102_1_midterm_practice_solution

102_1_midterm_practice_solution - Systems and Signals EE102...

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Systems and Signals Lee, Spring 2009-10 EE102 Midterm Practice Problems Solutions Problem 1. Computing Fourier Transforms The signal f ( t ) is plotted below: f ( t ) t 1 1 - 1 - 2 - 3 0 2 3 2 - 1 a) Find an expression for f ( t ) in terms sums and convolutions of signals defined for all t such as rect( t ) and Δ ( t ) ( i.e. do not express it as a piecewise signal, defined over intervals). Solution: There are several solutions. Two simple examples are f ( t ) = 2 Δ ( t ) - Δ ( t - 1) - Δ ( t + 1) and f ( t ) = 4 Δ ( t ) - 2 Δ ( t/ 2) b) Find the Fourier transform of f ( t ). Simplify your answer. Solution: The Fourier transform of the first solution is F ( j ω ) = 2sinc 2 ω 2 π - e j ω sinc 2 ω 2 π - e - j ω sinc 2 ω 2 π which can be simplified to F ( j ω ) = 2sinc 2 ω 2 π (1 - cos( ω )) . The second solution has the Fourier transform F ( j ω ) = 4sinc 2 ω 2 π - 4sinc 2 ω π . 1
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Problem 2. Fourier Series a) You compute the Fourier series of a signal f ( t ) using a period T 0 . You find that all of the odd Fourier series coe ffi cients are zero (i.e. D 1 , D - 1 , D 3 , D - 3 , · · · are all zero).
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