Quiz3 sol - δ t-1 2 δ t-1 ² e-jπkt 2 dt = 1 4 ± 1-1 2...

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ECEN 314 Quiz # 3 Georghiades November 23, 2009 Note: Quiz is open-book, 15 minutes long. 1. Compute the complex Fourier series, as well as the alternative form ( a k , b k , a v ) for the signal x ( t ) shown in the figure below: ... ... 1 1 -.5 2 3 -1 -2 -3 t x(t) Solution 1. The period of x ( t ) is T = 4. We can express x ( t ) in a complex Fourier series as: x ( t ) = X k x k e j 2 πkt/T = X k x k e jπkt/ 2 , where x k = 1 T Z T/ 2 - T/ 2 x ( t ) e - j 2 πkt/T dt = 1 4
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Unformatted text preview: δ ( t )-1 2 δ ( t-1) ² e-jπkt/ 2 dt = 1 4 ± 1-1 2 ³ e jπk/ 2 + e-jπk/ 2 ´ ² = 1 4 (1-cos( πk/ 2)) . Using the relation between the complex Fourier coefficients and the alternative form derived in class, we have a v = x = 0 a k = 2 < [ x k ] = 1 2 (1-cos( πk/ 2)) , b k =-2 = [ x k ] = 0 . 1...
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