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Engineering · Electrical Engineering
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Consider a general LTI system with input x(t), impulse response h(t) and output y(t) - r(t) h(t) Xy()) X(j ω) Y(Go) 2W 2W Figure 1: Left: Spectrum of the input signal, X(ju), in part 5(b). Right: feedback system in part 5(c) (a) Assuming that x(t) exp(-at}a(t), for some constant a > 0, andl h (t) 6δ(t-2) (b) Assume now that the spectrum of r(t) is the real and odd signal X(jw) displayed in Find an impulse response h(ț), different from the zero function h (t) 0, such that the (c) Let the LTI system be given by the feedback scheme depicted in Figure 1 (right), where i. Obtain the frequency response of the overall system, H(jw)- FT(h(t), and write i. If Hi(jw) and H2(jw) -k, with constants a 0 and k >0, obtain a calculate Y(jw), the Fourier transform of y(t). Figure 1 (left) output y(t) is a real and even signal H1(jw) and H2(jw) are frequency response functions of two LTI sub-systems it as an explicit function of Hi(jw) and H2 (jw) differential equation that relates the output signal y(t) and the input signal r(t). (d) If the LTI system is specified by the differential equation d.r obtain the frequency response H(ju) and the impulse response h(t) of the system. Hint: The four parts of this question, (a), (b), (c) and (d), can be answered independently

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