Question

X(z) = 3-ᶻ⁻¹/1-3zᶻ⁻¹+2ᶻ⁻²

(a) List all of the poles and zeros of X(z).

Answer 1

The zeros of the transfer function X(z) are found by solving the numerator, resulting in one zero at z=1/3. The poles are found by solving the denominator, yielding two poles at z=1 and z=2.

Step-by-step explanation:

The transfer function given is X(z) = (3-z-1)/(1-3z-1+2z-2). To find the poles and zeros of X(z), we need to solve for the values of z that make the numerator (zeros) and the denominator (poles) equal to zero.

The numerator is 3-z-1 which is equal to zero when z-1=3. Taking the reciprocal, we find that the zero occurs at z=1/3.

The denominator is 1-3z-1+2z-2 which can be rewritten as z2-3z+2. This is a quadratic equation that can be factored as (z-1)(z-2), thus the poles are at z=1 and z=2.