Final-2016-Solutions - EE 261 The Fourier Transform and its...

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EE 261 The Fourier Transform and its Applications Fall 2016 Final Exam Solutions
1. (10 points max) True or False State whether the following statements are True or False. You do not need to provide explanations. (Itemize the statements in your exam book as they are here and write the words “True” or “False.”) Each correct answer will get 2 points and each wrong answer will get -2 points (mini- mum total score 0) so you are advised not to randomly guess if you are not sure. (a) If FF f = f then f is an even function. (b) If L and M are LTI systems then both the sum and difference, L + M , L - M are LTI systems. (c) If ( f * g )( x ) = 0 for all x then at least one of f and g has to be the 0 signal. (d) f ( t - 1) * g ( t + 1) = ( f * g )( t ) (e) If f ( t ) is a bandlimited signal with bandwidth a then f ( t/ 2) has bandwidth a/ 2. Solutions :
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2. (10 points) Let f ( t ) = cos( πt ) + sin(2 πt ). What is the period of f ( t )? Find its Fourier series in terms of appropriate complex exponentials. Which complex exponentials? (This should not require any integration.) Calling the Fourier coefficients c k , k = 0 , ± 1 , ± 2 , . . . , list them.
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