20115ee102_1_hw1_solution

20115ee102_1_hw1_solution - 1 EE102 Fall Quarter 2011...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #1 Solution 1. Find the even and odd decomposition of this signal: 1 2 0 -1 -2 t 1 2 x ( t ) Solution: 1 2 0 -1 -2 t 1 2 1 2 0 -1 -2 t 1 2 1 2 0 -1 -2 t 1 2 1 2 0 -1 -2 t 1 2 x ( t ) x ( t ) x e ( t ) x o ( t ) 2. Given the signal x ( t ) shown below -2 -1 0 1 2 1 -1 t x ( t ) draw the following signals:
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2 (a) x (2(1 - t )) Solution: This is centered at t = 1 , is time reversed, and compressed by a factor of 2. -2 -1 0 1 2 1 -1 t x (2(1 t )) (b) x ( t 2 - 1 ) Solution: This is x ± t 2 - 1 ² = x ± t - 1 2 ² which is centered at t = 2 , and is expanded by a factor of 2. -2 -1 0 1 2 1 -1 3 4 t 5 6 x t 2 1 3. In class we showed that any signal can be written as the sum of an evan and odd compo- nent, x ( t ) = x e ( t ) + x o ( t ) . Show that the energy of x(t) is the sum of the energies of the even and odd components Z -∞ x 2 ( t ) dt = Z -∞ x 2 e ( t ) dt + Z -∞ x 2 o ( t ) dt. Solution: Z -∞ x 2 ( t ) dt = Z -∞ ( x e ( t ) + x o ( t )) 2 dt = Z -∞ x 2 e ( t ) dt + 2 Z -∞ x e ( t
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This note was uploaded on 12/13/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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20115ee102_1_hw1_solution - 1 EE102 Fall Quarter 2011...

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