信号与系统知识要点new

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∞=,t0δ()t=1,t≥00,t≠0ᓫ▤ε()t=ut()=ᓫ0,t<0∞δ()1t=∫−∞dε()ttδ()t=∫δτ()dτ=ε()tdt−∞δ()tḄឋft()()δt=f(0)()δtft()(δtt−)=ft()(δtt−)000∞∫ft()()δtdt=f(0)−∞∞ft()(δttdt−)=ft()∫−∞00δ()t=δ()−tδ(tt−)=δ[(−−tt)]001δ()at=δ()ta1t0δ(att−)=δ(t−)0aaᓫᏔδ′(

2、)tdδ()tδ′()t=dtδ′()t=−δ′()−tδ′(tt−)=−δ′[(−−tt)]00∞∫δ′()tdt=0−∞t∫δτ′()dτ=δ()t−∞ft()()δ′t=f(0)()δ′t−f′(0)()δtft()(δ′tt−)=ft()(δ′tt−)−ft′()(δtt−)00000∞∫ft()()δ′tdt=−f′(0)−∞∞ft()(δ′ttdt−)=−ft′()∫−∞00
ᦪsgn()t1,t>0sgn()t=0,t=0ᡈᡈᡈsgn()t=ut()−u()−=t2()1ut−−1,t<0ᓫᚁrt()0

3、,t<0tdrt()rt()=tut()=rt()=∫u()ττd=ut()tt,≥0−∞dtᦪgt()ττ1,t

4、s=σ+jωKᦪᦪ஺᪷σωḄf(t)⊤"#ᑡ%&'(Ḅ
஺当s=0时f(t)=K(直流信号)（即σ=0，ω=0时）σt当s=实数时f(t)=Ke(实指数信号)（即σ≠0，ω=0时）f(t)=Kest⇒当s=虚数时f(t)=Kcosωt+jsinωt（即σ=0，ω≠0时）(正弦信号与余弦信号)−σt当s=复数时f(t)=Ke(cosωt+sinωt)（即σ≠0，ω≠0时）(振幅变化的正、余弦信号)222᜻᜻'(Ḅ)*+,᜻.ᨵᓫ123Ꮤδ′(t)

5、ஹᓫ123δ(t)ஹᓫ1▤7u(t)89ᚁr(t)஺;<Ḅ)*f(t)ᵨ23δ(t)23δ(t)=>?+@ᑖ᪆ḄCD஺23ḄEFGAδ(t)=,0t≠0Aδ(t)→∞,t≠0∞∫Aδ(t)dt=A−∞Aᦪ஺HA=123δ(t)Iᓫ123δ(t)஺Ḅ⌕ឋ"###①J⌱ᱯឋf(t)δ(t)=f)0(δ(t)f(t)δ(t−t)=f(t)δ(t−t)t'ᦪ0000②N᪵ᱯឋ∞∞f(t)δ(t)dt=f)0(f(t)δ(t−t)dt=f(t)∫−∞∫−∞00③P

6、Qᱯឋ1bδ(at+b)=δ(t+)ab'ᦪaa④23ஹ▤7ஹ9ᚁ823ᏔR,ᐵTdddtδ′(t)=[δ(t)]δ(t)=[u(t)]u(t)=[r(t)]∫−∞δ′(τ)dτ=δ(t)dtdtdttt∫−∞δ(τ)dτ=u(t)∫−∞u(τ)dτ=r(t)23ᏔḄEFGdδ(t),t=0δ′(t)=dt,0t≠0
ᏔḄ⌕ᱯឋ①J⌱ᱯឋf(t)δ′(t−t)=f(t)δ′(t−t)−f′(t)δ(t−t)t'ᦪ000000②N᪵ᱯឋ∞f(t)δ′(t−t)dt=−f′(t)t'ᦪ∫

7、000−∞③PQᱯឋ1bδ′(at+b)=δ′(t+)ab'ᦪδ′(−t)=−δ′(t)aaa2ஹ)*+,Ḅ+@ᑖ᪆ḄUCVWGXஹYஹZᑖஹ[ᑖஹ]ஹ^_ஹPQஹᑖ`஺3ஹᔁ[[ᑖ∞b1cEFf(t)∗f(t)=f(τ)f(t−τ)dτ12∫12−∞b2cឋdeᣚgf(t)∗f(t)=f(t)∗f(t)1221ᑖh᳛f(t)∗[f(t)+f(t)]=f(t)∗f(t)+f(t)∗f(t)1231213jᔠg[f(t)∗f(t)]∗f(t)=f(t)∗[f(t)∗f(t)]123123(−)1ᔁ[ḄZ[ᑖឋdf′(t)∗

8、g(t)=f(t)∗g(t)(n)(n)f(t)∗g(t)=f(t)∗g(t)(−n)(−n)f(t)∗g(t)=f(t)∗g(t)᜻Ḅᔁឋf(t)∗δ(t)=f(t)δ(t−t)=t