Wiley signals and systems e book TLFe BO 404

16.2. Causal Stable LW-Systenis

3813

system function W ( z ) for a causal, stable,
discrete IXLsystem must lie within t,he rwi( circle of the z-pli3ne.

f
The genctal statemeiits CRU IN>inndc more precise for svsteizis for which the transfer
function i s rational. Their sixigularities are siinplc or inultiple poles ilial cfelint.
tlie charact eristic fr ecluencies of the system. We caii describe t l m e systems with
pole-zcro diagrams.

functioii H (s) for 8 caiisal and stable roritinuous
left half of t lit. s-plane.

1x1

--

_.1

'PO illustrate this property w e m i s t recall the interilitl term ylzlT( 6 ) of (be output,
sigiial from Chapter 7 . 3 3 . It describes the part of the output sigiial tltat is caused
by the initial slate of the system. The Laplace transform of the int,ernal tcrrri can
be represented (set. (7.92)) by pwti
tioris with poles s,. In the time-tlornein
this coiresponds t i t the sum of the
1's characterishave only written uiit here for N simple poles:

(16.1.3)
a-1

= (T, -/- J W ? , tlic real part CT, < 0 is in the lef%h d f of thc plane,
ing characteri~~,ic
frequency decays

liln

t-ec

c'h'

= lirrl p a ~ .f c)"''
t-ca

z0

I

the

(16.14)

The condiliorr for stability, that all poles lie in the left half of the s-pimc, means
that, in the t i i ~ ~ - d ~ the
~ I ~response
a i ~ ~ ,to the iiiitial coIiditions decay5 with time,
just mhat we expect for a stable system. This IS ctcar from Figure 3.3 which shows
that only poles in the left half of the s-plane coxrespond to decaying cqmrcxntial

illalions. The decay of ttie inttcrrral tcruri mc
that ttie iiiital state of a system
dors riot define the system Feknviour, a s long 22s one is prepaxecl to w a i t long
enough.
The location of the zcros in the cornplcx plane does not influence t Irt. charitcteristic frequencies mtl thus hi$%no influence on the stability of t h system.
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