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Unformatted text preview: re eliminated. F igure P 5.1-1 shows Fourier s pectra o f s ignals f 1(t) a nd h (t). D etermine t he N yquist
s ampling r ates for signals f 1(t), h (t), ! f(t), f~(t), a nd f 1(t)h(t). 5 3 58 S ampling Pr.oblems 359 ... 0 0 n 0 f"O 0 00.::·'
4 ms 8 ms t~ F ig. P 5.1-6
F ig. P 5.1-1 will b e a ble t o r econstruct j (t) from these samples. I f t he s ampled signal is p assed
t hrough a n ideal lowpass filter of b andwidth 100 Hz a nd u nit gain, find t he filter
o utput. W hat is t he filter o utput if its b andwidth is B Hz, where 100 < B < 150?
W hat will h appen if t he b andwidth exceeds 150 Hz? D etermine t he N yquist sampling r ate a nd t he N yquist sampling interval for t he
signals ( a) s inc 2 (1001l"t) ( b) 0.01 sinc 2 (1001l"t) ( c) sinc (IOO1l"t) + 3 s inc (601l"t) ( d)
sinc (501l"t)sinc (IOO1l"t).
>.1-3 A signal j (t) = sinc (2001l"t) is sampled (using uniformly spaced impulses) a t a r ate of
(a) 150 Hz ( b) 200 Hz (c) 300 Hz. For each of t he t hree cases (i) sketch t he s pectrum
o f t he s ampled signal, (ii) explain if you can recover t he signal j (t) from t he s ampled
signal, (iii) i f t he s ampled signal is passed t hrough a n i deallowpass filter of b andwidth
100 Hz, s ketch t he s pectrum of t he o utput signai.
5 .1-4 O ne r ealization of a practical zero-order hold circuit is presented in Fig. P5.1-4.
( a) F ind t he u nit impulse response of this circuit. Hint: Recall t hat t he impulse
response h (t) is t he o utput o f t he c ircuit in Fig. P5.1-4 when t he i nput j (t) = o (t).
( b)Find t he t ransfer function H (w), aEd s ketch IH(w)l·
( c)Show t hat when a sampled signal j (t) is applied a t t he i nput o f this circuit, t he
o utput is a s taircase a pproximation of j (t). T he sampling interval is T . >.1-2 Input ~ ~~
F ig. P 5.1-4 5 .1-5 ( a) A f irst-order hold circuit can also be used t o r econstruct a signal j (t) from its
samples. T he impulse response of this circuit is h (t) = 6. (2~) where T is t he s ampling interval. Consider a typical sampled signal ! (t) a nd show
t hat t his c ircuit performs t he l inear interpolation. In other words, t he filter o utput
consists of s ample t ops connected b y s traight line segments. Follow t he p rocedure
discussed i n Sec. 5.1-1 (Fig. 5.3b).
( b) D etermine t he t ransfer function of this filter, a nd its amplitude response, and
compare i t w ith t he ideal filter required for signal reconstruction.
( c) T his filter, being noncausal, is unrealizable. B y delaying its impulse response,
t he filter c an b e m ade realizable. W hat is t he minimum delay required t o make
i t r ealizable? How would this delay affect t he r econstructed signal a nd t he filter
( d) Show t hat t he filter in p art ( c) c an b e realized by a filter depicted in Fig. P5.1-4
followed b y a n identical filter in cascade. Hint: show t hat t he impulse response of
this circuit is 6.( fT) delayed by T seconds.
5.1-6 A signal j et) = sinc(2001l"t) is s ampled by a periodic pulse t rain PT(t) r esented in
Fig. P 5.1-6. F ind a nd sketch t he s pectrum of t he sampled signal. Explain...
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