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

# To count t he n umber o f c omputations required in t

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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&quot;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 &lt; B &lt; 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 2 signals ( a) s inc 2 (1001l&quot;t) ( b) 0.01 sinc 2 (1001l&quot;t) ( c) sinc (IOO1l&quot;t) + 3 s inc (601l&quot;t) ( d) sinc (501l&quot;t)sinc (IOO1l&quot;t). &gt;.1-3 A signal j (t) = sinc (2001l&quot;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 . &gt;.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 frequency response? ( 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&quot;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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## 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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