4. Systems Si and S, are two linear time-invariant systems with impulse responses: hi(t) = e fu(t) ha(t) = 4e-2* cos(26)u(t) Find the impulse responses for the following systems formed by interconnections of Si and S2. (Hint: What input signal (t) should you choose, to obtain the impulse response?) -xct hich (a) System A is the parallel interconnection of S, and S2. Yach +XCAW -hi(t)31 (t) DUA(t) Sco -ha(t) yz(t) (b) System B is the series interconnection of Si and S2. 3(1), haceri(t) . na,(e) pe() 3

function for system B, and then Use Laplace transforms to determine the transfer function for system , take the inverse Laplace transform to find the impulse response. em C is a feedback interconnection, with system S, in the feedback path, t(e) @ halt) yc(t). hz(t) Determine the impulse response for system C by following these steps: (i) Show that the following general result is true for the transfer function of any feedback interconnection, Hi(s) Hc(8)= 1 – HN(s)H2(s) (ii) Evaluate Hc(8) using the transfer functions for systems S, and S2. (iii) Take the inverse Laplace transform to obtain the impulse response ho(t).

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# 4. Systems Si and S, are two linear time-invariant systems with impulse responses: hi(t) = e fu(t) ha(t) = 4e-2* cos(26)u(t)

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