信号公式汇总之傅里叶变换

傅里叶级数
f(t)=a0+∑n=1∞[ancos(nω1t)+bnsin(nω1t)]f(t)=a_0+\sum_{n=1}^{\infty}{[a_ncos(n\omega_1t)+b_nsin(n\omega_1t)]}f(t)=a0+n=1∑∞[ancos(nω1t)+bnsin(nω1t)]

其中：a0=1T1∫t0t0+T1f(t)dta_0=\frac{1}{T_1}\int_{t_0}^{t_0+T_1}f(t)dta0=T11∫t0t0+T1f(t)dt

an=2T1∫t0t0+T1f(t)⋅cos(nω1t)dta_n=\frac{2}{T_1}\int_{t_0}^{t_0+T_1}f(t)\cdot cos(n\omega_1t)dtan=T12∫t0t0+T1f(t)⋅cos(nω1t)dt

bn=2T1∫t0t0+T1f(t)⋅sin(nω1t)dtb_n=\frac{2}{T_1}\int_{t_0}^{t_0+T_1}f(t)\cdot sin(n\omega_1t)dtbn=T12∫t0t0+T1f(t)⋅sin(nω1t)dt

傅里叶系数推导

合并同频项：f(t)=A0+∑n=1∞Ansin(nω1t+φn)f(t)=A_0+\sum_{n=1}^{\infty}{A_nsin(n\omega_1t+\varphi _n)}f(t)=A0+∑n=1∞Ansin(nω1t+φn)

其中：A0=a0,An=an2+bn2,φn=arctananbnA_0=a_0 , A_n=\sqrt{ a_n^2+b_n^2 },\varphi _n=arctan\frac{a_n}{b_n}A0=a0,An=an2+bn2,φn=arctanbnan

指数形式：f(t)=∑n=−∞∞Fn⋅ejnω1tf(t)=\sum_{n=-\infty}^{\infty}{F_n\cdot e^{jn\omega_1t}}f(t)=∑n=−∞∞Fn⋅ejnω1t

其中：Fn=1T1∫t0t0+T1f(t)e−jnω1tdt=F_n=\frac{1}{T_1}\int_{t_0}^{t_0+T_1} f(t)e^{-jn\omega_1t}dt=Fn=T11∫t0t0+T1f(t)e−jnω1tdt=

12(an−jbn)=12an2+bn2ejφn,φn=arctan(−bnan)\frac{1}{2}(a_n-jb_n)=\frac{1}{2}\sqrt{a_n^2+b_n^2}e^{j\varphi_n},\varphi_n=arctan(-\frac{b_n}{a_n})21(an−jbn)=21an2+bn2ejφn,φn=arctan(−anbn)

帕塞瓦尔定理：P=∑n=−∞∞∣Fn∣2P=\sum_{n=-\infty}^{\infty}{|F_n|^2}P=∑n=−∞∞∣Fn∣2

傅里叶变换
正变换：F(jω)=F[f(t)]=∫−∞∞f(t)e−jωtdtF(j\omega)=\mathscr{F}[f(t)]=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dtF(jω)=F[f(t)]=∫−∞∞f(t)e−jωtdt

逆变换：f(t)=F−1[F(jω)]=12π∫−∞∞F(jω)ejωtdωf(t)=\mathscr{F}^{-1}[F(j\omega)]=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t}d\omegaf(t)=F−1[F(jω)]=2π1∫−∞∞F(jω)ejωtdω

基本信号的傅里叶变换：

信号类型	傅里叶变换
单边指数：	e−αtε(t)←→e^{-\alpha t}\varepsilon(t)\leftarrow\rightarrowe−αtε(t)←→ 1α+jω\frac{1}{\alpha+j\omega}α+jω1
双边指数：	e−αt(绝对值)e^{-\alpha t(绝对值)}e−αt(绝对值)←→\leftarrow\rightarrow←→ 2αα2+ω2\frac{2\alpha}{\alpha^2+\omega^2}α2+ω22α
门信号：	gτ(t)g_{\tau}(t)gτ(t) ←→\leftarrow\rightarrow←→ τSa(ωτ2)\tau Sa(\frac{\omega \tau }{2})τSa(2ωτ)
符号函数：	sgn(t)sgn(t)sgn(t) ←→\leftarrow\rightarrow←→ 2jω\frac{2}{j\omega}jω2
冲激函数：	δ(t)\delta (t)δ(t) ←→\leftarrow\rightarrow←→ 111
常数1：	111 ←→\leftarrow\rightarrow←→ 2πδ(ω)2\pi\delta(\omega)2πδ(ω)
冲激偶：	δ′(t)\delta{\prime}(t)δ′(t) ←→\leftarrow\rightarrow←→ jω{j\omega}jω
阶跃函数：	ε(t)\varepsilon(t)ε(t) ←→\leftarrow\rightarrow←→ πδ(ω)+1jω{\pi\delta(\omega)+\frac{1}{j\omega}}πδ(ω)+jω1

性质：

线性： a1f1(t)+a2f2(t)a_1f_1(t)+a_2f_2(t)a1f1(t)+a2f2(t) ←→\leftarrow\rightarrow←→ a1F1(ω)+a2F2(ω)a_1F_1(\omega)+a_2F_2(\omega)a1F1(ω)+a2F2(ω)

对称性： F(t)F(t)F(t) ←→\leftarrow\rightarrow←→ 2πf(−ω)2\pi f(-\omega)2πf(−ω)

尺度变换： f(at)f(at)f(at) ←→\leftarrow\rightarrow←→ 1∣a∣F(jωa)\frac{1}{|a|}F(j\frac{\omega}{a})∣a∣1F(jaω)

虚实特性： F(ω)=R(ω)+jX(ω),R(ω)偶,X(ω)奇F(\omega)=R(\omega)+jX(\omega),R(\omega)偶,X(\omega)奇F(ω)=R(ω)+jX(ω),R(ω)偶,X(ω)奇
f(t)实偶，F(ω)实偶；f(t)实奇，F(ω)虚奇f(t)实偶，F(\omega)实偶；f(t)实奇，F(\omega)虚奇f(t)实偶，F(ω)实偶；f(t)实奇，F(ω)虚奇

位移：时移：f(t−t0)f(t-t_0)f(t−t0)←→\leftarrow\rightarrow←→ F(jω)e−jωt0=>F(j\omega)e^{-j\omega t_0}=>F(jω)e−jωt0=> δ(t−t0)\delta(t-t_0)δ(t−t0)←→\leftarrow\rightarrow←→ e−jωt0e^{-j\omega t_0}e−jωt0
频移：f(t)ejω0tf(t)e^{j\omega_0 t}f(t)ejω0t ←→\leftarrow\rightarrow←→ F[j(ω−ω0)]=>F[j(\omega-\omega _0)] =>F[j(ω−ω0)]=> ejω0te^{j\omega_0 t}ejω0t ←→\leftarrow\rightarrow←→ 2πδ(ω−ω0)2\pi\delta(\omega-\omega _0)2πδ(ω−ω0)

==>{cosω0t=12(ejωt+e−jωt)sinω0t=12j(ejωt−ejωt)=>{f(t)cos(ω0t)←→12[F(ω−ω0)+F(ω+ω0)]f(t)sin(ω0t)←→12j[F(ω−ω0)+F(ω+ω0)]\begin{cases} {}cos\omega_0 t =\frac{1}{2}(e^{j\omega t}+e^{-j\omega t})\\{sin\omega_0 t =\frac{1}{2j}(e^{j\omega t}-e^{j\omega t})} \end{cases}=>\begin{cases} {f(t)cos(\omega_0 t)\leftarrow\rightarrow\frac{1}{2}[F(\omega-\omega_0)+F(\omega+\omega_0)]}\\ {f(t)sin(\omega_0 t) \leftarrow\rightarrow \frac{1}{2j}[F(\omega-\omega_0)+F(\omega+\omega_0)]} \end{cases}{cosω0t=21(ejωt+e−jωt)sinω0t=2j1(ejωt−ejωt)=>{f(t)cos(ω0t)←→21[F(ω−ω0)+F(ω+ω0)]f(t)sin(ω0t)←→2j1[F(ω−ω0)+F(ω+ω0)]

==>{cosω0t←→π[δ(ω−ω0)+δ(ω+ω0))]sinω0t←→πj[δ(ω−ω0)−δ(ω+ω0))]\begin{cases} {cos\omega_0t \leftarrow\rightarrow \pi [\delta(\omega-\omega_0)+\delta(\omega+\omega_0))] }\\ {sin\omega_0t \leftarrow\rightarrow \frac{\pi}{j} [\delta(\omega-\omega_0)-\delta(\omega+\omega_0))]}\end{cases}{cosω0t←→π[δ(ω−ω0)+δ(ω+ω0))]sinω0t←→jπ[δ(ω−ω0)−δ(ω+ω0))]

调制定理：若信号f(t)乘以cosω0t或sinω0t，等效于f(t)的频谱一分为二，沿数轴向左或向右各平移ω0调制定理：若信号f(t)乘以cos\omega_0t或sin\omega_0t，等效于f(t)的频谱一分为二，沿数轴向左或向右各平移\omega_0调制定理：若信号f(t)乘以cosω0t或sinω0t，等效于f(t)的频谱一分为二，沿数轴向左或向右各平移ω0

推导：

f(t)⋅Cos(ω0t)f(t)\cdot Cos(\omega_0t)f(t)⋅Cos(ω0t)
傅里叶变换：
12πF(jω)∗π[δ(ω+ω0)+δ(ω−ω0))]\frac{1} {2\pi}F(j\omega)*\pi[\delta(\omega+\omega_0)+\delta(\omega-\omega_0))]2π1F(jω)∗π[δ(ω+ω0)+δ(ω−ω0))]
=12F(jω)∗δ(ω+ω0)+12F(jω)∗δ(ω−ω0)\frac{1}{2}F(j\omega)*\delta(\omega+\omega_0)+\frac{1}{2}F(j\omega)*\delta(\omega-\omega_0)21F(jω)∗δ(ω+ω0)+21F(jω)∗δ(ω−ω0)
由冲激函数性质 f(t)∗δ(t−t0)=f(t−t0)f(t)*\delta(t-t_0)=f(t-t_0)f(t)∗δ(t−t0)=f(t−t0)
=12F(j(ω+ω0))+12F(j(ω−ω0)\frac{1}{2}F(j(\omega+\omega_0))+\frac{1}{2}F(j(\omega-\omega_0)21F(j(ω+ω0))+21F(j(ω−ω0)

卷积：时域卷积：f1(t)∗f2(t)←→F1(ω)⋅F2(ω)f_1(t)*f_2(t) \leftarrow\rightarrow F_1(\omega)\cdot F_2(\omega)f1(t)∗f2(t)←→F1(ω)⋅F2(ω)

频域卷积：f1(t)⋅f2(t)←→12πF1(ω)∗F2(ω)f_1(t)\cdot f_2(t) \leftarrow\rightarrow \frac{1}{2\pi}F_1(\omega)*F_2(\omega)f1(t)⋅f2(t)←→2π1F1(ω)∗F2(ω)

微积分：时域：{微分:df(t)dt←→jωF(jω)积分:∫−∞tf(τ)dτ←→1jωF(jω)+πF(0)δ(ikω)\begin{cases} {微分:\frac{df(t)}{dt} \leftarrow\rightarrow j\omega F(j\omega )}\\{积分:\int_{-\infty}^{t} f(\tau)d\tau \leftarrow\rightarrow \frac{1}{j\omega} F(j\omega)+\pi F(0)\delta(ik\omega)} \end{cases}{微分:dtdf(t)←→jωF(jω)积分:∫−∞tf(τ)dτ←→jω1F(jω)+πF(0)δ(ikω)

频域：{微分:−jtf(t)←→dF(jω)dω积分:f(t)−jt←→∫−∞ωF(x)dx\begin{cases} {微分:-jtf(t) \leftarrow\rightarrow \frac{dF(j\omega)}{d\omega}}\\ {积分:\frac{f(t)}{-jt} \leftarrow\rightarrow \int_{-\infty}^{\omega}F(x)dx} \end{cases}{微分:−jtf(t)←→dωdF(jω)积分:−jtf(t)←→∫−∞ωF(x)dx

欧拉公式:
ejθ=cosθ+jsinθe^{j\theta}=cos\theta+jsin\thetaejθ=cosθ+jsinθ

==>{ejωt=cosωt+jsinωte−jωt=cosωt−jsinωt==>\begin{cases} {e^{j\omega t}=cos\omega t+jsin\omega t}\\{e^{-j\omega t}=cos\omega t-jsin\omega t} \end{cases}==>{ejωt=cosωt+jsinωte−jωt=cosωt−jsinωt

==>{cos(nωt)=ejnωt+e−jnωt2sin(nωt)=ejnωt−e−jnωt2j==>\begin{cases} {cos(n\omega t)=\frac{e^{jn\omega t}+e^{-jn\omega t}}{2}}\\ {sin(n\omega t)=\frac{e^{jn\omega t}-e^{-jn\omega t}}{2j}} \end{cases}==>{cos(nωt)=2ejnωt+e−jnωtsin(nωt)=2jejnωt−e−jnωt

阶跃信号和冲激信号:

冲激函数：
   加权特性：f(t)δ(t)=f(0)δ(t),f(t)δ(t−t0)=f(t0)δ(t−t0)f(t)\delta(t)=f(0)\delta(t),f(t)\delta(t-t_0)=f(t_0)\delta(t-t_0)f(t)δ(t)=f(0)δ(t),f(t)δ(t−t0)=f(t0)δ(t−t0)
   抽样特性：∫−∞∞f(t)δ(t)dt=∫−∞∞f(0)δ(t)dt=f(0)\int_{-\infty}^{\infty}f(t)\delta(t)dt=\int_{-\infty}^{\infty}f(0)\delta(t)dt=f(0)∫−∞∞f(t)δ(t)dt=∫−∞∞f(0)δ(t)dt=f(0)
          ∫−∞∞f(t)δ(t−t0)dt=f(t0)\int_{-\infty}^{\infty}f(t)\delta(t-t_0)dt=f(t_0)∫−∞∞f(t)δ(t−t0)dt=f(t0)
   尺度变换：δ(at)=1∣a∣δ(t),δ(at−t0)=1∣a∣δ(t−t0a)\delta(at)=\frac{1}{|a|}\delta(t),\delta(at-t_0)=\frac{1}{|a|}\delta(t-\frac{t_0}{a})δ(at)=∣a∣1δ(t),δ(at−t0)=∣a∣1δ(t−at0)

冲激偶：
   抽样特性：∫−∞∞f(t)δ′(t)dt=f(t)δ(t)∣−∞∞−∫−∞∞δ(t)f′(t)dt=\int_{-\infty}^{\infty}f(t)\delta{\prime}(t)dt=\left. f(t)\delta(t) \right| _{-\infty}^{\infty}-\int_{-\infty}^{\infty}\delta(t) f\prime(t)dt=∫−∞∞f(t)δ′(t)dt=f(t)δ(t)∣−∞∞−∫−∞∞δ(t)f′(t)dt=−∫−∞∞δ(t)f′(0)dt=−f′(0)-\int_{-\infty}^{\infty}\delta(t)f{\prime}(0)dt=-f{\prime}(0)−∫−∞∞δ(t)f′(0)dt=−f′(0)
   加权特性：f(t)δ′(t)=f(0)δ′(t)−f′(0)δ(t)f(t)\delta{\prime}(t)=f(0)\delta{\prime}(t)-f{\prime}(0)\delta(t)f(t)δ′(t)=f(0)δ′(t)−f′(0)δ(t)
             f(t)δ′(t−t0)=f(t0)δ′(t−t0)−f′(t0)δ(t−t0)f(t)\delta{\prime}(t-t_0)=f(t_0)\delta{\prime}(t-t_0)-f{\prime}(t_0)\delta(t-t_0)f(t)δ′(t−t0)=f(t0)δ′(t−t0)−f′(t0)δ(t−t0)

卷积运算:f(t)=f1(t)∗f2(t)=∫−∞∞f1(τ)f2(t−τ)dτf(t)=f_1(t)*f_2(t)=\int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tauf(t)=f1(t)∗f2(t)=∫−∞∞f1(τ)f2(t−τ)dτ

性质：微分：f(t)=f1′(t)∗f2(t)=f1(t)∗f2′(t)f(t)=f_1{\prime}(t)*f_2(t)=f_1(t)*f_2{\prime}(t)f(t)=f1′(t)∗f2(t)=f1(t)∗f2′(t)
      积分：f−1(t)=f1−1(t)∗f2(t)=f1(t)∗f2−1(t)f^{-1}(t)=f_1^{-1}(t)*f_2(t)=f_1(t)*f_2^{-1}(t)f−1(t)=f1−1(t)∗f2(t)=f1(t)∗f2−1(t)
      微积分：f(t)=f1−1(t)∗f2′(t)=f1(t)∗f2(t)f(t)=f_1^{-1}(t)*f_2{\prime}(t)=f_1(t)*f_2(t)f(t)=f1−1(t)∗f2′(t)=f1(t)∗f2(t)
      =======>f(i)(t)=f1(j)(t)∗f2(i−j)(t)f^{(i)}(t)=f_1^{(j)}(t)*f_2^{(i-j)}(t)f(i)(t)=f1(j)(t)∗f2(i−j)(t)
      交换律：f1(t)∗f2(t)=f2(t)∗f1(t)f_1(t)*f_2(t)=f_2(t)*f_1(t)f1(t)∗f2(t)=f2(t)∗f1(t)
      分配律：f1(t)∗[f2(t)+f3(t)]=f1(t)∗f2(t)+f1(t)∗f3(t)f_1(t)\ast [f_2(t)+f_3(t)]=f_1(t)\ast f_2(t)+f_1(t)\ast f_3(t)f1(t)∗[f2(t)+f3(t)]=f1(t)∗f2(t)+f1(t)∗f3(t)
       ==>==>==>并联系统h(t)=h1(t)+h2(t)并联系统h(t)=h_1(t)+h_2(t)并联系统h(t)=h1(t)+h2(t)
      结合律：[f1(t)∗f2(t)]∗f3(t)=f1(t)∗[f2(t)∗f3(t)][f_1(t)\ast f_2(t)]\ast f_3(t)=f_1(t)\ast [f_2(t)\ast f_3(t)][f1(t)∗f2(t)]∗f3(t)=f1(t)∗[f2(t)∗f3(t)]
       ==>==>==>串联系统h(t)=h1(t)∗h2(t)串联系统h(t)=h_1(t)*h_2(t)串联系统h(t)=h1(t)∗h2(t)

与冲激函数、冲激偶、阶跃函数
      冲激函数δ(t)\delta(t)δ(t)：f(t)∗δ(t)=f(t)f(t)*\delta(t)=f(t)f(t)∗δ(t)=f(t)
                      f(t)∗δ(t−t0)=f(t−t0)f(t)*\delta(t-t_0)=f(t-t_0)f(t)∗δ(t−t0)=f(t−t0)
                           f(t−t1)∗δ(t−t2)=f(t−t2)∗δ(t−t1)=f(t−t1−t2)f(t-t_1)\ast \delta(t-t_2)=f(t-t_2)\ast \delta(t-t_1) =f(t-t_1-t_2)f(t−t1)∗δ(t−t2)=f(t−t2)∗δ(t−t1)=f(t−t1−t2)
      冲激偶δ′(t)\delta{\prime}(t)δ′(t)：f(t)∗δ′(t)=f′(t)∗δ(t)=f′(t)f(t)*\delta{\prime}(t)=f{\prime}(t)*\delta(t)=f{\prime}(t)f(t)∗δ′(t)=f′(t)∗δ(t)=f′(t)
                  f(t)∗δ′′(t)=f′′(t)∗δ(t)=f′′(t)f(t)*\delta{\prime\prime}(t)=f{\prime\prime}(t)*\delta(t)=f{\prime\prime}(t)f(t)∗δ′′(t)=f′′(t)∗δ(t)=f′′(t)
      阶跃函数ε(t)\varepsilon(t)ε(t)：f(t)∗ε(t)=f(t)∗δ−1(t)=f−1(t)=∫−∞tf(τ)dτf(t)*\varepsilon(t)=f(t)*\delta^{-1}(t)=f^{-1}(t)=\int_{-\infty}^{t}f(\tau)d\tauf(t)∗ε(t)=f(t)∗δ−1(t)=f−1(t)=∫−∞tf(τ)dτ
                  f(t)∗ε(t−t0)=∫−∞tf(τ−τ0)dτ=∫−∞t−t0f(τ)dτf(t)*\varepsilon(t-t_0)=\int_{-\infty}^{t}f(\tau-\tau_0)d\tau=\int_{-\infty}^{t-t_0}f(\tau)d\tauf(t)∗ε(t−t0)=∫−∞tf(τ−τ0)dτ=∫−∞t−t0f(τ)dτ
（与阶跃函数卷积就是变上限积分，阶跃函数是个理想的积分器）

时移特性：f1(t−t1)∗f2(t−t2)=f(t−t1−t2)f_1(t-t_1)*f_2(t-t_2)=f(t-t_1-t_2)f1(t−t1)∗f2(t−t2)=f(t−t1−t2)

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