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exam1_practice_solutions

exam1_practice_solutions - 1(10 points Suppse that the...

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1. (10 points) Suppse that the impulse response of a LTI system is given by the following: t h(t) -1 0 1 2 3 4 0 0.5 1 Draw a plot(include values on both axes) of the output of this system when the following input is applied: x ( t ) = 2 X k =0 δ t - 3 k 2 2
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2. (10 points) Suppose that the input to a LTI system is given by u ( t + 1) - u ( t - 1), and that the impulse response of this system is h ( t ) = e - 2 t ( u ( t ) - u ( t - 3)). Determine the output of the system in response to this input by directly evaluating the convolution integral. 3
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3. (6 points) Suppose that a continuous time, linear time invariant system is defined by a differential equation of the form d 2 y ( t ) dt 2 + 6 dy ( t ) dt + Ky ( t ) = g ( t ) For what values of K is this system asymptotically stable? 4. (6 points) For each of the following systems, determine whether the system is linear, time-invariant, or both: (a) y ( t ) = t 2 x ( t - 1) (b) y [ n ] = x 2 [ n - 2] (c) y [ n ] = x [ n + 1] - x [ n - 1] 4
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5. (10 points) Suppose that a LTI system has the properties that it turns the unit pulse into a triangle as illustrated below: Sketch the output signal (label both axes) when the following signal is applied to this system:
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