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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 49 Copyright © Orchard Publications Solutions to End of Chapter Exercises 8.11 Solutions to End of Chapter Exercises 1 . and since at , whereas at , , we replace the limits of inte- gration with and . Then, 2 . From tables of integrals Then, With the upper limit of integration we obtain To evaluate the lower limit of integration, we apply L’Hôpital’s rule, i.e., and thus Check: and since u 0 t ()δ t () t d u 0 t t d d u 0 t t d u 0 t u 0 t d == t + = u 0 t 1 = t = u 0 t 0 = 10 u 0 t u 0 t d 0 1 u 0 2 t 2 ----------- 0 1 12 F ω ft e j ω t t d te at e j ω t t d 0 j ω a + t t d 0 = xe ax x d e a 2 ------ ax 1 = F ω e j ω a + t j ω a + t 1 [] j ω a + 2 ----------------------------------------------------------------- 0 j ω a + t1 + e j ω a + t j ω a + 2 --------------------------------------------- 0 F ω t0 = 1 j ω a + 2 --------------------- = j ω a + + e j ω a + t j ω a + 2 d dt ---- j ω a + + d ---- e j ω a + t j ω a + 2 -------------------------------------------------------- t lim j ω a + j ω a + e j ω a + t j ω a + 2 t lim 0 = F ω 1 j ω a + 2 = F ω Fs sj ω = =

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Chapter 8 The Fourier Transform 8 50 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications it follows that 3 . From Subsection 8.6.4, Page 8 30, and using the MATLAB script below, fplot('cos(x)',[ 2*pi 2*pi 1.2 1.2]) fplot('sin(x)./x',[ 20 20 0.4 1.2]) we obtain the plots below. 4 . From Subsection 8.6.1, Page 8 27, and from the time shifting property, te at u 0 t () 1 sa + 2 ------------------ F ω 1 + 2 sj ω = 1 j ω a + 2 --------------------- == A ω 0 tu 0 tT + u 0 [] cos AT ωω 0 T sin 0 T ------------------------------------- 0 + T sin 0 + T + F ω ft A 0 T T 0 ω 0 ω 0 2 π T ----- t ω Au 0 t3 T + u 0 + u 0 u 0 T + = t 0 T T A 3T 2T 0 + u 0 2AT ω T sin ω T --------------- ft t 0 F ω e j ω t 0
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 51 Copyright © Orchard Publications Solutions to End of Chapter Exercises Then, or 5 . From tables of integrals or integration by parts, Then, and multiplying both the numerator and denominator by we obtain We observe that since is real and odd, is imaginary and odd.

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