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Unformatted text preview: cted in Fig. P1.42a. ( d) ( h) b (t  2) sin 1rtdt e (xl) cos 5)]8(x  3) dx 1 . 7 2 CD (a) (b) 2 f (t) ~~ + (sin t )y(t) = ~ + 2 f(t) ( g) 2
t f (t) + 2y(t) = ~~ + 2y(t) = f(t)~ ( h) y (t) = 2 ( a) y(t) 1 1 ~ED = f (t  ( b) y(t) 1 = f (  t) ( e) y (t) = = f lat) ( f) y (t) F ig. P l.47 1 .47 Using t he g eneralized function definition, show t hat 8(t) is a n even function of t. f (r) d r 2) ( d) y (t) = t f(t  2) ISs f(r)dr = (~) 2 F ind a nd s ketch J~oo f (x) dx for t he signal f (t) i llustrated in Fig. P l.47. 1 .48 loo For t he s ystems described by t he e quations below, w ith t he i nput f (t) a nd o utput
y (t), d etermine which of t he s ystems are timeinvariant p arameter s ystems a nd which
are timevarying p arameter s ystems. ( c) y (t) o + 2 = f (t) ( f) ~~ + y2(t) = f (t) ( d) 1 .46 ( c) 3 y(t) = f (t) Hint: S tart w ith Eq. (1.24a) as t he definition of 6(t). Now change variable t
show t hat I: 1 .49 P rove t hat =  x to q ,(t)6(t)dt = q,(O)
1 8(at) = ~6(t)
Hint: Show t hat 1 1 00  00 . . 4 10 Show t hat q,(t)8(at) dt I: = ~q,(0) 8(t)q,(t) dt = 4>(0) 1 . 7 3 F or a certain LTI system with t he i nput f (t), t he o utput y (t) a nd t he two initial
conditions Xl(O) a nd X2(O), following observations were made: f (t) o
o
u (t) 1 e2t cos 3t ( d) e  2t ( e) e2t ( f) 5. e tu(t) 2 e  t (3t 1 1 + 2)u(t) 2u(t) D etermine y (t) when b oth t he i nitial conditions are zero a nd t he i nput f (t) is as
shown in Fig. P l.73.
Hint: T here a re t hree causes: t he i nput a nd each of t he two initial conditions. Because
of linearity property, if a cause is increased b y a f actor k, t he r esponse t o t hat cause
also increases by t he s ame factor k. Moreover, if causes are added, t he c orresponding
responses add. w here q,(t) a nd 4>(t) a re continuous a t t = 0, a nd q,(t)  > 0 a s t  > ± oo. T his integral
defines 8( t) a s a generalized function. Hint: Use i ntegration by p arts .
.. 4 11 A sinusoid eat cos wt c an be expressed as a s um o f exponentials est a nd e · t (Eq.
( l.30c) w ith complex frequencies s = a + j w a nd s = a  jw. L ocate in t he complex plane t he frequencies of t he following sinusoids: ( a) cos 3t ( b) e  3t cos 3t ( c) y (t) Xl(O) 5 1 F ig. P 1.73 f {t) t __ ~ 102 1 .14 1 I ntroduction t o S ignals a nd S ystems 1 .78 A system is specified by its i nputoutput relationship as = loo ( c) yet) J(T)dT = ret) n , integer ( d) yet) = cos [J(t)] ( b) yet) = J(3t  6) Show t hat t he circuit in Fig. P1.75 is zerostate linear b ut is not zeroinput linear.
Assume all diodes to have identical (matched) characteristics.
Hint: In zero s tate (when the initial capacitor voltage vc(O) = 0), t he circuit is linear.
If t he i nput J (t) = 0, a nd vc(O) is nonzero, t he c urrent yet) does not exhibit linearity
with respect to its cause vc(O).
y (t) For t he systems described by t he e quations below, with the i nput J(t) a nd o utput
yet), d etermine which of t he s...
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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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