Impulse invariance and the bilinear transformation are two methods for designing dis...

Impulse invariance and the bilinear transformation are two methods for designing discrete time filters. Both methods transform a continuous-time system function H_c(s) into a discrete time system function H(z). Answer the following questions by indicating which method(s) will yield the desired result:

(a) A minimum-phase continuous-time system has all its poles and zeros in the left-half s plane. If a minimum-phase continuous-time system is transformed into a discrete-time system, which method(s) will result in a minimum-phase discrete-time system?

(b) If the continuous-time system is an all-pass system, its poles will be at locations s_k in the left-half s-plane, and its zeros will be at corresponding locations −s_k in the right-half s-plane. Which design method(s) will result in an all-pass discrete-time system?

(c) Which design method(s) will guarantee that

(d) If the continuous-time system is a bandstop filter, which method(s) will result in a discrete-time bandstop filter?

(e) Suppose that H₁(z), H₂(z), and H(z) are transformed versions of H_c1(s), H_c2(s), and H_c(s), respectively. Which design method(s) will guarantee that H(z) = H₁(z)H2(z) whenever H_c(s) = H_c1(s)H_c2(s)?

( f ) Suppose that H₁(z), H₂(z), and H(z) are transformed versions of H_c1(s), H_c2(s), and H_c(s), respectively. Which design method(s) will guarantee that H(z) = H₁(z)+H₂(z) whenever H_c(s) = H_c1(s) + H_c2(s)?

(g) Assume that two continuous-time system functions satisfy the condition

If H₁(z) and H₂(z) are transformed versions of H_c1(s) and H_c2(s), respectively, which design method(s) will result in discrete-time systems such that

(Such systems are called “90-degree phase splitters.”)