The derivation of the sampling theorem involves the operations of impulse train sampling a...

The derivation of the sampling theorem involves the operations of impulse train sampling and reconstruction as shown in Fig. P-1.

(a) For the input with Fourier transform depicted in Fig. P-2, use the sampling theorem to choose the sampling rate ωs = 2π/Ts so that x1(t)= x(t)when

 (12.1)

Plot Xs(jω)for the value of ωs = 2π/Ts that is equal to the Nyquist rate:

(b) If ωs = 2π/T, = 100π in the above system and X(jω)is as depicted above, plot the Fourier transform Xs(jω)and show that aliasing occurs. There will be an infinite number of shifted copies of X(jω), so indicate the periodic pattern as a function of ω.

(c) For the conditions of part (b), determine and sketch the Fourier transform of the output Xr(jω)if the frequency response of the LTI system is given by

Figure P-12.1