Wiley signals and systems e book TLFe BO 403

'We will look back a t the ROC of the Laplace transform in Chapter 4.5.3.. \/Vc havet alreadj said !hat, this nicans the irvragiiiary axis of tlw s-plane iiiust be part of the regioii of

388 16 Stability ttrid Feedhack Systcrxis

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liTli-systenis (hat, itre causal and s t a l k at the s;me time can be recognised by the location of their siiigiilarities in the corriplex s-plane 5% will first give thcir properties in a gmcrel f c m n and then sprcidisr the iesnlt for Iil'I-systeins with

riitwna 1 traiisfer Smictitms

.I ~~~~~~~~ ~~~~~~~~~S

The 1Cc)C: for thr trarisfczr function of a ctmtiiiuour; LTI-systeni gives 11s important

~ ~ ~ f o r ~ ~ a t i o ~ ahoiit$ the camaiity artcl stability of tlic system 'We will look back a t the ROC of the Laplace transform in Chapter 4.5.3

Chrawlity mems that the i m p l s e rcsponse is a sight-sirled sigzial The RO@

of' the Laplace transform for right-sided signals lies to the riglit of' a line in the

s - p h e , wliich is parallel lo the im;tginary axis Therefore, the traiisfw fmction

of a causal Ll'l-system converges in a regioii of tlie s-plane that lies to tlre right

of the s ~ ~ ~with ~the gmctest ~ ~ a rreal part ~ t ~ ~

Now we look ag&iii at tfip property of stability It ~ ~ ~ i ~ ) l i e s that it must br pos-

sible t o ~ 1 ~ s ~ ) f ~ ~ t e f ~ iritegrde tlic iiilpulse response \/Vc havet alreadj said !hat, this nicans the irvragiiiary axis of tlw s-plane iiiust be part of the regioii of corivcrgence

of the traiisfer function As LEiib must lir to the riglit of a border paidlel t,o the imagilimy axis, howcvrr this mcaiis tlmt the bortlei rrriist lie to the lclt 01 the

imaginary axis in the left, half of the s-plane (see Figure 3.5)

Whcre then, do t'tie s ~ ~ i ~ i ~ ~ ~ r ~ t i ~ ~ ~ of the trarisfer functiort lie? Since lhr ROC is free frorri s ~ n ~ ~ ~ ~ a r i ~ i e s ~ they must lie t o the left of the Linc, in t k left half of the

s-plane

Wr can summarise this insight with t hc following statcmerrt

16.2.1.2 Discrete Systems

For discrete systems we look at the properties of the ROC of 12ir 2-lraiisform to find simiiar results

F~JT a causal discrete LTI-system, the ROC of the transfer ~ r i ~ t ~ ~ n H ( z ) lies cmtsitle a circlc arouud the origin of the z-plane Necm the irnit circie iiitxst be par (, of lhe ROG fur a, st&k syskrrn, the radius of the bolder must be less t l m I,

and since the sirigularities caniiol lie iu the ROC, thcy m i s t I)c inside this circle 'rhereforet