hw04su10_soln

hw04su10_soln - Problem 4.1 a) We have x [ n ] = 3cos 15 π...

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Unformatted text preview: Problem 4.1 a) We have x [ n ] = 3cos 15 π 14 n- 3 π/ 7 + 4cos 5 π 7 n + π/ 3 + 2cos 3 π 28 n + π/ 12 = 3cos 13 π 14 n + 3 π/ 7 + 4cos 5 π 7 n + π/ 3 + 2cos 3 π 28 n + π/ 12 The discrete spectrum is shown below b) The ideal D-C simply multiplies the discrete frequencies in [- π,π ] by the sampling frequency, and so y ( t ) = 3cos 13 π · 2800 14 t + 3 π/ 7 + 4cos 5 π · 2800 7 t + π/ 3 + 2cos 3 π · 2800 28 t + π/ 12 = 3cos(2600 πt + 3 π/ 7) + 4cos(2000 πt + π/ 3) + 2cos(300 πt + π/ 12) Notice that the component that was originally at 1500 Hz in the input has been aliased, as in this case 1500 > f s / 2. c) x [ n ] = 7 + 0 . 5cos(2 πn ) + 3 . 5cos(4 πn ) = 11 ∀ n ∈ Z and so y ( t ) = 11 ∀ t ∈ R . d) x ( t ) has frequency components (in radians/sec) at 3333 π , 8888 π , and 2222 π . Thus we need f s ≥ 8888 Hz. 1 ECE 2025 HW #4 Solutions, Summer 2010 – June 26, 2010 Problem 4.2 To have a discrete frequency of 0 . 65 π , the (continuous) input frequency ω must obey ω 5555 = 0 . 65 π + 2 π‘ ⇒ ω = 3610 . 75 π + 11110 π‘ for some integer ‘ . Taking ‘ = 0 gives us...
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This note was uploaded on 10/21/2010 for the course ECE 2025 taught by Professor Juang during the Summer '08 term at Georgia Tech.

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hw04su10_soln - Problem 4.1 a) We have x [ n ] = 3cos 15 π...

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