hw2sol - ECE220 Signals and Information Spring 2007...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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).

Page1 / 6

hw2sol - ECE220 Signals and Information Spring 2007...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online