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Suppose I have an LTI system $$H(z)=\frac{z}{(z-2)(z-\frac{1}{2})}$$ and I want to know its response to the step function $u[n]$.

The LTI system $H(z)$ has three possible ROCs: $$|z|<\frac{1}{2}$$ $$\frac{1}{2}<|z|<2$$ $$|z|>2$$

The response to the step function can be expressed as $$S(z)=H(z)U(z)=\frac{z^{2}}{(z-2)(z-\frac{1}{2})(z-1)}$$

with the ROC of $U(z)$ being $$|z|>1$$

Now, my question is: what is the response to the step function in each case (i.e. for every different ROC of $H(z)$)? Is the ROC of $S(z)$ the intersection of the other two? And what happens when the intersection is empty (when the ROC of $H(z)$ is $|z|<\frac{1}{2}$)?

Thank you for your time

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  • $\begingroup$ Note there should be a $z^2$ in the numerator of $S(z)$ because $U(z)=z/(z-1)$. $\endgroup$ – Matt L. Jan 26 '16 at 8:48
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Note that for any impulse response $h[n]$, the step response is given by the cumulative sum of $h[n]$:

$$a[n]=\sum_{k=-\infty}^nh[k]\tag{1}$$

Its $\mathcal{Z}$-transform is then

$$A(z)=\sum_{n=-\infty}^{\infty}a[n]z^{-n}=\sum_{n=-\infty}^{\infty}z^{-n}\sum_{k=-\infty}^nh[k]\tag{2}$$

The region of convergence (ROC) of $(2)$ is the range of values of $|z|$ for which the sum in $(2)$ converges.

For your example, you have three different impulse responses with the same expression for their $\mathcal{Z}$-transform, each with a different ROC. You are right that the ROC of $A(z)$ is the intersection of the ROC of $h[n]$ and the ROC of the unit step $u[n]$.

For the ROC $|z|<\frac12$ you get an empty intersection. This simply means that the sum $(2)$ converges nowhere in the $z$-plane, i.e. the $\mathcal{Z}$-transform of the step response $a[n]$ doesn't exist in this case. The reason is that the corresponding impulse response is anti-causal and exponentially increasing towards negative values of $n$, so from $(1)$, $a[n]$ is infinite for any value of $n$.

For the ROC $\frac12<|z|<2$ you have the intersection $1<|z|<2$, i.e., the step response $a[n]$ is a two-sided sequence with a $\mathcal{Z}$-transform $A(z)$ that converges in that range of values of $|z|$.

For the ROC $|z|>2$ the intersection is $|z|>2$, and, consequently, the step response $a[n]$ is an exponentially increasing right-sided sequence with a $\mathcal{Z}$-transform $A(z)$ that converges for $|z|>2$.

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