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

# Sampling a continuous time sinusoid cos wt 0 a t

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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 ime-delay e lement. ( e) R edo t he p roblem if t here is a 10% sales t ax. 2 (e) 8 .5-2 (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 .5-3 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 8.2-9 +H 8 .2-10 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 .2-11 F ind t he powers o f t he s ignals illustrated in Fig. PlO.1-4 a nd P I0.1-5 ( Chapter 10). 8 .2-12 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 o-l f[kJ =L r =O N o-l 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 iscrete-time 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 .5-4 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 long-term (secular) trend. We can discern the long-term trend by smoothing or averaging the past N values of the variable. For the stock market average, we may consider a five-day 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 time-delay elements, realize the five-day 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. P8.5-5 8 .5-5 The voltage a t t he kth node of a resistive ladder in Fig. PB.5-5 is v[kJ (k = 0, 1, 2, . .. , N ). Show t hat v[kJ satisfies the second-order 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 ime-Domain Analysis o f D iscrete-Time Systems I n t his c hapter we discuss t ime-domain a nalysis o f L TID (linear time-invariant discrete-time systems). T he p rocedure is parallel t o t hat for continuous-time systems, w ith m inor differences. 9.1 Discrete-Time 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...
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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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