Only question D and E.

Please, only answer this question if you are able to.

Thank you,

Consider the signal x(t) = u(t). cos 2nfot.e-at shown in the figure with fo = 2, and a = 0.25, and its Fourier transform X(f). x(t) 1 -1 0.0 2.5 7.5 10.0 5.0 ts) a. [3pts] Show that the Fourier transform of this signal is 1 x() =} (a+j28 la +j2n(f - fo) *a + j27(f+fo)) b. [3pts] We are interested in the Energy of this signal. Show that this is approximately Ex-1 The amplitude of X (f) is shown in the plot below: [X() 2 X(f) 1 -4 -2 0 2 f[Hz] We have a device that taking the signal x(t) as an input generates the following signal y(t) = (x(t)e-12%) hip(t) Let as assume that the filter given by hup(t) is an ideal low-pass filer, with frequency response H(A) = n(4) c. (3pts) Sketch Y. d. 3pts] Give and approximate expression for the energy of y(t) Since we cannot implement ideal filters, let us assume that we use the following one: t-T/2 h_p(t) = 11 ( =16:9) e. (3pts] Determine the frequency response, HP(). f. [3pts) Find a value of Ta such that the filter behaves as a low-pass filter with a cut-off frequency similar to that of the ideal one. Consider the signal x(t) = u(t). cos 2nfot.e-at shown in the figure with fo = 2, and a = 0.25, and its Fourier transform X(f). x(t) 1 -1 0.0 2.5 7.5 10.0 5.0 ts) a. [3pts] Show that the Fourier transform of this signal is 1 x() =} (a+j28 la +j2n(f - fo) *a + j27(f+fo)) b. [3pts] We are interested in the Energy of this signal. Show that this is approximately Ex-1 The amplitude of X (f) is shown in the plot below: [X() 2 X(f) 1 -4 -2 0 2 f[Hz] We have a device that taking the signal x(t) as an input generates the following signal y(t) = (x(t)e-12%) hip(t) Let as assume that the filter given by hup(t) is an ideal low-pass filer, with frequency response H(A) = n(4) c. (3pts) Sketch Y. d. 3pts] Give and approximate expression for the energy of y(t) Since we cannot implement ideal filters, let us assume that we use the following one: t-T/2 h_p(t) = 11 ( =16:9) e. (3pts] Determine the frequency response, HP(). f. [3pts) Find a value of Ta such that the filter behaves as a low-pass filter with a cut-off frequency similar to that of the ideal one