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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 ω ) Synthesis equation: ∑ ∞ −∞ = = k t jk k e a t x ) ( ω Analysis equation: ∫ > < − = T t jk k dt e t x T a ) ( 1 ω 1. Find the Fourier Series (FS) coefficients of: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + + = 4 2 cos ) cos( 2 ) sin( 1 ) ( π t t t t x Signals ) sin( t and ) cos( 2 t have period π 2 1 = T . Signal ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + 4 2 cos π t has period 2 / 1 2 T T = = π . The fundamental period of ) ( t x is π 2 1 = = T T , and the fundamental frequency is 1 / 2 = = T π ω . (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 ) ( ω . Using Euler’s Formula, we can expand ) ( t x in terms of complex exponentials: 2 2 2 2 1 ) ( 4 2 4 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − + + + + − + = π π 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 ) ( − − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + = π π 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 π π 2. Determine the time-domain signal represented by the following coefficients ⎪ ⎪ ⎩...
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3015_Discussion_6_Solutions - University of Minnesota Dept...

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