102_1_hw01

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 classi±cation 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 ( ωt + φ ) 1+ j (1 - j ) and leave the result in polar form. (b) Simplify (cos ωt + j sin ωt ) (cos 2 ωt - j sin 2 ωt ) and leave the result in polar form.
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This note was uploaded on 12/09/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

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

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