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

With yao 4 yao 2 2 7 repeat prob 22 1 if 3 according

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Unformatted text preview: he convolution o f two odd or two even functions is a n even function. Hint: Use t ime-scaling property of convolution in P roblem 2.4-2 . .4 -4 Using direct integration, find e -a'u(t) . 4-5 167 P roblems Using direct integration, find u (t) . 4-6 . 4-7 I (r) s in t Fig. P 2.4-14 'I ,'" * e -b'u(t). ( c) e - 2'u(t) 1 __ F ig. P 2.4-15 T he u nit i mpulse response of a n LTIC s ystem is h (t) = e -'u(t). F ind t his system's (zero-state) response yet) if t he i nput f (t) is: ( b) e -'u(t) 21t 1-- * u(t), e -a'u(t) * e -a'u(t), a nd t u(t) * u(t). Using direct i ntegration, find sin t u(t) * u (t) a nd cos t u(t) * u (t) . ( a) u (t) 1-- 1-- 2 .4-12 A first-order allpass filter impulse response is given by ( d) s in 3 tu(t). h (t) = - 8(t) + 2 e-'u(t) Use t he c onvolution table to find your answers. .4 -8 ( a) F ind t he z ero-state response of this filter for t he i nput e 'u( - t) . R epeat P rob. 2.4-7 if h (t) a nd if t he i nput f (t) is: ( a) u (t) . 4 -9 = [2e- 3' - ( b) Sketch t he i nput a nd t he c orresponding zero-state response. e- 2'] u(t) ( b) e -'u(t) 2 .4-13 S ketch t he functions f (t) = ( c) e - 2'u(t). 2 .4-15 F ind a nd s ketch c(t) R epeat P rob. 2.4-7 if ~-1l = 4 e- 2' cos 3 tu(t) and if t he i nput f (t) = f (t) * get) * get). for t he f unctions depicted in Fig. P 2.4-l5 . 2 .4-17 For a n LTIC s ystem, if t he ( zero-state) response t o a n i nput f (t) is yet), show t hat t he ( zero-state) response t o t he i nput j et) is yet) a nd t hat for t he i nput J~oo f (T) dT a nd if t he i nput f (t) = u (t). R epeat P rob. 2.4-7 if h (t) e -'u(t) . a nd u (t). Now find f (t) * u(t) a nd s ketch t he result. 2 .4-16 F ind a nd s ketch c(t) = h (t)*h(t) for t he pairs offunctions illustrated in Fig. P2.4-16. h (t) = ( 1- 2t)e- 2'u(t) ~-10 $ 2 .4-14 F igure P2.4-14 shows f (t) a nd get). F ind a nd s ketch c(t) = f (t) R epeat P rob. 2.4-7 if is: ( a) u (t) ( b) is J~ooY(T)dT. Hint: Recognize t hat j et) = I imT_o ,jolf(t) - f(t - T)J. Now use linearity a nd t ime f (T) dT = f (t) * u(t). invariance t o find t he response t o j et). Also, recognize t hat too h (t) = e -'u(t) 2 .4-18 I f f (t) a nd i f t he i nput f (t) is: ( a) e - 2'u(t) ( b) e - 2 ('-3)u(t) ( c) e - 2'u(t - 3) g ate pulse d epicted in Fig. P2.4-11. For ( d), sketch yet). * get) = c(t), t hen show t hat ( d) t he Hint: T he i nput in ( d) c an be expressed as u (t) - u(t - 1). For p arts ( c) a nd ( d), use t he s hift p roperty (2.34) of convolution. (Alternatively, you may want t o invoke t he s ystem's time-invariance a nd s uperposition properties.) j et) * get) = f (t) * get) = c(t) E xtend t his result to show t hat J 'm)(t) * g(n)(t) = c(m+n)(t) w here x(m)(t) is t he m th d erivative of x (t), a nd all t he derivatives of f (t) a nd get) in this integral exist. Hint: Use t he first p art o f t he h int in P rob. 2.4-17 a nd t he t ime-shift property of convolution. o 1-- Fig. P 2.4-11 2 .4-19 As mentioned in C hapter 1 (Fig. 1.27b), i t is possible to express a n i nput in term...
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