HW 02_examples - 1 ME360 - Solved Examples for Homework #2...

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1 ME360 - Solved Examples for Homework #2 Problem 1: Classify the following signal as periodic, nonperiodic, or almost periodic and find the signal power if appropriate. If the signal is periodic, find the fundamental frequency and the common period: (a) x ( t )=cos(10 πt )cos(10 t ); (b ) x ( t )=cos(6 t )+cos(15 t ); Solution: (a) x ( t ) cos(10 t ) x ( t ) cos(10 t )= 1 2 (cos [(10 π +10) t ]+cos[(10 π 10) t ]) = 1 2 cos[(10 π | {z } ω 1 t ]+ 1 2 cos[(10 π 10) | {z } ω 2 t ] T 1 = 2 π ω 1 = 2 π 10 π +10 T 2 = 2 π ω 2 = 2 π 10 π 10 & T 2 T 1 is not rational: there is no common period. This signal is nonperiodic. Signal power: P x = 1 2 ¡ 1 2 ¢ 2 + 1 2 ¡ 1 2 ¢ 2 = 1 4 (b) x ( t t t ω 1 =6 ω 2 =15 & ω 1 ω 2 = 3 5 rational fraction. This signal is periodic . fundamental frequency = ω o =GCD (6, 15) = 3 rad/sec common period = 2 π ω o = 2 π 3 sec Signal power: P z = 1 2 (1) 2 + 1 2 (1) 2 =1 Problem 2: For the signal in Figure 1., (a) express x(t) by intervals, (b) express x(t) as a linear combination of steps and/or ramps, (c) express x(t) as a linear combination of rect and/or tri functions, (d) sketch the first derivative x’(t), (e) find the signal energy in x(t). 2 4 6 t 4 x(t) Figure 1. Solution: (a) express x(t) by intervals
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2 x ( t )= 0 ,t< 0 2 t, 0 t 2 4 , 2 <t 4 2( t 6) , 4 6 0 ,t> 6 (b) express x(t) as a linear combination of steps and/or ramps The slope of r ( t ramp ( t )=1 ; r ( t ramp ( t t, t 0 0 ,else r(t) t 1 1 x ( t )=2 r ( t ) 2 r ( t 2) 2 r ( t 4) + 2 r ( t 6) ( c) express x(t) as a linear combination of rect and/or tri functions rect(t) t -0.5 0.5 tri(t) t -1 1 1 1 x(t) can be seen as the superposition of two triangles:
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3 x ( t )=6 tri ( 1 3 ( t 3)) 2 tri ( t 3) ( d) sketch the first derivative x’(t) x 0 ( t )= 0 ,t< 0 2 , 0 <t< 2 0 , 2 4 2 , 4 6 0 ,t> 6 2 -2 0 2 4 6 t x’ (t) (e) find the signal energy in x(t) E x = R −∞ | x ( t ) | 2 dt = R 2 0 (2 t ) 2 dt + R 4 2 4 2 dt + R 6 4 ( 2( t 6)) 2 dt = 128 3 =42 . 6667 Problem 3: Sketch the following signals. Note that the comb function is a periodic train of unit impulses with unit spacing defined as comb ( t P k = −∞ δ ( t k ) ;(a ) x ( t )=3 δ (2 t 2) = 3 2 δ ( t 1) (b) x ( t comb (0 . 5 t ) rect ¡ t 5 5 ¢ .
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This note was uploaded on 04/18/2010 for the course ME 360 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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HW 02_examples - 1 ME360 - Solved Examples for Homework #2...

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