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HW 02_examples

# 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 πt ) cos(10 t ) x ( t ) = cos(10 πt ) cos(10 t ) = 1 2 (cos [(10 π + 10) t ] + cos [(10 π 10) t ]) = 1 2 cos[(10 π + 10) | {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 ) = cos (6 t ) + cos (15 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 < t 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:
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 < t < 4 2 , 4 < t < 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

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

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