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 φ φ φ + + = + + + linear 0 0 ( ) ( )sin( ) y t t x t t wt φ - - + time variant b) ( ) ( ) ( 1) y t x t x t = - 1 2 1 2 1 1 2 2 ( ( ) ( ))( ( 1) ( 1)) ( ) ( 1) ( ) ( 1) x t x t x t x t x t x t x t x t + - + - - + - not linear 0 0 0 ( ) ( ) ( 1) 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 0 0 ( ) 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 0 0 ( ) cos( ( )) y t t wt x t t - + - time variant e) ( ) ( ) t y t x d τ τ -∞ = 1 2 1 2 ( ( ) ( )) ( ) ( ) t t t x x d x d x d τ τ τ τ τ τ τ -∞ -∞ -∞ + = + linear 0 0 0 ( ) ( ) ( ) t t t y t t x d x t d τ τ τ τ - -∞ -∞ - = = - time invariant f) /2 ( ) ( ) t y t x d τ τ -∞ = /2 /2 /2 1 2 1 2 ( ( ) ( )) ( ) ( ) t t t x x d x d x d τ τ τ τ τ τ τ -∞ -∞ -∞ + = + linear 0 ( )/2 /2 0 0 ( ) ( ) ( ) t t t y t t x d x t d τ τ τ τ - -∞ -∞ - = - 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 ( ) ( ) ( ) ( / ) ( / ) ( / ) s s y t
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