Wiley signals and systems e book TLFe BO 403

16. Stability ttrid Feedhack Systcrxis

388
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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

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 ~ 1 ~ 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, do 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