Figure P4.31-1 shows the overall system for filtering a continuous-time signal using a d...

Figure P4.31-1 shows the overall system for filtering a continuous-time signal using a discrete-time filter. The frequency responses of the reconstruction filter He(jΩ)and the discrete-time filter H(e^jω) are shown in Figure P4.31-2.

(a) For Xc(jΩ) as shown in Figure P4.31-3 and 1/T = 20 kHz, sketch Xs (jΩ) and X (e^jω).

For a certain range of values of T, the overall system, with input x_c(t) and output y_c(t), is equivalent to a continuous-time lowpass filter with frequency response H_eff(jΩ) sketched in Figure P4.31-4.

(b) Determine the range of values of T for which the information presented in (a) is true when X_c(jΩ) is bandlimited to |Ω| ≤ 2π × 10⁴ as shown in Figure P4.31-3.

(c) For the range of values determined in (b), sketch Ω_c as a function of 1/T.

Note: This is one way of implementing a variable-cutoff continuous-time filter using fixed continuous-time and discrete-time filters and a variable sampling rate.