sol5 - ECE 351 Systems I Mar 2 2009 OSU Winter 2009...

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Unformatted text preview: ECE 351, Systems I Mar. 2, 2009 OSU, Winter 2009 Solutions - Problem Set 5 Problem 1 (a) Let v ( t ) = e (- 2 t ) u ( t ) and s ( t ) = e (- 20 t ) u ( t ). Then V ( jω ) = 1 jω +2 and S ( jω ) = 1 jω +20 . Now x ( t ) = 2 v ( t ) cos(4 t ) + s ( t ) cos(20 t ). Applying the property of multiplication by cosine in time domain we get the following: X ( jω ) = 1 j ( ω + 4) + 2 + 1 j ( ω- 4) + 2 + . 5 j ( ω + 20) + 20 + . 5 j ( ω- 20) + 20 . (b) Note that p 2 ( t ) ↔ 2sinc( 2 ω 2 π ). Now applying the property of time shifting we obtain: X ( jω ) = 4sinc( ω π ) e (- j 3 ω ) . Problem 2 The ideal filter, H ( jω ), gives a gain of 1 ∠ 0 to any frequency component in the band 2 6 | ω | 6 7 and gives a gain of 0 ∠ 0 ( i.e. , infinite attenuation) to all other frequency components. (a) For x ( t ) = 2 + 3 cos(3 t )- 5 sin(6 t- 30 ◦ ) + 4 cos(13 t- 20 ◦ ), by superposition we obtain y ( t ) = 3 cos(3 t )- 5 sin(6 t- 30 ◦ ) ....
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This note was uploaded on 05/05/2010 for the course ECE 351 taught by Professor Staff during the Spring '10 term at Ohio State.

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sol5 - ECE 351 Systems I Mar 2 2009 OSU Winter 2009...

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