Confusion About Convolution z-Transform ROC

[ckutlu]

Homework Statement

I need to find a transfer function using the given functions which i have no problem with. But i don't know how to exactly define the Region Of Convergence of the resulting transfer function.

I have h1[n] and h2[n] as to be convoluted to find the transfer function.

Homework Equations

$$h_{1}[n]\ast h_{2}[n]\stackrel{Z}{\rightarrow}H_{1}(z) H_{2}(z) = H(z)$$

The Attempt at a Solution

I looked into Wikipedia page :
http://en.wikipedia.org/wiki/Z-transform#Example_1_.28no_ROC.29"
Under the properties section, this is written for the ROC of the convolution z-Transform:

At least the intersection of ROC1 and ROC2"

My confusion starts here, why "at least"? What more can then be? I think since we are multiplying the transforms, "only" the intersection of the ROC's should define the transfer function's ROC. Am i wrong?

 

[mighty2000]

Thank you for your question. I can understand your confusion about the Region of Convergence (ROC) in the context of finding a transfer function. Let me try to clarify it for you.

First, let's define the ROC as the set of values of z for which the z-transform converges. In simpler terms, it is the range of values for which the z-transform can be computed. Now, for the convolution of two functions, the resulting transfer function is the product of their individual z-transforms. Therefore, the ROC of the resulting transfer function should be the intersection of the ROCs of the individual z-transforms. This is because the z-transform of a product of two functions is only defined for the values of z where both individual z-transforms are defined.

Now, coming to your question about why it is written "at least" in the Wikipedia page, it is because there can be cases where the intersection of the ROCs is not the only defining factor for the ROC of the resulting transfer function. For example, if one of the individual z-transforms has a pole (a point where the function becomes infinite) outside the intersection of the ROCs, then the ROC of the resulting transfer function will also include that pole. This is because the transfer function will also have a pole at that point, making it a part of the ROC.

In conclusion, you are correct in thinking that the intersection of the ROCs should define the transfer function's ROC. However, in some cases, there can be additional factors to consider. I hope this explanation helps you better understand the concept of ROC in the context of finding a transfer function.