20105ee102_1_hw2_soln

20105ee102_1_hw2_soln - 1 EE102 Systems and Signals Fall...

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Unformatted text preview: 1 EE102 Systems and Signals Fall Quarter 2010 Jin Hyung Lee Homework #2 Solutions Due: Wednesday, Oct 13, 2010 at 5 PM. 1. State whether the following systems are linear or nonlinear; time invariant or time variant; and why. (a) y ( t ) = x ( t )sin( ωt + φ ) Solultion: Let x ( t ) = ax 1 ( t ) + bx 2 ( t ) , and check y ( t ) = x ( t )sin( ωt + φ ) = ( ax 1 ( t ) + bx 2 ( t ))sin( ωt + φ ) = ax 1 ( t )sin( ωt + φ ) + bx 2 ( t )sin( ωt + φ ) = ay 1 ( t ) + by 2 ( t ) , so the system is linear . If we delay the input x ( t ) by τ x ( t- τ )sin( ωt + φ ) However, the delayed output would be x ( t- τ )sin( ω ( t- τ ) + φ ) which is not the same, so the system is time variant . (b) y ( t ) = x ( t ) x ( t- 1) We first check homogeneity, if we input x 1 ( t ) = ax ( t ) we get y 1 ( t ) = x 1 ( t ) x 1 ( t- 1) = ( ax ( t ))( a ( x ( t- 1)) = a 2 x ( t ) x ( t- 1) = a 2 y ( t ) so this system is non-linear . Next we check delay the input, x ( t- τ ) x ( t- τ- 1) = y ( t- τ ) so the system is time invariant . (c) y ( t ) = 1 + x ( t ) First we check homogeneity, 1 + a ( x ( t )) 6 = a (1 + x ( t )) 2 so the system is non-linear . Next, if we delay the input 1 + x ( t- τ ) = y ( t- τ ) so the system is time invariant . (d) y ( t ) = cos( ωt + x ( t )) Solultion: We first check homogeneity cos( ωt + ax ( t )) 6 = a cos( ωt + x ( t )) so scaling the input by a does not result in a scaled output. This system is non-linear . Next, if we delay the input by τ cos( ωt + x ( t- τ )) 6 = cos( ω ( t- τ ) + x ( t- τ )) = y ( t- τ ) so this system is time variant . (e) y ( t ) = Z t-∞ x ( τ ) dτ This time we’ll check superposition and homogeneity together, Z t-∞ ( ax 1 ( τ ) + bx 2 ( t )) dτ = a Z t-∞ x 1 ( τ ) dτ + b Z t-∞ x 2 ( t ) dτ = ay 1 ( t ) + by 2 ( t ) so the system is linear . Next, if we delay the input by T (since τ is already used) Z t-∞ x ( τ- T ) dτ = Z t- T-∞ x ( τ ) dτ = y ( t- T ) where we have made the change of variables τ = τ- T . So this system is time invariant . (f) y ( t ) = Z t/ 2-∞ x ( τ ) dτ Solultion: Again we’ll check superposition and homogeneity together, Z t/ 2-∞ ( ax 1 ( τ ) + bx 2 ( t )) dτ = a Z t/ 2-∞ x 1 ( τ ) dτ + b Z t/ 2-∞ x 2 ( t ) dτ = ay 1 ( t ) + by 2 ( t ) so the system is linear . Next, if we delay the input by T (since τ is already used) Z t/ 2-∞ x ( τ- T ) dτ = Z t/ 2- T-∞ x ( τ ) dτ = Z ( t- 2 T ) / 2-∞ x ( τ ) dτ = y ( t- 2 T ) where we have made the change of variables τ = τ- T . Delaying the input by T delays the output by 2 T , so this system is time variant . 3 2. A periodic signal x ( t ) , with a period T , is applied to a linear, time-invariant system H ....
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This note was uploaded on 04/20/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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20105ee102_1_hw2_soln - 1 EE102 Systems and Signals Fall...

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