# How to find H(z) from just zeros and poles

I have a system with a DC gain of 8, poles at z = +- j/2 and zeroes at e^+-j5, I need to find the H(z). I have tried this but not sure if it is right. $$H(z) = G_o * z^{-1} \frac{(z-z_0)(z-z_1)}{(z-p_0)(z-p_1)}$$ which would be $$H(z) = G_o * z^{-1} \frac{(z-e^{j5})(z-e^{-j5})}{(z-\frac{j}{2})(z+\frac{j}{2})}$$

Am I going about finding H(z) right and would the ROC be between two "circles"?

• You don't need H(z) for the ROC, you have the roots. For H(z) you might get away slightly cheaper if you used $z^{-1}$, directly. Or not. But something doesn't look right. Also, what two circles? – a concerned citizen Mar 17 at 22:36
• Where does the first $z^{-1}$ come from? I think $H(z) = G_o * \frac{(z-e^{j5})(z-e^{-j5})}{(z-\frac{j}{2})(z+\frac{j}{2})} = G_o *\frac{1 - 0.5673z^{-1} + z^{-2}}{1+0.25z^{-2}}$ would be right. – ZR Han Mar 18 at 1:21