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

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

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: 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...
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.

Ask a homework question - tutors are online