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

However t he coefficient of y k n is normalized t o

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Unformatted text preview: T he s econd-order 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 th-order 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 n-l E n-l + ... + 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 th-order p olynomial operators (9.9b) 578 9 Time-Domain Analysis of Discrete-Time Systems 9.2 System response t o I nternal C onditions: T he Z ero-Input 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 continuous-time 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 time-invariant discrete-time (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 ero-input a nd z ero-state 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 th-order 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 ero-Input R esponse T he z ero-input 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 n-lYo[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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