Signal Processing and Linear Systems-B.P.Lathi copy

# T he r emaining p art of t he t otal response t hat

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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 =&quot;2 (3) k 3(3) k~0 • J ust as in a continuous-time system, we define a discrete-time system t o be a symptotically s table if, a nd only if, t he z ero-input response approaches zero as k -+ 0 0. I f t he z ero-input response grows w ithout b ound a s k -+ 0 0, t he s ystem is u nstable. I f t he z ero-input 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 zero-input response approaches a constant o r oscillates with a constant amplitude. Recall t hat t he z ero-input 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 &quot;1 is 'Yk. To be more general, let &quot;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&quot;11 &lt; 1, if a nd if b l&gt; bl = 1, 1, 'Yk - t 0 a sk-+oo 'Yk -+ a sk-too bl k 00 =1 for all k 6 04 9 T ime-Domain Analysis of Discrete-Time 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 &lt; 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 ero-input response is of t he form F -Lyk, a nd if h'l = 1, t hen I kr-I-ykl = 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.

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