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R epeat E xercise E2.5 if t he i nput I tt) = e tu(t).
Answer; 6 te t u(t)
'V
£:, The Convolution Table
T he task o f convolution is considerably simplified by a readymade convolution
table (Table 2.1). This table, which lists several pairs of signals a nd their resulting
convolution, c an conveniently determine y(t), a system response t o a n input f (t),
w ithout performing the tedious j ob o f integration. For instance, we could have
readily found t he convolution in Example 2.4 using pair 4 (with ), 1 =  1 a nd
t
2t
A 2 =  2) to be ( e  e )u(t). T he following example demonstrates the utility o f
t his table. 12 ec>t c os ({3t + 9)u(t) eAtu(t) cos (9  <I»e At  ec>t cos ({3t J (a
<I> 13 14 = +9  + ),)2 + {32 t an 1 [{3/(a + )')l <1» u (t) 126 2 TimeDomain Analysis of ContinuousTime S ystems • E xample 2.5
Find the loop current yet) of the R LC circuit in Example 2.2 for the input f (t) =
3
1 0e 'u(t), when a ll t he initial conditions are zero.
The loop equation for this circuit [see Example 1.11 or Eq. (1.55)] is
(D2 + 3 D + 2) yet) = * [2e 2'  e'] u (t) Using the distributive property of the convolution [Eq. (2.32)], we obtain
y et) * 2 e 2'u(t)  lOe 3'u(t) * e 'u(t)
= 20 [ e 3'u(t) * e  2'u(t)]10 [ e 3'u(t) * e 'u(t)]
= 3 l Oe 'u(t) (2.44) Now the use of P air 4 in Table 2.1 yields
y(t ) 20
=  3(2) [e3'  e 2'] u(t)  = 10
[ 3,
']
3(1) e
 e u(t) (e 3'  e 2') u (t) + 5 (e 3'
( 5e' + 2 0e 2'  15e 3') u (t) =  20  e') u (t)
(2.43) •
6 E xercise E 2.7 Rework Probs. E2.5 and E2.6 using the convolution table.
6 Graphical Understanding o f Convolution T o have a p roper g rasp of convolution operation, we should u nderstand t he
g raphical i nterpretation of convolution. Such a c omprehension also helps in evaluating t he c onvolution integral of more complicated signals. I n a ddition, graphical
convolution allows us t o g rasp visually or mentally t he c onvolution integral's result, which c an b e o f g reat h elp in sampling, filtering, a nd m any o ther p roblems.
Finally, m any signals have no exact m athematical d escription, so t hey c an b e described only graphically. I f two such signals are t o b e convolved, we have no choice
b ut t o p erform t heir c onvolution graphically.
We shall now explain t he c onvolution o peration b y convolving t he signals f (t)
a nd get), i llustrated i n Figs. 2.7a a nd 2.7b respectively. I f c(t) is t he c onvolution of
f (t) w ith get), t hen = f (t) * h (t)
3 O ne o f t he c rucial points t o r emember here is t hat t his i ntegration is performed w ith
r espect t o r , so t hat t is j ust a p arameter (like a c onstant). T his c onsideration is
especially i mportant w hen we s ketch t he g raphical representations o f t he f unctions
f (r) a nd get  r) a ppearing i n t he i ntegrand o f Eq. (2.44). B oth o f t hese f unctions
should b e s ketched as functions of r , n ot of t.
T he f unction f er) is identical t o f (t), w ith r r eplacing...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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