Wiley signals and systems e book TLFe BO 180

(8.31) Figure 8.1 I : D(41ciimpulse and step function Figure 8.12 shows a iigrial ~ ( t(top ) left), thitt c-arinot be clif€eirrrli a t e d brcausc it is tliscvntiiiiious at oiir point Forming A cleri\ative in the iiorrnal way i s riot possiblc, 11 t8hr right-sided m d left,-sidrd limits of tlic diffeientia.1 yuot icnt (not the function itself !) i i i c tqiial at tho point of tliscontinuit,v, the signal r ( t )can, iiowevrr, be rtywscnted cis lie sum of a fiinctioii that can bii tliffrreiitinted arid U, function (Figiirc % top riglrt) Tlw diffeieirce at tlic tliscoiitinuit> 'C;liitli diffcrriiticztion o f ilie continuous part aiitl drriv;ition of klie step fiinc-tiorr, the desivativr of the con1 inuous p x t a n d it delta impulse with wciglit (I is obtairid (Figiirc 8.12 buttcnri right) Both t,eirxis l x put together to givc the drrimtioii of the discontinuous sigii,tl, nhicli bns 110 deLivativc in thc iioririal sensti rlie discontiimitj wii li s t c y liriglit U nppears in thc ckvivatiori as a &>hairiipulsr with wcight ( I (Figuic 8.12 bottom lcit) I i a m