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Unformatted text preview: 1 EE102 Spring 2009-10 Lee Systems and Signals Homework #1 Due: Tuesday April 13, 2010 at 5 PM. 1. Find the even and odd decomposition of this signal: 1 2-1-2 t 1 2 x ( t ) 2. Given the signal x ( t ) shown below-2-1 1 2 1-1 t x ( t ) draw the following signals: (a) x (2(1- t )) (b) 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. 2 4. Periodic Signals (a) Assume that the signal x ( t ) is periodic with period T , and that x ( t ) is odd ( i.e. x ( t ) =- x (- t ) ). What is the value of x ( T ) ? (b) Two continuous-time sequences x 1 ( t ) and x 2 ( t ) are periodic with periods T 1 and T 2 . Find values of T 1 and T 2 such that x 1 ( t ) + x 2 ( t ) is aperiodic. 5. Power and Energy Signals Plot these signals, and classify them as energy or power signals. Support your classification with an explicit calculation or an argument. In each case-∞ < t < ∞ . (a) x ( t ) = e- 2 | t | (b) x ( t ) = 1 √ t t ≥ 1 t < 1 (c) x ( t ) = ( t ≥ e t t < (d) x ( t ) = e |- t | 6. Review of Complex Numbers (a) Simplify the following expression e i ( ωt + φ ) 1 + j (1- j ) and leave the result in polar form. (b) Simplify (cos ωt + j sin ωt )(cos2 ωt- j sin2 ωt ) and leave the result in polar form....
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This note was uploaded on 10/21/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.
- Spring '09