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MIT6_003S10_assn11

# MIT6_003S10_assn11 - 6.003 Homework 11 Please do the...

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6.003 Homework 11 Please do the following problems by Wednesday, April 28, 2010 . You need not submit your answers: they will NOT be graded. Solutions will be posted. Problems 1. Impulsive Input Let the following periodic signal x ( t ) = δ ( t 3 m ) + δ ( t 1 3 m ) δ ( t 2 3 m ) m = −∞ be the input to an LTI system with system function H ( s ) = e s/ 4 e s/ 4 . Let b k represent the Fourier series coeﬃcients of the resulting output signal y ( t ) . Deter- mine b 3 . 2. Fourier transform Part a. Find the Fourier transform of x 1 ( t ) = e −| t | . Part b. Find the Fourier transform of 1 x 2 ( t ) = . 1 + t 2 Hint: Try duality. 3. Fourier representations of the pulse In this problem you connect the four Fourier representations, as we did for you in Lecture 20, where we used a triangle as the canonical function. Here, you use a pulse: f ( t ) = 1 for 1 < t < 1 ; 0 otherwise. Part a. Find F ( ) , the continuous-time Fourier transform of f ( t ) . State Parseval’s theorem for the Fourier transform, and check that it works by applying it to f ( t ) and F ( ) . Part b. Sketch f p ( t ) , a periodic version of f ( t ) with period T = 4 . Find the Fourier transform of f p ( t ) and compare it to the Fourier-series coeﬃcients f k for f p ( t ) . State Parseval’s theorem for Fourier series.

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