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

# 36 6 the width property i f t he d urations widths of

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Unformatted text preview: 6 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 ready-made 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 e-c&gt;t c os ({3t + 9)u(t) eAtu(t) cos (9 - &lt;I»e At - e-c&gt;t cos ({3t J (a &lt;I&gt; 13 14 = +9 - + ),)2 + {32 t an- 1 [-{3/(a + )')l &lt;1» u (t) 126 2 Time-Domain Analysis of Continuous-Time 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) [e-3' - 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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