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# hw11 - Could the system whose input is x 1 n and output is...

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6.003 Homework 11 Due at the beginning of recitation on Wednesday, November 25, 2009 . Problems 1. DT Fourier Series 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 Find the Fourier transforms of the following signals, which are 0 for | n | > 7 . 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 11 / Fall 2009 2 4. Pulsatile input Let the following periodic signal x [ n ] = m = −∞ δ [ n 3 m ] + δ [ n 1 3 m ) δ [ n 2 3 m ] be the input to an LTI system with frequency response H ( e j ) = j 2 sin(Ω) Determine the Fourier series coefficients b k of the resulting output signal y [ n ] . 5. Input/output pairs The following signals are all periodic in N = 15 . n 15 0 15 x 1 [ n ]= x 1 [ n + 15] 1 n 15 0 15 x 2 [ n ]= x 2 [ n + 15] 1 n 15 0 15 x 3 [ n ]= x 3 [ n + 15] 1 a. Could the system whose input is x 1 [ n ] and output is x 2 [ n ] be linear and time-invariant? b. Could the system whose input is
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Unformatted text preview: Could the system whose input is x 1 [ n ] and output is x 3 [ n ] be linear and time-invariant? c. Could the system whose input is x 2 [ n ] and output is x 1 [ n ] be linear and time-invariant? d. Could the system whose input is x 2 [ n ] and output is x 3 [ n ] be linear and time-invariant? e. Could the system whose input is x 3 [ n ] and output is x 1 [ n ] be linear and time-invariant? f. Could the system whose input is x 3 [ n ] and output is x 2 [ n ] be linear and time-invariant? 6.003 Homework 11 / Fall 2009 3 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....
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