20095ee102_1_EE102_HW5_Solution

20095ee102_1_EE102_HW5_Solution - n F . Then show that the...

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EE102 Fall 2009 HW 5 1. Let f(t) be a periodic square pulse function defined over its period as: 1, 1 0 ( ) 1, 0 1 t f t t - - ≤ ≤ = ≤ ≤ Find its Fourier-series coefficients n F . 0 1 0 1 1 1 0 2 2 1 1 2cos( ) 2 ( ) 2 2 j nt j nt j nt n w n F f t e dt e dt e dt j n π - - - - - = = - = = - + = 2. Find the fundamental frequency 0 ϖ and the Fourier-series coefficients n F of the following functions: a) ( ) cos(3 ) sin(5 ) f t t t = + 0 1 2 1 2 0 3 3 5 5 3 5 5 2 2 , 2 ( ) 3 5 2 2 1 1 1 1 ( ) 2 2 2 2 1 1 1 , , 2 2 2 0, jw nt j t j t j t j t n n n T T T least common multiple of T T w f t e e e e F e j j F F F j j and F for other n - - =-∞ ± - = = = = = = + + - = = = = - = b) 2 ( ) cos (3 ) f t t = 6 6 0 0 1 1 cos(6 ) 1 1 1 ( ) , 2 2 4 4 1 1 1 6 , 3 2 4 j t j t t f t e e T w F F - ± + = = + + = = = =
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3. Parseval’s relation: For the function ( ) sin(2 ) f t t π = , find its Fourier-series coefficients
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Unformatted text preview: n F . Then show that the Parsevals relation is satisfied: 2 /2 2 /2 1 ( ) T n T n f t dt F T -=- = 2 2 1 1 1/2 1/2 1/2 2 2 1/2 1/2 1/2 2 2 2 1/2 2 2 1/2 1 1 1 1 sin(2 ) , 2 2 2 2 1 1 ( ) sin(2 ) (1 cos(4 )) 2 2 1 1 1 2 2 2 ( ) j t j t n n n n t e e F F j j j j f t dt t dt t dt F j j F f t----- =- =---=- = = = =-=-= + = =...
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20095ee102_1_EE102_HW5_Solution - n F . Then show that the...

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