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

# Observe t hat a nonrecursive s ystem is a special

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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 ero-input 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 r-l"(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 ero-input 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 continuous-time systems, t he c omplex roots o f a d iscrete-time 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 continuous-time 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 ime-Domain A nalysis o f D iscrete-Time 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 ero-Input 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 ero-input 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 = ~e-J8 [e (/3 k+8J + e- (/3k+8J] j d etermine york], t he z ero-input 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.

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