Question

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.

1. (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 system relating the z-transform Y (z) of the output sequence {yk} to the z-transform X(z) of the input sequence {xk} ? Express your answer as a ratio of polynomials in z.

2. (b) What are the number of poles and the number of (finite) zeros of G(z) ?

3. (c) Given that G(z) has a pole at z = −0.25, find the remaining poles, and the finite zeros of

   G(z). Clearly indicate how you obtained the poles.

4. (d) Is the system G(z) bounded-input, bounded-output stable ? Explain your answer making

   reference to the stability boundary in the z-plane.

5. (e) In your answer booklet, neatly sketch the pole-zero-gain plot of G(z) remembering to

   include the leading co-efficient of the numerator polynomial.

6. (f) What are the transient response modes for G(z) ? Note : You are NOT required to determine residues, just the general form of each transient response mode. The response modes must be real-valued sequences.

Answer 1