Consider designing a discrete-time filter with system function H(z) from a continuous-ti...

Consider designing a discrete-time filter with system function H(z) from a continuous-time filter with rational system function H_c(s) by the transformation

where α is a nonzero integer and β is real.

(a) Ifα > 0, for what values of β does a stable, causal continuous-time filter with rational H_c(s) always lead to a stable, causal discrete-time filter with rational H(z)?

(b) Ifα < 0, for what values of β does a stable, causal continuous-time filter with rational H_c(s) always lead to a stable, causal discrete-time filter with rational H(z)?

(c) For α = 2 and β = 1, determine to what contour in the z-plane the j Ω-axis of the s-plane maps.

(d) Suppose that the continuous-time filter is a stable lowpass filter with passband frequency response such that

If the discrete-time system H(z) is obtained by the transformation set forth at the beginning of this problem, with α = 2 and β = 1, determine the values of ω in the interval |ω| ≤ π for which