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

n lying within t he u nit circle under these

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Unformatted text preview: e c annot b e e xpressed as a n e xplicit function o f t he i nput. T hese l imitations make i t useless in t he t heoretical s tudy o f systems. Problems 9.1-1 Solve iteratively (first three terms only): ( a) y[k + 1 ]- 0.5y[k] (b)y[k + 1] = 0, with y [-l] + 2y[k] = J[k + 1], = 10 with J[k] = e-ku[k] and y [-l] =0 9.1-2 Solve the following equation iteratively (first three terms only): y [k]- 0.6y[k - 1]- 0.16y[k - 2] = 0 with y [-l] = - 25, y [-2] =0 9.1-3 Solve iteratively the second-order difference Eq. (8.26b) in chapter 8 (first three terms only), assuming y [-l] = y [-2] = 0 and f [k] = 100u[k]. 9.1-4 Solve the following equation iteratively (first three terms only): y[k + 2] + 3y[k + 1] with f [k] = (3)k u [k], y [-l] + 2y[k] = J[k + 2] + 3J[k + 1] + 3f[k] = 3, and y [-2] = 2 9.1-5 Repeat Prob. 9.1-4 if y[k] with f [k] + 2y[k - 1] + y[k - 2] = 2 f[k]- f[k - 1] = (3)-ku[k],y[-1] = 2, and y [-2] = 3. 9.2-1 Solve y[k + 2] T he s tability criterion in t erms o f t he l ocation of characteristic roots of t he s ystem c an be summarized as follows: 1. An LTID system is asymptotically stable if a nd only if all t he c haracteristic roots are inside t he u nit circle. T he r oots may b e r epeated o r u nrepeated. + 3y[k + 1] + 2y[k] = 0 if y [-l] = 0, y [-2] = 1 + 2] + 2y[k + 1] + y[k] = 0 if y [-l] = 1, y [-2] = 1 if y [-l] = 1, y [-2] = 0 9.2-2 Solve 9.2-3 Solve 2. An LTID system is unstable if a nd o nly if e ither one or b oth o f t he following tThere is a possibility of an impulse ark] in addition to characteristic modes. y[k y[k + 2 ]- 2y[k + 1] + 2y[k] =0 9.2-4 Find v[k], t he voltage at the kth node of the resistive ladder depicted in Fig. P8.5-5 in Chapter 8, if V = 100 volts and a = 2. 9 6 12 T ime-Domain A nalysis o f D iscrete-Time S ystems 613 P roblems Hint: T he difference e quation for v[k] is given i n P rob. 8.4-5. T he a uxiliary conditions a re v[O] = 100 a nd v[N] = O. 9 .3-1 y[k 9 .3-2 + 1] + 2y[k] = F ind t he t otal r esponse of a s ystem specified by t he e quation y[k = E f[k] 9 .4-7 R epeat P rob. 9.3-1 i f = J[k + 1] 9 .4-8 I n t he s avings a ccount p roblem d escribed i n E xample 8 .5 ( chapter 8 ), a p erson deposits $500 a t t he b eginning o f e very m onth, s tarting a t k = 0 w ith t he e xception a t k = 4, w hen i nstead o f d epositing $500, s he w ithdraws $1000. F ind y[k] i f t he i nterest r ate is 1 % p er m onth ( r = 0 .01). H int: Because t he d eposit s tarts a t k = 0, t he i nitial condition y [-l] = O. W ithdrawal is a d eposit o f n egative a mount. 9 .4-9 T o p ayoff a l oan o f M d ollars in N p ayments u sing a fixed m onthly p ayment o f P d ollars, show t hat ao = a l = a2 = . .. = a n-l = 0 t he r esulting e quation is called a n onrecursive difference e quation. ( a) Show t hat t he i mpulse response o f a s ystem d escribed b y t his e quation is + bn -l6[k - 1] + ... + bl6[k - n R epeat P rob. 9.4-1 i f t he i mpulse response h[k] = (0.5)ku[k], a nd t he i nput f [k] is ( a) 2ku[k] ( b) 2(k-3)u[k] ( c) 2ku[k - 2]. H...
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