Unformatted text preview: em, a nd
(9.19)
Q bJ = 0
is t he c haracteristic e quation o f t he s ystem. Moreover, " (1, "(2, . .. , "(n, t he r oots
o f t he c haracteristic equation, a re called c haracteristic r oots o r c haracteristic
v alues (also e igenvalues) o f t he s ystem. T he e xponentials "(f(i = 1 ,2, . .. , n ) a re
t he c haracteristic m odes o r n atural m odes o f t he s ystem. A characteristic
m ode c orresponds t o e ach characteristic r oot o f t he s ystem, a nd t he z eroinput
response is a linear c ombination o f t he c haracteristic m odes o f t he s ystem.
R epeated R oots
I n t he discussion so far we have a ssumed t he s ystem t o have n d istinct c haracteristic roots " (1,"(2, . .. , "(n w ith c orresponding characteristic modes "(f,"(~,· . . ,"(~. I f
two o r m ore r oots coincide ( repeated r oots), t he form of characteristic modes is m odified. Direct s ubstitution shows t hat if a r oot "( r epeats r t imes ( root o f multiplicity
r ), t he c haracteristic modes corresponding t o t his r oot a re "(k, k"(k, k2"(k, . .. , k rl"(k.
T hus, i f t he c haracteristic e quation o f a system is (9.14)
To determine c a nd ,,(, we s ubstitute t his solution in Eq. (9.12). E quation (9.14)
yields
Eyo[kJ = york
E2YO[kJ = york Enyo[kJ + IJ =
+ 2J (9.20)
t he z eroinput response o f t he s ystem is q k+l = c ·/+ 2 = york + nJ = c ·l+ (9.15) n S ubstitution o f t hese results in Eq. (9.12b) yields
(9.16) C omplex R oots
As in t he c ase of continuoustime systems, t he c omplex roots o f a d iscretetime
s ystem m ust o ccur i n p airs o f conjugates so t hat t he s ystem e quation coefficients
are real. Complex roots c an b e t reated e xactly as we would t reat r eal roots. However, j ust a s in t he c ase of continuoustime systems, we c an e liminate dealing w ith
c omplex numbers by using t he real form o f t he s olution. 580 9 T imeDomain A nalysis o f D iscreteTime S ystems F irst w e e xpress t he c omplex c onjugate r oots 'Y a nd 'Y* i n p olar f orm. I f
t he m agnitude a nd f3 i s t he a ngle o f 'Y, t hen hi is 9 .2 S ystem r esponse t o I nternal C onditions: T he Z eroInput R esponse T he r eader c an verify t his s olution by c omputing t he first few t erms u sing t he i terative
m ethod (see E xamples 9.1 a nd 9.2). ( b) A s imilar p rocedure m ay b e followed for r epeated r oots.
s ystem s pecified by t he e quation a nd 581 F or i nstance, for a T he z eroinput r esponse i s g iven b y (E2 + 6 E + 9)y[kJ = (2E2 + 6 E)f[kJ + C2("(*)k
kjk
= cd'Yl e /3 + c 2hl k e j /3k York] = cn k F or a r eal s ystem, q a nd
Ie. L et
T hen C2 m ust b e c onjugates s o t hat = ~ej8 q York] =~I'Ylk a nd C2 York] i s a r eal f unction o f = ~eJ8 [e (/3 k+8J + e (/3k+8J]
j d etermine york], t he z eroinput c omponent o f t he r esponse i f t he i nitial c onditions a re
y o[I] = ~ a nd yo[2] = ~.
T he c haracteristic p olynomial is ')'2 + 6'Y + 9 = (')' + 3)2, a nd we h...
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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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