# Compute the Z-transform of a^{-n} and step function

1. Dec 13, 2007

### l46kok

1. The problem statement, all variables and given/known data
Compute the Z-transform of

$$x[n] = (0.5)^{-n} * u[n-1]$$

And find the ROC (Region of Convergence)

2. Relevant equations
Z-Transform for discrete time
$$x(z) = \sum_{n=0}^\infty x[n] * z^{-n}$$

3. The attempt at a solution

To solve this, I used the definition of Z-transform

$$x(z) = \sum_{n=1}^\infty (0.5)^{-n} * z^{-n}$$

Note that the summation starts from n = 1 due to the time shift to the right by 1.

Simplifying this, we get

$$x(z) = \sum_{n=1}^\infty (0.5z)^{-n}$$

Here's where I'm confused as hell. Apparently, we can't apply the geometric series to simplify this further since the expression is powered to the negative n.

Any help will be appriciated.

2. Dec 14, 2007

### wildman

Try a little rewrite:

$$x(z) = \sum_{n=1}^\infty (0.5^{-1}z^{-1})^{n}$$ or

$$x(z) = \sum_{n=1}^\infty (2z^{-1})^{n}$$ Now use your geometric series....