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Unformatted text preview: ving differential equations (p. 147) also apply t o difference equations . 3
5
= (3)2 _ +2 + 2 (3) k ="2 (3) k
3(3) k~0 • J ust as in a continuoustime system, we define a discretetime system t o be
a symptotically s table if, a nd only if, t he z eroinput response approaches zero as
k + 0 0. I f t he z eroinput response grows w ithout b ound a s k + 0 0, t he s ystem
is u nstable. I f t he z eroinput response neither approaches zero nor grows w ithout
b ound, b ut r emains within a finite limit as k + 0 0, t he s ystem is m arginally
s table. I n t he l ast case, t he zeroinput response approaches a constant o r oscillates
with a constant amplitude. Recall t hat t he z eroinput response consists o f t he
c haracteristic modes of t he s ystem. T he m ode corresponding t o a c haracteristic
root "1 is 'Yk. To be more general, let "1 b e complex so t hat
a nd • E xample 9 .12
For a n L TID system described by t he e quation 2
(E  E + 0.16)y[k] = ( E + 0.32)f[k] d etermine t he forced response y.p[k] i f t he i nput is f [k] = cos (2k + ~ )u[k] Since t he m agnitude o f ejf3k is always unity regardless of t he value o f k, t he magnitude of 'Yk is bl k . T herefore
if 1"11 < 1,
if a nd if b l>
bl = 1,
1, 'Yk  t 0 a sk+oo 'Yk + a sktoo bl k 00 =1 for all k 6 04 9 T imeDomain Analysis of DiscreteTime Systems 9.6 System Stability 605 F ig. 9 .6 C haracteristic r oots l ocation a nd s ystem s tability. I t i s c lear t hat a s ystem is a symptotically stable i f a nd only if
i = 1 ,2,··· , n T hese results can be grasped more effectively in t erms o f t he l ocation of characteristic r oots i n t he c omplex plane. Figure 9.6 shows a circle of u nit r adius, centered
a t t he o rigin in a complex plane. O ur discussion clearly shows t hat if all characteristic r oots of t he s ystem lie inside t his circle ( unit c ircle), hi I < 1 for all i a nd
t he s ystem is asymptotically stable. O n t he o ther h and, even if one characteristic
r oot lies outside t he u nit circle, t he s ystem is u nstable. I f n one of t he c haracteristic
r oots lie outside t he u nit circle, b ut s ome simple (unrepeated) r oots lie o n t he circle
itself, t he s ystem is marginally stable. I f two or more characteristic roots coincide
o n t he u nit circle (repeated roots), t he s ystem is unstable. T he r eason is t hat for
r epeated r oots, t he z eroinput response is of t he form F Lyk, a nd if h'l = 1, t hen
I krIykl = F  1  + 0 0 a s k  + oo.t N ote, however, t hat r epeated r oots inside
t he u nit circle d o n ot c ause instability. Figure 9.7 shows t he c haracteristic modes
corresponding t o c haracteristic roots a t various locations in t he complex plane. To
s ummarize:
1. A n LTID system is asymptotically stable i f a nd only if all t he c haracteristic
roots are inside t he u nit circle. T he r oots may b e simple o r r epeated. 2. A n L TID system is unstable if a nd o nly if either one o r b oth of t he following
cond...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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