Signals and Systems, Final ExamSolutions Spring 2005, Edited by bypeng 1. [21 points] Consider a continuous-time system with impulse response 31( )( )( )2( )tth tteu te u tδ−−=++a)  Determine the transfer function of the inversesystem of h1. b)  Is the inverse system causal and stable? c)  Sketch the asymptotic approximation to the gain and phase of the Bode plot for the transfer function of the inverse system. d)  Draw a direct-form representation of the inverse system using a minimum number of integrators and multipliers. e)  Repeat Step d) and draw a parallel-form representation of the inverse system. f)  Repeat Step d) and draw a cascade-form representation of the inverse system. g)  Suppose the continuous-time system 1( )h tis cascaded with a sampler, a converter that converts an impulse train to a sequence, and an discrete-time LTI system 2[ ]hn, as shown in the following block diagram. Determine the frequency response 2()jHeωsuch that [ ][ ]w nnδ=when the input ( )x tis a unit impulse. h1(t) Conversion of impulse train to a sequence h2[n]w[n] y[n] y(t) x(t)( )()np ttnTδ+∞=−∞=−∑Solution: a) We need the transfer function of the inverse system 1( )IHssatisfying 11( )( )1IHs Hs=, and we have 2112(3)(1)(1)2(3)107(2)(5)( )131(3)(1)(3)(1)(3)(1)ssssssssHsssssssss++++++++++=++===++++++++so 1(3)(1)( )(2)(5)ssHsss++=++; There are three possible ROCs Re[ ]5s< −; 5Re[ ]2s−<< −; Re[ ]2s> −. But we need the ROC includes Re[ ]1s> −, so the ROC is Re[ ]2s> −. b) Since 1( )IHsis rational and the ROC is right-hand-sided, the inverse system is causal. Since 1( )IHshas 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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