Z-transform

Hello All,

I would like to get some clarifications on Z-transform. 

1) In Z-transform, what is the exact value of variable z.

Since z is a complex variable, what is its relation with complex
exponential.

2) I am not able to understand the practical use of ROC. A systems
definition is what is is. So, in what way ROC is useful?

3) Value of Z in polar form is given as, 

     z = r*exp(j*theta), where r = magnitude(z) etc.

Thanks a lot ...

	 

_____________________________		
Posted through www.DSPRelated.com

On Sunday, January 5, 2014 11:54:11 AM UTC-5, Sharan123 wrote:
> Hello All,
> 
> 
> 
> I would like to get some clarifications on Z-transform. 
> 
> 
> 
> 1) In Z-transform, what is the exact value of variable z.
> 
> 
> 
> Since z is a complex variable, what is its relation with complex
> 
> exponential.
> 
> 
> 
> 2) I am not able to understand the practical use of ROC. A systems
> 
> definition is what is is. So, in what way ROC is useful?
> 
> 
> 
> 3) Value of Z in polar form is given as, 
> 
> 
> 
>      z = r*exp(j*theta), where r = magnitude(z) etc.
> 
> 
> 
> Thanks a lot ...
> 
> 
> _____________________________		
> 
> Posted through www.DSPRelated.com

The variable z can take on any value you want. Taking the z-transform you end up with a function F(z). You plug in a value of z and get a value out.

In general z=r*exp(jw). Most of the time we are interested in the case when we evaluate z around the unit circle, so z=exp(jw) - when we do we get the same result as the discrete Fourier transform.

ROC - this determines the area or region over which we want the Z-transform to converge. Usually the different choices of ROC choose between stability and causality. Choose one ROC the system is causal but it is not stable. Choose the other ROC and the system is stable but it is not causal.

Hope that helps.
Dave

On Sunday, January 5, 2014 11:54:11 AM UTC-5, Sharan123 wrote:
> Hello All,
> 
> 
> 
> I would like to get some clarifications on Z-transform. 
> 
> 
> 
> 1) In Z-transform, what is the exact value of variable z.
> 
> 
> 
> Since z is a complex variable, what is its relation with complex
> 
> exponential.
> 
> 
Stability and convergence aren't the same thing.  An LTI filter that has all its poles inside the unit circle and is causal is stable.  A causal LTI filter that has a pole outside the unit circle isn't stable.

> 
> 2) I am not able to understand the practical use of ROC. A systems
> 
> definition is what is is. So, in what way ROC is useful?
> 
> 
> 
> 3) Value of Z in polar form is given as, 
> 
> 
> 
>      z = r*exp(j*theta), where r = magnitude(z) etc.
> 
> 
> 
> Thanks a lot ...
> 
> 
> 
> 	 
> 
> 
> 
> _____________________________		
> 
> Posted through www.DSPRelated.com