Typical signals and their representation ☆ Gate signal p(t) 0( 2 2 The gate signal can be represented by unit step signals P(t)=u(t+2)-u(t-T2)
Typical signals and their representation Gate signal = < < 2 1 || 2 0 || ( ) τ τ τ t t p t The gate signal can be represented by unit step signals: Pτ (t) = u(t + τ/2) – u(t – τ/2) -τ/2 τ/2 1
Typical signals and their representation ☆ Unit Impulse Signal 8(tdt=1 6()=0t≠0 a.d(t is non-zero only at t=0, otherwise is o a(t)could not be represented by a constant even at t=0 but by an integral Regular function has exact value at exact time Obviously, 8(t) is not a Regular fu inction
Typical signals and their representation Unit Impulse Signal ( ) 0 0 ( ) 1 = ≠ = ∫ ∞ −∞ t t t dt δ δ • δ(t) is non-zero only at t=0，otherwise is 0 • δ(t) could not be represented by a constant even at t=0， but by an integral. • Regular function has exact value at exact time. Obviously，δ(t) is not a Regular function
Unit impulse function o(t) o with a gate signal p(t), short the duration t and keep the unit area 4/r 2/τ 1/t 2 τ/8τ8 Whenτ-→)0, the amplitude tends to, which means it is impossible to define d(t by a regular function
Unit impulse function δ(t) With a gate signal pτ (t), short the duration τ and keep the unit area When τ→0, the amplitude tends to ∞, which means it is impossible to define δ(t) by a regular function. -τ/2 τ/2 1/τ -τ/4 1/τ τ/4 2/τ -τ/8 4/τ τ/8
Properties of o(t) ☆ Sampling Property f(8(t)dt=f(O) Briefly understanding .When t+0, 8(t=0, then f(t)od(t=0 .When t=0, f(t)=f(O) is a constant. Based on the definition of o(t), it is easy to get: (0800080(0)j8(0=:(
Properties of δ(t) Sampling Property ∫ ∞ −∞ f (t)δ (t)dt = f )0( Briefly understanding： •When t ≠ 0, δ(t)=0, then f(t)●δ(t)=0 •When t = 0, f(t) = f(0) is a constant. Based on the definition of δ(t), it is easy to get: ∫ ∫ ∫ ∞ −∞ ∞ −∞ ∞ −∞ f (t)δ (t)dt = f )0( δ (t)dt = f )0( δ (t)dt = f )0(
Properties of o(t) 冷8(t) shift 6(tT) Bo(t)times a constant A: A8(t) a is called impulse intension which is the area of the integral
Properties of δ(t) δ(t) shift δ(t) 0 t δ(t- τ) 0 t τ δ(t) times a constant A: Aδ(t) A is called impulse intension which is the area of the integral