102_1_hw01 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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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 0 -1 -2 t 1 2 x ( t ) 2. Given the signal x ( t ) shown below -2 -1 0 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 -∞ x 2 ( t ) dt = -∞ x 2 e ( t ) dt + -∞ x 2 o ( t ) dt.
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2 4. Periodic Signals (a) Assume that the signal x ( t ) is periodic with period T 0 , and that x ( t ) is odd ( i.e. x ( t ) = - x ( - t ) ). What is the value of x ( T 0 ) ? (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 0 t < 1 (c) x ( t ) = 0 t 0 e t t < 0 (d) x ( t ) = e | - t | 6. Review of Complex Numbers (a) Simplify the following expression e i ( ω t + φ ) 1 + j (1 - j ) and leave the result in polar form. (b) Simplify (cos ω t + j sin ω t ) (cos 2 ω t - j sin 2 ω t ) and leave the result in polar form. (c) Use the relationship cos θ = 1 2 e j θ + e - j θ to determine how many frequency components there are in the signal s ( t ) = (cos ω t ) 2 .
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