Hw4sol - HW3 solution February 8 2008 1 g(t = A1 cos(1 t 1...

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HW3 solution February 8, 2008 1 g ( t ) = A 1 cos( ω 1 t + φ 1 ) + A 1 cos(2 ω 1 t + φ 2 ) + A 1 cos(3 ω 1 t + φ 3 ) . . . (1) = X n =1 A n cos( 1 t + φ n ) (2) In order to prove that g ( t ) is a periodic function with period T , we need to show that g ( t + T ) = g ( t ). g ( t + T ) = X n =1 A n cos( 1 ( t + T ) + φ n ) (3) since ω 1 = 2 π T (4) = X n =1 A n cos( 2 T ( t + T ) + φ n ) (5) = X n =1 A n cos( 2 nπt T + 2 nπt + φ n ) (6) = X n =1 A n cos( 1 t + φ n ) (7) = g ( t ) (8) * You do not have to show general proof like above. You can show that first few terms satisfy g ( t + T ) = g ( t ) condition and by induction we can generalize it to n th term. 2 y ( t ) = 0 . 5 cos( ω 1 t ) + 0 . 5 sin( ω 1 t ) + 0 . 1 cos(2 ω 1 t ) + 0 . 05 sin( ω 1 t ) (9) In order to plot the power spectrum graph, we need to convert all sin terms into cos terms.(you can also change all cos terms into sin term. You will get same power spectrum either way.) you can use the trigonometry indentities on the lecture note. sin( x ) + sin( y ) = 2 sin(( x + y ) / 2) cos(( x - y ) / 2) (10) cos( x ) + cos( y ) = 2 cos(( x + y ) / 2) cos(( x - y ) / 2) (11) sin( x ± y ) = sin( x ) cos( y ) ± cos( x ) sin( y ) (12) cos( x ± y ) = cos( x ) cos( y ) sin( x ) sin( y ) (13) You can also use a trick like below.
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