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Unformatted text preview: lkT] f lk] f [(kI)T] (a) t F ig. 8 .19 R ealization o f a secondorder discretetime s ystem i n E xample 8.6 .
• E xample 8 .6 I n t he k th s emester, J[k] n umber o f s tudents enroll in a course requiring a certain
t extbook. T he p ublisher sells y[k] n ew copies o f t he b ook i n t he k th s emester. O n t he
a verage, one q uarter o f s tudents w ith books in saleable condition resell t heir b ooks a t t he
e nd o f t he s emester, a nd t he b ook life is three semesters. Write t he e quation r elating y[k],
t he n ew books sold b y t he p ublisher, t o J[k]' t he n umber o f s tudents e nrolled in t he k th
s emester, a ssuming t hat every s tudent b uys a book.
In t he k th s emester, t he t otal books J[k] sold t o s tudents m ust b e e qual t o y[k] (new
b ooks from t he p ublisher) plus used books from s tudents enrolled in t he t wo previous
semesters (because t he b ook life is only three semesters). T here a re y[k  1] n ew books
sold in t he {k  l)st s emester, a nd o ne q uarter of these books; t hat is, h [k  1] will b e
r esold in t he k th s emester. Also, y[k  2] new books a re sold in t he {k  2)nd s emester,
a nd o ne q uarter o f t hese; t hat is, h [k  2] will be resold in t he {k  l)st s emester. Again
a q uarter o f these; t hat is, f ty[k  2] will b e resold in t he k th s emester. Therefore, J[k]
m ust b e e qual t o t he s um o f y[k], h [k  1], a nd f ty[k  2]. ( b) y I k] + h [k  1] + ftY[k  2] = J[k] y[k + 2] + h [k + 1] + fty[k] = J[k + 2] (B.26b) T his is t he a lternative form of Eq. (B.26a).
F or a r ealization o f a s ystem w ith t his i nputoutput e quation, we r ewrite Eq. (B.26a)
as y[k] =  h[k  1] fty[k  2] + J[k] E quations (8.25) a nd (8.26) a re e xamples o f d ifference e quations; t he f ormer is a
f irstorder a nd t he l atter is a s econdorder d ifference e quation. D ifference e quations
a lso a rise i n n umerical s olution o f d ifferential e quations. (I) • E xample 8 .1: D igital D ifferentiator
Design a discretetime system, like t he o ne in Fig. 8.2, t o d ifferentiate continuoustime
signals. Determine t he s ampling interval if t his d ifferentiator is used i n a n a udio s ystem
w here t he i nput signal b andwidth is below 20 kHz.
I n t his case, t he o utput yet) is r equired t o b e t he d erivative o f t he i nput J{t). T he
d iscretetime processor (system) G processes t he s amples o f J{t) t o p roduce t he d iscretetime o utput y[k]. L et J[k] a nd y[k] r epresent t he s amples T s econds a part o f t he s ignals
J {t) a nd yet), respectively; t hat is, J[k] = J {kT) (B.26c) F igure B.19 shows a h ardware r ealization o f Eq. (B.26c) using two time delays (here the
time delay T is a semester). To u nderstand t his realization, assume t hat y[k] is available.
T hen, b y delaying i t successively, we g enerate y[k  1] a nd y[k  2]. N ext we g enerate y[k]
from J[k], y[k  1], a nd y[k  2] a ccording t o E q. (B.26c). • t F ig. 8 .20 D igital differentiator a nd i ts realization. (B.26a) E quation (B.26a) c an a lso b e e xpressed in a n a lternative form by realizing...
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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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