8 15-2 The unit-impulse function We define the unit impulse as a function of time which is zero when its argument, generally(t-to), is less than zero; which is also zero when argument is greater than zero; which is infinite when its argument is zero; and which has unit area (t-t0) 6(t-t)=0t≠t0 ayt-4)=10m8(-4)M=1 (t) δ(t)=0t≠0
§15-2 The unit-impulse function We define the unit impulse as a function of time which is zero when its argument, generally (t-t0 ), is less than zero; which is also zero when argument is greater than zero; which is infinite when its argument is zero; and which has unit area. = = = + − + − 0 0 ( ) 1 ( ) 1 ( ) 0 0 and t dt or t dt t t 0 t t 1 ( ) 0 t − t 0 t 1 (t) 0 − = − = − = + − + − 0 0 ( ) 1 ( ) 1 ( ) 0 0 0 0 0 t t and t t dt or t t dt t t t t
The strength of the impulse (t) 5 5δ(1)→5 2.58(t-3)->-25 2.5 If the unit impulse is multiplied by a function of time, then the strength of the impulse must be the value of that function at the time for which the impulse argument is zero. In other words,e-8(t-2)->0.368, and the strength of the impulse sin(5m+/4)δ(t)→>0.707 ∫ f(to(tdt=f(O The mathematical form + 广f()(-t0)d=f(t0)
The strength of the impulse 5 (t) → 5 − 2.5 (t − 3) → −2.5 t 5 (t) 0 − 2.5 3 The mathematical form + − f (t) (t)dt = f (0) + − ( ) ( − ) = ( ) 0 0 f t t t dt f t ( 2) 0.368, e −t / 2 t − → sin(5t + / 4) (t) → 0.707. If the unit impulse is multiplied by a function of time, then the strength of the impulse must be the value of that function at the time for which the impulse argument is zero. In other words, and the strength of the impulse
df(t)/dt △ △ △t △ (b) (a)a rectangular pulse of unit area which approaches a unit impulse as△→>0 (b)A modified (YE)unit-step function. (c) The derivative of the modified unit step The unit impulse may be regarded as the time derivative of the unit step function. du(t) (t)= or()=o()dt(r>0) dt
(a) A rectangular pulse of unit area which approaches a unit impulse as → 0. (b) A modified(准) unit-step function. (c) The derivative of the modified unit step. The unit impulse may be regarded as the time derivative of the unit step function. dt du t t ( ) ( ) = − = t or u t t dt t 0 ( ) ( ) ( 0) − 2 1 2 1 1 (a) t f (t) − 2 1 2 1 1 (b) t f (t) − 2 1 2 1 1 (c) t df (t)/ dt
