Consider the system for discrete-time filtering of a continuous-time signal shown in Fig....

Consider the system for discrete-time filtering of a continuous-time signal shown in Fig. P-12.14. The Fourier transform of the input signal is shown in Fig. P- 12.15(a), and the frequency response of the discrete-time system in Fig. P-12.14 is shown in Fig. P-12.15(b).

(a) Assume that the input signal x(t)has a bandlimited Fourier transform X(jω)as depicted in Fig. P-12.15(a). For this input signal, what is the smallest value of the sampling frequency fs = 1/Ts such that the Fourier transforms of the input and output satisfy the relation Y(Jω)= Heff(jω)X(jω)?

(b) Assume that the discrete-time system is an ideal lowpass discrete-time filter with frequency response  defined by the plot in Fig. P-12.15(b). Recall that  is periodic with period 2π as shown.

Now, if fs = 100 samples/sec, make a carefully labeled plot of Heff(jω), the effective frequency response of the overall system. Also plot Y(jω), the Fourier transform of the output y(t) , when the input has Fourier transform X(jω)as depicted in the graph Fig. P-12.l5 (a).

(c) For the input depicted in Fig. P-12.15(a) and the system defined in Fig. P-12.15(b), what is the smallest sampling rate such that the input signal passes through the lowpass filter unaltered; i.e., what is the minimum fs such that Y(jω)= X (jω)?