hw 7 - y 00 t 4 y t 13 y t = 13 x t where the initial...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #7 Due: Tuesday, June 1, 2010 5PM 1. Find the Laplace transform of the following signals: (a) f ( t ) = (1 - t 2 ) e - 2 t . (b) One cycle of a sinusoid, f ( t ) = ( sin(2 πt ) 0 t < 1 0 1 t which is plotted below: 0 1 2 t f ( t ) 1 - 1 Hint: Use the delay theorem, and the fact that a delayed signal is padded with zeros. (c) Find the Laplace transform of the following signal, t t 2 1 1 0 2 3 4 5 f ( t ) Hint: Differentiate once or twice, and then use the integral theorem. (d) Find the Laplace transform of the following signal t 1 2 0 3 f ( t ) e - t 1 This is a cosine with an envelope that bounds the cosine below by 0 and above by e - t . Combine terms over a common denominator, and simplify your answer. Hint: Express the signal as a sum of a decaying exponential, and an exponentially decaying cosine.
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2 2. Find the Inverse Laplace transforms of the following functions (a) 4 s 3 + 4 s (b) s + 3 ( s + 1) 2 ( s + 2) (c) 10 ( s + 1)( s 2 + 4 s + 13) (d) s 2 + 8 s s 2 + 8 s + 25 3. Solving Differential Equations A system is described by the differential equation
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Unformatted text preview: y 00 ( t ) + 4 y ( t ) + 13 y ( t ) = 13 x ( t ) where the initial conditions are all zero, y 00 (0) = 0 , y (0) = 0 , and y (0) = 0 . The input is the unit step x ( t ) = u ( t ) . Find y ( t ) . 4. For each of these assertions, determine whether they are true or false. Provide an argument for your conclusion. Remember that an unstable system has poles either in the right-half plane (diverging solutions) or on the jω axis (oscillating or constant solutions). a) Let h ( t ) be the impulse response of a stable, causal system. Then d dt h ( t ) is also stable. b) Let h ( t ) be the impulse response of a stable, causal system. Then Z t-∞ h ( τ ) dτ must be unstable. c) H ( s ) is the transfer function of a stable, causal system. The zeros of H ( s ) must be in the right-half plane for the inverse system H inv ( s ) to be stable....
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