Typical signals and their representation ☆ Gate signal (t) 2 0 τ/2 τ/2 The gate signal can be represented by unit step signals: Pr(t)=u(t+τ/2)-u(t-T/2)
Typical signals and their representation ❖Gate signal = 2 1 | | 2 0 | | ( ) 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(t)lt=1 (t)=0t≠0 o(t) is non-zero only at t=0, otherwise is O o(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 d(t) is not a Regular function
Typical signals and their representation ❖Unit Impulse Signal ( ) 0 0 ( ) 1 = = − t t t dt • 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 8(t) o With a gate signal p(t), short the duration t and keep the unit area 4/T 2/τ 1/ 2 τ/2 4/4 τ/8Tτ/8 When t-0, the amplitude tends to oo, 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 ofδ(t) Sampling property f(t)6(dt=f(0) Briefly understanding When t≠0,8(t)=0, then f(t)●δ(t)=0 .When t=0, f(t)=f(O)is a constant. Based on the definition of s(t), it is easy to get: ∫/()d=J0o8(o)=0)Jlo=f(0)
Properties of δ(t) ❖Sampling Property − f (t) (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) (t)dt = f (0) (t)dt = f (0) (t)dt = f (0)
Properties ofδ(t) ☆6(t) shift 6(t) δ(t-τ 0 s o(t)times a constantA: Aδ(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 τ t ❖δ(t) times a constant A: Aδ(t) A is called impulse intension which is the area of the integral