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# Lesson_6 - Lesson6 Challenge5 Lesson6: Challenge6...

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Lesson 6 Lesson 6 Challenge 5 Lesson 6:  Shannon’s Sampling Theorem       Challenge 6 No class Monday

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Lesson 6 Challenge 5 The Fourier series representation of x(t)=cos(k ϖ 0 t) is given to be: where a k =0 for all k= ± n, and a ± k  otherwise. Express the Fourier series of  in terms of the coefficients a k . Hint: What is the differential relationship between sine and cosine? ( 29 ( 29 ( 29 -∞ = = = n t jn k e a t k t x 0 0 cos ϖ ϖ ( 29 ( 29 ( 29 -∞ = = = n t jn k e b t k t y 0 0 sin ϖ ϖ
Lesson 6 Challenge 5 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,  Upon differentiating: ( 29 ( 29 ( 29 2 1 ; ) cos( 0 0 0 = + = = - - k t jn k t jn k a e a e a t t x ϖ ϖ ϖ ( 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 e a d dt e a d t k k dt t dx 0 0 0 0 0 0 0 0 0 0 sin ϖ ϖ ϖ ϖ ϖ ϖ ϖ φ ϖ ϖ - - - - -∞ = - + = + = = + - =

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Lesson 6 Challenge 5 Collecting terms Simplifying or Finally: b k =-ja k ; b -k =ja -k The same result could be obtained using analysis based on Euler’s  equation  ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k e jk a e jk a t k k 0 0 0 0 0 0 sin ϖ ϖ ϖ ϖ ϖ ϖ - - - + = - ( 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 0 0 0 0 0 0 0 sin ϖ ϖ ϖ ϖ ϖ ϖ ϖ - - - - + - = ( 29 ( 29 ( 29 ( 29 ( 29 t jk k t jk k e ja e
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Lesson_6 - Lesson6 Challenge5 Lesson6: Challenge6...

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