Wiley signals and systems e book TLFe BO 403

16 Stability ttrid Feedhack Systcrxis 388 - 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 traiisfer Smictitms .I ~~~~~~~~ ~~~~~~~~~S 16.2J I Continuous Systems The 1Cc)C: for thr trarisfczr function of a ctmtiiiuour; LTI-systeni gives 11s important ~ ~ ~ f o r ~ahoiit$ ~ a tthe i o 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&iiiat tfip property of stability It ~ ~ ~ i ~that ) l i eits must br possible to ~ ~ s ~ ) firitegrde ~ ~ t e f 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, t'tie s ~ ~ i ~ of i the ~ ~trarisfer ~ r ~ functiort t i ~ ~lie? ~ Since lhr ROC is free frorri s ~ n ~ ~ ~ ~ they a r i must ~ i e slie~ 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 ~ Ht( z~) lies ~ n 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