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Chapter 2 The Laplace Transformation 2 32 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 8 . Derive the Laplace transform for the sawtooth waveform below. 9 . Derive the Laplace transform for the full rectified waveform below. Write a simple MATLAB script that will produce the waveform above. f ST t () A a 2a t f t 3a f FR t f t π 2 π 3 π 4 π

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 2 33 Copyright © Orchard Publications Solutions to End of Chapter Exercises 2.8 Solutions to End of Chapter Exercises 1 . From the definition of the Laplace transform or from Table 2.2, Page 2 22, we obtain: a b. c. d. e. 2 . From the definition of the Laplace transform or from Table 2.2, Page 2 22, we obtain: a. b. c. d. e. 3 . a. From Table 2.2, Page 2 22, and the linearity property, we obtain b. and c. d. e. This answer for part (e) looks suspicious because and the Laplace transform is unilateral, that is, there is one to one correspondence between the time domain and the complex frequency domain. The fallacy with this procedure is that we assumed that if and , we cannot conclude that . For this exercise , and as we’ve learned, multiplication in the time domain corre- sponds to convolution in the complex frequency domain. Accordingly, we must use the Laplace transform definition and this requires integration by parts. We skip this analytical derivation. The interested reader may try to find the answer with the MAT- LAB script syms s t; 2*laplace(sin(4*t)/cos(4*t)) 4 . From (2.22), Page 2 6, a. 12 s 6s e 12s 24 s ----- 5s 2 4 5 ! s 6 ---- j8 s 5 s5 + ----------- 8 7 ! + () 8 ------------------ 15e 4s 3 ! s 4 32 ! × s 3 ------------- 4 s 2 3 s -- ++ + t 3 δ t3 = δ 9 δ == 9 δ 9e 3s 3 5 s 2 5 2 + ---------------- 5 s s 2 3 2 + 24 t tan 2 4t sin cos 2 2 2 2 + ss 2 2 2 + --------------------------- 8 s = = 8s 8u 0 t f 1 t F 1 s f 2 t F 2 s f 1 t f 2 t ---------- F 1 s F 2 s ------------ f 1 t f 2 t 1 cos sin = t tan e st t d 0 t n ft 1 n d n ds n -------Fs 31 1 d 5 s 2 5 2 +   3 52 s s 2 25 + 2 ----------------------- 30s s 2 25 + 2
Chapter 2 The Laplace Transformation 2 34 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications b.

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