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5. As you know from class and the homework to this point, the following simple statement is very important:If a system with input 2(t) and output y(t) is linear and time-invariant (LTI), then Y (f)- H(/)x(f), where Y(f) = F(y(t), X(f)-F(x(t)), and H(/) is the Fourier transform of the impulse response h(t). We wll find that this statement is even more useful than what appears at first glance, because it has implications for all kind of design. This problem starts you down one interesting application path purposes beyond simple LTI system analysis and (a) For input (t) 2sinc(4t), an LTI system has output sinc (2t). Find h(t) for this LTI system. (Hint: Use the Fourier transform.) (b) Suppose, for input (t)2sinc(4t), your boss asks you to design a system that has output (t)sin (4t). Can this be an LTI system? (Hint: Try to follow the path you used in (a).) What you have seen in (b) is that an LTI system cannot expand the bandwidth of a signal; thus . If you need to expand the bandwidth of a system, you will need to employ a non-LTI system e If a system (or component) expands the bandwidth of a system, is not an LTI system. The degree to which bandwidth is expanded can be used to characterize non-linear systems (or com ponents); we will have a problem studying this in more detail soon

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A T K. x(F) ylt) sinc (at) YCe) -r/k toǐ LTI System-H(f): △

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