I cannot figure out what is going on here. I have an example problem in my book that says the ROC of a certain function is $$ 0.5 < |z| < \infty$$ The function's denominator is $$ 1 - z^{-1} + 0.75z^{-2} -0.25z^{-3} + 0.0625z^{-4} $$ By properties of the ROC, there should be a pole on the boundary of the ROC. Since the example given was a right sided signal, the ROC will be the area outside of a circle boundary. So, the signal should have a pole on the boundary of that circle, which has a radius of 0.5 correct?

But when I plug in 0.5 to the denominator equation, I get 1, not zero, which makes 0.5 not a pole in this case. I have tried graphs, and placing different parenthesis, and calculated it multiple times, but the denominator equation above never reaches zero. According to ROC properties, it has to though!

Where am I thinking wrong here?




The ROC $0.5<|z|<\infty$ does not imply that there's a pole at $z=0.5$. What it does say is that there's at least one pole satisfying $|z|=0.5$ (and no other pole with a radius larger than $0.5$), and this is also the case for the given denominator polynomial. The roots of the polynomial are

$$z_{1,2}=0.5\, e^{j\pi/3}\tag{1}$$


$$z_{3,4}=0.5\, e^{-j\pi/3}\tag{1}$$

I.e., there are two double poles, and all four poles lie on a circle with radius $0.5$.


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