Region of Convergence (ROC) of z-transforms

Hi,

I have a problem determining the ROC of a z transform.
In the first place, I can't understand fully the definition of ROC.
My teacher says it is the value for which x(z) should not be zero or negative.

I've been applying this equation: x(z) > 0 in my quizzes and exams
From the equation I have above, it is so clear that I get values for z that makes x(z) not zero and non-negative.

But I always get ROC wrong. Can someone please clarify what ROC means and how to get it.
Thank you.

Hi,

As you probably know, a continuous time system is stable when all of the poles lie in the left half side of the complex plane. Well, the s plane maps to the z plane where the imaginary axis in the s plane maps to the unit circle in the z plane. Therefore the poles of the transfer equation must all lie within the unit circle, which means the roots of the characteristic equation must all lie within the unit circle.

As an example, you might be asked to find the values of K where the following characteristic equation is stable:
z^2-1.3678*z+0.3678*z*K+0.2644*K+0.3678

for K=1 we get:
z^2-z+0.6322=0

Solving for the roots we get:
z=0.5+0.618*i
z=0.5-0.618*i
approximately.

These roots both lie within a circle of radius 1 at the origin so this system is stable (converges).

With K=2.5 we get:
z=0.22415+0.989220*i
z=0.22415-0.989220*i

and since at least one of these roots lies just outside the unit circle this system is unstable (doesnt converge).

So the value of K where the system goes unstable is somewhere between 1 and 2.5 (closer to 2.5 than 1). See if you can find the value for K where the system goes from stable to unstable if you like.

We might also look at K being negative or zero. We might find that the system again goes unstable when K<=0. This would mean the system converges when 0<K<n where we know n is a number somewhere between 1 and 2.5 so far. A little more work and we would find out what n was so we might get something like this:
0<K<2.2
(note it is not actually 2.2 as i left that for you to find if you like).

Hi,

The maximum value of K that will make the characteristic equation stable is 2.39
Above that, the roots for the equation would already be outside the unit circle.

I did it by trial and error. Substituting values for K. I tried my algebra but it doesn't work for me

Hi again,


Hey that's good Now you should be able to do any of these.