Explanation Given data: Refer to Figure 15.36 in the textbook. Formula used: Write the general expression to find the convolution integral. y ( t ) = h ( t ) ∗ x ( t ) = ∫ 0 t x ( λ ) h ( t − λ ) d λ (1) Calculation: The equation (1) can be rewritten as, z ( t ) = ∫ 0 t y ( λ ) x ( t − λ ) d λ (2) For t < 0 : Figure 1 shows the convolution of x ( t ) and y ( t ) for t < 0 . In Figure 1, for t < 0 there is no overlap of the two functions so that z ( 0 ) = 0 . For 0 < t < 2 : Suppose the given Figure of x ( t ) is fold and shift it by t as shown in Figure 2. Figure 2 shows the convolution of x ( t ) and y ( t ) for 0 < t < 2 . From Figure 2, x ( t − λ ) and y ( λ ) overlap from 0 to t . Therefore, x ( t − λ ) = 2 and y ( λ ) = 4 . Substitute 2 for x ( t − λ ) and 4 for y ( λ ) in equation (2) to find z ( t ) for 0 < t < 2 . z ( t ) = ∫ 0 t ( 2 ) ( 4 ) d λ = 8 ( λ ) 0 t = 8 ( t − 0 ) = 8 t For 2 < t < 6 : Figure 3 shows the convolution of x ( t ) and y ( t ) for 2 < t < 6 . The overlap shown in Figure 3. Therefore, z ( t ) = ∫ 0 2 ( 2 ) ( 4 ) d λ + ∫ 0 t ( 2 ) ( − 4 ) d λ = 8 ( λ ) 0 2 − 8 ( λ ) 0 t = 8 ( 2 − 0 ) − 8 ( t − 0 ) = 16 − 8 t For 6 < t < 8 : Figure 4 shows the convolution of x ( t ) and y ( t ) for 6 < t < 8 . The overlap shown in Figure 4

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Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is …

Question

Chapter 15, Problem 41P

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Graphically convolve x(t) and y(t) shown in Figure 15.36. Find z(t)=x(t)∗y(t).

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Chapter 15, Problem 40P

Chapter 15, Problem 42P

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Chapter 15 Solutions

EBK FUNDAMENTALS OF ELECTRIC CIRCUITS

Ch. 15.2 - Prob. 1PP Ch. 15.2 - Prob. 2PP Ch. 15.3 - Prob. 3PP Ch. 15.3 - Prob. 4PP Ch. 15.3 - Prob. 5PP Ch. 15.3 - Prob. 6PP Ch. 15.3 - Obtain the initial and the final values of...Ch. 15.4 - Prob. 8PP Ch. 15.4 - Find f(t) if F(s)=48(s+2)(s+1)(s+3)(s+4)Ch. 15.4 - Obtain g(t) if G(s)=s3+2s+6s(s+1)2(s+3)

Ch. 15.4 - Find g(t) given that G(s)=20(s+1)(s2+4s+13)Ch. 15.5 - Graphically convolve the two functions in Fig....Ch. 15.5 - Given g(t) and f(t) in Fig. 15.20, graphically...Ch. 15.5 - Use convolution to find vo(t) in the circuit of...Ch. 15.6 - Prob. 15PP Ch. 15.6 - Prob. 16PP Ch. 15 - Prob. 1RQ Ch. 15 - Prob. 2RQ Ch. 15 - Prob. 3RQ Ch. 15 - Prob. 4RQ Ch. 15 - Prob. 5RQ Ch. 15 - If F(s) = 1/(s + 2), then f(t) is (a) e2t u(t) (b)...Ch. 15 - Prob. 7RQ Ch. 15 - Prob. 8RQ Ch. 15 - Prob. 9RQ Ch. 15 - Prob. 10RQ Ch. 15 - Prob. 1P Ch. 15 - Prob. 2P Ch. 15 - Prob. 3P Ch. 15 - Prob. 4P Ch. 15 - Prob. 5P Ch. 15 - Prob. 6P Ch. 15 - Prob. 7P Ch. 15 - Prob. 8P Ch. 15 - Prob. 9P Ch. 15 - Prob. 10P Ch. 15 - Find F(s) if: (a) ft=6etcosh2t (b) ft=3te2tsinh4t...Ch. 15 - If g(t) = 4e 2t cos 4t, find G(s).Ch. 15 - Prob. 13P Ch. 15 - Prob. 14P Ch. 15 - Prob. 15P Ch. 15 - Prob. 16P Ch. 15 - Prob. 17P Ch. 15 - Prob. 18P Ch. 15 - Prob. 19P Ch. 15 - Prob. 20P Ch. 15 - Prob. 21P Ch. 15 - Prob. 22P Ch. 15 - Prob. 23P Ch. 15 - Design a problem to help other students better...Ch. 15 - Let F(s)=18(s+1)(s+2)(s+3) (a) Use the initial and...Ch. 15 - Determine the initial and final values of f(t), if...Ch. 15 - Prob. 27P Ch. 15 - Prob. 28P Ch. 15 - Prob. 29P Ch. 15 - Prob. 30P Ch. 15 - Find f(t) for each F(s): (a) 10ss+1s+2s+3 (b)...Ch. 15 - Prob. 32P Ch. 15 - Prob. 33P Ch. 15 - Prob. 34P Ch. 15 - Obtain f(t) for the following transforms: (a)...Ch. 15 - Prob. 36P Ch. 15 - Prob. 37P Ch. 15 - Prob. 38P Ch. 15 - Determine f(t) if: (a)...Ch. 15 - Show that...Ch. 15 - Prob. 41P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 43P Ch. 15 - Prob. 44P Ch. 15 - Given h(t) = 4e2tu(t) and x(t) = (t) 2e 2tu(t),...Ch. 15 - Given the following functions...Ch. 15 - A system has the transfer function...Ch. 15 - Find f(t) using convolution given that: (a)...Ch. 15 - Prob. 49P Ch. 15 - Prob. 50P Ch. 15 - Given that v(0) = 5 and dv(0)/dt = 10, solve...Ch. 15 - Prob. 52P Ch. 15 - Prob. 53P Ch. 15 - Design a problem to help other students better...Ch. 15 - Prob. 55P Ch. 15 - Solve for v(t) in the integrodifferential equation...Ch. 15 - Prob. 57P Ch. 15 - Given that dvdt+2v+50tv()d=4u(t) with v(0) = 1,...Ch. 15 - Solve the integrodifferential equation...Ch. 15 - Prob. 60P

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