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PracticeExam%232sol_05

PracticeExam%232sol_05 - ECE 108 Exam 2 Solution 1 Given...

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ECE 108 Exam 2 3/25/05 — Solution 1. Given the following periodic signals (30pts) (a) Show how you would determine the Fourier coefficients of x(t) using the integral formula. Make sure you properly define the limits of integration and appropriately express x(t) mathematically. Do not complete the integration in this part. ( ) ( ) ( ) ( ) 0 0 0 2 0 2 0 0 2 3 3 2 0 1 1 2 2 for 6 6 3 1 1 2 2 6 6 jk t jk t k jk t jk t X t e dt t e dt t e dt t e dt ω ω π π π ω = + + − + = = + (b) It can be shown that the Fourier coefficients of the signal x ( t ) are given by the equation 2 1 3 3 k X Sa k π = Express g ( t ) in terms of x ( t ). Use this result and the properties of the Fourier series to determine the Fourier coefficients, , of signal g(t). Make sure that the coefficient k G G 0 is correct. ( ) ( ) 3 2 g t x t =− + Using multiplication and time shifting property 0 3 2 0 1 4 = for 0 and 3 3 3 jk jk k k G X e Sa k e k G ω π π =− =
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