ECE108Hw%234sol

ECE108Hw%234sol - ECE108 HW#4 Solution 1 Given the periodic...

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ECE108 HW#4 — Solution 1. Given the periodic, time-continuous signal shown below. Determine its exponential, trigonometric, and cosine Fourier coefficients. 0 2 and 2 T T π ω == Therefore 0 1 = Then () 0 22 00 11 2 1 2 jk t jkt jkt k X xte d t te d t e d t T ππ −− + ∫∫ Now 2 1 at at t te dt e aa  =−   Thus 2 2 2 1 0 111 1 1 1 1 1 jkt jkt k jk jk jk jk jk jk t Xe e tt jk k jk ej e jk k k k jj j ee e kk k k k j e  =+   =+− + + + Now observe that ( ) 1 This is a real value k jk e Then ( ) 1 k k j X +
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2 These are the Fourier coefficients of the exponential Fourier series expansion. The Fourier series coefficients of the trigonometric Fourier series can be obtained from these as follows. Using the notation in the notes () ( ) 22 11 a n d k kk ab π ′′ =− = Then the Fourier coefficients of the trigonometric Fourier are and 2 21 1 2 2 k aa k bb k == From these coefficients we can find the coefficients as follows and A θ 2 44 2 2 1 2 tan tan tan 2 k k k k k k k Aa b b k k a k ππ −− =+ = +   =  2.
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ECE108Hw%234sol - ECE108 HW#4 Solution 1 Given the periodic...

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