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

# The signal therefore may be denoted as f kt

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Unformatted text preview: r esulting s ampled s ignal? 8 .3-2 A s ampler w ith s ampling i nterval is T = 0.1 second samples a c ontinuous-time sinusoids 10 cos (1l7rt + f) a nd 5 cos (2971"t - f). F ind t he e xpressions for t he r esulting discrete-time sinusoids. Hint: R educe a ll discrete-time frequencies t o t he s mallest v alue (in t he r ange 0 t o 71"). 8 .3-3 ( a) A s ignal 10 cos 200071"t + V2 s in 300071"t + 2 cos (500071"t + ~) is s ampled a t a r ate o f 4000 Hz (4000 s amples/second). F ind t he r esulting s ampled signal. Does this s ampling r ate c ause a ny aliasing? E xplain. ( b) D etermine t he m aximum s ampling i nterval T t hat c an b e u sed t o s ample t he s ignal in ( a) w ithout aliasing. 8 .3-4 A s ampler w ith s ampling i nterval is T = 1 0- 4 s econds samples continuous-time s inusoids o f t he following frequencies: : F = (i) 1500 Hz (ii) 8,500 H z (iii) 10 kHz (iv) 11.5 kHz (v) 32 kHz (vi) 9600 Hz. D etermine o f w hat ( continuous-time) frequency t hese s amples a ppear i n e ach case. For t he s ignal shown i n Fig. PB.2-9b, sketch t he signals: ( a) f [-kJ ( b) f [k + 6J ( c) f [k - 6J ( d) f[3kJ ( e) f [~l w ithout t he i nterpolation o f t he m issing samples. e (I+j,,)k 8 .2-4 R epeat P rob. 8.2-3 for t he signals ( a) cos (0.671"k + 0 .2) ( b) cos (0.871"k + 1.2) ( e) cos (0.671"k + 0.3) + 3 sin (0.571"k + 0.4) + 8 cos (0.871"k - i) H int: For p arts ( a) a nd ( b), use Eq. (B.9b) t o d etermine t he p eriod. F or p art ( e), i f N N 2, a nd N3 a re t he p eriods o f t he t hree s inusoids, t he w idths o f m " m 2, a nd " m3 n umber o f cycles ( mI, m 2, m3 integers) o f t he t hree s inusoids m ust b e e qual for t he s ignal t o b e p eriodic. 8 .2-6 S how t hat cos (0.671"k 571 8 .4-1 e -(I-j,,)k 8 .2-3 S tate w ith r eason(s) i f t he following signals a re p eriodic: ( a) cos (0.571"k + 0.2) ( b) cos (V271"k + 1.2) ( e) s in (0.5k + 1) ( d) e 3tk. I f p eriodic, d etermine t he p eriod. 8 .2-5 P roblems f). w here 0 :'0 f l < 271": ( e) e - j 1. 95k ( d) e - jlO .7 . . k . 8 .2-8 I f f[kJ = (O.B)ku[kJ, find t he energies o f f [k], - f[k], a nd cf[kJ. 8 .2-9 F ind t he energies o f t he s ignals d epicted in Figs. PB.2-9. R epeat P rob. B.4-1 for t he s ignal d epicted in Fig. PB.2-9c. S ketch t he s ignals ( a) u[k - 2J - u[k - 6J ( b) k{u[kJ - u[k - 7]} ( e) (k - 2){u[k2J - u[k - 6]) ( d)( - k+ B){u[k - 6J - u[k - 9]} ( e)(k - 2){u[k - 2J - u[k - 6]) + ( -k + B){u[k - 6J - u[k - 9]) 8 .4-4 f [k) 8 .4-2 8 .4-3 D escribe each o f t he s ignals in Fig. PB.2-9 b y a single expression valid for all k. H int: Use t he e xpressions o f t he f orm in P rob. B.4-3. f [k] 3 3 3 6 (a) 8 .4-5 I f t he power o f a p eriodic signal J[kJ is P f, find t he powers a nd t he r ms values o f ( a) - f[kJ ( b) f [-kJ ( c) f [k-mJ ( d) cf[kJ. C omment. 8 .5-1 k- I f t he e nergy o f a s ignal f[kJ is E f, t hen s how t hat t he e nergy o f f [k - mJ is also E f· 8 -4-6 3 A c ash r egister o utput y[kJ r epresents t he t otal c ost o f k i te...
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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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