S&S Midterm Quiz 1112-1 (Q) (Solutions).pdf

S&S Midterm Quiz 1112-1 (Q) (Solutions).pdf

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Page 1 of 4 EE2023 Signals & Systems Quiz Semester 1 AY2011/12 Date: 6 October 2011 Time Allowed: 1.5 hours Q.1 Consider the periodic signal ( ) x t given by the expression ( ) ( ) ( ) 3 2 2 3 2 2 3 5 3 2 2 j t j t j t j t x t j e je je j e = + + + + (a) What is the fundamental period and fundamental frequency of ( ) x t ? ANSWER: Fundamental frequency: { } ( ) 1 2,3 1 or 2 HCF rad s Hz π = Fundamental period: 2 s π (b) Sketch the amplitude and phase spectra of ( ) x t . ANSWER: ( ) 45 3 90 2 90 2 45 3 8 3 5 3 8 j j t j j t j j t j j t x t e e e e e e e e ° − ° − ° ° = + + + + (c) Is ( ) x t a real signal ? Justify your answer. ANSWER: ( ) x t is a real signal because its magnitude and phase spectra are even and odd function of frequency, respectively. The Fourier series coefficients satisfy the condition that k k X X = . (d) What is the power of ( ) x t ? ANSWER: Using Parseval’s theorem: ( ) ( ) ( ) 2 2 2 Power of 2 8 2 3 5 16 18 25 59 x t = × + × + = + + = k X 3 5 8 3 8 0 -3 -1 2 3 or k ω -2 1 0 -3 -1 2 3 or k ω -2 1 k X 45 ° 45 ° 90 ° 90 ° 0 °
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Page 2 of 4 Q.2 Derive the Fourier transform of the signal ( ) x t shown in Figure Q2-1.
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