MIT6_003S10_assn10

MIT6_003S10_assn10 - System B x 1 ( t ) x 3 ( t ) System C...

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6.003 Homework 10 Problems 1. DT Fourier Series Due at the beginning of recitation on Wednesday, April 21, 2010 . Find the Fourier transforms of the following signals, which are 0 for | n | > 7. Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. n 1 x 1 [ n ] 1 / 2 n 1 x 2 [ n ] n 1 x 3 [ n ] n 1 x 4 [ n ] 1 / 2 2. Inverse DT Fourier Series Determine the DT signals with the following Fourier series coefficients. Assume that the signals are periodic in N = 8. k 1 a k 1 / 2 k 1 b k 3. DT Fourier transforms n x 1 [ n ] 1 n x 2 [ n ] 1 n x 3 [ n ] 1 1 n x 4 [ n ] 1
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6.003 Homework 10 / Spring 2010 2 4. Which are True? For each of the DT signals x 1 [ n ] through x 4 [ n ] (below), determine whether the conditions listed in the following table are satisfied, and answer T for true or F for false. X ( e j 0 ) = 0 s π π X ( e j ) d Ω = 0 X ( e j ) is purely imaginary e jk X ( e j ) is purely real for some integer k x 1 [ n ] x 2 [ n ] x 3 [ n ] x 4 [ n ] 1 n x 1 [ n ] 1 n x 2 [ n ] 1 n x 3 [ n ] 1 n x 4 [ n ]= x 4 [ n + 5]
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3 6.003 Homework 10 / Spring 2010 5. Input/Output Pairs The following signals are all periodic with period T = 1. 1 x 1 ( t )= x 1 ( t + 1) t 0 1 4 1 1 x 2 ( t )= x 2 ( t + 1) t 0 1 4 1 1 x 3 ( t )= x 3 ( t + 1) t 0 π 10 1 Indicate which of the systems could or could not be linear and time-invariant. System A x 1 ( t ) x 2 ( t )
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Unformatted text preview: System B x 1 ( t ) x 3 ( t ) System C x 2 ( t ) x 1 ( t ) System D x 2 ( t ) x 3 ( t ) System E x 3 ( t ) x 1 ( t ) System F x 3 ( t ) x 2 ( t ) 4 6.003 Homework 10 / Spring 2010 Engineering Design Problem 6. Image reconstruction The rows and/or columns of the following images have been blurred. Figure out a way to sharpen each image, and identify the building. Here are thumbnails of the images: a1 a2 b1 b2 c1 c2 The images are available in machine-readable form (buildings.zip) on our website. MIT OpenCourseWare http://ocw.mit.edu 6.003 Signals and Systems Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT6_003S10_assn10 - System B x 1 ( t ) x 3 ( t ) System C...

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