A continuous-time filter with impulse response h_c(t) and frequency-response magnitude is to be used as the prototype for the design of a discrete-time filter. The resulting discrete-time system is to be used in the configuration of Figure to filter the continuous-time signal x_c(t).

(a) A discrete-time system with impulse response h₁[n] and system function H₁(z) is obtained from the prototype continuous-time system by impulse invariance with T_d = 0.01; i.e., h₁[n] = 0.01 h_c(0.01n). Plot the magnitude of the overall effective frequency response H_eff(jΩ) = Y_c(jΩ)/ X_c(jΩ) when this discrete-time system is used in Figure.

(b) Alternatively, suppose that a discrete-time system with impulse response h₂[n] and system function H₂(z) is obtained from the prototype continuous-time system by the bilinear transformation with T_d = 2; i.e., 

H₂(z) = H_c(s)|_{s= (1 – z –1) / (1 + z –1)}’

Plot the magnitude of the overall effective frequency response H_eff(jΩ) when this discrete-time system is used in Figure.

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|21. I2| < 10T, |2| > 107, |11.(j2)| = 0, Ideal Ideal hi[n] or h3[n] H(2) or H2(z) CID converter DIC converter (1)’a HeGN) T = 10 's T = 10-4s