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

# I f t he s peed is i ncreased beyond some critical

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Unformatted text preview: f m is positive; t he shift is t o t he right (delay). I f m is negative, t he shift is t o t he left (advance). Thus, f [k - 2] is f[k] delayed (right-shifted) by 2 units, a nd f [k + 2] is f[k] advanced (left-shifted) by 2 units. T he signal fd[k] in Fig. 8.l6b, being t he signal in Fig. 8 .l6a delayed by 5 units, is t he same as f[k] w ith k replaced by k - 5. Now, f[k] = (0.9)k for 3 :s; k :s; 10. Therefore, f dk] = (0.9)k-5 for 3 :s; k - 5 :s; 10 or 8 :s; k :s; 15, as i llustrated i n Fig. 8.l6b. 8 .4-2 T ime Inversion (or Reversal) Following t he a rgument used for continuous-time signals, we c an show t hat t o t ime i nvert a signal f[k], we replace k w ith - k. T his operation rotates the signal about the vertical axis. Figure 8 .l6c shows fr[k], which is t he time-inverted signal f[k] i n Fig. 8.16a. T he expression for fr[k] is t he same as t hat for f[k] with k replaced b y - k. Because f[k] = (0.9)k for 3 :s; k :s; 10, Jr[k] = ( 0.9)-k for 3:S; - k :s; lO; t hat is, - 32: k 2: - 10 , as shown in Fig. 8.l6c. (8.23) T he signal fe[k] is t he signal f [k] e xpanded by a factor 2. According t o E q. (8.23), fe[O] = frO]' fe[l] = f[1/2], fe[2] = f [l], fe[3] = f[3/2], fe[4] = 1[2], fe[5] = f[5/2], fe[6] = f[3], a nd so on. Now, f[k] is defined only for integral values of k, a nd is zero (or undefined) for all fractional values of k. Therefore, for fe[k]' its o dd n umbered samples f ell]' fe[3], fe[5], . .. a re all zero (or undefined), as depicted in Fig. 8.17c. In general, a function fe[k] = f [k/m] (m integer) is defined for k = 0, ±m, ±2m, ±3m, . .. , a nd is zero (or undefined) for all remaining values of k. Interpolation I n t he t ime-expanded signal in fig. 8.17c, t he missing o dd n umbered samples can be reconstructed from t he nonzero valued samples using some suitable interpolation formula. Figure 8.17d shows such a n i nterpolated function fdk], where t he missing samples are constructed using a n i deallowpass filter interpolation formula (5.lOb). In practice, we may use a realizable interpolation, such as a linear interpolation, where fd1] is t aken as t he m ean of fdO] a nd J;[2]. Similarly, J;[3] is t aken as t he m ean of fd2] a nd f;[4], a nd so on. This process of time expansion a nd i nserting t he missing samples using a n i nterpolation is called i nterpolation or u psampling. I n this operation, we increase t he n umber of samples. 6 E xercise E 8.6 Show t hat for a linearly i nterpolated f unction I i [k] = ! [k/2]' t he o dd n umbered samples i nterpolated values a re /;[k] = ~ { i[k;!] + ![~]}. \1 t Odd n umbered samples of I [k] c an be r etained ( and even numbered samples o mitted) by using t he t ransform I olk] = 1 [2k + 1] 562 8 D iscrete-time S ignals a nd S ystems 8 .5 f lk] 5 63 E xamples o f D iscrete- Time S ystems I n t his case, t he signals a re i nherently discrete-time. L et ( a) 246 WU M 8 f fi g f[kJ = t he d eposit made a t t he kth discrete i nstant y[kJ = t he a ccount balance a t t he kth i nstant c omputed immediately a fter t he kth d eposit f[kJ is rec...
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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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