Z-Transform of a Restricted Function

Question

$H[n]=(1/2)^n (u[n+9]-u[n-10])$ I want to find the z-transform and its zeros and poles. But when I try to do that there will be high power of z's. It is between -9 and 10 (There are 20 terms.). So, I cannot find its zeros and poles. What should I do?

Answer 1

We are using the bilateral z transform. Then (since $u(k+9)-u(k-10)$ is 1 for $ |k|\le 9$ and zero otherwise) we have $\hat{H}(z) = {\cal Z}(H)(z) =\sum_{k=-\infty}^\infty {1 \over 2^k}(u(k+9)-u(k-10))z^{-k} =\sum_{k=-9}^9 {1 \over 2^k} z^{-k} $.

For $z \neq {1 \over 2}$, we can sum the geometric series to get $\hat{H}(z) = {1 \over {1\over 2^{9}}z^{-9}}{ 1-{1\over 2^{19}}z^{-19}\over 1-{1\over 2}z^{-1} }$.