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p2.5 - 98 2.5 2.6 2.7 Continuous-Time Systems Chapter 2 Use...

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Unformatted text preview: 98 2.5. 2.6. 2.7. Continuous-Time Systems Chapter 2 Use the convolution integral to find the response y(t) of an LT] system with impulse response h(t) to input x(!): (a) x(1) = exp[—t]u(t) h(!) = exp[—-21]u(t) (b) x(t) = texp[-t]u(t) h(t) = u(!) (c) X(I) = exp[-t]u(r) + u(t) h(!) = u(r) (d) x(t) = u(l) h(!) = exp[-2t]u(!) + 6(1) (e) x(t) = exp[—at]u(1) h(t) = u(t) — exp[—at]u(t - b) (D x(t) = 6(1 ~ 1) + exp[—t]u(t) h(t) = exp[-2I]u(t) The cross correlation of two different signals is defined as a: 'x RMO=I fiflfl1—0k=f fir+0flflfi —:x (a) Show that Rx,(1)= IO) * y(-!) (b) Show that the cross correlation does not obey the commutative law. (c) Show that R”,(I) is symmetric (ny(t) = Ryx(—t)). Find the cross correlation between a signal x(t) and the signal y(t) = x(t — 1) + n(t) for B/A = 0, 0.1, and 1, where x(t) and n(t) are as shown in Figure P2.7. Jr“) Mr) Figure P2.7 The autocorrelation is a special case of cross correlation with y(t) = x(t). In this case. a: mm=RAo=Jxonn+nw —: (a) Show that R,(0) = E, the energy of x(t) (b) Show that R,(t) s Rx(0) (use the Schwarz inequality) (c) Show that the autocorrelation of z(r) = x(t) + y(t) is R,(t) = R4!) + Rv(t) + Rum + Ru“) ...
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