Question on convergence in Z transform calculation

Imagine a system with an impulsive response with infinite terms say h = u . a^(-n) then calculate its Z transform. I find that the region of convergence is |z| > 1/a, what is the meaning of this in practice?

Re: Z Transform question

the Z-transform is a complex transform deals with the explanation on impedance. here in your que z<|1\a| means your system transfer function will be convergent only untill your impedance value remains between the values of -1\a to 1\a. that means your system's ROC i.e. stability is inside the circle with radius 1\a.

Re: Z Transform question

If you want to explore on your own using jigsha's thoughts. Do this. Write down the condition for system stability given an IIR response sequence. Note what type of series does the given h form? What is the condition for convergence of such a series? You ll derive at the result. Now, the practical significance is trivial from stability condition you just wrote.

Re: Z Transform question

lebesgue said:

If you want to explore on your own using jigsha's thoughts. Do this. Write down the condition for system stability given an IIR response sequence. Note what type of series does the given h form? What is the condition for convergence of such a series? You ll derive at the result. Now, the practical significance is trivial from stability condition you just wrote.

Click to expand...

I know the system is only stable when |a| > 1. My problem is the practical meaning of Z, I know for instance that the discrete Fourier transform is a particular case of the Z transform when we make Z = exp(jw), aside from that I find difficult to find a meaning for Z.

---------- Post added at 01:38 ---------- Previous post was at 01:22 ----------

jigisha said:

the Z-transform is a complex transform deals with the explanation on impedance. here in your que z<|1\a| means your system transfer function will be convergent only untill your impedance value remains between the values of -1\a to 1\a. that means your system's ROC i.e. stability is inside the circle with radius 1\a.

Click to expand...

Hard to think of impedance on digital discrete systems, I assume it's not the ratio voltage/current..

Re: Z Transform question

Fine...but see as the Laplace transform converts your time domain signal to the freq domain, just like that the Z-transform allows you to make complex domain analysis of your poles & zeros of your transfer function h, where your index n is the nT{in time domain only}, while your index Z in H(z) will be complex. It means it maps your time domain to the complex domain...as per the best of my knowledge the Z-plain gives you the idea about stability on the basis of location/placement of poles n zeros. thats why z-transform deals with stability of the digital system.
And in some literature, u may find Z transform explained as impedance transform, bcoz basically your h is h(nT)=output/input, which directly means the ratio of Vout/Vin for any network.........!!!

---------- Post added at 05:17 ---------- Previous post was at 05:16 ----------

Fine...but see as the Laplace transform converts your time domain signal to the freq domain, just like that the Z-transform allows you to make complex domain analysis of your poles & zeros of your transfer function h, where your index n is the nT{in time domain only}, while your index Z in H(z) will be complex. It means it maps your time domain to the complex domain...as per the best of my knowledge the Z-plain gives you the idea about stability on the basis of location/placement of poles n zeros. thats why z-transform deals with stability of the digital system.
And in some literature, u may find Z transform explained as impedance transform, bcoz basically your h is h(nT)=output/input, which directly means the ratio of Vout/Vin for any network.........!!!

Re: Z Transform question

A piece of insight , I apologize if the content I am posting is redundant. I also accept with Jigsaw's argument. To add to more to it. Also note that Laplace transform generalizes the idea of Fourier transform in continuous domain which is capable of looking into stability through ROC. In case of discrete sense, the z transform generalizes the idea of discrete Fourier transform and also has the capability of doing pole-zero analysis. The main idea of representing the system in terms of poles and zero is that we can easily get an idea into the geometry of the frequency response of the system