Wlirii tl~alitig with cmupli- catecl signal?. difler- twtiation in the frecii~‘3rrc,y-doirIwin.. We will learn the rtiost iriiportmt propertics of the liitplncc~ transform in tliis secti
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For the simple sigiials we have wrisitIeicd so far, e v a h t i n g the i n t ~ g r d (4.1) Lias berii tlw shortcsl way to find thc Laplace trarisforru Wlirii tl~alitig with cmupli- catecl signal? it i s an mlraiitagc, hottevcr to be able to use tlic pxopcrties of the
1 and avoiding liavitig t o cvnluate the grnl clirrctly Oftcn
t o pcrtornr ail o p e r d h n 011 it t imc-dt 11 signal (e.g difler- twtiation) in the frecii~‘3rrc,y-doirIwin We will learn the rtiost iriiportmt propertics
of the liitplncc~ transform in tliis section arid foranulate thpoicxis Ihi tlicrii ‘I’hc tlicoreiris will b r rieedctl in l a k r calculalioris usiiig I hc Laplace traiisforiii,
hi all v;tli~>s on the co~npltlx fi.c~lnr.ncy plane as long ab both L { j ’ ( t ) } i d L{,9(t))
exist The region of corrvt‘rgence for the cwri1)incd funclion is tlir iritcrsectioii of I lir~
rcgtoiis of convergence for the individiial tmictioris Sirigiilarities irray br ic~iovcc-l
I y the addition, l i o ~ h
super set of the iritri
, so in generd the conibined region o j con
on bctwecn the indivitliial regions of convergence:
[ ROC‘{cif + h q ] T: ROCC(j {i R O r / y } l (4 2 1 )
To show lincwrity, n j ( t ) -+ b y ( f ) is ixisci ted into tlw tlcxfinition of tht J,aplacc
t r i t n s h m (4.1 j From the linearity of the intcy,rat,ioii, (4.20) i s iinmeditLte1y oh-
ta ivicrl