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Signals and Systems, Final Exam Solutions Spring 2005, Edited by bypeng 1. [21 points] Consider a continuous-time system with impulse response 3 1 ( ) ( ) ( ) 2 ( ) t t h t t e u t e u t δ = + + a) [3] Determine the transfer function of the inverse system of h 1 . b) [3] Is the inverse system causal and stable? c) [3] Sketch the asymptotic approximation to the gain and phase of the Bode plot for the transfer function of the inverse system. d) [3] Draw a direct-form representation of the inverse system using a minimum number of integrators and multipliers. e) [3] Repeat Step d) and draw a parallel-form representation of the inverse system. f) [3] Repeat Step d) and draw a cascade-form representation of the inverse system. g) [3] Suppose the continuous-time system 1 ( ) h t is cascaded with a sampler, a converter that converts an impulse train to a sequence, and an discrete-time LTI system 2 [ ] h n , as shown in the following block diagram. Determine the frequency response 2 ( ) j H e ω such that [ ] [ ] w n n δ = when the input ( ) x t is a unit impulse. h 1 ( t ) Conversion of impulse train to a sequence h 2 [ n ] w [ n ] y [ n ] y ( t ) x ( t ) ( ) ( ) n p t t nT δ +∞ =−∞ = Solution: a) We need the transfer function of the inverse system 1 ( ) I H s satisfying 1 1 ( ) ( ) 1 I H s H s = , and we have 2 1 1 2 (3 )(1 ) (1 ) 2(3 ) 10 7 (2 )(5 ) ( ) 1 3 1 (3 )(1 ) (3 )(1 ) (3 )(1 ) s s s s s s s s H s s s s s s s s s + + + + + + + + + + = + + = = = + + + + + + + + so 1 (3 )(1 ) ( ) (2 )(5 ) s s H s s s + + = + + ; There are three possible ROCs Re[ ] 5 s < − ; 5 Re[ ] 2 s < < − ; Re[ ] 2 s > − . But we need the ROC includes Re[ ] 1 s > − , so the ROC is Re[ ] 2 s > − . b) Since 1 ( ) I H s is rational and the ROC is right-hand-sided, the inverse system is causal. Since 1 ( ) I H s has all poles in the left half side of the s-plane, the inverse system is stable. c) The sketch of the Bode plot is as the following figure. [Note that the dashed line is the actual response while the solid line is the approximation using the technique in Section 6.5/9.4 in the textbook. One should use the technique in Section 6.5/9.4 to get the full credit.]

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