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Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 27 Copyright © Orchard Publications Solutions to End of Chapter Exercises 4 . From tables of integrals, and thus Check: , , syms s t; ilaplace(4/(s^3+2*s^2)) ans = 2*t-1+exp(-2*t) 5 . To use the convolution integral, we must first find the impulse response. It was found in Exer- cise 1 as and with the given values, 1 t 0 0 1 0 v 2 τ () τ τ v 2 t τ t 0 v 2 t 4t v 1 t *v 2 t τ v 1 t 2 t v 1 τ v 2 t τ τ d 0 t 4 τ e 2t τ τ d 0 t 4e τ e 2 τ τ d 0 t == = xe ax x d e a 2 ------ ax 1 = v 1 t 2 t e 2 τ 2 τ 1 4 -------------------------- 0 t e e 1 e 0 1 [] e 0 2t 1 e + e 1 + = = v 1 t 2 t V 1 s V 2 s V 1 s 4s 2 = V 2 s 1s2 + = V 1 s V 2 s 4 s 2 s2 + --------------------- 4 s 3 2s 2 + ------------------ ht it 1L e RL t u 0 t e t u 0 t =

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Chapter 6 The Impulse Response and Convolution 6 28 Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition Copyright © Orchard Publications When the input is the unit step , 6 . We will first compute the impulse response, that is, the output when the input is the delta func- tion, i.e., . Then, by KVL and with + 1 H R i L t () u 0 t L 1 Ω u 0 t it v in u 0 t = u 0 t τ h τ ()τ d = 1 1 t 0 0 ht 1 0 u 0 τ τ τ τ t e t 1 0 u 0 t τ τ v u 0 t = 1 e τ τ d 0 t e τ 0 t e τ t 0 1e t u 0 t == = = 1 u 0 t *h t t R L + 1 H + 1 Ω v t v out t v t δ t = L di L dt ------- Ri L + δ t = i L x = 1x · + δ t =
Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 29 Copyright © Orchard Publications Solutions to End of Chapter Exercises or By comparison with , we observe that and . From (6.5) Now, we compute when by convolving the impulse response with this input , that is, . The remaining steps are as in Exam- ple 6.5 and are shown below.

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