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hw3 - ASE 330M Linear System Analysis Unique Number 12495...

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ASE 330M Linear System Analysis Unique Number: 12495, Spring 2006 Homework #3 Impulse & Step Responses Convolution Integral Date given: February 16, 2006 Date Due: February 28, 2006 1. An LTI system has an impulse response h ( t ) = ( e - t + sin t ) u ( t ). (a) Evaluate the step response g ( t ) for this system. (b) Find the output of this system resulting from an input x ( t ) = u ( t + 2) - u ( t - 2). 2. An LTI system with zero initial energy exhibits an output y ( t ) = (1 - e - 2 t cos t ) u ( t - 1) when an input x ( t ) = u ( t - 1) is applied. Evaluate the impulse response h ( t ) for this system. 3. Consider the motion of an automobile on a level surface. The output variable of interest is the speed of the vehicle v ( t ) which is governed by the following dynamics: m ( t ) ˙ v ( t ) + k f v ( t ) = x ( t ) where m ( t ) is the instantaneous mass of the vehicle, k f is a friction related positive constant, and x ( t ) is the input force acting on the vehicle. Determine the impulse response. For your calculations, you may assume mass depletion from fuel consumption to be negligible when compared to the total mass of the vehicle, i.e.,
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