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Unformatted text preview: 33 k =m k=m a ssuming a ll frequencies t o b e d istinct, t hat is,
1 .16 ( a) 10 cos ( lOOt +
( c) (10 i) + 2 s in 3t) cos ( e) 10 sin 5 t c os lOt
1 .31 Wi i ' W k for all i i ' k
4 D etermine t he p ower a nd t he r ms value for each o f t he following signals:
( b) 10 cos (lOOt lOt + i) + 16 sin ( 150t+ (a) i)
1 ( d) 10 cos 5 t cos lOt ( f) ejo. t cos wot 4 I n Fig. P 1.31, t he s ignal /1(t) = f (  t). E xpress signals h (t), h (t), f4(t), a nd f5(t)
in t erms o f s ignals f (t), /1 (t), a nd t heir t imeshifted, timescaled o r t imeinverted
versions. F or i nstance h (t) = f (t  T) + / 1(t  T). f it) F ig. P l.42
1 .41 S ketch t he s ignals ( a) u (t5)u(t7) ( b) u (t5)+u(t7) ( c) t 2 [u(t1)u(t2)]
( d) (t  4)[u(t  2)  u(t  4)] 1 .42 E xpress each of t he s ignals in Fig. P1.42 by a single expression valid for all t. 1 .43 F or a n e nergy signal f (t) w ith e nergy E f, show t hat t he e nergy of a nyone o f t he
s ignals  f(t), f (  t) a nd f (t  T) is E f. Show also t hat t he e nergy o f f rat) as well
as f rat  b) is E fla. T his s hows t hat t imeinversion a nd t imeshifting does n ot affect
signal energy. O n t he o ther h and, t ime c ompression of a signal (a > 1) reduces t he
energy, a nd t ime e xpansion o f a signal (a < 1) i ncreases t he energy. W hat is t he
effect o n s ignal energy if t he s ignal is multiplied by a c onstant a? 1 .44 Simplify t he following expressions: o ( b) ( a)
(c) I o ( s in t ) 6(t)
t2 + 2
[ e t cos (3t  60 )]6(t) 2 0 ( lW + 2 ) 6(w)
w2 + 9 ( d) C in 1 .32 For t he s ignal f (t) d epicted in Fig. P 1.32, s ketch t he signals: ( a) f (t) ( b) f (t+6)
( c) f (3t) ( d) f(~). 1 .33 F or t he s ignalf(t) i llustrated in Fig. P1.33, s ketch ( a) f (t  4) ( b) f ( 1 \) ( c) f (  t)
( d) f (2t  4) ( e) f (2  t). ( e) ( _._ 1_) 6(w
JW + 2 + 3) ( f) (SinwkW) 6(w) [2 ~(t
t F ig. P 1.31 H int: Use Eq. (1.23). For p art ( f) u se L 'H6pital's rule. +4 2)]) 6 (t1) 100
1 .45 1 I ntroduction t o S ignals a nd S ystems
E valuate t he following integrals: ( a)
( b)
( c) I:
I:
I:
I: b (r)f(t  r)dr ( e) j (r)8(t  r)dr ( f) 6 (t)e jwt ( g) dt I:
I:
I:
I: 101 P roblems
1 .51
1 .61 (t 3 + 4)6(1  t) dt W rite t he i nputoutput r elationship for a n ideal integrator. Determine t he z eroinput
a nd z erostate components of t he response. 1 . 7 1 t
8(t + 3 )e dt F ind a nd s ketch t he o dd a nd t he even components of ( a) u (t) ( b) t u(t) ( c) sin wat u (t)
( d) cos w atu(t) ( e) s in wat ( f) cos wat. 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 a re l inear a nd which are nonlinear. f (2  t)8(3  t) dt [~(x  dy
( a) dt + 2y(t) 2 ( b) ~~ + 3ty(t) = Hint: 8(x) i s l ocated a t x = O. For example, 8(1  t) is located a t 1  t = 0, a nd so
on. ( e) (~~) ( a) F ind a nd sketch df / dt for t he signal f (t) shown in Fig. P1.33.
( b) F ind a nd sketch d2f / dt 2 for t he signal f (t) d epi...
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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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