Unformatted text preview: ms r ung u p b y a cashier.
T he i nput f[kJ is t he c ost o f t he k th i tem. (b) f [k) f [k) 9 o 2 k ( a) W rite t he difference e quation r elating y[kJ t o f[kJ.
( b) R ealize t his s ystem u sing a t imedelay e lement.
( e) R edo t he p roblem if t here is a 10% sales t ax. 2 (e) 8 .52 (d) L et p[kJ b e t he p opulation o f a c ertain c ountry a t t he b eginning o f t he k th y ear.
T he b irth a nd d eath r ates o f t he p opulation d uring a ny y ear a re 3.3% a nd 1.3%,
respectively. I f i[kJ is t he t otal n umber o f i mmigrants e ntering t he c ountry d uring t he
k th y ear, write t he difference e quation r elating p[k + IJ, p[kJ, a nd i[kJ.
H int: Assume t hat t he i mmigrants e nter t he c ountry t hroughout t he y ear a t a u niform
r ate, so t hat t heir a verage b irth a nd d eath r ates n eed t o b e a veraged. 8 .53 For a n i ntegrator, t he o utput y (t) is t he a rea u nder t he i nput f (t) from t = 0 t o t .
S how t hat t he e quation o f a d igital i ntegrator is F ig. P 8.29 +H 8 .210 F ind t he powers o f t he s ignals (1)k, ( I)k, u[k], ( I)ku[kJ, a nd c os[ik 8 .211 F ind t he powers o f t he s ignals illustrated in Fig. PlO.14 a nd P I0.15 ( Chapter 10). 8 .212 Show t hat t he power o f a s ignal
s ignal j 1 )e *ok is 11)1 2 H ence, show t hat t he p ower o f a y[kJ  y[k  IJ "" T f[k  IJ
N ol f[kJ =L r =O N ol 1 )r e jr *ok is Pf = L l1)rl2
r=O I f a n i nput u[kl is applied t o s uch a n i ntegrator, show t hat t he o utput is a r amp
k T u[kJ. H int: f ~t k = 0, 1, 2, 3, . .. successively in this e quation t o find y[kJ. 572 8 D iscretetime S ignals a nd S ystems
In Exercise E8.8, using a slightly different approach, we found the integrator equation
to be y[kJ  y[k  1J = T l[kJ. Show t hat t he response of this integrator to a unit step
input u[kJ is kT u[kJ + T u[kJ, which approaches the ramp kT u[kJ as T + O. 8 .54 A moving average is used to detect a trend of a rapidly fluctuating variable such as
t he stock market average. A variable may fluctuate (up and down) daily, masking
its longterm (secular) trend. We can discern the longterm trend by smoothing or
averaging the past N values of the variable. For the stock market average, we may
consider a fiveday moving average y[kJ to be the mean of the past five days' market
closing values I [k], I [k  1], . .. , I [k  4J.
( a) Write the difference equation relating y[kJ t o t he input l[kJ.
( b) Using timedelay elements, realize the fiveday moving average filter. R R 1
R "" " J J J Vv[ k v[k+I I. . " r n V [ N  1vl
[kl
1
[N] ITE R R R R R V~~aR::
F ig. P8.55 8 .55 The voltage a t t he kth node of a resistive ladder in Fig. PB.55 is v[kJ
(k = 0, 1, 2, . .. , N ). Show t hat v[kJ satisfies the secondorder difference equation v[k + 2J  Av[k + 1J + v[kJ = 0
Hint: Consider the node equation a t the kth node with voltage v[kJ. T imeDomain Analysis o f
D iscreteTime Systems
I n t his c hapter we discuss t imedomain a nalysis o f L TID (linear timeinvariant
discretetime systems). T he p rocedure is parallel t o t hat for continuoustime systems, w ith m inor differences. 9.1 DiscreteTime System equations Difference Equations
E quations (8.25), (8.26), a nd (8.29) a re e xamples o f difference equations. E quations (8.25) a n...
View
Full
Document
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

Click to edit the document details