Z- Transform - Poles , Transfer Function , ROC

Ask

For the LTI system described by the equation:
$y(n)= y(n-1)+y(n-2)+x(n-1)$

(i) Determine the transfer function $G(z)= \frac{Y(z)}{X(z)}$
(a)$\frac{z}{(z^2-z-1)}$
(b)$\frac{z-1}{(z^2+z-1)}$
(c)$\frac{z+2}{(z^2-z-2)}$
(d)none of above

(ii)The poles of the transfer function G(z) are
(a)1/2 - 1/2√5, 1/2 +1/2√5
(b)1-√5, 1+√5
(c)1,-2
(d) insufficient data

(iii) Determine the ROC and derive the consequent information regarding the system stability
(a)ROC is |z| > 0.5+0.5√5, system is unstable
(b)ROC is |z|<0.5+0.5√5 system is stable
(c)ROC is |z| > 2 , system is unstable
(d)cannot determine

Answer for (i) is ( a )
Answer for (ii) is ( a )
Answer for (iii) is ( a )

Concept

Solution for (i)

Given  LTI system described by the equation:

$y(n)= y(n-1)+y(n-2)+x(n-1)$

Taking Z- Transform
$\implies Y(z) = z^{-1}Y(z) + z^{-2}Y(z)+ z{-1}X(z)$
$\implies Y(z) - z^{-1}Y(z)-  z^{-2}Y(z) =  z{-1}X(z)$
$\implies G(z) =  \frac{z}{z^{2} -z -  1}$

Solution for (ii)
The poles of Transfer Function =Roots of Equation $( z^{2} -z -  1) =  \frac{(1-\sqrt 5)}{2} , \frac{(1+ \sqrt 5)}{2}$

It can be seen from the figure that the ROC does not contain the unit circle
Hence, the System is UNSTABLE.