3015_Discussion_6_Solutions

3015_Discussion_6_Solutions - University of Minnesota Dept....

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Unformatted text preview: University of Minnesota Dept. of Electrical and Computer Engineering EE3015 Signals and Systems Discussion Session #6: CT Fourier Series FS equations for CT signals ( ) ( t x periodic with fundamental frequency ) Synthesis equation: = = k t jk k e a t x ) ( Analysis equation: > < = T t jk k dt e t x T a ) ( 1 1. Find the Fourier Series (FS) coefficients of: + + + + = 4 2 cos ) cos( 2 ) sin( 1 ) ( t t t t x Signals ) sin( t and ) cos( 2 t have period 2 1 = T . Signal + 4 2 cos t has period 2 / 1 2 T T = = . The fundamental period of ) ( t x is 2 1 = = T T , and the fundamental frequency is 1 / 2 = = T . (See also discussion session #2 for more information on finding the period of the sum of periodic signals.) We want to find coefficients k a so that we can express ) ( t x as = = = = k jkt k k t jk k e a e a t x ) ( . Using Eulers Formula, we can expand ) ( t x in terms of complex exponentials: 2 2 2 2 1 ) ( 4 2 4 2 + + + + + + + = t j t j jt jt jt jt e e e e j e e t x Collecting terms, we have that t j j t j j jt jt e e e e e j e j t x 2 4 2 4 2 1 2 1 2 1 1 2 1 1 1 ) ( + + + + + = By inspection, the FS coefficients are = = = = + = = otherwise k e k e k j k j k a j j k , 2 , 2 1 2 , 2 1 1 , 2 1 1 1 , 2 1 1 , 1 4 4 2. Determine the time-domain signal represented by the following coefficients...
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This note was uploaded on 12/20/2009 for the course EE 3015 taught by Professor Staff during the Fall '08 term at Minnesota.

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3015_Discussion_6_Solutions - University of Minnesota Dept....

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