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Unformatted text preview: ECE220 Signals and Information Spring 2007 Homework 2 Solutions Problem 1 Consider the signal x ( t ) = 2 + 4 cos(10 t + 2) cos(120 t 2) 1. Sketch the spectrum of this signal as in Figure 3.1 in the textbook. Show your calculations. From Eulers formula we have x ( t ) = 2 e j + 2 e j(10 t +2) + e j(10 t +2) 1 2 e j(120 t 2) + e j(120 t 2) = 2 e j + 2 e 2j e j(2 ) 5 t + 2 e 2j e j(2 ) 5 t 1 2 e 2j e j(2 ) 60 t 1 2 e 2j e j(2 ) 60 t The above form allows us to sketch the spectrum in the form as in Figure 3.1 in the textbook. The spectrum is shown in Figure 1.5/ pi 5/ pi 60/pi 60/pi 2 1 2 exp(2j) 1 2 exp(2j) 2exp(2j) 2exp(2j) Hz Figure 1. 2. Is this signal periodic? If so, what is its fundamental period? Let us consider the two cosine waves individally. We need to show that there exists T s > such that for all t cos(10 t + 2) = cos(10( t + T s ) + 2) cos(120 t 2) = cos(120( t + T s ) 2) As the cosine function is periodic with the period 2 , we want to find T s such that 10 T s = 2 k 120 T s = 2 l for some integers k and l . It is clear that T s = 5 is the smallest positive number which satisfies the two conditions. Thus T s = 5 is the fundamental period. 3. Now consider the signal y ( t ) = x ( t ) + 5 sin(15 t 1 2 ) and answer questions (1) and (2) for this signal. How do the answers change? Clearly we need to express 5 sin(15 t 1 2 ) as a linear combination of complex exponentials, 5 sin(15 t 1 2 ) = 5 2j e j((2 ) 7 . 5 t 1 2 ) e j((2 ) 7 . 5 t 1 2 ) = 5 2 e j 3  1 2 e j(2 ) 7 . 5 t 5 2 e j 3  1 2 e j(2 ) 7 . 5 t Thus y ( t ) = 2 e j + 2 e 2j e j(2 ) 5 t + 2 e 2j e j(2 ) 5 t 1 2 e 2j e j(2 ) 60 t 1 2 e 2j e j(2 ) 60 t + 5 2 e j 3  1 2 e j(2...
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This note was uploaded on 04/17/2008 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 JOHNSON

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