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Unformatted text preview: T he s econdorder difference e quation (8. 26b )
y[k + 2] + h [k + 1] + fsy[k] = I [k + 2] c an be expressed in operational n otation as
(E2 + ~E + fs) y[k] = E 2f[k] A general n thorder difference Eq. (9.2a) can b e expressed as c 91 (En
1 .0000
2 .0000
1 .7600
2 .2800
0 .8576 (9.7) E y[k]  ay[k] = E f[k] a ns=  2.0000
1.0000
0
1 .0000
2 .0000 = I [k + a nl E nl + ... + a lE + ao)y[k] =
(bnEn + b n_IE n 1 + ... + b lE + bo)f[k] (9.9a) or 0 Q[E]y[k] = P [E]f[k]
where Q[E] a nd p rE] a re n thorder p olynomial operators (9.9b) 578 9 TimeDomain Analysis of DiscreteTime Systems 9.2 System response t o I nternal C onditions: T he Z eroInput Response 579 For a nontrivial solution of t his e quation + a n_lE n l + ... + a lE + aO
bnEn + b n_lE n l + .,. + b lE + bo Q[EJ = E n
P[EJ = (9.10)
(9.17a)
(9.11) or
Q bJ = 0 R esponse o f l inear D iscrete Time S ystems
Following t he p rocedure used for continuoustime systems on p. 105 ( footnote),
we can show t hat Eq. (9.9) is a linear equation (with c onstant coefficients). A system
described by such a n e quation is a linear timeinvariant discretetime (LTJD) system.
We can verify, as in t he case of LTJC s ystems (see footnote o n p. 105), t hat t he
g eneral solution of Eq. (9.9) consists o f z eroinput a nd z erostate components. (9.17b) O ur s olution q k [Eq. (9.14)J is correct, provided t hat "( satisfies Eq. (9.17). Now,
Q bJ is a n n thorder p olynomial a nd c an b e expressed i n t he f actorized form (assuming all d istinct r oots):
(9.17c)
Clearly, "( h as n s olutions " (1, " (2, . .. , " (n a nd, t herefore, Eq. (9.12) also has n s olutions 9 .2 S ystem r esponse t o I nternal C onditions: T he Z eroInput
R esponse
T he z eroinput response yo[kJ is t he s olution of Eq. (9.9) w ith I [kJ = 0; t hat is,
(9.12a) Q[EJYo[kJ = 0 or (9.12b) or
york + nJ + a nlYo[k + n  IJ + ... + alYo[k + IJ + aoyo[kJ =0 (9.12c) We can solve this equation systematically. B ut even a cursory e xamination o f this
equation points t o i ts s olution. T his e quation s tates t hat a l inear combination
of yo[kJ a nd a dvanced Yo[kJ is zero n ot for s ome values o f k, b ut for a ll k . Such
s ituation is possible i f a nd o nly ifYo[kJ a nd a dvanced yo[kJ h ave t he s ame form. O nly
a n e xponential function " (k h as t his p roperty a s t he following equation indicates.
(9.13)
T his e quation shows t hat t he delayed " (k is a c onstant t imes
s olution o f Eq. (9.12) m ust b e o f t he form " (k. T herefore, t he cl"(f,C2"(~,···,Cn"(~. I n s uch a case we have shown (see footnote o n p . 106) t hat
t he g eneral solution is a linear combination o f t he n s olutions. T hus (9.18)
where " (1, " (2, . .. , "(n a re t he r oots o f Eq. (9.17) a nd C l, C2, . .. , C a re a rbitrary conn
stants d etermined from n a uxiliary conditions, generally given in t he form o f i nitial
conditions. T he p olynomial Q bJ is called t he c haracteristic p olynomial o f t he
s yst...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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