102_1_hw2_sol

102_1_hw2_sol - EE102 HW 2 Solution 1 a y(t = x(t sin wt...

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EE102 HW 2 Solution 1. a) ( ) ( )sin( ) y t x t wt φ =+ 1 2 1 2 ( ( ) ( ))sin( ) ( )sin( ) ( )sin( ) x t x t wt x t wt x t wt + + = + + + linear 00 ( ) ( )sin( ) y t t x t t wt - - + time variant b) ( ) ( ) ( 1) y t x t x t = - 1 2 1 2 1 1 2 2 ( ( ) ( ))( ( ( 1)) ( ) ( ( ) ( x t x t x t x t x t x t x t x t + - + - - + - → not linear 0 0 0 ( ) ( ) ( y t t x t t x t t - = - - - → time invariant c) ( ) 1 ( ) y t x t 1 2 1 2 1 ( ) ( ) 1 ( ) 1 ( ) x t x t x t x t + + ≠ + + + not linear ( ) 1 ( ) y t t x t t - = + - time invariant d) ( ) cos( ( )) y t wt x t 1 2 1 2 cos( ( ) ( )) cos( ( )) cos( ( )) wt x t x t wt x t wt x t + + + + + not linear ( ) cos( ( )) y t t wt x t t - + - time variant e) ( ) ( ) t y t x d ττ -∞ = 1 2 1 2 ( ( ) ( )) ( ) ( ) t t t x x d x d x d τ -∞ -∞ -∞ + = + linear 0 0 0 ( ) ( ) ( ) tt t y t t x d x t d - -∞ -∞ - = = -  time invariant f) /2 ( ) ( ) t y t x d -∞ = 1 2 1 2 ( ( ) ( )) ( ) ( ) t t t x x d x d x d -∞ -∞ -∞ + = + linear 0 ( )/2 0 0 ( ) ( ) ( ) t y t t x d x t d - -∞ -∞ - = - time variant 2. ( ) ( ( )) y t H x t = ( ) ( ( )) ( ( )) ( ) y t T H x t T H x t y t + = + = = y(t) is also periodic with period T 3. S/H system: ( ) ( / ) y t x h t h =   1 2 1 2 ( ) ( ) ( ) ( / ) ( / ) ( / ) ss y t x t x t y h t h x h t h x h t h = + = + linear
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This note was uploaded on 12/09/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

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102_1_hw2_sol - EE102 HW 2 Solution 1 a y(t = x(t sin wt...

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