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

Sampling a continuous time sinusoid cos wt 0 a t

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online