Wiley signals and systems e book TLFe BO 415

16. Stability aird Feedback Systems

400

Exercise 16.1
Ttnc stability of a system with irnpiilse rcsponsc? h(t) = t”-O

It

sin(5nt),c(f) is to be

iriwstigated.

a) Can the impulse response be absolutely iril egra,tetl’?
b) Check tlir location of tlie poles.

xercise 16.2
Show that (16.7) is a iiecessarg and sufficient condition for the BIBO-stability of
a disrrtlte ILTI-sysLern.

Zxercise 16.3
ten1 is described by the following diffec.rrnitia1equation:
- 61 with 0 < \ U \< 00.

33

a ) Dcteriniiir If,( 2 ) =
and the corresponding polr-wro diagram. Give 1 he
ROC of I I , ( z ) . For what values of a is the
b)

miiic the trariskr lunction H Z ( z ) of a second d
IIz( cc.) = 1. Givc both p c ible KOCs of N J
each whether it is st;tblc.
s

system so that
cl clet wmine for

Exercise 18.4

A system is giver1 by thc differcntial equation yrk] - ;v[k - I] -t- $y[k - 21 = ~ [ k ] .
a ) Giw t h c transfer function N ( z ) =

x(z)

of the system.

b) Tr thc spslcm stable‘? This slionld be tlcttrrriined

a ) with tthe pole-zero diagram,

a) with the biliizear traiisforiiiation z = 5 and a. stability test for continuous loystcms.

c ) Is the system minimal p h w ?
tl) 1s the systcrri causal:“
Exercise 16.5
Uriiig tlic internal coiiipoyicnt of the out piit, sigital. giw a mothratioii that r2 discrctc
system is stablc if its poles lie within the w i t circle of tlic complex z-plme (see
Section 16.2.2).