Consider the following linear difference equation
yk −0.55yk−1 +0.12yk−2 +0.08yk−3 =xk−1 +1.5xk−2 ,
defined for integers k ≥ 0.
(a) Assuming that the initial conditions y−1 = y−2 = y−3 = 0,what is the z-domain transfer function G(z) of the linear systemrelating the z-transform Y (z) of the output sequence {yk} to thez-transform X(z) of the input sequence {xk} ? Express your answeras a ratio of polynomials in z.
(b) What are the number of poles and the number of (finite)zeros of G(z) ?
(c) Given that G(z) has a pole at z = −0.25, find the remainingpoles, and the finite zeros of
G(z). Clearly indicate how you obtained the poles.
(d) Is the system G(z) bounded-input, bounded-output stable ?Explain your answer making
reference to the stability boundary in the z-plane.
(e) In your answer booklet, neatly sketch the pole-zero-gainplot of G(z) remembering to
include the leading co-efficient of the numeratorpolynomial.
(f) What are the transient response modes for G(z) ? Note : Youare NOT required to determine residues, just the general form ofeach transient response mode. The response modes must bereal-valued sequences.
