Wiley signals and systems e book TLFe BO 174

xlrere F ( t ) is an :uititlrrivi~tivPof j ( ) (b”(t)= f ( $ ) ) The limit To the differential quotient of F ( t ) at t = ant1 yields leads to w f ( t ) z ( t )tlt = liin F ( l b )- I ’ ( ) ~ I -0 -x, G (8.12) - f =o W i t h (8.9) we ohtain I I (8.13) tztic property of the tlelta impulse It i i i ~ m s This r c k i ionship describes tlic s intrgral of tlrr prodtict of a fiinctiion ;iacl the clclta inzpulsr renzuves all values f ( i ) l o ~t # arid selects tlie value f ( ) Note that f ( t ) utiiist be writiniitms at C - it’ rlrlta impdsc is rcprrseiited by ail Figrirc 8.7descrilrw this sititation arrow poiiitiiig iqwaxdh a t t = It nieans that 6(t) rlisapprari for i J and grows I)eyoiiti all liiriiti at = O (compare with (8.JU)) t Figure 5.7: Sclectiori t lic fiinvtioii value f ( ) If f i t ) = b’t, we lmow imnicdiately from t h selective ~ propcrty (8.13) that (8.14) This result is clrar if the delta impulse is iniagiricxl to be ilie limit vahw of o unit thc delta impulse ari arcn or bet7pr a ~ ~ q t i f The clcl-ta, inipulw i s riot restricted to sel iiig a \rii,ltic at t = 0, iri fact, it can be i w d to wlcct any poiut In thr samc way :U, in equations (8 11) t o (8.1:i) fctllow.; E; - (8.15)