Practice+Exam+3_ans - Practice Exam#3 Fourier series and...

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Practice Exam #3 1 Practice Exam #3 Fourier series and transform Lessons 24-33 No Challenge 33 Some answers are derived using 1 st principle, others tables or class notes. Unless specified, your choice. HKN Circuits 2 Review on the 15th from 6:15 to 8:20 PM in Benson 328.
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Practice Exam #3 2 Problem 1 X(t) R=1 C=1 y(t) a) If x(t) = 15 e -5t u(t), what is the Fourier transform of x(t) (i.e., X( ))? Laplace transform method (Lesson 30, Slide 23 or Table 17.1) X(s) = 15/(s+5) or X( ) = 15/(j +5). R=10k C=10 F b) What is H( )? Voltage Divider: Laplace transform method H(s) = (1/RC)/(s+(1/RC))=10/(s+10), or H( ) = 10/(j +10).
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Practice Exam #3 3 X(t) R=1 C=1 y(t) c) What is Y( )? Convolution theorem: Y( )=H( )X( ) = 150/((j +5)(j +10)). R=10k C=10 F d) What is y(t)? Y( )=150/((j +5)(j +10))= ?? (not in Tables) Y( )= -30/(j +10) + 30/(j +5) (Heaviside expansion) y(t) = -30 e -10t u(t) + 30 e -5t u(t) (Table 17.1 or textbook back cover)
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Practice Exam #3 4 Problem 2 Given an at-rest linear system characterized by: H( ) = {1 for | | 1 r/s, 0 otherwise} and an input x(t) = |sin(t/2)| (full wave rectified sinewave) having a known trigonometric Fourier series representation (back cover table): Even symmetry – cosine series. 0 2 0 1 1 H( ) Ideal 0 phase lowpass filter ) cos( ) ( ; ; / ; nt n t x f T n 1 4 1 4 2 1 2 1 2 2 1 0 0 0 π π ω π π
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Practice Exam #3 5 Problem 2 a) What is the trigonometric Fourier series of the output signal y(t)? 0 2 ) cos( ) cos( ) ( t t t y π π π π 3 4 2 3 1 4 2 1 {a 0 , a 1 } >> t=0:.001:2*pi; >> x=(2/pi)-(4/(3*pi))*cos(t); >> plot(t,abs(sin(t/2)),t,x) Output Input The ideal filter H( ) can only pass the 0 th and 1 st harmonic. Therefore:
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Practice Exam #3 6 ) ( ) sin( ) ( ) sin( ) ( π π t u t t u t t x b) What is the Fourier transform of x(t)?
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