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Midterm1_Solution

# Midterm1_Solution - Name Page 1 of 4 Midterm 1(Fall 2007 of...

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Unformatted text preview: Name: ______________________________________ Page 1 of 4 Midterm 1 (Fall 2007) of EE421G: 1. This midterm consists of four single-sided pages. The last page is an extra worksheet. You can tear out any page but make sure all the pages you turn in have your name on them. 2. There are three problems in this exam. You have 50 minutes to finish this exam. 3. You are allowed to use one double-sided page of cheat sheet. Good luck! 1. (Outcome 2, 34 points) Consider the signal + + + = 2 5 sin 2 3 cos ) ( π π t t t f . a. Is the signal ) ( t f periodic? If yes, what is its period? The period of the cosine function is 2 π /3 and that of the sine function is 2 π /5. If their sum is periodic with period T, T must be an integral multiples of the period of the constituent components, or T=m2 π /3 and T=n2 π /5. The smallest integers that satisfy these equations are m=3 and n=5. Since such multipliers exist, f(t) is periodic with period T=2 π . b. Write ) ( t f in terms of complex exponentials. f(t) = cos(3t + π /2) + sin(5t + π /2) = 0.5{exp[j(3t+ π /2)]+exp[-j(3t+ π /2)]} - 0.5j{exp[j(5t+ π /2)]-exp[-j(5t+ π /2)]} = 0.5[jexp(j3t) – jexp(-j3t)] – 0.5j[jexp(j5t) + jexp(-j5t)] = 0.5jexp(j3t) – 0.5jexp(-j3t) + 0.5exp(j5t) + 0.5exp(-j5t) = 0....
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Midterm1_Solution - Name Page 1 of 4 Midterm 1(Fall 2007 of...

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