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HW 01_examples

# HW 01_examples - 1 ME360 Solved Examples for Homework#1...

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1 ME360 - Solved Examples for Homework #1 Problem 1: For each signal x(t) of Figure 1, 2 3 t x(t) t x(t) -2 2 4 -2 2 -2 2 t x(t) Signal 1 Signal 2 Signal 3 Figure 1. sketch the following: (a) p ( t ) = x [0 . 5 ( t 2)] ; (b) h ( t ) = x (2 2 t ) ; (c) s ( t ) = x ( 0 . 5 t 1) ; Solution: (a) p ( t ) = x [0 . 5 ( t 2)] t x(t) Signal 1 Signal 2 Signal 3 t x(t) t x(t) 2 8 2 -2 2 10 2 -2 2 2 -2 6 (b) h ( t ) = x (2 2 t ) = x ( 2 t + 2) = x [ 2 ( t 1)] t t t Signal 1 Signal 2 Signal 3 -1/2 1 2 h(t) h(t) h(t) -1 1 2 2 -2 2 2 1

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2 (c) s ( t ) = x ( 0 . 5 t 1) = x [ 0 . 5 ( t + 2)] = x [ 0 . 5 ( t ( 2))] t t t Signal 1 Signal 2 Signal 3 2 s(t) s(t) s(t) -8 -2 2 -10 -2 2 -6 -2 2 2 -2 Problem 2: Classify the following signal(s) as a power signal, energy signal, or neither and find its power or energy as appropriate: (a) x ( t ) = e t ; (b) x ( t ) = te | t | ; Solution: (a) x ( t ) = e t E x = R −∞ ¯ ¯ x 2 ( t ) ¯ ¯ dt = R −∞ e ( 2 t ) dt = e 2 t 2 | −∞ = P x = lim T →∞ 1 T R T 2 T 2 | x ( t ) | 2 dt = lim T →∞ 1 T R T 2 T 2 e ( 2 t ) dt = lim T →∞ 1 T h e ( 2 t ) 2 i T 2 T 2 = Since this signal has infinite energy and power signals, it is neither an energy nor a power signal. (b) x ( t ) = te | t | E x = R −∞ ¯ ¯ te | t | ¯ ¯ 2 dt = 2 R 0 ( te t ) 2 dt = = 2 R 0 t 2 e 2 t dt = 2 h e 2 t ³ t 2 2 2 t ( 2) 2 + 2 ( 2) 3 ´i 0 = = 2 ¡ 0 e 0 ¡ 0 0 2 8 ¢¢ = 2 ¡ 1 ¡ 1 4 ¢¢ = = 1 2 P x = lim T →∞ 1 T R T 2 T 2 ¡ te | t | ¢ 2 dt = lim T →∞ 2 T R T 2 0 t 2 e 2 t dt = = lim T →∞ 2 T h e 2 t ³ t 2 2 t 2 1 4 ´i T 2 0 = = lim T →∞ 2 T £ e T ¡
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HW 01_examples - 1 ME360 Solved Examples for Homework#1...

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