Lesson_50 - a e jk a t k k sin =-Simplifying 29 29 29 29 29...

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EEL 3135 Dr. Fred J. Taylor, Professor Lesson #5 Fourier series Challenge The Fourier series representation of x(t)=cos(k ϖ 0 t) is given to be: ( 29 ( 29 ( 29 -∞ = = = n t jn k e a t k t x 0 0 cos ϖ where a k =0 for all k= ± n, and a ± k otherwise. Express the Fourier series of ( 29 ( 29 ( 29 -∞ = = = n t jn k e b t k t y 0 0 sin in terms of the coefficients a k of x(t). Response Your previous studies should lead you to the conclusion that sine and cosine waves are related to each other through a differential operator. From the given data, ( 29 ( 29 ( 29 t jn k t jn k e a e a t x 0 0 - - + = . Upon differentiating: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k t jk k t jk k n t jn k e jk a e jk a dt e a a e a d dt e a d t k k dt t dx 0 0 0 0 0 0 0 0 0 0 0 sin - - - - -∞ = - + + = + + = = - = Therefore: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k e jk
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Unformatted text preview: a e jk a t k k sin---+ =-Simplifying: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k e k jk a e k jk a t k sin----+-= or ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k e ja e ja t k sin --+-= + Therefore: InvestiGATOR 1 EEL 3135 Dr. Fred J. Taylor, Professor b k =-ja k b-k =ja-k The same result could be obtained using analysis based on Eulers equation. 2...
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.

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Lesson_50 - a e jk a t k k sin =-Simplifying 29 29 29 29 29...

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