}{L}dx\end{array}$$

常见函数傅里叶变换

因子12π" role="presentation" style="position: relative;">12π$\frac{1}{2\pi }$统一放在逆变换前，gτ(t)" role="presentation" style="position: relative;">gτ(t)$g_{\tau }(t)$指的是关于y" role="presentation" style="position: relative;">y$y$轴对称宽度为τ" role="presentation" style="position: relative;">τ$\tau$的门函数

(9)gτ(t)↔τSa(ωτ2)" role="presentation">gτ(t)↔τSa(ωτ2)(9)$$\begin{array}{}\text{(9)}& g_{\tau }(t)\leftrightarrow \tau Sa\left(\frac{\omega \tau }{2}\right)\end{array}$$

(10)e−atε(t)↔1a+iω" role="presentation">e−atε(t)↔1a+iω(10)$$\begin{array}{}\text{(10)}& e^{-at}\epsilon (t)\leftrightarrow \frac{1}{a+i\omega }\end{array}$$

(11)e−a|t|ε(t)↔2aa2+ω2" role="presentation">e−a|t|ε(t)↔2aa2+ω2(11)$$\begin{array}{}\text{(11)}& e^{-a|t|}\epsilon (t)\leftrightarrow \frac{2a}{a^2+{\omega }^2}\end{array}$$

(12)e−at2↔πae−ω24a" role="presentation">e−at2↔πa−−√e−ω24a(12)$$\begin{array}{}\text{(12)}& e^{-a{t}^2}\leftrightarrow \sqrt{\frac{\pi }{a}}{e}^{-\frac{{\omega }^2}{4a}}\end{array}$$

(13)δ(t)↔1" role="presentation">δ(t)↔1(13)$$\begin{array}{}\text{(13)}& \delta (t)\leftrightarrow 1\end{array}$$

(14)ε(t)↔πδ(ω)+1iω" role="presentation">ε(t)↔πδ(ω)+1iω(14)$$\begin{array}{}\text{(14)}& \epsilon (t)\leftrightarrow \pi \delta (\omega )+\frac{1}{i\omega }\end{array}$$

(15)cos(ω0t)↔π[δ(ω+ω0)+δ(ω−ω0)]" role="presentation">cos(ω0t)↔π[δ(ω+ω0)+δ(ω−ω0)](15)$$\begin{array}{}\text{(15)}& cos({\omega }_{0}t)\leftrightarrow \pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]\end{array}$$

(16)sin(ω0t)↔iπ[δ(ω+ω0)−δ(ω−ω0)]" role="presentation">sin(ω0t)↔iπ[δ(ω+ω0)−δ(ω−ω0)](16)$$\begin{array}{}\text{(16)}& sin({\omega }_{0}t)\leftrightarrow i\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]\end{array}$$

(17)tn↔2π(i)nδ(n)(ω)" role="presentation">tn↔2π(i)nδ(n)(ω)(17)$$\begin{array}{}\text{(17)}& t^n\leftrightarrow 2\pi (i{)}^n{\delta }^{(n)}(\omega )\end{array}$$

(18)1t↔−iπsgn(ω)" role="presentation">1t↔−iπsgn(ω)(18)$$\begin{array}{}\text{(18)}& \frac{1}{t}\leftrightarrow -i\pi sgn(\omega )\end{array}$$

(19)δT(t)↔ΩδΩ(ω)" role="presentation">δT(t)↔ΩδΩ(ω)(19)$$\begin{array}{}\text{(19)}& {\delta }_T(t)\leftrightarrow \mathrm{\Omega }{\delta }_{\mathrm{\Omega }}(\omega )\end{array}$$

性质

时域微分

(20)f(n)(t)↔(iω)nF(ω)" role="presentation">f(n)(t)↔(iω)nF(ω)(20)$$\begin{array}{}\text{(20)}& f^{(n)}(t)\leftrightarrow (i\omega {)}^{n}F(\omega )\end{array}$$

时域积分

(21)∫−∞tf(τ)dτ↔πF(0)δ(ω)+F(ω)iω" role="presentation">∫t−∞f(τ)dτ↔πF(0)δ(ω)+F(ω)iω(21)$$\begin{array}{}\text{(21)}& {\int }_{-\mathrm{\infty }}^{t}f(\tau )d\tau \leftrightarrow \pi F(0)\delta (\omega )+\frac{F(\omega )}{i\omega }\end{array}$$

频域微分

(22)(−it)nf(t)↔F(n)(ω)" role="presentation">(−it)nf(t)↔F(n)(ω)(22)$$\begin{array}{}\text{(22)}& (-it{)}^{n}f(t)\leftrightarrow F^{(n)}(\omega )\end{array}$$

频域积分

(23)πf(0)δ(t)+f(t)−it↔∫−∞ωF(ν)dν" role="presentation">πf(0)δ(t)+f(t)−it↔∫ω−∞F(ν)dν(23)$$\begin{array}{}\text{(23)}& \pi f(0)\delta (t)+\frac{f(t)}{-it}\leftrightarrow {\int }_{-\mathrm{\infty }}^{\omega }F(\nu )d\nu \end{array}$$

对称性

(24)F(t)↔2πf(−ω)" role="presentation">F(t)↔2πf(−ω)(24)$$\begin{array}{}\text{(24)}& F(t)\leftrightarrow 2\pi f(-\omega )\end{array}$$

尺度变换

(25)f(at)↔1|a|F(ωa)" role="presentation">f(at)↔1|a|F(ωa)(25)$$\begin{array}{}\text{(25)}& f(at)\leftrightarrow \frac{1}{|a|}F\left(\frac{\omega }{a}\right)\end{array}$$

时移

(26)f(t±t0)↔e±iωt0F(ω)" role="presentation">f(t±t0)↔e±iωt0F(ω)(26)$$\begin{array}{}\text{(26)}& f(t\pm t_0)\leftrightarrow e^{\pm i\omega t_0}F(\omega )\end{array}$$

频移

(27)f(t)e±iω0t↔F(ω∓ω0)" role="presentation">f(t)e±iω0t↔F(ω∓ω0)(27)$$\begin{array}{}\text{(27)}& f(t)e^{\pm i{\omega }_{0}t}\leftrightarrow F(\omega \mp {\omega }_0)\end{array}$$

卷积的微分性质

设f(t)=g(t)∗h(t)" role="presentation" style="position: relative;">f(t)=g(t)∗h(t)$f(t)=g(t)\ast h(t)$，则f′(t)=g′(t)∗h(t)=g(t)∗h′(t)" role="presentation" style="position: relative;">f′(t)=g′(t)∗h(t)=g(t)∗h′(t)$f^{\prime }(t)=g^{\prime }(t)\ast h(t)=g(t)\ast h^{\prime }(t)$

卷积定理

时域f(t)=g(t)∗h(t)" role="presentation" style="position: relative;">f(t)=g(t)∗h(t)$f(t)=g(t)\ast h(t)$，频域有F(ω)=G(ω)H(ω)" role="presentation" style="position: relative;">F(ω)=G(ω)H(ω)$F(\omega )=G(\omega )H(\omega )$

时域f(t)=g(t)h(t)" role="presentation" style="position: relative;">f(t)=g(t)h(t)$f(t)=g(t)h(t)$，频域有F(ω)=12πG(ω)∗H(ω)" role="presentation" style="position: relative;">F(ω)=12πG(ω)∗H(ω)${\displaystyle F(\omega )=\frac{1}{2\pi }G(\omega )\ast H(\omega )}$

周期函数fT(t)" role="presentation" style="position: relative;">fT(t)$f_T(t)$傅里叶变换

由指数形式的傅里叶级数，两边取傅里叶变换，所以周期函数的傅里叶变换时受到2πFn" role="presentation" style="position: relative;">2πFn$2\pi F_n$调制的梳状脉冲(T" role="presentation" style="position: relative;">T$T$代表周期，Ω=2πT" role="presentation" style="position: relative;">Ω=2πT$\mathrm{\Omega }=\frac{2\pi }{T}$)

(28)fT(t)↔2π∑n=−∞∞Fnδ(ω−nΩ)" role="presentation">fT(t)↔2π∑n=−∞∞Fnδ(ω−nΩ)(28)$$\begin{array}{}\text{(28)}& f_T(t)\leftrightarrow 2\pi \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{F}_n\delta (\omega -n\mathrm{\Omega })\end{array}$$

拉普拉斯变换

因果信号f(t)" role="presentation" style="position: relative;">f(t)$f(t)$可以显式地写为f(t)ε(t)" role="presentation" style="position: relative;">f(t)ε(t)$f(t)\epsilon (t)$，一个因果信号及其单边拉普拉斯变换是一一对应的。每个非因果信号都对应唯一一个双边拉普拉斯变换，但是一个双边拉普拉斯变换在不同收敛域条件下，可以对应不同的非因果信号。

常见的单边拉普拉斯变换

(29)gτ(t−τ2)ε(t)" role="presentation">gτ(t−τ2)ε(t)(29)$$\begin{array}{}\text{(29)}& g_{\tau }(t-\frac{\tau }{2})\epsilon (t)\end{array}$$

(30)∑n=0∞δ(t−nT)↔11−e−Ts" role="presentation">∑n=0∞δ(t−nT)↔11−e−Ts(30)$$\begin{array}{}\text{(30)}& \sum _{n=0}^{\mathrm{\infty }}\delta (t-nT)\leftrightarrow \frac{1}{1-e^{-Ts}}\end{array}$$

(31)ε(t)↔1s" role="presentation">ε(t)↔1s(31)$$\begin{array}{}\text{(31)}& \epsilon (t)\leftrightarrow \frac{1}{s}\end{array}$$

(32)tε(t)↔1s2" role="presentation">tε(t)↔1s2(32)$$\begin{array}{}\text{(32)}& t\epsilon (t)\leftrightarrow \frac{1}{s^2}\end{array}$$

(33)e−atε(t)↔1s+a" role="presentation">e−atε(t)↔1s+a(33)$$\begin{array}{}\text{(33)}& e^{-at}\epsilon (t)\leftrightarrow \frac{1}{s+a}\end{array}$$

(34)sin(βt)ε(t)↔βs2+β2" role="presentation">sin(βt)ε(t)↔βs2+β2(34)$$\begin{array}{}\text{(34)}& sin(\beta t)\epsilon (t)\leftrightarrow \frac{\beta }{s^2+{\beta }^2}\end{array}$$

(35)cos(βt)ε(t)↔ss2+β2" role="presentation">cos(βt)ε(t)↔ss2+β2(35)$$\begin{array}{}\text{(35)}& cos(\beta t)\epsilon (t)\leftrightarrow \frac{s}{s^2+{\beta }^2}\end{array}$$

(36)sinh(βt)ε(t)↔βs2−β2" role="presentation">sinh(βt)ε(t)↔βs2−β2(36)$$\begin{array}{}\text{(36)}& sinh(\beta t)\epsilon (t)\leftrightarrow \frac{\beta }{s^2-{\beta }^2}\end{array}$$

(37)cosh(βt)ε(t)↔ss2−β2" role="presentation">cosh(βt)ε(t)↔ss2−β2(37)$$\begin{array}{}\text{(37)}& cosh(\beta t)\epsilon (t)\leftrightarrow \frac{s}{s^2-{\beta }^2}\end{array}$$

性质

（单边）尺度变换

(38)f(at)↔1aF(sa),a>0" role="presentation">f(at)↔1aF(sa),a>0(38)$$\begin{array}{}\text{(38)}& f(at)\leftrightarrow \frac{1}{a}F(\frac{s}{a}),a>0\end{array}$$

（双边）尺度变换

(39)f(at)⟷b1|a|Fb(sa)" role="presentation">f(at)⟷b1|a|Fb(sa)(39)$$\begin{array}{}\text{(39)}& f(at)\stackrel{b}{⟷}\frac{1}{|a|}{F}_b(\frac{s}{a})\end{array}$$

（单边）时移

(40)f(t−t0)ε(t−t0)↔e−st0F(s)" role="presentation">f(t−t0)ε(t−t0)↔e−st0F(s)(40)$$\begin{array}{}\text{(40)}& f(t-t_0)\epsilon (t-t_0)\leftrightarrow e^{-s{t}_0}F(s)\end{array}$$

（双边）时移

(41)f(t−t0)⟷be−st0Fb(s)" role="presentation">f(t−t0)⟷be−st0Fb(s)(41)$$\begin{array}{}\text{(41)}& f(t-t_0)\stackrel{b}{⟷}{e}^{-s{t}_0}{F}_b(s)\end{array}$$

（单边，双边）复频移

(42)es0tf(t)ε(t)↔F(s−s0)" role="presentation">es0tf(t)ε(t)↔F(s−s0)(42)$$\begin{array}{}\text{(42)}& e^{s_{0}t}f(t)\epsilon (t)\leftrightarrow F(s-s_0)\end{array}$$

（单边）时域微分，可递推

(43)f′(t)↔sF(s)−f(0−)" role="presentation">f′(t)↔sF(s)−f(0−)(43)$$\begin{array}{}\text{(43)}& f^{\prime }(t)\leftrightarrow sF(s)-f(0_{-})\end{array}$$

(44)f″(t)↔s2F(s)−sf(0−)−f′(0−)" role="presentation">f′′(t)↔s2F(s)−sf(0−)−f′(0−)(44)$$\begin{array}{}\text{(44)}& f^{″}(t)\leftrightarrow s^{2}F(s)-sf(0_{-})-f^{\prime }(0_{-})\end{array}$$

（双边）时域微分，可递推

(45)f′(t)⟷bsFb(s)" role="presentation">f′(t)⟷bsFb(s)(45)$$\begin{array}{}\text{(45)}& f^{\prime }(t)\stackrel{b}{⟷}s{F}_b(s)\end{array}$$

（单边）时域积分，可递推

(46)∫0−tf(τ)dτ↔1sF(s)" role="presentation">∫t0−f(τ)dτ↔1sF(s)(46)$$\begin{array}{}\text{(46)}& {\int }_{0_{-}}^{t}f(\tau )d\tau \leftrightarrow \frac{1}{s}F(s)\end{array}$$

(47)∫−∞tf(τ)dτ↔1sF(s)+1s∫−∞0−f(τ)dτ" role="presentation">∫t−∞f(τ)dτ↔1sF(s)+1s∫0−−∞f(τ)dτ(47)$$\begin{array}{}\text{(47)}& {\int }_{-\mathrm{\infty }}^{t}f(\tau )d\tau \leftrightarrow \frac{1}{s}F(s)+\frac{1}{s}{\int }_{-\mathrm{\infty }}^{0_{-}}f(\tau )d\tau \end{array}$$

（双边）时域积分，可递推

(48)&#x222B;0tf(&#x03C4;)d&#x03C4;&#x27F7;b1sFb(s),&#x03B1;&lt;&#x03C3;&lt;max(&#x03B2;,0)" role="presentation">∫t0f(τ)dτ⟷b1sFb(s),α<σ<max(β,0)(48)$$\begin{array}{}\text{(48)}& {\int }_0^{t}f(\tau )d\tau \stackrel{b}{⟷}\frac{1}{s}{F}_b(s),\alpha <\sigma <max(\beta ,0)\end{array}$$

(49)&#x222B;&#x2212;&#x221E;tf(&#x03C4;)d&#x03C4;&#x27F7;b1sFb(s),max(&#x03B1;,0)&lt;&#x03C3;&lt;&#x03B2;" role="presentation">∫t−∞f(τ)dτ⟷b1sFb(s),max(α,0)<σ<β(49)$$\begin{array}{}\text{(49)}& {\int }_{-\mathrm{\infty }}^{t}f(\tau )d\tau \stackrel{b}{⟷}\frac{1}{s}{F}_b(s),max(\alpha ,0)<\sigma <\beta \end{array}$$

（单边，双边）s" role="presentation" style="position: relative;">s$s$域微分，可递推

(50)(&#x2212;t)f(t)&#x2194;dF(s)ds" role="presentation">(−t)f(t)↔dF(s)ds(50)$$\begin{array}{}\text{(50)}& (-t)f(t)\leftrightarrow \frac{dF(s)}{ds}\end{array}$$

（单边）s" role="presentation" style="position: relative;">s$s$域积分，可递推

(51)f(t)t&#x2194;&#x222B;s&#x221E;F(&#x03BD;)d&#x03BD;" role="presentation">f(t)t↔∫∞sF(ν)dν(51)$$\begin{array}{}\text{(51)}& \frac{f(t)}{t}\leftrightarrow {\int }_s^{\mathrm{\infty }}F(\nu )d\nu \end{array}$$

对于傅里叶变换和拉普拉斯的积分性质，成立的条件是积分确实收敛，否则不成立。对于拉普拉斯变换的s" role="presentation" style="position: relative;">s$s$域积分性质而言，积分变量&#x03BD;" role="presentation" style="position: relative;">ν$\nu$仅具有形式上的意义，可以不认为这是一个具有特定路径的复积分.

（单边，双边）时域卷积

(52)f1(t)&#x2217;f2(t)&#x2194;F1(s)F2(s)" role="presentation">f1(t)∗f2(t)↔F1(s)F2(s)(52)$$\begin{array}{}\text{(52)}& f_1(t)\ast f_2(t)\leftrightarrow F_1(s)F_2(s)\end{array}$$

（单边）拉普拉斯逆变换

可以将一个有理分式拆成多项式+真分式的形式F(s)=P(s)+B(s)A(s)" role="presentation" style="position: relative;">F(s)=P(s)+B(s)A(s)${\displaystyle F(s)=P(s)+\frac{B(s)}{A(s)}}$，其中多项式部分的拉普拉斯逆变换是&#x03B4;" role="presentation" style="position: relative;">δ$\delta$函数的各阶导数，真分式部分可以作部分分式分解，再分别做逆变换。先求分母A(s)" role="presentation" style="position: relative;">A(s)$A(s)$的根:

（1）若s0" role="presentation" style="position: relative;">s0$s_0$为一个单实根，则在部分分式中加入Ks&#x2212;s0" role="presentation" style="position: relative;">Ks−s0${\displaystyle \frac{K}{s-s_0}}$这一项，其中

(53)K=[(s&#x2212;s0)B(s)A(s)]|s=s0" role="presentation">K=[(s−s0)B(s)A(s)]∣∣∣s=s0(53)$$\begin{array}{}\text{(53)}& {\displaystyle K=\left[(s-s_0)\frac{B(s)}{A(s)}\right]{|}_{s=s_0}}\end{array}$$

（2）若s0,s0&#x2217;" role="presentation" style="position: relative;">s0,s∗0$s_0,s_0^{\ast }$为一个单重共轭复根，则在部分分式中加入K1s&#x2212;s0+K2s&#x2212;s0&#x2217;" role="presentation" style="position: relative;">K1s−s0+K2s−s∗0${\displaystyle \frac{K_1}{s-s_0}+\frac{K_2}{s-s_0^{\ast }}}$，系数K" role="presentation" style="position: relative;">K$K$的计算方法同上，其中K1" role="presentation" style="position: relative;">K1$K_1$和K2" role="presentation" style="position: relative;">K2$K_2$必然是共轭关系

（3）若s0" role="presentation" style="position: relative;">s0$s_0$为一个r重实根，则在部分分式中加入

(54)K1(s&#x2212;s0)r+K2(s&#x2212;s0)r&#x2212;1+&#x22EF;+Krs&#x2212;s0" role="presentation">K1(s−s0)r+K2(s−s0)r−1+⋯+Krs−s0(54)$$\begin{array}{}\text{(54)}& {\displaystyle \frac{K_1}{(s-s_0{)}^r}+\frac{K_2}{(s-s_0{)}^{r-1}}+\cdots +\frac{K_r}{s-s_0}}\end{array}$$

其中

(55)Ki={1(i&#x2212;1)!di&#x2212;1dsi&#x2212;1[(s&#x2212;s0)rB(s)A(s)]}|s=s0" role="presentation">Ki={1(i−1)!di−1dsi−1[(s−s0)rB(s)A(s)]}∣∣∣s=s0(55)$$\begin{array}{}\text{(55)}& {\displaystyle K_i=\left\{\frac{1}{(i-1)!}\frac{d^{i-1}}{d{s}^{i-1}}\left[(s-s_0{)}^r\frac{B(s)}{A(s)}\right]\right\}{|}_{s=s_0}}\end{array}$$

因果信号的单边拉普拉斯变换和傅里叶变换的关系

收敛域&#x03C3;&gt;&#x03C3;0" role="presentation" style="position: relative;">σ>σ0$\sigma >{\sigma }_0$，若有&#x03C3;0&lt;0" role="presentation" style="position: relative;">σ0<0${\sigma }_0<0$，由于因果信号的单边拉普拉斯变换的收敛域在复平面上是一条竖线的右半开平面，此时直接令拉普拉斯变换中的s" role="presentation" style="position: relative;">s$s$为i&#x03C9;" role="presentation" style="position: relative;">iω$i\omega$即可得到傅里叶变换，若有&#x03C3;0&gt;0" role="presentation" style="position: relative;">σ0>0${\sigma }_0>0$，则此时收敛域不包括虚轴，傅里叶变换不存在，若有&#x03C3;0=0" role="presentation" style="position: relative;">σ0=0${\sigma }_0=0$，则此时将拉普拉斯变换作部分分式分解，必然有分式的极点位于虚轴上，此时可将其余部分作代换s&#x2192;i&#x03C9;" role="presentation" style="position: relative;">s→iω$s\to i\omega$，而将极点位于虚轴上的部分，做逆变换求得时域形式，再作傅里叶变换

反因果信号的双边拉普拉斯变换

现有反因果信号f(t)&#x03B5;(&#x2212;t)" role="presentation" style="position: relative;">f(t)ε(−t)$f(t)\epsilon (-t)$，则f(&#x2212;t)&#x03B5;(t)" role="presentation" style="position: relative;">f(−t)ε(t)$f(-t)\epsilon (t)$为一个因果信号。求f(&#x2212;t)&#x03B5;(t)" role="presentation" style="position: relative;">f(−t)ε(t)$f(-t)\epsilon (t)$的单边拉普拉斯变换F(s)" role="presentation" style="position: relative;">F(s)$F(s)$且设收敛域&#x03C3;&gt;&#x03C3;2" role="presentation" style="position: relative;">σ>σ2$\sigma >{\sigma }_2$.则有f(t)&#x03B5;(&#x2212;t)&#x27F7;bF(&#x2212;s)" role="presentation" style="position: relative;">f(t)ε(−t)⟷bF(−s)$f(t)\epsilon (-t)\stackrel{b}{⟷}F(-s)$，收敛域&#x03C3;&lt;&#x2212;&#x03C3;2" role="presentation" style="position: relative;">σ<−σ2$\sigma <-{\sigma }_2$

反因果信号的双边拉普拉斯逆变换

反因果信号及其双边拉普拉斯变换是一一对应的。现有Fb(s)" role="presentation" style="position: relative;">Fb(s)$F_b(s)$为某反因果信号的双边拉普拉斯变换，则先求出Fb(&#x2212;s)" role="presentation" style="position: relative;">Fb(−s)$F_b(-s)$的单边拉普拉斯逆变换f(t)" role="presentation" style="position: relative;">f(t)$f(t)$. 有f(&#x2212;t)&#x03B5;(&#x2212;t)&#x27F7;bFb(s)" role="presentation" style="position: relative;">f(−t)ε(−t)⟷bFb(s)$f(-t)\epsilon (-t)\stackrel{b}{⟷}{F}_b(s)$

一般非因果信号的双边拉普拉斯正/逆变换

正变换：将f(t)" role="presentation" style="position: relative;">f(t)$f(t)$分成因果信号与反因果信号的和，分别作双边变换，需要注意的是，收敛域为因果信号与反因果信号各自的收敛域的交集.

逆变换：将Fb(s)" role="presentation" style="position: relative;">Fb(s)$F_b(s)$进行部分分式分解，根据给定的收敛域区分哪些分式是因果的，哪些是反因果的，分别对它们进行双边拉普拉斯逆变换.

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