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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 (rightshifted) by 2 units, a nd f [k + 2] is f[k] advanced (leftshifted) 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)k5 for 3 :s; k  5 :s; 10 or 8 :s; k :s; 15, as
i llustrated i n Fig. 8.l6b. 8 .42 T ime Inversion (or Reversal)
Following t he a rgument used for continuoustime 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 timeinverted
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 imeexpanded 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 iscretetime 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 discretetime. 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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