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

15 i fkw k oo t his definition is valid for real or

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Unformatted text preview: eived r = interest p er dollar per period T W T he b alance y[kJ is t he s um o f (i) t he previous balance y[k - 1], (ii) t he i nterest o n y[k - lJ d uring t he p eriod T , a nd (iii) t he d eposit f[kJ k --- . flk] f l k] = fI2k] c y[kJ = y[k - 1J + ry[k - 1J + f[kJ = (1 + r )y[k - 1J + f[kJ Decimation (Downsampling) (b) (8.24) or y[kJ - ay[k - 1J = f[kJ 2468Wk- In this example t he d eposit f[kJ is t he i nput (cause) a nd t he b alance y[kJ is t he o utput (effect). We c an e xpress Eq. (8.25a) in a n a lternate form. T he choice of index k in Eq. (8.25a) i s c ompletely arbitrary, so we c an s ubstitute k + 1 for k t o o btain ~lk]=f[1] k lk ] lllliTl~ I I I ('~ 246 8 W 12 14 16 18 W 22 24 26 28 30 y[k + 1J T 32 34 36 38 40 k- - ay[kJ = f [k + 1J Interpolation (Upsampling) 2 4 6 8 W 12 14 16 18 W 22 24 26 28 30 (d) 32 34 36 38 40 k- (8.25b) We also could have obtained Eq. (8.25b) directly by realizing t hat y[k + 1J, t he b alance a t t he (k + l )st i nstant, is t he s um o f y[kJ plus ry[kJ ( the i nterest ony[k]) p lus t he d eposit (input) f [k + 1J a t t he (k + l )st i nstant. For a hardware realization o f s uch a system, we r ewrite Eq. (8.25a) as y[kJ = ay[k - IJ f; l k] (8.25a) a =l+r + f[kJ (8.25c) Figure 8.18 shows t he h ardware realization of this equation using a single t ime delay of T u nits.t To u nderstand t his realization, assume t hat y[kJ is available. Delaying i t by T, we generate y[k - 1J. Next, we generate y[kJ from f[kJ a nd y[k - 1J a ccording t o E q. (8.25c). y l k] Fig. 8 .17 T ime compression (decimation) a nd t ime expansion (interpolation) of a signal. 8 .5 Examples o f D iscrete-Time Systems W e s hall g ive h ere t hree e xamples o f d iscrete-time s ystems. I n t he f irst t wo e xamples, t he s ignals a re i nherently d iscrete-time. I n t he t hird e xample, a c ontinuoustime s ignal is p rocessed b y a d iscrete-time s ystem, a s i llustrated i n F ig. 8 .2, b y d iscretizing t he s ignal t hrough s ampling. • E xample 8 .5 A p erson makes a deposit ( the i nput) in a b ank r egularly a t a n interval of T (say, 1 m onth). T he b ank pays a certain interest on t he a ccount balance during t he p eriod T a nd m ails o ut a p eriodic s tatement of t he a ccount balance ( the o utput) t o t he depositor. Find t he e quation relating t he o utput y[kJ ( the balance) t o t he i nput f[kJ ( the d eposit). F ig. 8 .18 R ealization of t he savings account system. A withdrawal is a negative deposit. Therefore, this formulation c an h andle deposits a s well as withdrawals. I t also applies t o a loan payment problem w ith t he initial value y[OJ = - M, w here M is t he a mount o f t he loan. A loan is a n i nitial deposit w ith a n egative value. Alternately, we may t reat a l oan of M dollars taken a t k = 0 a s a n i nput o f - M a t k = 0 [see P rob. 9.4-9J. • t The time delay in Fig. 8.18 need not be T . The use of any other value will result in a time-scaled output. 564 8 D iscrete-time S ignals a nd S ystems 8.5 565 E xamples o f D iscrete- Time S ystems f (t) f...
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