20115ee102_1_hw1

# 20115ee102_1_hw1 - with an explicit calculation or an...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #1 Due: Wednesday October 05, 2011 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 Z -∞ x 2 ( t ) dt = Z -∞ x 2 e ( t ) dt + Z -∞ x 2 o ( t ) dt. 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 ) ?

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2 (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
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Unformatted text preview: 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 j ( 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. 7. Write the following signals as a combination (sums or products) of scaled and shifted unit rectangles rect ( t ) and unit triangles Δ( t ) . 2-1 1-2 1 2-2-1 x(t) a) 2-1 1-2 1 2-2-1 t x(t) b)...
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20115ee102_1_hw1 - with an explicit calculation or an...

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