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信号与线性系统分析 Analysis of Signals and Linear Systems

信号与线性系统分析 Analysis of Signals and Linear Systems

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信号与线性系统分析

Items (61)

  • 冲激函数

    δ(t)

  • 阶跃函数

    ε(t)

  • 周期信号求的公式

    2π/β

  • 门函数

    gτ=0时 |t|>τ/2; gτ=1时 |t|< τ/2

  • δ(t)=

    dε(t)/dt

  • ε(t)=

    ∫δ(τ)dτ, -∞ to t

  • δ(-t)=

    δ(t)

  • δ(t-T)=

    δ[-(t-T)]

  • δ(at)=

    1/|a|·δ(t)

  • f(t)δ(t-T)=

    f(T)δ(t-T)

  • ∫f(t)δ(t-T), -∞ to ∞=

    ∫f(T)δ(t-T), -∞ to ∞

  • f(t)*δ(t±T)=

    f(t±T)

  • f(t)δ'(t±T)=

    f(T)δ'(t±T)-f'(T)δ(t±T)

  • f(t)*δ'(t±T)=

    f'(t±T)

  • 设f(t)→y(t), 线性指

    A1f1(t)+A2f2(t)→A1y1(t)+A2y2(t)

  • 设f(t)→y(t), 时不变性指

    f(t-τ)→y(t-τ)

  • 全响应分解

    y(t)=yzi(t)+yzs(t)

  • 时不变系统

    f(t-τ)→y(t-τ)

  • LTI系统满足微分性和积分性,即若f(t)→yzs(t),则:

    df(t)/dt→dyzs(t)/dt; ∫f(τ)dτ, -∞ to t →∫yzs(τ)dτ, -∞ to t

  • f1(t)*f2(t)=

    ∫f1(τ)f2(t-τ)dτ, -∞ to +∞

  • f(t)*δ(t)=

    f(t)

  • f(t)*δ'(t)=

    f'(t)

  • yzs(t)=? (利用卷积)

    yzs(t)=f(t)*h(t)

  • 卷积运算规律

    交换律,结合律,分配率

  • ∫[f1(τ)*f2(τ)]dτ, -∞ to t

    f1(t)*∫f2(τ)dτ, -∞ to t; =f2(t)*∫f1(τ)dτ, -∞ to t

  • [f1(t)*f2(t)]'=

    f1(t)*f2'(t)=f1'(t)*f2(t)

  • f1'(t)*∫f2(τ)dτ, -∞ to t; =

    f2'(t)*∫f1(τ)dτ, -∞ to t; =f1(t)*f2(t)

  • ε(t)*ε(t)=

    tε(t)

  • e^-αt·ε(t)=

    1/α·(1-e^-αt)ε(t)

  • yzs=? (卷积求零状态响应)

    f(t)*h(t)=∫f(τ)h(t-τ)dτ, -∞ to +∞; =∫f(τ)h(t-τ)dτ, 0 to t (因果系统下);

  • 傅里叶级数(三角函数形式)

    f(t)=a0/2+Σan·cos(nΩt)+Σbn·sin(nΩt), n=1 to ∞, n∈Z+

  • 傅里叶系数an=

    2/T·∫f(t)cos(nΩt)dt,-T/2 to T/2, n∈N

  • 傅里叶系数bn=

    2/T·∫f(t)sin(nΩt)dt, -T/2 to T/2, n∈Z+

  • 傅里叶级数 (复指数形式)

    f(t)=ΣFn·e^jnΩt, n=-∞ to ∞, n∈Z

  • 复傅里叶系数Fn

    Fn=1/T·∫f(t)e^(-jnΩt)dt, -T/2 to T/2, n∈Z

  • 傅里叶变换

    F(jω)=∫f(t)e^(-jωt)dt, -∞ to ∞

  • 傅里叶逆变换

    f(t)=1/2π·∫F(jω)e^(jωt)dω, -∞ to ∞

  • f(t)=gτ; F(jω)=

    τ·Sa(ωτ/2)

  • Fn=?(绝对值乘以…)

    |Fn|e^jφₙ

  • |Fn|=

    1/2An=1/2√(aₙ²+bₙ²)

  • φₙ=

    -arctan(bₙ/aₙ)

  • f(t)=f(-t)偶函数,aₙ=

    4/T·∫f(t)cos(nΩt)dt, 0 to T/2, n∈N

  • f(t)=-f(-t)奇函数,bₙ=

    4/T·∫f(t)sin(nΩt)dt, 0 to T/2, n∈N+

  • f(t)=Σgτ(t-nT),n=-∞ to ∞, n∈Z;Fn=

    τ/T·Sa(nΩτ/2)=τ/T·Sa(nπτ/T), n∈Z

  • 矩形周期信号零分量频率

    ω=2kπ/τ, k=±1,±2…

  • 矩形周期信号频带宽度

    ΔF=1/τ

  • 非周期信号的频谱函数F(jω)=|F(jω)|e^jφ(ω),|F(jω)|为

    f(t)的振幅频谱,ω的偶函数

  • 非周期信号的频谱函数F(jω)=|F(jω)|e^jφ(ω),φ(ω)为

    f(t)的相位频谱,ω的奇函数

  • 非周期信号的频谱函数F(jω)=R(ω)+j·X(ω),R(ω)为

    ω的偶函数

  • 非周期信号的频谱函数F(jω)=R(ω)+j·X(ω),X(ω)为

    ω的奇函数

  • 门函数零分量频率

    ω=2kπ/τ, k=±1,±2…

  • 门函数频带宽度

    ΔF=1/τ

  • F(jt)→?(对称性)

    2π·f(-ω)

  • F(t)→?(实偶函数下)

    2π·f(ω)

  • 对任意连续信号,如果是周期信号,则频谱是

    离散的

  • 对任意连续信号,如果是非周期信号,则频谱是

    连续的

  • f(a·t)→

    1/|a|·F(jω/a)

  • f(t±t₀)→

    e^(±jωt₀)·F(jω)

  • f(t)e^(±jω₀t)→

    F[j(ω-±ω₀)]

  • f₁(t)*f₂(t)→

    F₁(jω)·F₂(jω)

  • f₁(t)·f₂(t)→

    1/2π·F₁(jω)*F₂(jω)