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

[asd1249jf]

Homework Statement

Compute the Z-transform of

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

And find the ROC (Region of Convergence)

Homework Equations

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

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.

Furthermore, what's contradicting about this problem is that the series diverges. Does this indicate that the z-transform does not exist at all?

Any help will be appriciated.

 

[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...

 

[piercas]

Firstly, it is important to note that the Z-transform is only defined for signals that are absolutely summable, i.e. the sum of the absolute values of the signal must converge in order for the Z-transform to exist. In this case, since the series (0.5)^{-n} diverges, the Z-transform of x[n] does not exist.

However, we can still discuss the Region of Convergence (ROC) for this signal. The ROC is the set of values of z for which the Z-transform converges. In this case, since the series diverges, the ROC will be empty, i.e. there is no region of convergence for x[n].

Now, moving on to the step function u[n-1]. The Z-transform of a step function is given by:

U(z) = \sum_{n=0}^\infty u[n-1] * z^{-n}

= \sum_{n=1}^\infty z^{-n}

= \frac{1}{1-z^{-1}}, |z| > 1

Here, we can apply the geometric series since the expression is powered to the negative n. The ROC for the step function is |z| > 1, which means that the Z-transform exists for all values of z outside the unit circle in the z-plane.

In conclusion, the Z-transform of x[n] does not exist, while the Z-transform of u[n-1] exists for all values of z outside the unit circle.