The Continuous-Time Unit Impulse Function ·Definition δ(t)=0,t≠0 ∫o(t)dt=1 -00 Aka,Dirac delta function
The Continuous-Time Unit Impulse Function • Definition ( ) 1 ( ) 0, 0 t dt t t • Aka, Dirac delta function
(a)Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e.,unit impulse).(b)Graphical symbol for unit impulse. (c)Representation of an impulse of strength a that results from allowing the duration A of a rectangular pulse of area a to approach zero. x(t) 6t) a6(t) Area=1 Strength a Area =l Area=1 Strength=1 alA Area =a 1/4 -412 0 4/2 0 -412 0 4/2 (a) b (c) o)=lima)）
(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero. ( ) lim ( ) 0 t x t
The Continuous-Time Unit Step Function We view the unit impulse as the limiting form of any pulse x(t)that is an even function of time t with duration and unit area. δ(t)=limx(t) Λ→0
The Continuous-Time Unit Step Function • We view the unit impulse as the limiting form of any pulse xΔ(t) that is an even function of time t with duration Δ and unit area. ( ) lim ( ) 0 t x t
Sampling Property of Impulse The unit impulse can be used to sample the value of a signal at t=0 x(t)δ(t)=x(0)δ(t) x(t)δ(t-to)=x(to)δ(t-t)
Sampling Property of Impulse • The unit impulse can be used to sample the value of a signal at t=0 ( ) ( ) ( ) ( ) ( ) ( ) (0) ( ) 0 0 0 x t t t x t t t x t t x t
Sifting Property of Impulse The integral sifts out the value of x(t)at time t=to 「x(t)6(t-to)dt=x(t)
Sifting Property of Impulse • The integral sifts out the value of x(t) at time t=t0 ( ) ( ) ( ) 0 0 x t t t dt x t