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18Spring_hw09_sol.pdf - University of Illinois Spring 18...

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University of Illinois Spring 18 ECE210 / ECE211 - Homework 09 Solution 1. Obtain the average power of the following signals: (a) f ( t ) = 5 + 2 e j 6 t + 2 e - j 6 t + j 4 e j 8 t + j 4 e - j 8 t (b) f ( t ) = 2 - 3 sin (4 t ) + 4 cos (6 t - 0 . 1) Solution (a) P = 5 2 + 2 2 + 2 2 + 4 2 + 4 2 = 65 (b) P = 2 2 + 3 2 / 2 + 4 2 / 2 = 16 . 5 2. Consider the periodic function f ( t ) = t + sin (2 πt ) for t [0 , 1) , where the signal period is T = 1 s . Its corresponding Fourier series in exponential form is given by: f ( t ) = 1 2 - j 2 e j 2 πt + j 2 e - j 2 πt + X n = -∞ ,n 6 =0 j n 2 π e j 2 nπt and in compact form: f ( t ) = 1 2 + sin (2 πt ) + X n =1 ,n 6 =0 1 cos ( n 2 πt + π/ 2) Let f ( t ) be the input to an LTI system with frequency response H ( ω ) = j ω 2 π e - for | ω | > 7 π, rad / s , and zero elsewhere. (a) Obtain the corresponding steady state response y ss ( t ) . (b) Is y ss ( t ) periodic? If so, obtain its fundamental frequency. (c) Calculate the ratio between the input signal power and the output signal power. Solution: (a) Since H ( ω ) = 0 ω 7 π
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