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hw6 - 5 An LTI causal system with zero initial energy is...

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ASE 330M Linear System Analysis Unique Number: 12880, Spring 2007 Homework #6 Date Given: Wednesday, April 11, 2007 Due Date: Monday, April 23, 2007 For this homework, when necessary, you are permitted use of MATLAB and/or calculator in evaluating roots of polynomials order higher than 2. 1. Determine the unilateral Laplace transform of the following signals: (a) x ( t ) = e - t ( t - 2) u ( t - 2) (b) x ( t ) = e - t u ( t ) * cos( t - 2) u ( t - 2) 2. Consider a signal x ( t ) satisfying x ( t ) = 0 for all t < 0 . Determine the signal initial value x (0 + ) and final value x ( ) if the unilateral Laplace transform X ( s ) is given by: (a) X ( s ) = - 2 e - 5 s / [ s ( s + 2)] (b) X ( s ) = (2 s + 3) / ( s 2 + 5 s + 6) 3. Find the inverse Laplace transform of the following functions: (a) X ( s ) = s/ ( s 4 + 2 s + 3) (b) X ( s ) = ( s 2 + s - 3) / ( s 2 + 5 s + 6) (c) X ( s ) = (4 s 2 + 6) / ( s 3 + s 2 - 2) 4. The transfer function of an LTI causal system is given by H ( s ) = 4 s 2 + 8 s + 10 2 s 3 + 8 s 2 + 18 s + 20
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Unformatted text preview: 5. An LTI causal system with zero initial energy is subjected to an input signal x ( t ) = e-3 t cos(2 t ) u ( t ) . The resulting output is given by y ( t ) = t 3 e-2 t sin( t ) u ( t ) . Your tasks are as follows: (a) Find the transfer function. (b) Find the system’s impulse response function. (c) Obtain the step response of the system. (d) Determine the output of this system for an input x ( t ) = sin( ωt ) u ( t ) for any (real-valued) frequency ω . Again assume zero initial energy at the time of input application. 6. The step response of an LTI system is given by g ( t ) = t 2 cos(Ω t ) u ( t-1) . Determine the transfer function for this system....
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