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convolution o f two odd or two even functions is a n even function.
Hint: Use t imescaling property of convolution in P roblem 2.42 . .4 4 Using direct integration, find e a'u(t) . 45 167 P roblems Using direct integration, find u (t) . 46
. 47 I (r) s in t Fig. P 2.414 'I ,'" * e b'u(t). ( c) e  2'u(t) 1 __ F ig. P 2.415 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
(zerostate) 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 .412 A firstorder 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 erostate response of this filter for t he i nput e 'u(  t) . R epeat P rob. 2.47 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 zerostate response. e 2'] u(t) ( b) e 'u(t) 2 .413 S ketch t he functions f (t) = ( c) e  2'u(t). 2 .415 F ind a nd s ketch c(t) R epeat P rob. 2.47 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 2.4l5 . 2 .417 For a n LTIC s ystem, if t he ( zerostate) response t o a n i nput f (t) is yet), show t hat
t he ( zerostate) 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. 2.47 if h (t)
e 'u(t) . a nd u (t). Now find f (t) * u(t) a nd s ketch t he result. 2 .416 F ind a nd s ketch c(t) = h (t)*h(t) for t he pairs offunctions illustrated in Fig. P2.416. h (t) = ( 1 2t)e 2'u(t) ~10 $ 2 .414 F igure P2.414 shows f (t) a nd get). F ind a nd s ketch c(t) = f (t) R epeat P rob. 2.47 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 .418 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. P2.411. 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 timeinvariance 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. 2.417 a nd t he t imeshift property of
convolution. o 1 Fig. P 2.411 2 .419 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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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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